Minimising the Decoherence of Rare Earth Ion Solid State ......Minimising the Decoherence of Rare Earth Ion Solid State Spin Qubits Elliot Fraval A thesis submitted for the degree

Minimising the Decoherence

of Rare Earth Ion

Solid State Spin Qubits

Elliot Fraval

A thesissubmitted for the degree

ofDoctor of Philosophy

of theAustralian National University

The Australian National UniversityJuly 2005

Statement of authorship

This thesis contains no material which has been accepted for the award

of any other degree or diploma in any university. To the best of the author’s

knowledge and belief, it contains no material previously published or written

by another person, except where due reference is made in the text.

Elliot Fraval

August 22, 2006

iv

“In the beginning the Universe was created. Thismade a lot of people very angry and has been widelyregarded as a bad move.”

– Douglas Adams

vi

Acknowledgements

Firstly, I would like to thank my primary supervisor Matt Sellars. I havethoroughly enjoyed working with Matt and have come to respect his talentfor problem solving and enthusiasm for this area of physics. I’d also like tothank Neil Manson who as my supervisor and the head of our group hasbeen fundamental in creating the excellent working environment within theSolid State Spectroscopy group. Their interest in not just physics but life ingeneral has contributed greatly to the experience of my study.

Particular thanks go to the Laser Physics Centre technicians, withoutwhom the project would simply not have happened. Many thanks to IanMcRae, the god of all cryostats, John Bottega for his award winning singing,comedy, liquid helium and, like Craig McLeod, his invaluable hands. Thanksfor helping me out in the workshop to get my head around the vast array oftools.

It has been a pleasure to share an office with Jevon Longdell and JoHarrison. I thank both of them for putting up with my atonal singing,extreme musical taste and slapping out the occasional funk bass line to keepme sane in the lab. Big thanks to the lunch time crew for their laughs andbizarre conversational tangents pursued well beyond their reasonable ends.

Big thanks and group hugs to all of my muso friends that have helpedme keep the correct balance of sanity and lack of it with the free psychiatrythat playing original music is. In particular Nic, Marky, Dan and Candiethat make up the rest of eyTis as well as the indominable Red Rocko andThe Whalebone from The Milk. Youse guys rock! Thanks for keepin it realin the arts end of town :)

Last but by no means least thanks to all of my family and friends thathave had to put up with me over the last few years. Definitely thanks toIrma for her editorial prowess and Amelie for her abstract corrections. Tothose friends I haven’t had the spare time for I will call.... no really, I will!

Elliot FravalJuly 2005

vii

Abstract

This work has demonstrated that hyperfine decoherence times sufficiently

long for QIP and quantum optics applications are achievable in rare earth ion

centres. Prior to this work there were several QIP proposals using rare earth

hyperfine states for long term coherent storage of optical interactions [1, 2, 3].

The very long T1 (∼weeks [4]) observed for rare-earth hyperfine transitions

appears promising but hyperfine T2s were only a few ms, comparable to rare-

earth optical transitions and therefore the usefulness of such proposals was

doubtful.

This work demonstrated an increase in hyperfine T2 by a factor of ∼7 × 104 compared to the previously reported hyperfine T2 for Pr3+:Y2SiO5

through the application of static and dynamic magnetic field techniques.

This increase in T2 makes previous QIP proposals useful and provides the

first solid state optically active Λ system with very long hyperfine T2 for

quantum optics applications.

The first technique employed the conventional wisdom of applying a small

static magnetic field to minimise the superhyperfine interaction [5, 6, 7], as

studied in chapter 4. This resulted in hyperfine transition T2 an order of

magnitude larger than the T2 of optical transitions, ranging fro 5 to 10 ms.

The increase in T2 was not sufficient and consequently other approaches were

required.

Development of the critical point technique during this work was crucial

to achieving further gains in T2. The critical point technique is the applica-

tion of a static magnetic field such that the Zeeman shift of the hyperfine

transition of interest has no first order component, thereby nulling decoher-

ing magnetic interactions to first order. This technique also represents a

global minimum for back action of the Y spin bath due to a change in the

Pr spin state, allowing the assumption that the Pr ion is surrounded by a

thermal bath. The critical point technique resulted in a dramatic increase of

the hyperfine transition T2 from ∼10 ms to 860 ms.

Satisfied that the optimal static magnetic field configuration for increas-

ing T2 had been achieved, dynamic magnetic field techniques, driving ei-

ix

ther the system of interest or spin bath were investigated. These tech-

niques are broadly classed as Dynamic Decoherence Control (DDC) in the

QIP community. The first DDC technique investigated was driving the Pr

ion using a CPMG or Bang Bang decoupling pulse sequence. This sig-

nificantly extended T2 from 0.86 s to 70 s. This decoupling strategy has

been extensively discussed for correcting phase errors in quantum computers

[8, 9, 10, 11, 12, 13, 14, 15], with this work being the first application to solid

state systems.

Magic Angle Line Narrowing was used to investigate driving the spin

bath to increase T2. This experiment resulted in T2 increasing from 0.84 s to

1.12 s. Both dynamic techniques introduce a periodic condition on when QIP

operation can be performed without the qubits participating in the operation

accumulating phase errors relative to the qubits not involved in the operation.

Without using the critical point technique Dynamic Decoherence Control

techniques such as the Bang Bang decoupling sequence and MALN are not

useful due to the sensitivity of the Pr ion to magnetic field fluctuations.

Critical point and DDC techniques are mutually beneficial since the critical

point is most effective at removing high frequency perturbations while DDC

techniques remove the low frequency perturbations. A further benefit of

using the critical point technique is it allows changing the coupling to the

spin bath without changing the spin bath dynamics. This was useful for

discerning whether the limits are inherent to the DDC technique or are due

to experimental limitations.

Solid state systems exhibiting long T2 are typically very specialised sys-

tems, such as 29Si dopants in an isotopically pure 28Si and therefore spin free

host lattice [16]. These systems rely on on the purity of their environment

to achieve long T2. Despite possessing a long T2, the spin system remain

inherently sensitive to magnetic field fluctuations. In contrast, this work has

demonstrated that decoherence times, sufficiently long to rival any solid state

system [16], are achievable when the spin of interest is surrounded by a con-

centrated spin bath. Using the critical point technique results in a hyperfine

state that is inherently insensitive to small magnetic field perturbations and

therefore more robust for QIP applications.

x

Contents

Acknowledgements vii

Abstract ix

1 Introduction: From Classical to Quantum Information 1

1.1 Classical Information Processing . . . . . . . . . . . . . . . . . 2

1.1.1 Theoretical Developments . . . . . . . . . . . . . . . . 2

1.1.2 Towards Quantum Hardware . . . . . . . . . . . . . . . 4

1.1.3 Information goes Quantum . . . . . . . . . . . . . . . . 5

1.1.4 The Power of Hilbert Space . . . . . . . . . . . . . . . 7

1.2 Quantum Computing Requirements . . . . . . . . . . . . . . . 8

1.3 The Two Level Atom . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 The Density Matrix . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Dynamics on the Bloch Sphere . . . . . . . . . . . . . 14

1.3.3 Quantum Process Tomography . . . . . . . . . . . . . 16

1.4 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.1 System Bath Interactions . . . . . . . . . . . . . . . . 20

1.4.2 Decoherence on the Bloch Sphere . . . . . . . . . . . . 22

1.5 Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5.1 Quantum Error Correction Codes . . . . . . . . . . . . 24

1.5.2 Decoherence Free Subspaces . . . . . . . . . . . . . . . 25

1.5.3 Dynamic Decoherence Control . . . . . . . . . . . . . . 25

1.5.4 A Quiet Corner of Hilbert Space . . . . . . . . . . . . . 26

1.6 Why the Rush? . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7 Other application and fundamental interest . . . . . . . . . . . 27

2 Rare Earth Ion Spectroscopy 29

2.1 Introducing The Lanthanides . . . . . . . . . . . . . . . . . . 30

2.2 4f Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Hyperfine Interaction . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Yttrium Orthosilicate: The Gracious Host . . . . . . . 36

xi

2.3.2 Hyperfine Interaction in Praseodymium Doped Y2SiO5 37

2.3.3 M and Q . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Homogeneous and Inhomogeneous Broadening . . . . . . . . . 41

2.5 Optically Detected NMR and Coherent Transients . . . . . . . 42

2.5.1 Raman Heterodyne . . . . . . . . . . . . . . . . . . . . 42

2.5.2 Spin Echos . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.3 Spin Echo Decays . . . . . . . . . . . . . . . . . . . . . 46

3 QC Benchmarks and Benefits of Rare Earth QC 50

3.1 Rare Earth Ion ODNMR Quantum Computing . . . . . . . . . 51

3.2 Rare Earth Quantum Computing Architecture . . . . . . . . . 52

3.3 Rationale for System Comparison . . . . . . . . . . . . . . . . 57

3.4 Liquid Phase NMR . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Limitations of Liquid State NMR . . . . . . . . . . . . 61

3.5 The Case for Solids . . . . . . . . . . . . . . . . . . . . . . . . 62

3.6 Limitations of Rare Earth QC due to Hyperfine Decoherence . 64

4 Hyperfine Decoherence with Small Applied Mangetic Field 66

4.1 Pr3+:Y2SiO5 Hyperfine Decoherence . . . . . . . . . . . . . . 66

4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Maximising Hyperfine T2 using Static Magnetic Fields 79

5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


5.2.1 Finding a Critical Point . . . . . . . . . . . . . . . . . 86

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4.1 Future Improvements . . . . . . . . . . . . . . . . . . . 97

6 Dynamic Decoherence Control 99

6.1 Application to Pr3+:Y2SiO5 . . . . . . . . . . . . . . . . . . . 100


6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4.1 Bang Bang Process Tomography . . . . . . . . . . . . 110

7 Extending T2 Through Driving the Environment 116

7.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xii

8 Future Decoherence Challenges 124

8.1 Exchanging Praseodymium for Europium . . . . . . . . . . . . 124

8.2 Stoichiometric Materials . . . . . . . . . . . . . . . . . . . . . 125

8.3 Considerations for QIP in Stoichiometric Materials . . . . . . 126

8.4 Minimising Decoherence in Stoichiometric Defect QIP Systems 127

9 Conclusions and Future Work 129

9.1 Strategies for Further Increases in Decoherence Time . . . . . 132

9.1.1 Improved RF Control . . . . . . . . . . . . . . . . . . . 132

9.1.2 Rabi Frequency and Inhomogeneous Broadening . . . . 132

9.1.3 Eulerian Decoupling . . . . . . . . . . . . . . . . . . . 133

9.2 Other Applications For Long T2 Optically Active Solids . . . . 133

9.2.1 Slow and Stopped Light . . . . . . . . . . . . . . . . . 133

9.2.2 Stark Echo Quantum Memory . . . . . . . . . . . . . . 134

Appendices 135

A Y2SiO5 site position calculation 136

B Full Critical Point List 150

C Published Papers 153

Bibliography 175

xiii

xiv

Chapter 1

Introduction: From Classical to

Quantum Information

“I think there is a world market for maybe five computers.”

- Thomas Watson (1874-1956), Chairman of IBM, 1943

“There is no reason for any individual to have a computer in his home.”

–Ken Olsen, President, Digital Equipment, 1977

The time we live in is often referred to as the ‘information age’ since the

ability to gather and process information is crucial for the decision making

process of any entity. Modern computing enables information processing

tasks that were previously impossible as a result of the dramatic advances

made by computing technology. The rate at which information processing

technology has advanced is unprecedented in human experience, prompting

the question: what are the fundamental limits to processing information in

our Universe? Our knowledge of physics tells us this limit is defined by

quantum physics. Exploring these limits will result in advances to both

information science and quantum physics.

The first generation of quantum communication systems that exploit the

measurement principals of quantum mechanics to secure the transmitted in-

formation, known as quantum cryptography, have already been deployed.

Current research is trying to create the quantum analogue of other required

building blocks of information processing, being quantum memories and

quantum computers.

2 Introduction: From Classical to Quantum Information

Fundamental to the new information processing applications of quantum

mechanics is the ability to preserve superposition states. Superposition states

can be observed in a vast range of systems, such as polarisation states of light,

electronic states of atoms, nuclear spin orientation in a magnetic field and

vibrational state of trapped ions. The loss of superposition information of

large quantum systems is what gives rise to classical physics [17, 18]. This

work focuses on methods to preserve superposition states of hyperfine nuclear

spin states such that they are useful for implementing quantum information

processing technology.

1.1 Classical Information Processing

Algorithmic processes and their implementation have been important for

many cultures throughout human history for tasks as diverse as calculat-

ing the seasons to economics and construction. Physical systems were often

sought to implement algorithms that were difficult to perform mentally, par-

ticularly if some measurement was required. In this context, many ancient

structures, from Stonehenge to the pyramids form in part a physical imple-

mentation to aid the calculation of seasonal variables. Such structures are

clearly not reprogrammable and implementation of the algorithm was still

performed by the user. For the development of engineering and economics it

quickly became important for numeric representations where several parties

could agree on the outcome. Flexible structures such as the abacus were

developed, however the algorithm was always implemented by the user.

1.1.1 Theoretical Developments

While there were several attempts to create early mechanical computers de-

signed for specific calculations, the basis of modern computer science was

created by Alan Turing in 1936 [19]. Turing developed the concept of a

Turing Machine, a machine that implements a calculation by following an

algorithmic process. Theoretical investigations centre on the Universal Tur-

ing Machine, one that could simulate any other Turing machine.

Turing was not alone in considering this problem with significant contri-

butions being made by Alonso Church, Kurt Godel and Emil Post. Despite

each description initially appearing different, due to the different perspec-

tives on algorithmic computing, they were shown to be equivalent models

[20]. This reinforced the universality of the models, and since the Turing

Machine provided a common reference for both hardware and software it

1.1 Classical Information Processing 3

became the standard conceptual reference.

Turing considered that any physical system that performed an algorithmic

process was in some sense a Turing machine. Therefore a universal Turing

machine could simulate any physical algorithmic process. This strengthened

assertion is known as the Church-Turing Principle, stated as:

Every ‘function which could be naturally computable’ can be computed by

the universal Turing machine.

Computer science changed very quickly from a mathematical curiosity to

a fundamentally important area of technology in the years following 1936.

It became important to understand whether a computer could provide an

answer in a reasonable amount of time, how that time changed as the com-

plexity of the problem increased and consequently how efficient the algorithm

was.

An algorithm’s efficiency is the scaling of the resources required to achieve

a solution as the size of the problem is increased. Resources in this context

are the number of steps required in the computation and the amount of in-

formation required to be stored by the algorithm. The definition that arose

was that if the resources required to implement the algorithm were bounded

by a polynomial in the complexity of the problem then it is ‘efficient’. Many

algorithms require an exponential increase in resources and are consequently

considered ‘inefficient’. Algorithms are typically compared by the resources

they consume as the problem tends to large size. In this limit efficient al-

gorithms will always outperform inefficient algorithms. Discussions of this

nature resulted in a strengthening of the Church-Turing assertion that any

algorithm could be implemented on a universal Turing machine efficiently.

Shortly after the strengthened Church-Turing principal was presented,

stochastic algorithms were developed that solved problems more efficiently

than deterministic algorithms [21]. Although each iteration of the algorithm

has a finite probability of failing, stochastic methods will typically converge

to a solution much faster than analogous deterministic algorithms. There-

fore the definition of the conceptual efficient universal computing machine

required modification to a probabilistic Turing machine.

The universality of the Turing machine approach taken by computer sci-

entists is verified by the vast array of computing applications. There are,

however, many processes that do not have distinct algorithmic states. Con-

sequently measuring and processing information in a discreet state space such

that it mimics a continuous process presents many limitations and necessi-

tates approximations.


1970 1975 1980 1985 1990 1995 2000 200510

3

104

105

106

107

108

109

Nu

mb

er

of

tra

nsi

sto

rs o

n a

ch

ip

year

4004

80088080

8086286

386

486

Pentium

Pentium II

Pentium III

Pentium 4

Itanium

Itanium 2

Figure 1.1: Moores law has been a remarkably simple description of the increase incomplexity of computer chips. Data courtesy of the Intel Website, www.intel.com

1.1.2 Towards Quantum Hardware

In unison with the theoretical developments the physical systems used to

implement Turing machines underwent rapid change. Mechanical systems

quickly became electro-mechanical and the all electrical, initially using valves

and then transistors. In 1965 Gordon Moore [22] noted that the number of

transistors contained in a integrated circuit had doubled roughly every 18

months for the previous three years despite occupying the same physical

space. If this progression were to continue Moore predicted that by 1975

integrated circuits would contain in excess of 50,000 elements. This was

an accurate prediction and what has become known as ‘Moore’s Law’ has

continued to be a good estimate to this day, as shown in figure 1.1.

There are many areas of research hidden under the exponentially increas-

ing complexity of Moores Law. Miniaturisation has been a major driver of

Moores Law. By reducing the size of a transistor the benefit is twofold: the

time required for a signal to propagate through the device is reduced and the

amount of material required to change state is reduced. Therefore the device

can operate faster and consume less energy. In the context of information

storage and manipulation, this results in less atoms being required to store

the same amount of information. Given our current knowledge of physics this


progression is limited by information being represented on a single quantum

object.

Quantum effects will, however, dominate the behaviour of a computing

structure before miniaturisation attains single atom feature size. Electrons

tunnelling between conductors is one of the many quantum effects that will

cause the flow of signals to deviate from classical descriptions in a counter

productive manner. At this stage the classical computing concepts are invalid

and the design philosophy must change to one that begins with considering

the quantum nature of how the desired signals will propagate and interact.

Some devices already exploit these effects such as tunnelling diodes, however

the signals external to the device are still classical.

In conjunction with miniaturisation the materials used to fabricate the

devices has been an area of unrelenting research. This is important because

while the power requirements are somewhat reduced by miniaturisation, in-

creasing the density of computing elements would eventually result in power

consumption that would melt the device as Moores Law continued. From the

initial 12V transistor logic we are now below 1V with a similar reduction in

required current resulting in massive reduction in power consumption. With-

out this progress any portable information processing device would not be

possible in it’s current form. If the automotive industry had made equivalent

efficiency gain in combustion engine technology as the electronics industry

did from 1975 to 1995 you could drive a car around Australia on one tank

of fuel. Eventually this process also leads to a fundamental limit of what is

the minimum amount of energy that can be used to store and manipulate

information. This again leads to quantum mechanics.

1.1.3 Information goes Quantum

Computation on a Turing machine is typically irreversible. Given a compu-

tational result and knowing the algorithm that processed it is not generally

sufficient to reconstruct the input states. Therefore information has been lost

during the calculation. Irreversibility was considered to be intrinsic to com-

puting until, in 1973, Bennett, a researcher at IBM, proved this conjecture

false [23]. Reversibility is a fundamental feature of quantum computing.

Despite rigorous definitions of efficiency and extensive investigation of

numerical methods the perceived limits of algorithmic performance did not

rely on physical arguments. This was unsatisfying since it did not permit any

argument that a Turing machine was the optimal framework permitted by

physics. Furthermore, it was becoming apparent that a Turing machine could

not efficiently simulate a quantum system as highlighted by Feynmann [24].


During the mid-1980s David Deutsch began the search for a derivation based

on physical laws of something analogous to the Church-Turing Principle.

The first phase of this was to rephrase the Church-Turing principle such

that the physical implications were clearer [25].

Every finitely realisable physical system can be perfectly simulated by a

universal model computing machine operating by finite means.

Deutsch was able to show that while Quantum mechanics obeyed this

principal, classical physics does not. Classical physics, due to its continuity

will always require a continuum of input and output states, however there

is only a finite mapping of input to output states in the computer and con-

sequently fails the ‘perfectly simulated’ part of the Church-Turing principle.

Therefore, while numerical techniques allow good approximations to physical

systems they cannot perfectly simulate them. Deutsch showed the Universal

Quantum Computer could simulate any Turing machine and finite physical

system that has equal or lower complexity than the computer.

A quantum computer as proposed by Deutsch is a register of two state

quantum systems initialised into some known state, typically |0〉, the ground

state of the system (|1〉 being the excited state). The system than has some

time reversible operations, termed unitary operations, performed on one or

many qubits at a time to implement the quantum algorithm. On completion

of the algorithm the state of the register is measured. Quantum algorithms

are structured such that despite qubits being in a superposition of basis states

during the algorithm, once the algorithm has finished the state of the qubits

are pure basis states. This avoids the problems associated with measuring an

arbitrary state and the associated quantum tomography procedures required.

Unitarity of the quantum information processing requires that no information

is lost during the computation, an essential feature of quantum computing.

Unitarity allows the input state to be recovered by applying the conjugate

of the algorithm to the output state.

Perhaps the most obvious situation that a quantum computer would out-

perform a classical computer is in simulating a quantum system. Accurately

simulating quantum physics is what led Feynmann to consider problems as-

sociated with using a classical computer and propose a quantum computer

[24]. Already an algorithms to simulate many-body fermionic systems [26]

have been developed, answering Feynmann’s uncertainty as to whether his

description of a quantum computer could simulate fermionic systems. An

algorithm for finding eigenvalues and eigenvectors of atomic Hamiltonians

[27] have also been developed.


Conceptually, the simplest advantage is that a quantum computer can

generate truly random numbers efficiently. A classical computer cannot gen-

erate true random numbers since classical physics is deterministic. Therefore

only pseudo random numbers can be generated and this is done so ineffi-

ciently [24]. In a quantum computer this is achieved by measuring a super-

position state, a very simple quantum algorithm. The same qubit cannot be

used to generate a series of random numbers without correlations, however

Deutsch also showed this can be used to advantage to build up more complex

stochastic systems. Bell’s theorem states that classical systems, even if given

perfect random number generator hardware, cannot reproduce the statistics

of consecutive measurements of a quantum state.

Deutsch also introduced the notion of Quantum Parallelism whereby in

the many worlds interpretation of quantum mechanics quantum computer

performs a computation in many near universes and combines the answers

at the output. Only one result is accessible in each universe, however this

result can be the product of many interfering universes.

1.1.4 The Power of Hilbert Space

The clearest method to visualise the benefit of quantum over classical com-

puting is to consider the state space in which it operates. To completely

describe the state space S of the quantum computer we need to describe

both the population of the states and the coherence between them. Using a

density matrix representation the population of the states are the diagonal

elements of a matrix (αnn) while the coherence terms are the off diagonal

elements (βmn). In classical computation knowledge of the value of each el-

ement is sufficient to completely describe the system. Therefore the state

space of the classical computer is described by the diagonal elements of S.

S =

α11 β12 · · · β1n

β21 α22 · · · β2n

......

. . .

βn1 βn2 αnn

(1.1)

It is clear that the state space is larger for a quantum system and as such

a quantum computer is able to represent more information given the same

number of elements. Considering the increase in system complexity due to

adding another element to the computer we can see that in the classical case

adding a bit to the system results in a linear increase in complexity. Adding

a qubit to a quantum computer results in an exponential increase in system


complexity since this new qubit can have a coherent relationship with any

other qubit. Therefore the computational space grows exponentially, leading

to an exponential ability to process information.

1.2 Quantum Computing Requirements

While the work by Deutsch showed that quantum computing had enormous

potential it did not specify any rigorous criterion for candidate systems. Any

system with coherent interactions was a potential physics system. David

DiVincenzo [28] proposed five requirements for that the physical system must

meet if it is to be useful for quantum computing.

1. A scalable physical system with well characterised qubits.

2. The ability to initialise the state of the qubits to a simple starting state,

such as |000...〉.

3. Very long coherence times relative to the time required for quantum

gates.

4. A ‘universal’ set of quantum gates.

5. A qubit-specific measurement capability.

All of these requirements are inextricably linked with each other. Scala-

bility is very important to satisfy since a quantum computer is only worth

building if it outperforms a classical computer. Since we can already build

complex classical computers a useful quantum computer must span a large

state space.

Scaling up a quantum computing system is challenging. Each qubit must

be only weakly coupled to it’s environment, to allow long decoherence times.

Qubits, however, must also be strongly coupled to each other to allow quan-

tum logic gates to be performed well within the decoherence time of the

system. Therefore when adding qubits to make our quantum computer more

complex we are adding noise sources that, while weakly coupled to their

environment, are strongly coupled to all other qubits in their locality.

Although methods to correct for some decohereing interactions have been

developed (reviewed in section 1.5) many require more qubits to implement

and therefore must introduce more decoherence to the system as a whole.

1.2 Quantum Computing Requirements 9

Unfortunately we cannot settle for a simple trade-off between the number

of elements and their decoherence since both the number of individually

controllable quantum elements and the decoherence time are required to be

pushed significantly further than currently available.

The ability to manipulate the qubit state fast compared to the decoher-

ence rate necessitates that there is a strong interaction with a driving field,

and therefore limits the isolation from the rest of the universe, providing po-

tential decoherence mechanisms. Decoherence is also directly introduced by

imperfect driving fields, an unavoidable consequence of amplitude and phase

noise in the driving electronics.

While manipulating large ensembles of identical quantum systems is rou-

tinely performed in spectroscopy, atom optics and many other areas of physics,

achieving individual control of each qubit is a more complex challenge. The

approaches are necessarily different for each candidate system, however the

general trend is to use spatial confinement, inhomogeneities in transition fre-

quencies or specifically placed nanostructures to address individual qubits.

All of these have limitations: spatial selectivity of the driving field is lim-

ited by the wavelength; inhomogeneous broadening implies disorder; nanos-

tructues distort the local environment and cannot be totally decoupled from

their assigned qubit. Again there are decoherence implications to all ap-

proaches.

It should also be noted that detection of single quantum systems is a

current area of intense research. Single molecule or single site detection

is of interest to the biotechnology [29] and quantum computing fraternities

[30, 31]. Very coherent systems must be very isolated and consequently they

are some of the hardest systems to achieve reliable measurement of a single

quantum system. Again we are faced with a problem for which a simple

compromise does not exist. The more interactive our system is, the faster it

will decohere and therefore it is less useful for QIP.

The physical requirements of isolation and interaction are paradoxical and

will be hard to satisfy. Consequently the predictions of a useful quantum

computer requiring ∼106 qubits [32] should be taken with a grain of salt.

Having complete control over a Hilbert space millions of dimensions large is

an extremely ambitious task.

Assuming we can eventually meet the physical requirements the universal-

ity criterion of quantum logic operations is comparatively simple. In classical

computing any boolean logic function can be realised with only NAND gates

and consequently NAND gates form a universal set of gates. A “universal”

set of quantum gates is a set of unitary quantum operations that can be


concatenated to produce an arbitrary unitary transformation of the input

state. DiVincenzo [33] was able to show that single qubit operations and

the conditional-not (CNOT) operation between two qubits was sufficient to

form a universal set. The CNOT gate inverts the “target” qubit conditional

on the state of a “control” qubit. Using the basis |00〉 , |01〉 , |10〉 , |11〉 this

has the matrix representation:

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

(1.2)

The CNOT and single qubit rotations are typically demonstrated by

quantum computing proposals to show that the system is a strong candi-

date. In fact it has been shown that almost any quantum gate forms a

universal set [34]. Consequently this approach is somewhat an artefact of

the CNOT and single qubit being the first demonstrated universal set and

having a direct analogy with Boolean logic gates.

One of the hardest requirements is that decoherence time of the system

must be very long compared to the time required to perform gate operations.

This necessitates that the system is well isolated. However, to implement a

gate you require strong interaction between the qubits of interest and some

driving field such that it can be performed quickly. Therefore we have a

paradox since the system cannot be well isolated and strongly interacting.

This is only compounded by the last requirement of being able to measure

the final state of the system, since this also must be done on the time scale

of the gate operations [35]. In order to discuss this point in detail some basic

two level atom theory is required.

1.3 The Two Level Atom

Quantum computing is generally considered using an array of interacting two

level quantum systems. While quantum systems with more than two states

per qubit are considered by some researchers [36] it significantly complicates

the discussion. This treatment of two level atom theory is focused at giving

the reader sufficient tools to consider the state manipulations used in this

thesis.

The Bloch equations provide the best tool to visualise the evolution of a

two level system interacting with a coherent driving field. These equations

were originally derived to describe NMR experiments in spin 1/2 systems

1.3 The Two Level Atom 11

but is formally equivalent to any closed two level systems. Visualisation

of processes described by these equations is achieved via the Bloch Sphere,

shown in figure 1.2, which is the primary conceptual tool for considering

single qubit rotations. The “poles” of the sphere represent the ground and

excited state with the rest of the surface representing possible phases of a

superposition state. Discussion of State representation and evolution on the

Bloch Sphere is restricted spin 1/2 systems for simplicity, with no loss of

generality.

If we consider our fictitious spin 1/2 particle to have |0〉 as the z = −1/2

state and |1〉 as the z = +1/2,

|0〉 =

[

0

1

]

, |1〉 =

[

1

0

]

(1.3)

then a vector describing an arbitrary state of this system can be mapped into

the visualisation space via projection onto the Pauli spin operators:

rψ = (〈ψ|X|ψ〉 , 〈ψ|Y |ψ〉 , 〈ψ|Z|ψ〉) (1.4)

where X, Y and Z are the Pauli spin operators:

X =

[

0 1

1 0

]

, Y =

[

0 −ii 0

]

, Z =

[

1 0

0 −1

]

(1.5)

Due to normalisation of the states |ψ〉 the projection rψ will always be a unit

vector. To demonstrate this let us consider the mapping of a arbitrary state

to the Bloch sphere with the state given by:

|ψ〉 = sin θ |0〉 + eiφ cos θ |1〉 (1.6)

which, as shown in figure 1.2, gets mapped to:

r = (cos φ sin θ, sin φ sin θ, cos θ) (1.7)

This mapping is clearly one to one with the Z axis poles of the sphere

representing the pure states and the angle φ represents the phase of the

superposition. It is clear that this allows the visualisation of an arbitrary,

and therefore in general, mixed state. For both mathematical and conceptual

clarity the rotating wave approximation is applied to the Bloch equations.

This results in the Bloch sphere effectively being spin around the Z axis

at the resonant frequency of the transition. Therefore a Bloch vector has a


| >0

| >1

| >0 | >1+i

| >0 | >1+

| >0 | >1i-

Z

X

Y

| >Ψ

φ

θ

Figure 1.2: Projection of an arbitrary state, given by equation 1.6 on the Bloch sphere

stationary projection on the sphere if it is on resonance. Detuning and driven

evolution is discussed in section 1.3.2

1.3.1 The Density Matrix

The density matrix is a useful tool to describe mixed states. A pure state can

be described by a single wavefunction and when measured will always yield

the same result. Superposition states are a linear combination of pure states

which when measured, collapse into a pure state in order to interact with

the measuring apparatus. Mixed states are ensemble states and therefore

can be a mixture of different superposition states, as well as pure states.

Consequently when measuring a mixed state some members of the ensemble

will have opposite projections on the measurement basis. Therefore the total

measured projection over all basis states will not necessarily be normalised.

The density matrix, traditionally denoted as ρ is used to describe mixed

states. If the system could be in any linear combination of states |ψi〉, with

the probability of being in a particular state being given by pi the density

matrix would be given by:

ρ =∑

i

pi |ψi〉〈ψi| (1.8)

Some important properties of the density matrix are that it has a trace of


one and the eigen values are real, resulting in det(p) ≥ 1. The expectation

value for a quantity associated with an operator M is

〈M〉 = Tr(ρM) (1.9)

where Tr denotes the trace. The evolution of the density matrix can be

derived from the Schrodinger equation as:

ρ(t) =−i~

[H, ρ(t)] (1.10)

The density matrix can be directly related to a vector on the Bloch sphere

by tracing the projection on the Pauli operators:

rρ = (Tr(ρX),Tr(ρY ),Tr(ρZ)) (1.11)

The one to one mapping of the density matrix to a Bloch vector is confirmed

by the following identity, which is true for all 2 × 2 Hermitian matrices.

ρ =1

2(Tr(ρ)I + Tr(ρX)X + Tr(ρY )Y + Tr(ρZ)) (1.12)

This identity is also useful to determine the density matrix from a series of

measurements on a quantum system and will be used in the process tomogra-

phy in section 1.3.3. The magnitude of the vector is, however, now bounded

by the Bloch sphere and need not strictly exist on the surface of the sphere.

This can be seen from the determinant:

det ρ = 1 −X2 − Y 2 − Z2 ≥ 0 (1.13)

⇒ X2 + Y 2 + Z2 ≤ 1 (1.14)

An incoherent state is mapped to a point at the origin of the Bloch sphere

and therefore any deviation from the surface of the sphere represents decay

of some form as will be discussed in section 1.4.

In circumstances where the Hamiltonian is also time dependant he equa-

tions of motion of the density matrix are described by the Louiville-Von

Neumann equation.d

dtρ(t) = −i [H(t), ρ(t)] (1.15)

This equation is required for evaluating the result of manipulating quantum

states by the application of pulsed driving fields.


Z

X

Y

Figure 1.3: Schematic representation of an inhomogeneously broadened line evolving intime showing the precession of the Bloch vectors around the Z axis at a rate proportionalto the detuning.

1.3.2 Dynamics on the Bloch Sphere

The Bloch sphere was designed as a visualisation tool for the Bloch equations

which describe a closed two level quantum system [37]. The real power of

the Bloch sphere lies in it’s ability to visualise dynamic processes. The Bloch

equations are given extensive treatment in a number of texts [38, 39, 40,

41, 42, 35] and consequently the following discussion is intended to provide

sufficient background to properly visualise the system dynamics described in

later chapters.

If we consider a fictitious spin 1/2 particle experiencing a strong, static

magnetic field aligned with the Z axis, B0, the degeneracy of the spin states

|0〉 and |1〉 is lifted transition frequency between them is:

ω0 =2π

~µB0Z (1.16)

In rare-earth doped crystals there will always be some static inhomoge-

neous broadening and the detuning from line centre will be represented as

∆ = (ω − ω0). The rotating wave approximation results in a Bloch vector

on resonance having a stationary projection. However if the detuning is

nonzero the Bloch vector will precess around the Z axis with a frequency of

∆ as shown in figure 1.3. The consequences for the projection of an inho-

mogeneously broadened ensemble on the Bloch sphere will be discussed in

section 2.5.2.


Z

XY

θ

Z

X

Y

a

bc

Figure 1.4: Shows the action of a driving field on a series of Bloch vectors. All Blochvectors are rotated around the rotation axis, defined by the phase of the driving field. TheBloch sphere on the right describes the action of resonant driving a) in phase, b) phaseshifted by 90 and c) phase shifted by 90 and detuned.

When the system is driven on resonance the resulting action is a rotation

of the Bloch vector as shown in figure 1.4. The axis of rotation is in the X,Y

plane and by convention a driving field with no phase shift rotates about the

X axis. Introducing a phase shift of θ into the driving field results in a shift

of the rotation axis in the X,Y plane by an angle of θ relative to the X axis.

This is described by the equations of motion for Bloch vectors:

X = −∆Y

Y = ∆X + ΩZ

Z = −ΩY

(1.17)

where Ω is the Rabi frequency, which describes rotation on the Bloch sphere

due to the action of the driving Hamiltonian, Hd. The Rabi frequency is

given by:

Ω =

√

Tr (HdX)2 + Tr (HdY )2 + Tr (HdZ)2 (1.18)

with the rotation axis being,

n(θ, φ) = (Tr (HdX) + Tr (HdY ) + Tr (HdZ)) (1.19)

In the case of our spin 1/2 particle in the magnetic field, B0 the rotation is

caused by driving the system with an RF field at the transition frequency

ω0, perpendicular to B0. Assuming the driving field is in phase this results

in a Hamiltonian of the form:

Hd =µB1

2cos (ωt)X (1.20)


If the driving field is close to the transition resonance (ω ≈ ω0) then the

contribution from cos (ωt)X has a non-zero average. This can be rewritten

in the usual rotating frame as:

H =∆

2Z +

Ω

2X (1.21)

This results in the rotation axis, n being the X axis as depicted in figure

1.4, path a. If the driving field is rotated by 90 it can be seen from equation

1.19 that the rotation axis n is also shifted by 90 as shown in figure 1.4.

Detuning from resonance results in n leaving the X,Y plane as per equation

1.19. In this case the Rabi frequency should be replacesd with the generalised

Rabi frequency, which includes detuning effects and for nuclear spins has the

form:

Ω =

√

∆2 +

(

µB1

~

)2

(1.22)

1.3.3 Quantum Process Tomography

When driving a quantum system the state manipulation will never be per-

fect. Consequently we require a mathematical tool to asses what the actual

state manipulation was performed by the experiment. This is of particular

importance to QIP applications since we require high fidelity manipulation

of an arbitrary state and therefore any non-ideal state manipulations in-

troduce errors. Quantum mechanics tells us it is impossible to completely

determine the state of the system given only one opportunity to measure the

system. Quantum Process Tomography is a method by which the process

superoperator O can be determined over a series of experiments by preparing

a number of known input states ρini and measuring the output state, ρfin.

This method was derived by Jevon Longdell [43] and is equivelent to methods

derived by Chuang and Nielsen [44, 43]. Using density matricies the action

of the process superoperator O is expressed as:

ρfin = O(

ρini)

(1.23)

where ρini and ρfin are the initial and final state of the system and O is the

process superoperator. The process operator is linear and can therefore be

decomposed as:

ρfink

=∑

j

λjkρini

j(1.24)

By preparing a set of known input states that span the input state space,

ρini, and measuring the projection on σi we can create a set of measurements


si.

si = Tr(

σiO(

ρinii))

= vec (σi) · vec(

O(

ρinii))

= σiTλρini

i(1.25)

where vec is the operation of flattening a matrix into a vector, defined as

implemented in fortran libraries:

A =

[

a b

c d

]

vec(A) =

a

b

c

d

(1.26)

If we flatten both sides this produces

si =(

ρi

T)

⊗(

σiT)

vec (λ) (1.27)

where ⊗ is the kronecker product, which maps two arbitrarily dimensioned

matrices into a larger matrix with a block structure, defined as

A =

a1,1 · · · a1,n

.... . .

...

am,1 · · · am, n

B =

b1,1 · · · b1,k...

. . ....

aj,1 · · · aj, k

A⊗ B =

a1,1B · · · a1,nB...

. . ....

am,1B · · · am,nB

(1.28)

In order to determine what λ is, and therefore O we need to solve the inverse

problem such that we are confident of a physically reasonable result. The


general linear inverse problem is characterised by the following equation [45]:

y = Cx+ n (1.29)

where x ∈ RM is the measured initial state, y ∈ R

N is the measured final

state, n N dimensional variable representing the noise present in the mea-

surements and C is the process that maps the input state to the output state.

The noise, n, is assumed to be a Gaussian distributed random variable with

zero mean and a covarience matrix Γ. The noise present in the measurement

can lead to unphysical results and therefrore we draw on Bayes’ theorem

ensure a physically reasonable result. Bayes’ theorem relates joint and con-

ditional probabilities between two events, which in theis case are the initial

and final states of the system. Given two events, x and y, Bayes’ theorem

states that:

p(x, y) = p(x|y)p(y) = p(y|x)p(x) (1.30)

and therefore

p(x|y) =1

p(y)p(y|x)p(x) (1.31)

where p(x) represent what we know about x before making the observation

y, known as the prior probability, p(x|y) represents what we know about x

after making the observation y, or posterior probability. Equation 1.31 states

that a prior probability is turned into a posterior probability by multiplying

by the forward probability, p(y|x) and a normalisation factor 1/p(y). In

this context p(y|x) is considered a likelyhood function, i.e. how likely is it

that the observation y was dependant on the prior state x. This equation is

interpreted as telling us how we should change our state of knowledge of x

as a result of making an observation which yields the result y.

When applied to linear inverse problems, as stated in equation 1.29, the

likelihood function becomes [45]:

p(y|x) = p(n = y − Cx) =1

(2π)N/2√detΓ

exp

[

−E (x; y)

2

]

(1.32)

where the misfit function, E is

E (x; y) = (y − Cx)TΓ( − 1)(y − Cx) (1.33)

When E is minimised, the mapping of observations x to y has a ’maximum

likleihood’ of being described by process C. The maximum likleihood solu-

tion is obtained when

‖y − Cx‖ (1.34)

1.4 Decoherence 19

which, restated in the context of the quantum process tomography is

‖s−Mλ‖ (1.35)

where,

M =(

ρi

T)

⊗(

σiT)

(1.36)

which is satisfied by

λ = pinv(M)s (1.37)

where pinv is the Morse-Penrose pseudoinverse. We can therefore determine

λ and therefore construct the processes superoperator O using equation 1.24.

In order for the mapping of λ to s to be completely specified we require that

Rank(M) ≥ dim(λ) (1.38)

This is simply that we require at least as many measurements as there are

parameters of the system. For a single qubit, or any two state quantum

system, this can be achieved by preparing the input states |0〉, (|0〉+ i |1〉)/2,

(|0〉 − i |1〉)/2 and |1〉 projecting on the σi basis states, which are the Pauli

spin matrices (σx, σy, σz, I). This results in 12 independent measurements

on the system with the remainder of the 16, or 2n2 measurements (where n is

the number of Hilbert space dimensions) being known due to the properties

of the density matrix, defined in equation 1.12.

1.4 Decoherence

Since the quantum computer is a finite physical system it will have inter-

actions with its surrounding environment. Therefore, the finite dimension

Hilbert space of the quantum computer HQ is a subset of the total semi

infinite dimension hilbert space of the quantum computer and rest of the

universe, HE . Performing unitary operations on the finite Hilbert space HQ

results in non-unitary dynamics of HQ due to its’ projection on the total

Hilbert space, HQ ×HE . Undesired interactions between the quantum com-

puter and it’s environs represent a loss of information from the quantum

computer to the environment, referred to as Decoherence. This loss of in-

formation from the system is exactly analogous to the effect of entropy on

the quantum computer [25]. Consequently Deutsch was able to show that

using the non-unitary dynamics of the quantum computer interacting with

it’s environment and the physical interpretation of the Church-Turing prin-

ciple the third law of thermodynamics can be derived. For the first time real


physical significance could be derived from computational theories. This also

enabled new perspectives on entropy and it’s relation to the computational

accumulation of knowledge.

Decoherence is the border between the quantum and classical worlds. It

is the loss of phase coherence between quantum objects that leads to classical

physics emerging from quantum mechanics [17, 18, 46]. Irrespective of the

final application reducing decoherence of quantum systems is important for

developing technology that utilises quantum mechanics.

Population decay and Decoherence, or longitudinal and transverse relax-

ation in NMR terminology has been an major area of study since the advent

of NMR. Reducing the rate at which a system relaxes allows greater spec-

tral resolution thereby allowing subtle interactions to be discerned. While a

detailed discussion of these developments is beyond the scope of this thesis,

several results of this work are fundamentally important for this work. There

are a number of models that have emerged that treat the interaction between

the quantum system of interest and the surrounding environment in different

levels of detail. These models can be put in three groups according to the

assumptions about the system-environment interaction.

1.4.1 System Bath Interactions

All decoherence that is not purely due to population decay results from in-

teractions between the quantum system of interest and the surrounding en-

vironment. In the context of the systems investigated in this work we are

interested in a nuclear spin system IS surrounded by a nuclear spin bath, IB.

The most general Hamiltonian to describe this is as follows:

H0 = HS ⊗ IB + IS ⊗HB + HSB

HSB =∑

γ

Sγ ⊗ Bγ (1.39)

where HS describes the quantum system of interest, HB describes the sur-

rounding “bath” and HSB is the interaction between them with Sγ and Bγ

being operators on the system and bath respectively. The Sγ operators are

responsible for decoherence of the system due to interaction with the bath.

The result of the system-environment interaction, HSB, is to induce a

time varying perturbation in the state energies[38, 39, 47, 41]. When con-

sidering the effect of HSB on a transition of interest in HS both the time

scale of the perturbation, τB, and the frequency shifts, ∆ω, that they in-

duce are important. The system-environment interaction is a homogeneous

1.4 Decoherence 21

process as the environment is considered to be equivalent between different

sites. The effect of the ∆ω is generally discussed as either spectral diffusion

or decoherence depending on the ratio of ∆ω to τB. Spectral diffusion refers

to large, slow shifts in transition frequency or τB∆ω ≫ 1, resulting in a slow

random walk. While these processes certainly contributes to dephasing by

accumulating a phase relative to the reference oscillator generally a faster,

lower energy process will have decohered the transition before a significant

amount of configurations of the spectral diffusion processes are progressed

through. This makes the homogeneous linewidth dependant on the period of

measurement. The short time dephasing is of most interest to this work and

will be examined in more detail.

In the regime where where the shifts are small and rapidly change, τB∆ω ≪1, the effect of the perturbations on a transition of HS acts like a homoge-

neous broadening mechanism. In this case a phenomenological description

of an ensemble of HS systems interacting with a bath can be constructed

using decay constants, T1 and T2, describing the population and coherence

decay respectively. In the basis of in phase superpositions, quadrature su-

perpositions and population (+,−, Z) the relaxation matrix can be written

as:

R =

1/T2 0 0

0 1/T2 0

0 0 1/T1

(1.40)

These constants can be applied to the Bloch equations as will be shown in

section 1.4.2.

This approach is convenient and conceptually simple, becoming standard

terminology in NMR and many other areas of spectroscopy. The ability to

apply it to the Bloch equations allows visualisation of the effect of dephasing

and it provides a simple parameter to describe homogeneous decay processes

and compare lifetimes of different quantum systems.

If, however, τB∆ω ≈ 1 or we want to modify the effect of the interaction

between the system and it’s environment then we need to understand the

perturbations in greater detail. For this we can use a semi-classical approach

where the effect of the environment is modelled as a time varying field due

to an array of particles in the lattice. The interaction of the particles in the

environment with each other is treated by statistical models with a charac-

teristic time for reconfiguration of the environment. This will be investigated

for the specific case of nuclear spins interacting with a nuclear spin ‘bath’ in

section 2.5.3.

Knowledge of the range of possible fields due to environmental configura-


tions and the time scale on which the field changes allows an assessment of

how the system or environment can be driven to minimise the effect of their

interaction. These techniques known as Dynamic Decoupling techniques can

drastically increase the observed T2 of the system [41, 39, 42, 38, 13, 48, 49,

50, 51, 52, 53, 54, 55, 56, 9, 8, 57, 58], often by several orders of magnitude.

This illustrates that T2 is a convenient conceptual tool, but the underlying

dynamics that it represents should always be carefully considered.

There are also cases where the system of interest is sufficiently strongly

coupled to the environment that driving the quantum system of interest also

drives the local environment, resulting in back action on the system of interest

[59]. In this case a full quantum mechanical description of the coupling to

the environment is required to adequately describe the response to a driving

field. In this circumstance T2 is far too simple a description to encapsulate

the dynamics of the system. Such a system is also not suitable for quantum

computing because the system cannot be approximated as a closed two level

system.

1.4.2 Decoherence on the Bloch Sphere

The effect of population decay and decoherence, if able to be approximated

by a T2 decay can be incorporated into the Bloch equations as follows:

X = −∆Y − X

T2

Y = ∆X − Y

T2+ ΩZ

Z =Z − Z0

T1− ΩY

(1.41)

where Z0 is the equilibrium state population, ∆ is the detuning (ω − ω0), Ω

is the Rabi frequency, T1 and T2 are the population lifetime and decoherence

time respectively and dotted variables denote the time derivative.

The action of T2 on the Bloch vector causes it to decay toward the Z axis.

The effect of T1 on a Bloch Vector is that the population component will decay

toward the mixture of states defined by the Boltzmann distribution for the

given temperature. The bahaviour is clearest when considering a system at

zero temperature, in which case the Bloch vector decays to the |0〉 state at the

“south pole”. The remainder of the discussion will consider zero temperature

systems for conceptual clarity. The effects of T1 and T2 on the evolution of

an ensemble spin vector in a 50-50 superposition state on the Bloch sphere

are shown in figure 1.5. First considering a spin ensemble on resonance as

1.5 Error Correction 23

Z

X

Y

aZ

X

Y

b

Z

X

Y

cZ

X

Y

d

Figure 1.5: Schematic representation of an inhomogeneously broadened line evolving intime showing the precession of the Bloch vectors around the Z axis at a rate proportionalto the detuning.

shown in figure 1.5a we see that if T2 < T1 the spin vector undergoes a T2

decay toward the origin before the T1 process becomes significant. If the only

coherence loss if from population decay, or T2 = T1 then the decay proceeds

toward |0〉 and the Z axis at an equal rate. If we follow a detuned ensemble,

T2 decay causes the spin vector to spiral toward the Z axis as shown for

T2 ≪ T1 and T2 < T1 in figure 1.5c and d.

1.5 Error Correction

As previously mentioned in section 1.4 Deutsch was able to show that using

the non-unitary dynamics of the quantum computer interacting with it’s

environment and the physical interpretation of the Church-Turing principle

the third law of thermodynamics can be derived [25]. Therefore entropy

will cause information to be lost to the environment via T1 and T2 processes,

with a corresponding accumulation of errors. The precision or fidelity of qubit

manipulations also contributes to the error accumulation during calculations.

This requires the computation to proceed with a finite probability of an

error occurring with strategies to correct the error broadly known as “Fault

Tolerant” quantum computing [60, 32, 61, 62, 63]. As with analog computers,

the possible errors form a continuum and accumulate over the course of a

calculation. Small errors of this kind are difficult to correct and, due to

entropy are impossible to completely remove.

Estimates of the tolerable error rate are of the order of 10−6 [32], requiring


extremely high fidelity (F > 0.999999) from all operations performed on the

quantum computer. If these fidelities are to be reached, the decoherence

time of the system needs to be significantly larger than the gate operation

time. Estimates of the required ratio of the gate operation time to the

decoherence time of the system are of the order of 105 [35]. These benchmarks

are extremely hard to reach and as yet no scalable system has achieved both

gate fidelity and coherence time to gate operation time ratio. Consequently

investigating the decoherence present in the candidate quantum system is

of paramount importance to understanding the potential of the system to

function as a quantum computer.

The emerging strategies for correcting errors in quantum computers draw

extensively from NMR decoupling methods and classical error correction

techniques. These strategies can be broadly grouped as Quantum Error Cor-

rection Codes (QECC) [60, 64, 65, 66], Decoherence Free Subspaces (DFS)

[67, 68, 69, 70] and Dynamic Decoherence Control [8, 10, 11, 12, 13, 71].

1.5.1 Quantum Error Correction Codes

Quantum Error Correction Codes adapt classical coding concepts to deal with

the more challenging problem of preserving phase and population informa-

tion. All codes draw from classical techniques of redundancy of information

by encoding information using several physical qubits to represent one log-

ical qubit. Active schemes can then be to correct the error. This can have

several forms either measure one physical qubit or partially measure several

physical qubits [72] to check for errors without collapsing the logical qubit.

The measurement scheme must be carefully constructed such that it only

measures the error and does not gain any information about the state of the

qubit before the error occurred [72]. Alternatively the correction can take

place as part of the decoding process of a logical qubit [66].

The encoding is important so that these quantum codes, like their classi-

cal counterparts, can discern between valid computational states and errors.

This requires that any superposition of code words is also a code word and

that there is a volume around each code word that is only occupied due to

an error. If an error is detected, a correcting operation can be performed

on the logical qubit before the calculation proceeds. These codes are typi-

cally limited to correcting a single error in population and phase of the state

[60, 64, 56]. Only correcting for single errors allows a minimum “distance”

between code words, thereby making efficient use of the code space. It is

important to note that the rate at which error correction must be performed

is minimally the rate at which errors occur, and therefore directly related to

1.5 Error Correction 25

the decoherence experienced by the system.

QECC codes have major implications for scalability as codes that can

correct for all errors generally require at 5 to 9 physical qubits [60, 64, 65, 73].

1.5.2 Decoherence Free Subspaces

Decoherence Free Subspaces (DFS) are a set of techniques that encode a

logical qubit using a number of physical qubits to protect against specific

decoherence mechanisms [67, 74, 70, 64, 68, 69]. Consequently they are

sometimes referred to as “noiseless” or “error avoiding schemes”. Like QECC

a single qubit is encoded using several physical qubits to create a DFS. This

encoding is designed to create multi qubit states that have special symmetries

with respect to the dephasing mechanisms. This results in a the definition

that a Hilbert space H is said to possess a decoherence free subspace H if

the evolution inside H ⊂ H is purely unitary.

Encoding information in this way is only effective at preventing decoher-

ence if the perturbing field is correlated for all physical qubits that constitute

the logical qubit. In certain liquid phase NMR systems and ion traps this has

been shown to be effective at reducing decoherence [73, 75]. In general, how-

ever, one cannot assume correlated, equivalent perturbations at two or more

different qubit sites. In solid state systems there may be several elements of

the dephasing bath between nearest neighbour qubits. To correct for asym-

metric perturbations or uncorrelated perturbations, other techniques such as

QECCs or Dynamic Decoherence Control need to be incorporated [55].

DFS codes can be effectively implemented with less physical qubits than

QECCs, the simplest requiring only 2 qubits [67]. Asymmetric or uncorre-

lated perturbations acting on the physical qubits still result in dephasing.

1.5.3 Dynamic Decoherence Control

Dynamic Decoherence Control (DDC) techniques, or phase cycling in NMR

terminology, were developed by NMR spectroscopists remove unwanted con-

tributions to the spin Hamiltonian [48, 57, 51, 52, 50]. This is achieved by

driving the system of interest or the bath such that they are decoupled and

the effect of system - bath interactions is minimised. These techniques have

been particularly useful in increasing the resolution of liquid phase NMR ex-

periments [76, 77, 78, 79, 41, 42, 80, 81, 48, 51, 82, 53, 83, 84]. Of this large

range of potential DDC techniques many are designed to decouple a partic-

ular interaction and often the technique is specifically designed to protect a

particular state and as such do not leave an arbitrary state unchanged. In


order to perform QIP we need to use arbitrary states which limits our choice

of DDC techniques.

There are two DDC techniques investigated in this work. The technique

given most attention by the quantum computing community for increasing

decoherence times is a variant of the Carr-Purcell Mieboom-Gill (CPMG)

pulse sequence [81], well known to NMR spectroscopists and renamed as

“Bang Bang” decoupling [85, 13, 62, 8, 9] due to analogy with the classical

error correction protocol [8]. This scheme ideally drives the system of interest

faster than the reconfiguration time of the bath such that the action of the

bath averages to zero. Application of this technique is studied in chapter 6.

While the Bang Bang decoupling sequence acts on the quantum system

of interest we can also act on the bath to increase the decoherence time. If

the bath is reconfiguring slowly, resulting in a random walk of the transition

frequency the transition linewidth can be reduced by driving the bath suf-

ficiently hard such that it’s contribution to transition frequency is averaged

out. This process, called Magic Angle Line Narrowing (MALN) [53, 54, 41]

is routinely used in NMR to decouple a spin system from a surrounding spin

bath and is investigated in chapter 7.

1.5.4 A Quiet Corner of Hilbert Space

All of these previously mentioned decoherence control methods are still lim-

ited by the underlying dynamics of their local environs and the sensitivity of

the transition used to those dynamics. QECCs must check for errors before

there is a significant probability of an error occurring. Decohering of DFS

codes is limited by asymmetry of the System-Bath interactions and Dynamic

Decoherence Control techniques must be applied faster than the characteris-

tic time for the bath to reconfigure to be effective [13]. Therefore all of these

techniques benefit from reducing the effect of system bath interactions. Fur-

ther, any resources used for error correction cannot be used for computation.

Therefore a useful quantum computer needs to be built in a quiet corner of

Hilbert space such that it does not spend most of it’s time error correcting

itself.

The increase in the number of qubits required for error correction is a

major obstacle to achieving Fault Tolerant quantum computation. Qubits

are a precious commodity and the difficulty of increasing the number of

available qubits by a factor of ∼5 is very challenging.

Consequently the approach of our group has been to find quantum sys-

tems that have are inherently low decoherence and determine how far de-

coherence can be reduced. Rare Earth ions have the narrowest optical

1.6 Why the Rush? 27

linewidths observed in solids [5] as well as displaying moderately long co-

herence times in hyperfine ground states. This is due to Rare Earth ions

being well isolated from their surrounding environment. In this work we in-

vestigate the limits of decoherence as experienced by Praseodymium ions in a

Yttrium Orthosilicate host to create a low noise environment for the storage

and processing of quantum information using nuclear spins.

1.6 Why the Rush?

While the theoretical development of quantum computing was fundamentally

interesting for physics and computer science quantum computation did not

gain widespread attention until in 1994 Peter Shor showed that a quantum

computer could efficiently solve a problem of great interest. This was finding

two prime factors given only their product [86]. No known classical algorithm

can perform this calculation efficiently and as a consequence it provides the

security of what is known as Public key cryptography.

Public key cryptography allows two parties to exchange secure messages

with all of their correspondence public and no prior exchange of code books.

The lack of need for exchanging code books fundamentally changed the

manner in which secure communications took place. Consequently it is

an enabling technology for secure personal or business communications, e-

commerce and heavily utilised in defence applications. Therefore the interest

in breaking this encryption has been intense since the advent of public key

cryptography.

Therefore a quantum computer, comparatively simple to the current su-

percomputers could unlock a significant amount of protected information

with obvious application in military and industrial espoiange. In combina-

tion with this is the desire by intelligence agencies and finance sector to

protect their information from hacking by a quantum computer. This has

resulted in frantic research to find a suitable system for scalable quantum

computing to take advantage of this opportunity. It also poses some serious

questions as the what will be the spoils for the victor.

1.7 Other application and fundamental inter-

est

The required parameters for a scalable quantum computer are still unsat-

isfied by any system and some would argue that the requirements are too


paradoxical to be satisfied. Despite this there are several areas that benefit

from the study of decoherence times and the extension thereof.

The most closely related is that of quantum memories. Quantum commu-

nication links are already in place, so irrespective of the future of quantum

computing, the ability to store and recall a quantum state is important. If

these networks are to extend beyond point to point links there will be a

need to cache and route quantum information. The ability to store optical

information using nuclear spins is clearly beneficial for integration into op-

tical networks. This memory could be based on either Electromagnetically

Induced Transparency (EIT) or stark echos as described by Alexander et al.

[87].

Most EIT experiments are performed in dilute gasses and as such the

coherence time is limited by diffusion and collisions. The atomic motion is

a problem, even at the limit of trapped dilute gas techniques since in a Bose

Einstein Condensate (BEC). For a Rubidium 87 BEC at 100nK atoms move

of the order of 5mm/s and consequently will move half an optical wavelength

in approximately 6×10−6s, moving through a 1mm laser beam in only 0.2s.

This creates some hard limits as to maximum time scales that can be expected

from EIT experiments, even after pushing dilute gas technology to it’s limit

with a BEC.

In a rare-earth doped crystal since the ions are not moving the upper limit

is the population lifetime of the hyperfine ground states, which can be several

weeks [4]. If techniques can be developed to achieve a significant fraction

of the population lifetime rare-earth doped crystals have great promise to

provide a very coherent Λ system for quantum optics experiments and non-

linear optics. EIT is one application that could achieve slower light and

stopping light for longer periods, thereby extending prior work in dilute gases.

In extending the decoherence time the goal is to remove the effect of

the dominant dephasing mechanism. In doing so more subtle interactions

can be discerned. The techniques presented herein allow the observation

of strain broadening in the hyperfine ground state levels of Pr3+:Y2SiO5 ,

which is usually dominated by magnetic dephasing interactions. This allows

for a future study of the correlation of strain broadening between optical and

hyperfine inhomogeneous linewidths.

Chapter 2

Rare Earth Ion Spectroscopy

“We all agree that your theory is crazy, but is it crazy enough?”

- Niels Bohr (1885-1962)

Rare-Earth ions exhibit rich spectroscopy making them both interesting

scientifically and an indispensable part of our modern technology. Spec-

troscopists were initially fascinated by the observation of optical spectral

features that are unusually narrow for solid state systems, which prior to

the invention of the laser required massive spectrometers to obtain data that

was not instrument limited [47, 88]. The vast array of transition energies

available from rare-earth compounds revealed by this work proved ideal for

application to phosphors for TV screens, X-ray intensifying screens and tri-

phosphor fluorescent lamps.

Spectroscopy of rare-earth ions, like many other areas of spectroscopy un-

derwent rapid development in the 1960s with the invention of the laser. This

initial phase focused on finding and studying the dynamics of lasing systems,

for which the narrow optical transitions of rare-earth ions were particularly

attractive. Materials containing Neodymium, such as Nd:YAG and Nd:YLF

are of particular note, becoming the de facto standard for many lasing appli-

cations due to the high output power and narrow linewidth. More recently

Erbium has become an indispensable part of long haul fibre optic commu-

nications with the development of in-line Erbium Doped Fibre Amplifiers

(EDFAs).

The development of tunable single frequency dye lasers in the 1970s dra-

matically increased the systems that could be studied. This development

provided researchers with sufficient resolution to study inhomogeneous broad-

ening effects. With the use of holeburning and time-domain techniques the

30 Rare Earth Ion Spectroscopy

homogeneous broadening of these transitions could also be probed. Homoge-

neous linewidths can be up to 107 times narrower than the inhomogeneous

linewidths with linewidths as small as 122Hz for optical transitions have been

reported [5].

These studies led to new applications using holograms which exploited the

inhomogeneous broadening to store [89, 90], process [91, 92, 93] and route

[94] classical information. Although not yet reaching commercial application

this demonstrates diverse range of applications that a detailed knowledge of

rare-earth spectroscopy allows.

This chapter provides a description of the interactions that create the

rich spectroscopy of rare-earth ions. The discussion focuses on details that

relate to hyperfine transitions of Pr3+:Y2SiO5 and as such is not a complete

discussion for all rare-earth ions.

2.1 Introducing The Lanthanides

The rare-earth elements are formally known as the Lanthanide group, con-

sisting of the 15 elements from Lanthanum (Z = 57) to Lutetium (Z = 71)

during which the 4f electron shell is filled. The term rare-earth was ascribed

to this group by Johann Gadolin in 1794. Rare due to initially only being

found in abundance in Ytterby mine in Sweden, in “earthy” coloured oxide

mixtures. The only member of the group that actually is rare is promethium

due to being radioactive and is often found in uranium ores since it is part

of the uranium decay process [95]. All stable rare-earth ions have similar

abundance to arsenic and mercury, neither of which are considered rare.

The rare-earth elements share the same bonding electrons, consisting of

the 6s, 5s and 5p shells (figure 2.1) and consequently share many chemical

properties. The elements all exhibit strong electropositvity and their bond

is often well approximated as ionic. The most stable oxidation state (M3+)

dominates chemistry of the rare-earth elements in a manner not seen in any

other group in the periodic table. Therefore the distinguishing feature of rare-

earth ions is their size, which uniformly reduces across the group. This is

know as the Lanthanide contraction. Yttrium, due to sharing this oxidation

state (M3+) and having a similar size is often included in a discussion of

rare-earth ions.

2.1 Introducing The Lanthanides 31

Figure 2.1: Radial distribution functions of the 4f, 5s, 5p and 6s states calculated forGd+[96]. The 4f electrons are partially shielded from perturbations by the 5s, 5p and 6sorbitals


2.2 4f Energy Levels

In high symmetry hosts the 4fn → 5dn transitions are forbidden by parity

conservation. It is only when rare-earth ions are placed in low symmetry hosts

that the orbital angular momentum is no longer a good quantum number and

optical 4fn → 5dn transitions are allowed.

The rich nature rare-earth spectroscopy stems from the purely electronic

4fn → 5dn transitions. The partial shielding due to the 5s and 5p electrons

results in narrow optical transitions and reduces the influence of crystal strain

and of lattice phonons.

There are a number of contributions to the energy levels of the 4fn → 5dn

transitions that result in the rich spectra observed in rare-earth spectroscopy.

In a purely coulombic potential the 4f states are degenerate. Rare Earth ions

have strong spin orbit coupling, typical of heavy atoms, which splits the 4f

states into manifolds. These manifolds consist of states with the same total

angular momentum, J . Since J is conserved by spin orbit coupling it is a

good quantum number to label the manifolds. The spin orbit coupling is the

largest contribution and therefore primarily defines the spectrum. The spec-

tra produced by the spin orbit coupling as the rare-earth group is progressed

through is best shown by the seminal work of Dieke and Carnall [97, 98] us-

ing a lanthanum trichloride host, shown in figure 2.2. Interactions between

the rare-earth ion and the host lattice can be treated as perturbations to the

spin orbit coupling. At liquid helium temperatures each J manifold is then

split into a maximum of 2J + 1 levels by interaction with the crystal field.

When considering their magnetic properties, rare-earth ions can be di-

vided into groups according to having an even or odd number of f electrons.

Those with an odd number of electrons have a large magnetic moment due

to the unpaired spin, which splits the energy levels into a doublet known as

a Kramers Doublet. For rare-earth ions with an even number of electrons

or non-Kramers ions, such as Pr3+ used in this work, placed in sites with

axial or lower symmetry the crystal field levels are singlets due to “quench-

ing” of the angular momentum. Quenching refers to all states having zero

angular momentum due to the Hamiltonian not commuting with an angular

momentum operator in any direction.

All of the work presented herein was performed using trivalent praseodymium

doped yttrium orthosilicate (Pr3+:Y2SiO5). Pr is a non-Kramers ions and

Y2SiO5 is a low symmetry host belonging to the C62h space group. Pr can sub-

stitute for Y at both Y sites, each of which has C1 symmetry. Consequently

to investigate the remaining perturbations the discussion will be restricted

to rare-earth non-Kramers ions in hosts with lower than axial symmetry.

2.2 4f Energy Levels 33

Figure 2.2: Energy levels of triply ionised rare earth ions in LaCl3. The semicirclesindicate fluorescing transitions


2.3 Hyperfine Interaction

Pr has nuclear spin of 5/2 with each crystal field split level further split into

three levels levels by the hyperfine interaction. The hyperfine splittings are of

the order of 10MHz with each level being doubly degenerate in zero applied

magnetic field. Manipulation of these hyperfine states is the basis of this

thesis. For the experiments performed in this thesis, rare-earth ions can be

described using the following Hamiltonian:

H = [HFI + HCF ] + [HHF + HQ + HZ + Hz] (2.1)

As described previously the spin orbit coupling and crystal field couplings

dominate the spectrum, represented as HFI (Free Ion) and HCF respectively.

The remaining terms in order of appearance are the hyperfine coupling be-

tween the 4f electrons and the nuclear spin, nuclear electric quadrupole

interaction, electronic Zeeman and nuclear Zeeman.

As previously mentioned angular momentum of non-Kramers ions in low

symmetry hosts is “quenched”. This results in the degeneracy within the J

manifolds being completely lifted and all states are electronic singlets with

only second order contributions from the last four terms in equation 2.1.

The last four terms result in approximately equal contributions to state en-

ergies, hence the grouping of terms in equation 2.1. Hyperfine and electronic

magnetic effects are therefore described by applying second order perturba-

tion theory. The effective Hamiltonian including the nuclear Zeeman and

quadrupole interactions is [47]:

Heff = g2Jµ

2BB ·Λ ·B − (2AJgJµBB ·Λ · I −Hz)− (A2

JI ·Λ · I −HQ) (2.2)

where µB is the Bohr magneton, g is the Landau g value, B is the magnetic

field vector, I is the nuclear spin state, AJ is given by

AJ = 2µB~γN〈r−3〉〈J‖N‖J〉 (2.3)

which is the hyperfine interaction for a particular LSJ state given the mean

4f electron-nuclear distance 〈r−3〉 and nuclear gyromagnetic ratio γN . The

tensor Λ is given by

Λ =

2J+1∑

n=1

〈0| Jα |n〉〈0| Jβ |n〉∆En,0

(2.4)

where 〈0| is the level described by Heff , |n〉 is the other crystal field level of

2.3 Hyperfine Interaction 35

the LSJ state involved and ∆En,0 is the energy difference between the states

En − E0.

Equation 2.2 can be rewritten as

Heff =≡ g2Jµ2BB · Λ · B + H′

z + H′Q (2.5)

where the first term is the quadratic Zeeman shift, H′z is the enhanced nu-

clear Zeeman and H′Q is the effective quadrupole, incorporating both the

quadrupole and pseudoquadrupole interaction. H′z has the form [47]

H′z = −~ [γxBxIx + γyByIy + γzBzIz] (2.6)

where x, y and z are the axes of the Λ tensor with the effective gyromagnetic

ratios given by

γα = γN + 2gJµBAJΛαα/~ (2.7)

The quadrupole interaction is described by

HQ = P[(

I2z′ − I(I + 1)/3

)

+ (η/3)(

I2x′ − I2

y′

)]

(2.8)

where P is the quadrupole coupling constant, η is the electric field gradient

assymetry parameter and x′, y′, z′ are the principal axes of the electric field

gradient tensor.

The second order magnetic hyperfine or pseudo-quadrupole interaction,

described by the term A2JI ·Λ ·I in equation 2.2 can be rewritten in the same

form as the quadrupole contribution HQ, i.e.,

Hpq = Dpq

[

I2z − I (I + 1) /3

]

+ Epq[

I2x − I2

y

]

(2.9)

The zero field pseudoquadrupole parameters can be determined from the

original Λ description as follows:

Dpq = A2J ·

[

1

2(Λxx + Λyy) − Λzz

]

(2.10)

Epq = A2J ·

1

2(Λxx + Λyy) (2.11)

When combining the pseudo-quadrupole and pure quadrupole terms to cre-

ate a single effective quadrupole Hamiltonian it should be noted that in low

symmetry sites the axes of Λ and HQ (x′, y′, z′) are in general different. When

the sum of these terms is diagonalised, a third set of principal axes is found

result, (x′′, y′′, z′′). In low symmetry sights the crystal field and effective


quadrupole principal axes will only coincide when the enhanced nuclear Zee-

man and pseudo-quadrupole interactions dominate the contribution of the

nuclear Zeeman and electric quadrupole.

H′Q = D

[

I2z′′ − I (I + 1) /3

]

+ E[

I2x′′ − I2

y′′

]

(2.12)

In order to accurately calculate the enhanced Zeeman and pseudo-quadrupole

interactions the crystal field wavefunctions that define Λ need to be known

precisely. Precise determination of Λ requires fitting these theoretical re-

sults to experimental data. For low symmetry sights fitting all the required

parameters is difficult due the principal axes being free parameters.

2.3.1 Yttrium Orthosilicate: The Gracious Host

Yttrium Orthosilicate (Y2SiO5) is a low symmetry (C62h) insulator host, the

crystallographic properties of which are summarised in table 2.1. The trans-

lational unit cell consists of with two formula units with the resulting crystal

structure shown in figure 2.4. This creates two crystallographically inequiv-

alent sites at which the Pr can substitute for Y, labelled ‘site 1’ and ‘site 2’

[6]. While the crystal has C62h symmetry the Y sites have C1 or no symmetry.

Each crystallographically identical site in fact consists of a pair of magneti-

cally inequivalent sites, related by the C2 axis, labelled ‘a’ and ‘b’. Only site

1 ions are used in this work.

While Y is a good substitutional ion for all rare-earth ions, Pr and Y

are a particularly good match with the ionic radius for the 3+ oxidation

state being 1.03 and 0.97 respectively. As previously mentioned both Pr and

Y have a stable 3+ oxidation state and consequently substitution does not

require any charge compensation in the crystal.

Despite the structure of Yttrium Orthosilicate being known since 1969

[99] the ability to grow Y2SiO5 crystals did not eventuate until the late

1980s. This was partially due to being a high temperature crystal which

poses significant fabrication challenges. Consequently Y2SiO5 is a relatively

new host for rare-earth ions [100, 88]. Due to the small magnetic moment

of Yttrium both Y2SiO5 and Y2O3 when doped with rare-earth ions result

in the some of the narrowest homogeneous optical transitions observed in a

solid [5, 100].


Space group C62h

Y site symmetry C1

Lattice constants

a (A) 14.371b (A) 6.71c (A) 10.388β 122.17

Table 2.1: Y2SiO5 Crystal parameters

2.3.2 Hyperfine Interaction in Praseodymium Doped

Y2SiO5

The complete description of the state energy within one LSJ state of equation

2.2 can be rewritten using an effective Zeeman M tensor and an effective

quadrupole tensor Q as [101]

H = B · M · I + I · Q · I (2.13)

This description is experimentally usefull since we can measure the zero field

aplittings produced by the I ·Q ·I interaction and the response to a magnetic

field and therefore the B ·M ·I interaction. The zero field splittings produced

by the quadrupole splittings are 10.2 and 17.3MHz [102] as shown in figure

2.3.

2.3.3 M and Q

In order to determine M we need to measure the spectrum of the hyperfine

ground state for a number of magnetic field configurations. The applied

magnetic field must be rotated such that it spans the space. Peaks in the

spectrum are then assigned to a particular state and a minimisation routine,

such as simulated annealing method [104, 105], is used to fit the M and Q

tensors. Knowledge about the site symmetry can be used to constrain the

axes of M and Q if the symmetry is sufficiently high and thereby reduce the

free parameters.

In axial or higher symmetry Λ and HQ share the same principle axes, re-

ducing the free parameters. Consequently the Y site being C1 or no symmetry

is the least constrained situation, resulting in 12 free parameters [105, 107].

This is further complicated since each crystallographic site consists of a pair

of magnetically inequivalent sites, related via the Y2SiO5 C2 axis. Therefore,

any y component of the applied field experienced by site ‘b’ ions will appear

rotated by 180 with respect to site ‘a’ ions. Consequently there are two sets


32/+-

12/

+-

52/

+-

32/+-

12/

+-

52/

+-

4.6 MHz

4.8 MHz

17.3 MHz

10.2 MHz

H

D1

2

34

605.7 nm

Figure 2.3: The energy levels of the Pr ground and excited state in Pr3+:Y2SiO5 labelledaccording to Ham et al [103, 102]

of spectral lines whenever the applied field has a projection on the y axis.

By performing a detailed field rotation study the M and Q tensors for site

1 were recently determined [105, 107] for Pr3+:Y2SiO5 to be:

Q = R(αQ, βQ, γQ)

−E 0 0

0 E 0

0 0 D

RT (αQ, βQ, γQ)

M = R(αM , βM , γM)

gx 0 0

0 gy 0

0 0 gz

RT (αM , βM , γM)

(2.14)

where E = 0.5624Mhz, D = 4.4450Mhz, g = (2.86, 3.05, 11.56)kHz/G, and

the Euler angles are (αQ, βQ, γQ) = (−94, 58.1,−20.7) and (αM , βM , γM) =

(−99.7, 55.7,−40). These values are for the crystal aligned with the C2 axis

in the y direction, and the z axis is the direction of linear polarization of

the Pr optical transitions. These tensors are highly anisotropic due the low

symmetry of the site. The agreement between experiment and the fitted M

and Q tensors is very good, as shown in figure 2.5.


a

c

b

o

YSi

O

Figure 2.4: Yttrium Orthosilicate structure. Figure originally printed in Ching et.al.[106]


0 500 1000 1500 20000

10

20

30

Magnetic Field (Gauss)

Freq

(M

Hz)

32/ 52/+- +-

12/

32/+- +-

52/12/+- +-

Figure 2.5: Shows part of the data used to fit M and Q with the spectrum predicted bythe fitted data

2.4 Homogeneous and Inhomogeneous Broadening 41

Figure 2.6: An inhomogeneously broadened transition consists of a continuum of ho-mogeneously broadened packets offset in frequency. The height represents the number ofions in each packet. This diagram is not to scale and in a typical rare-earth system theratio of inhomogeneous to homogeneous linewidth is of the order of107

2.4 Homogeneous and Inhomogeneous Broad-

ening

The broadening of transitions can be separated into two categories of Homo-

geneous and Inhomogeneous. Homogeneous broadening refers to broadening

that is common to all ions, while inhomogeneous broadening is experienced

differently for each member of the ensemble. The homogeneous linewidth

would be be the linewidth measured if a single ion was probed. The lower

limit of the homogeneous linewidth is determined by the excited state life-

time of the transition studied. In general, dynamic interactions with the

host broaden the transition significantly beyond this limit. At higher tem-

peratures phonon interactions dominate the homogeneous linewidth contri-

butions, while at liquid helium temperatures the exchange of energy between

spins tends to dominate.

Every crystal has some degree of strain due to solidifying at a finite tem-

perature. Combined with the strain are point defects, chemical impurity and

other effects which cause a spatially varying local potential. This results in

a different resonant frequency for each ion in the ensemble and is referred to

as inhomogeneous broadening. This detuning is static for the crystals used


in this work. For the systems considered in this work and rare-earth systems

in general, the inhomogeneous broadening is significantly greater than the

homogeneous broadening and consequently the transitions are referred to as

inhomogeneously broadened.

2.5 Optically Detected NMR and Coherent

Transients

All experiments performed in this work fall under the banner of Optically

Detected Nuclear Magnetic Resonance (ODNMR). Optical detection has sig-

nificant sensitivity advantages compared with direct detection of the RF

photons traditionally used in NMR due to the higher efficiency of optical de-

tectors. ODNMR therefore allows probing significantly more dilute samples

than traditional methods with a large Signal to Noise Ratio (SNR).

2.5.1 Raman Heterodyne

While Raman heterodyne is a useful technique for many areas of spectroscopy

the following discussion is limited to optical transitions split by the hyperfine

interaction. Raman heterodyne signals are only observed in systems which

have a common excited state for both hyperfine ground states, often referred

to as a Λ system. This only occurs in systems with sufficiently low symmetry

that spin is no longer a good quantum number and all transitions are allowed.

For the purpose of this discussion we will label the hyperfine ground states

as |1〉 and |2〉 with the common excited state being |3〉 as shown in figure

2.7. Driving the system at ω13 and ω23 a coherent relationship is set up

between all of the levels. If there is a population difference between |1〉 and

|2〉 then the system will emit coherent radiation at the frequency ω32. The

hyperfine splitting for Pr3+:Y2SiO5 are on the order of 10 MHz and as such

the associated wavelength is large compared to typical sample dimensions

of several mm. Therefore the phase matching is effectively satisfied over

the sample and as such the coherent emission at ω32 is collinear with ω23.

This results in a beat note on the transmitted light at the frequency of the

hyperfine transition ω12, the amplitude of which is defined by the population

difference between the levels.

The benefit of using Raman heterodyne detection of nuclear spin states is

the detection efficiency and flexibility. Optical detectors such as photodiodes

are commonly available with high quantum efficiency (> 80%). Direct RF

detection of spin resonances typically requires resonant detection coils placed

2.5 Optically Detected NMR and Coherent Transients 43

23

12

31

| >1

| >3

| >2

Figure 2.7: The Raman heterodyne interaction: By driving the system at ω13 and ω23

a coherent relationship is set up between all 3 levels, resulting in coherent emission at ω32

proportional lto the population difference between the ground states |1〉 and |2〉

close to the sample, limiting the experimental configurations, whereas optical

detection only requires a beam path through the sample. A further benefit

of using optical detection is that significantly lower concentration samples

can be used which is of particular importance for maximising T1, as will be

discussed in chapter 4.

2.5.2 Spin Echos

As previously discussed in section 1.3.2, detuning from resonance causes the

spin vector to rotate round the Bloch sphere at a rate equal to the detuning.

The different rate of precession around the Bloch sphere of each component of

an ensemble causes the total projection of an ensemble superposition state to

decay, termed a Free Induction Decay (FID) as shown in figure 2.8. An FID

is typically observed by driving the transition and observing the coherence

decay as the driving field is switched off and as such is maximised if the

transition is driven to a 50-50 superposition state. The FID decay rate is

proportional to the inhomogeneous linewidth of the transition, given by the

following relation.

ωinh =2π

T ∗2

(2.15)

where T ∗2 is the time taken for the ensemble projection to reach 1/e of the

value when the driving field is turned off and ωinh is the inhomogeneous

linewidth of the transition. If the inhomogeneous broadening is static then

despite the projection of the ensemble on the Bloch sphere quickly tending

to zero the coherence each individual component of the ensemble can persist

significantly longer. The measured decay, with the constant T ∗2 , is therefore

due to our inability to follow each member of the ensemble.


Z

X

Y

Z

X

Y

Z

X

Y

a cb

Time

Ense

mb

le C

oh

ere

nce a

c

b

Figure 2.8: Bloch Sphere representation of an ensemble placed in a 50-50 superpositionstate undergoing a Free Induction Decay (FID)

If we consider an experiment as described in figure 2.9 where we apply

a π/2 pulse to an ensemble initially in the ground state it will exhibit an

FID signal. Applying a π pulse after a period τ1 results the ensemble being

rotated by π radians around the X axis, thereby effectively time reversing the

evolution of the ensemble, causing the spin vectors to return to a macroscopic

coherence after a further time τ1. This macroscopic rephasing of the ensemble

is referred to as a “Spin Echo” and the delays before and after the π pulse

are referred to as the dephasing and rephasing periods respectively.

Any difference in transition frequency between dephasing and rephasing

period will result in the spin vector accumulating a phase shift. Changes in

transition frequency considered in this work are due to reconfiguration of the

local magnetic field resulting in a time varying Zeeman shift. For an ensem-

ble of ions, each interacting with it’s own local field there will therefore be

a distribution of accumulated phase resulting in attenuation of the echo am-

plitude. Changing the delay τ allows the phase shift due to the environment

to accumulate over different periods, building up a decay sequence which

describes the decoherence time of the ensemble, T2.

It is assumed in the previous discussion that the evolution of the state

during a pulse can be described by only considering the action of the pulse

on the state. Pulses that satisfy this assertion are referred to as “hard”


Z

X

Y

π

Z

X

Y

π/2

ππ/2

Z

X

Y

Z

X

Y

Z

X

Y

1

2

3

4

5

1

23

4

5

Time

Time

En

sem

ble

C

oh

ere

nce

Ap

plie

d R

f

τ τ

Figure 2.9: Bloch Sphere representation of a 2 pulse Spin Echo.


pulses in NMR terminology. For this approximation to be valid the Rabi

frequency must be significantly larger than the linewidth of the transition

such that for all ions the action of the driving field is the same. If the

detuning approaches the Rabi frequency then off resonant effects become

important and each member of the ensemble will respond differently to the

driving field.

Γ ≪ γB1 (2.16)

where B1 is the ampltide of the RF magnetic field, Γ is the linewidth of the

transition.

2.5.3 Spin Echo Decays

The decay of a spin echo is due to uncontrolled interaction of the spin of in-

terest with it’s local environment. An exact calculation of each ion’s possible

local environment configurations is prohibitively complex and often unknown.

Consequently statistical models are used to approximate the effect of local

environment reconfiguration. For solid state NMR experiments these models

were rigorously investigated theoretically using a semi classical approach by

Klauder et al. [108] and Herzog [109] and experimentally by Mims [110].

The complete derivations are lengthy and for this work we are primarily

interested in what different decay behaviours indicate about the dephasing

environment. These derivations assume certain dynamics of the bath and

then calculate the expected decay functions. In these discussions the spin

echo amplitude of the spins of interest, or “A ” spins, is studied in the

presence of a random magnetic field due to an array of “B ” spins within the

host. The analysis requires that the experiment is performed using “hard”

pulses (see equation 2.16). The concentration of A spins is assumed to be

sufficiently low that interactions between A spins can be neglected. Under

these conditions the dominant perturbation of the A frequencies is due to

dipolar interactions with the array of B spins. The total interaction between

an A spin, at point r, with a magnetic moment of mj = ~γAIj and a number

of other B spins, each with a moment of mk = ~γBSk is given by:

HD = γAγB~2Ij ·

∑

k

Sk

rjk3− 3rjk [Sk · rjk]

rjk5(2.17)

where rjk = rj − rk is the distance between the spin A and the jth B spin

and mj is it’s magnetic moment. Reconfiguration of the B spins produce a

time dependant magnetic field fluctuation which results in a frequency shift


of the A spin, ∆ω(t), given by

∆ω(t) = γA∑

k

mk(t)(

1 − 3 cos2 θjk)

/r3jk (2.18)

The spin echo amplitude for a given pulse separation τ can therefore be

described by

E(2τ) =

⟨⟨

exp

i∑

k

αjξk(2τ)

⟩

time

⟩

lattice

(2.19)

where the angled brackets represent averages in time and over all A spin

environments in the ensemble, ξj(t) are the magnetic field fluctuations, αj

includes the A spin gyromagnetic ratio and the geometric factor for the dipole

fields

αj = γA(

1 − 3 cos2 θjk)

/r3jk (2.20)

The magnetic field fluctuations, ξj(t) are given by

ξk(t) =

∫ t

0

s(t′) (mk(t′) − m(0)) dt′ (2.21)

where s(t′) is the π rotation around the y axis at time τ , represented by

s(t′) = 1, t′ < τ

= −1, t′ > τ (2.22)

Statistical models of how the B spin bath reconfigures are required to find

suitable averages for equation 2.19. All of the models involve a Markoffian

evolution of the B spin configurations such that the next configuration only

depends on the current configuration and the bath therefore has no memory.

Consequently there is a bath reconfiguration rate parameter, R, in all of

the models that is directly related to T2. R is equivalent to the inverse of

population lifetime, 1/T1 of the bath.

Herzog et al. [109] assume that the the average over all bath configu-

rations results in a Gaussian lineshape independent of time. They further

consider the instantaneous distribution of detunings to also be Gaussian and

derive a probability for a particular detuning at time t of

P (∆ω, t,∆ω0) =1

γA [2π (1 − e−2Rt)]1

2

exp

[

−(

∆ω − ∆ω0e−Rt

)2

2γA2 (1 − e−2Rt)

]

(2.23)


It is assumed that all B spin neighbourhoods are the same and therefore can

be averaged over using the same probability distribution. This results in the

spin echo amplitude being described by

E(2τ) = E0exp

(

−(γAR

)2[

Rτ − 1 + (1 +Rτ)e−2Rτ]

)

(2.24)

where E0 is the maximum echo amplitude. If the bath is rapidly flipping

with respect to τ , the time scale of the echo, such that Rτ ≫ 1 equation 2.24

approximates to

E(2τ) ≈ E0exp

(

−(

γA2τ

R

))

(2.25)

If the B spin bath is flipping slowly, such that Rτ ≪ 1, then over the time

scale of the echo only a small number of possible bath configurations are

experienced by the A. It is not possible to argue that a small number of

randomly selected elements of a Gaussian distribution also form a Gaussian

distribution [110]. If we average over a number of A spins, each experiencing

an independent small number of B configurations the resulting distribution is

not Gaussian but Lorentzian [110]. Therefore if we examine the broadening

of a group of spins, all starting the echo sequence with an initial transition

frequency ω0 at time t0 it will broaden into a distribution [108]

K(ω − ω0,∆t) =

(

2R∆ω1/2∆t)

/π

(ω − ω0)2 +(

2R∆ω1/2∆t)2 (2.26)

where ∆t = t − t0 and ∆ω1/2 is the half width half maximum defined by

the distribution of all possible bath configurations and ∆t. Klauder and

Anderson [108] show that this results in the echo decay function

E(2τ) = E0e−(2τ/TM )2 (2.27)

Where TM is the phase memory, rather than coherence time T2 is used when

the decay time dependence is non-linear, reflecting the non-Bloch behaviour

of the decay [110]. In this work we will however refer to the 1/e point of all

echo decays as T2 for clarity.

In between these two extremes when Rτ ≈ 1 there is no simple statistical

approximation [110] and consequently no decay functions have been derived.

From equations 2.25 and 2.27 it is clear that two distinctly different spin

echo decays will be observed depending on the relative time scale of the echo

and B spin bath reconfigurations. In the laboratory observing the decay

characteristics allows the time scale of the bath perturbation to be estimated.


If the echo decay fits well to equation 2.25 only a lower limit can be placed

on R. When the echo decay is described by equation 2.27 noting the point

at which the gradient starts to increase allows estimation of R. Having these

indicators of the dynamic processes that cause decoherence is important for

choosing the right approach to minimise decoherence as will be seen in this

work.

Chapter 3

QC Benchmarks and Benefits

of Rare Earth QC

“It is not worth an intelligent man’s time to be in the majority. By definition,

there are already enough people to do that.”

- G. H. Hardy (1877 - 1947)

As discussed in section 1.2 Quantum computing places conflicting re-

quirements on quantum systems. Long coherence times are characteristic of

systems that are well isolated from their environment while the ability to

perform fast state manipulation and readout requires that the system can be

strongly coupled to the driving field. Generally systems that can be strongly

coupled to driving fields interact strongly with their environment. A diverse

range of systems has been proposed as quantum computing candidates. The

main emerging approaches focus on NMR [20, 111, 35, 112], trapped atoms

[113, 114], quantized magnetic fields [115], charge fluctuation [116] and lin-

ear optics [117, 118]. DiVincenzo’s requirements are an extremely useful tool

to assess proposals and all, if the advocates are honest, are either seriously

challenged by at least one requirement (section 1.2) or currently fail outright

with no change in sight.

The comparison of candidate quantum computing systems is however an

extremely useful process. It has provided a motivation and method to com-

pare our ability to initialise, manipulate and measure quantum information

in a vast range of systems that are otherwise rarely contrasted due to very

different mechanisms of interaction and areas of application. Each proposal

has strengths that typically stem form long term research investigating a

specific quantum interaction in particular class of materials. Since the re-

3.1 Rare Earth Ion ODNMR Quantum Computing 51

quirements for quantum computing require a system that is a “Master of all

trades” it is not sufficient just to focus exclusively on the traditional line of

research of quantum interactions in the particular system. With the require-

ments of a scalable quantum computer being so challenging to achieve it is

also important to consider as many systems as possible as they all contribute

to our knowledge of both desirable and undesirable system properties.

3.1 Rare Earth Ion ODNMR Quantum Com-

puting

The ability to optically manipulate both the electronic state and nuclear spin

state of rare-earth ions results in an interesting solution to DiVincenzo’s re-

quirement, desiring an isolated system that also strongly interacts (section

1.2). Superpositions of nuclear spin states are typically long lived and are

therefore a good place to store information, reflecting their isolation from

their environs. Their isolation is of course a hinderence to implementing

control Hamiltonians and interacting information on different nuclear spins.

Optical transitions in contrast typically have short lifetimes and higher os-

cillator strengths providing us a mechanism to strongly interact with the

nuclear spin state.

Using Raman techniques, coherence and population of a nuclear spin tran-

sition can be transferred to an electronic transition [119, 120]. For rare-earth

ions such as Praseodymium (Pr) or Europium (Eu) a change in electronic

state is accompanied by a change in static electric dipole moment. This

change stark shifts optical transitions of nearby ions by up to several GHz

which is significantly larger than both the laser linewidth and hyperfine split-

tings. This effect, termed instantaneous spectral diffusion, is well known to

rare-earth spectroscopists and has been observed in numerous rare-earth sys-

tems [121, 122, 123, 124, 125, 126].

We can therefore use a electronic ground state hyperfine transition as

a “non-interacting” qubit and an electronic transition as an “interacting”

qubit. This gives strong switchable coupling between qubits. This use of the

electronic ground state hyperfine transitions as a “non-interacting” qubit

and an electronic transition as an “interacting” qubit are common features

of all rare-earth quantum computing proposals [1, 2, 3].

To consider how this enables quantum computation we will examine the

operation of a CNOT gate, discussed in section 1.2, with pure input states.

A CNOT, or Controlled NOT operation involves two qubits: the “control”

and “target”. For the purpose of this discussion the electronic ground state

52 QC Benchmarks and Benefits of Rare Earth QC

hyperfine levels are labelled |0〉 and |1〉 with a common electronic excited

state |e〉 as in figure 3.1. Since the states that form the control qubit change

during the gate operation these have been labelled |Q0〉 and |Q1〉.The first stage of the CNOT is to change the “non-interacting” qubit

into an “interacting” qubit by transferring everything in |1〉 to |e〉. If the

control qubit is initially in |1〉 it will be promoted to |e〉 and stark shift the

transition of the neighbouring target qubit as shown in figure 3.1. The Stark

shift is large enough such that it no longer interacts with the driving fields,

leaving the state of the target qubit unchanged. The control qubit is then

returned to “non-interacting” state by transferring population |e〉 in to |1〉.Conversely if the control qubit is initially in |0〉 it will not be promoted to

|e〉 and will not Stark shift the target qubit. Therefore the driving fields will

remain resonant with the target qubit and invert it’s state as shown in figure

3.1.

3.2 Rare Earth Quantum Computing Archi-

tecture

The simplest architecture using the interactions described in the previous

section was proposed by Ohlsson et al. [2]. This scheme relies on finding

pairs of rare-earth dopant ions that have sufficiently strong interactions in a

randomly doped sample. Due to the difficulty of detecting emission from a

single rare-earth ion this architecture was proposed as an ensemble approach.

Using holeburning techniques two narrow features, or spikes can be pre-

pared in the inhomogeneous line that will form the two physical qubits we

desire to interact as shown in figure 3.2. The spikes now need to have the

ions removed which do not appreciably stark shift a corresponding ion in

the other spike. To achieve this Ohlssen et al. proposed optically exciting

the spike which produces a stark shift on interacting ions in the other spike,

thereby broadening the spike. The unshifted ions can be holeburned away

leaving two spikes of interacting ions. The ratio of homogeneous linewidth to

nearest neighbour Stark shift for rare-earth ions can exceed 107 [127]. Given

that hyperfine splittings are on the order of ∼ 100MHz and a laser linewidth

of < 1kHz ions need not be nearest neighbours to be sufficiently shifted and

therefore many ions can interact with each other via this mechanism.

The problem with the scheme arises when the number of remaining ions

are considered. Ohlssen et al. calculated that optically exciting a spike

resulted in a frequency shift that was ∼4% of the width of the spike. The

random direction of the frequency shift results is > 99% of the ions in each

3.2 Rare Earth Quantum Computing Architecture 53

1| >|0>

| >e

1| >|0>

| >e

1

Initial

TargetControl

|Q >|Q >0

Control

Final

Target

|Q >1

|Q >0

Intermediate

TargetControl

|Q >1

|Q >0

Initial

TargetControl

|Q >1

|Q >0

Final

TargetControl

|Q >1

|Q >0

Intermediate

TargetControl

|Q >1

|Q >0

Figure 3.1: CNOT implementation using rare-earth ions showing the two extreme casesof the control in either a pure |0〉 or |1〉 state. Transitions to the excited state |e〉 areoptical. The detuning ∆ω is much larger than the difference in energy between states |0〉and |1〉


spike being thrown away due to insufficient interaction strength. Selecting

for 3 or more interacting ions results in removing an impractically large

proportion of the initial ensemble. Consequently this approach does not

scale to complex quantum computing demonstrations.

The only previous gate demonstration using rare-earth ions used a modi-

fied form of this approach [127]. This modified approach was able to resolve

shifts within the width of the spike and consequently allowed more ions to

be kept. The scaling problems still remained however and the distribution

of interaction strengths was shown to limit the operation of the gate [127].

A different approach, recently developed by Sellars [128], is to introduce

point defects into a stoichiometric crystal. A magnetic defect is which has a

magnetic dipole moment that significantly differs from that of the stoichio-

metric ion will be the the focus of the discussion. Charge defects can also

be considered and it is also important to note that with a dopant ion will

typically differ in both electric and magnetic dipole moments. A system that

would realise a predominantly magnetic defect this is a stoichiometric Eu

crystal into which we lightly dope Er such as Er3+ : Eu2SiO5. Due to the

unpaired electron spin of Er (γEr ≈ 103γEu) many Eu ions will be Zeeman

shifted out of resonance with the bulk Eu, thereby creating a large region of

ions with unique detunings.

Such a system would exhibit rich spectroscopy, behaving like an artificial

molecule. The Zeeman splitting can be investigated to map the distance

between a particular rare-earth ion and the magnetic defect. Studying the

stark shift due to selective excitation of a particular rare-earth ion detuning

can then be used to determine the spatial relationship of each unique rare-

earth ion spectral feature. As the bulk rare-earth ions will have a different

hyperfine splitting to the rare-earth ions of interest near the defect, the rare-

earth ions in the bulk can be optically pumped into a single spin state. This

stops any cross relaxation within the bulk, thereby suppressing the associated

magnetic field fluctuations and increasing the hyperfine T2 of the frozen core

rare-earth ions.

Therefore such a system can create a small region of uniquely addressable

qubits, enabling simple fabrication of many qubit solid state quantum com-

puting. It is expected that a > 10 qubit device would be realisable with this

approach [128]. Due to the low oscillator strength of the optical transitions

many magnetic defect features would be detected as an ensemble to achieve

sufficient SNR. It would therefore perform as an ensemble of small quantum

computers with the final result representing an ensemble average.

Each computational step of an algorithm generally involves interacting

3.2 Rare Earth Quantum Computing Architecture 55A

bso

pb

tio

n

Frequency

π

π

π

π

1

2

3

4

5

Figure 3.2: Schematic representation of the selection process for obtaining two ensemblesof interacting ions from a randomly doped sample. 1) two narrow spikes are prepared inlarge holes burnt into the inhomogeneously broadened optical transition. 2) one of thespikes is optically excited using a π pulse while the unshifted ions of the other spike areholeburned away. 3) the initial spike is returned to the ground state. 4) Unshifted ions inthe first peak are holeburned away. 5) Two less intense spikes containing interaction ionsremain.


different qubits. In quantum computing systems that are limited to nearest

neighbour interactions information needs to be moved around the computer

such that the required qubits can interact. This requires a number of swap

operations [129], in which the interacting qubits “swap” their state. Further

to this each swap operation must be fault tolerant and therefore error cor-

rected, taking more time, resulting in more decoherence and increasing the

number of qubits required. Many schemes are limited to, or favour nearest

neighbour interaction, such as Liquid Phase NMR and most solid state NMR

proposals [130, 131].

The ability to perform operations not restricted to nearest neighbour in-

teractions, as afforded by Sellars’ rare-earth Stark shift scheme is a significant

advantage. By allowing each qubit to interact with many others algorithms

can be implemented with less concern form the specific physical structure

of the quantum computer. When information is required to be moved, or

swapped, it can be moved many qubits in the desired direction. This pro-

vides flexibility as to how the information is routed within the computer

and the number of acceptable nanostructures for implementing a quantum

computer.

More complex structures are of course possible. Network analysis has

shown that with only a few interconnecting nodes between highly connected

local networks the resulting global network is still well connected [132].

Therefore an effective, scalable architecture could in principal be created by

introducing another rare-earth ion species inbetween these magnetic defect

features to act as an interconnect between local computation nodes. There-

fore limits to the number of qubits per magnetic defect is not a fundamental

limitation to the number of possible interacting qubits.

This approach is not a complete solution to scaling of Rare-Earth quan-

tum computing architectures up to the desired 106 qubits [32] it demonstrates

the flexibility inherent in this approach. When compared to other solid state

systems the fabrication required to reach a 10 qubit device is negligible since

a random positioning of the magnetic defect is adequate and therefore no

nanostructuring is required. It should also be noted that an infinitely scal-

able quantum computing system, as routinely discussed to demonstrate the

power of quantum computing is not physically reasonable. No natural sys-

tem continues exponential growth indefinitely. At some point the system

will not be able to sustain coherent interactions between the most distant

qubits and as such the system will reach a more classical regime where the

computational phase space does not increase exponentially with the addition

of qubits.

3.3 Rationale for System Comparison 57

While the final quantum computer will involve some degree of nanostruc-

turing it is not required for initial investigations. Crystals randomly doped

with rare-earth ions are sufficient to investigate some of the physical issues

with controlling optical transitions [105], the Stark shift interaction [127] and

decoherence of hyperfine transitions investigated in this thesis. Therefore ini-

tial quantum computing investigations of rare-earth systems can use existing

crystals, many of which have been examined in detail by spectroscopists and

are therefore well understood. This circumvents many of the problems in-

volved with proposals that require exotic nanostructures, many of which have

not been built before and require fabrication research and characterisation

before the QIP applications can be fully assessed.

3.3 Rationale for System Comparison

Quantum computing demonstrations using Liquid state NMR systems have

to date set the benchmark for candidate quantum computing systems. Liq-

uid phase NMR exhibits very long coherence times [133] and was the first

to implement an algorithm [134]. It has demonstrated the largest number

of interacting qubits [135], 7 as well as the most sophisticated algorithm im-

plementations [111, 112, 135]. It has therefore played a fundamental role in

verifying the theory underlying quantum computing and become a bench-

mark for quantum computing. There are however well recognised limitations

to the approach which our approach does not suffer from. Consequently I will

compare the system we have chosen to investigate with Liquid phase NMR

quantum computing. Furthermore, the approach we are persuing can be

classed as Optically Detected Nuclear Magnetic Resonance (ODNMR) and

therefore has many similarities, allowing for direct comparison of important

parameters.

It should be noted that trapped ions are also a particularly attractive

physical system [114, 113]. Like NMR quantum computing investigations in

the area have benefited substantially from prior work maturing the experi-

mental apparatus and techniques. The atom-optics community has developed

numerous trapping structures and methods of manipulating trapped atoms

that have served to fast track quantum computing experiments.

3.4 Liquid Phase NMR

Liquid Phase NMR experiments typically consist of a sample placed in a

large, homogeneous static magnetic field applied by a solenoid such that


the strength of the applied field can be precisely varied. Great attention

is paid to the homogeneity of the applied magnetic field with the sample

space, typically of the order of 1cm3 having a variation of less than one part

in 109. Both driving and detecting nuclear spin transitions is accomplished

using a coil pair perpendicular to the applied static field. By convention the

coordinate system is chosen such that the static magnetic field lies along the

z axis. This results in a Hamiltonian of the form:

H = µ (B0Z +BRF (t)X) (3.1)

where X and Z are the Pauli spin operators while B0 and BRF are the static

and RF magnetic fields respectively.

Changes in the nuclear spin state are detected via a change in the magne-

tization of the sample. This change is very small due to the correspondingly

small nuclear magnetic moment and consequently would be difficult to di-

rectly observe over the static magnetic field. Magnetization perpendicular to

the z axis precesses at the Lamor frequency around the z axis at a rate of

µB/~. Detecting the magnetization perpendicular to the z axis removes the

large offsets and any associated fluctuations of B0 from the measurement.

Another benefit is that the desired signal is moved away from DC with the

benefits of removing 1/f noise. The detection coil pair is part of a high Q

resonant circuit at a fixed frequency. This frequency is chosen such that the

range of static magnetic fields that can be generated is sufficient to shift the

spins of interest into resonance with the detection coils.

Ideally, B0 should be as large as possible for a number of reasons. Signal

amplitude is determined by the magnetization or population difference within

the sample. Therefore to maintain the population difference the transition

energy should be much greater than the thermal energy kBT . Even for

the largest realistic fields the energy difference between spin states, ≈ µNB

is smaller than kBT due to the small magnetic moment of nuclei and the

high temperatures that working with a fluid requires. Increasing the applied

field also increases the resolution of the system since changes in the nuclear

magnetic moment result in larger energy differences.

Nuclear spin interactions are typically dominated by magnetic dipole -

dipole interactions [41], given by the following Hamiltonian.

HD1,2 =

γ1γ2~

4r3[−→σ1 · −→σ2 − 3 (−→σ1 · −→n ) (−→σ2 · −→n )] (3.2)

where γ1 and γ2 are the gyromagnetic ratios of the two spins of interest, −→nis the vector between the spins and σ is the Pauli spin matrix.

3.4 Liquid Phase NMR 59

In a fluid the molecules are reorienting with no particular directional-

ity and consequently the dipole- dipole interaction is described using a time

average. For low viscosity fluids the molecules are reorienting significantly

faster than the nuclear spin transition energy, resulting in the time aver-

age of the dipole - dipole interaction being vanishingly small [41]. This is

clearly demonstrated by observing the homogeneous linewidth of nuclear spin

transitions as the liquid is frozen. In a frozen liquid the mean position of

molecules is fixed, resulting in −→n ≈ n(t0) and therefore resulting in a non-

zero average to the inter molecular dipolar coupling. As the liquid is cooled

dramatic increases in the homogeneous linewidth are observed [41]. There-

fore, surprisingly, the chaotic nature of molecules in a liquid is a benefit

for achieving long nuclear spin coherence times by removing inter molecular

magnetic dipole interactions due to the vastly different time scales.

With the use of a suitable solvent the molecule of interest can be diluted

such that the spectrum of the NMR signal is dominated by intra molecu-

lar interactions. Nuclear spin transitions are shifted by the specifics of the

local molecular bonding due to coupling between the electron and nuclear

spins. Such shifts are termed chemical shifts since the shift determined by

the chemical bonds. This shift is of the order of tens to several hundred

kilohertz, which is much larger the typical transition linewidth. A change in

nuclear spin state will change the energy of other nuclear spin states within

the molecule. The energy change is mainly due to J-coupling whereby a

change in a nuclear spin state is coupled to other nuclei in the molecule by

overlap in bonding electron wavefunctions. This interaction is given by

HJ1,2 =

~J

4−→σ1 · −→σ2

=~J

4Z1Z2 +

~J

8[σ−σ+ + σ+σ−] (3.3)

where J is typically scalar (most generally it is a tensor), Zi are the Pauli Z op-

erator for the ith spin, σ+ and σ− are in phase and quadrature superposition

states. Using the rotating wave approximation the terms ~J8

[σ−σ+ + σ+σ−]

can be ignored. If the coupling, J, is weak or the resonant frequencies of the

two nuclei considered are very different this can be approximated as

HJ1,2 ≈

~

4JZ1Z2 (3.4)

This is a very good approximation for molecules with different nuclear species,

large chemical shifts or the same nuclear spins separated by a number of

bonds. It is clear that this effect will not be averaged out when a large


number of liquid molecules is considered.

The shift due to the J-coupling is typically larger than the direct dipole-

dipole coupling and consequently is very useful for determining the relative

position of nuclear spins in a molecule as well as the bonding between them.

This application has made liquid phase NMR indispensable in the fields of

chemistry and biology and remains the only method to extract detailed struc-

tural information about complex molecules. An impressive library of spec-

tral “fingerprints” have been accumulated so that simple molecules, whose

structure is simple and known can be instructive in discerning more complex

molecular structures. Combined with this is a formidable array of techniques

to drive the system to manipulate the spin states and enhance or suppress

particular interactions between spins. Of particular interest to this work are

those techniques that are designed to extend T2 beyond the undriven value.

These techniques are referred to as Phase Cycling or decoupling sequences

by the NMR community and Dynamic Decoherence Control (DDC) by the

QIP community. Two of these techniques are investigated experimentally in

chapters 6 and 7.

To investigate the general concept consider the Hamiltonian that de-

scribes a multiple spin system interacting with an RF driving field:

H =∑

k

ωkZk +∑

j,k

HJj,kZk + HRF +

∑

j,k

HDj,kZk + Henv (3.5)

where ωk are the resonant frequencies for each nuclei due to the applied

magnetic field and the chemical shifts, Zk is the Pauli Z operator for the ith

spin and α2Z1Z2 is the coupling between the nuclei. HRF describes the action

of the RF field on the nuclear spins and Henv is the uncontrollable interaction

of the environment with the system that results in decoherence.

To extend T2 by driving the system the first step is always understanding

which interaction currently limits T2. Once this is established generally one

of two approaches are taken. Firstly the system can be driven such that

the time scales of the dephasing process and transition of interest differ suf-

ficiently that they become decoupled. This is directly analogous with the

rapid reorientation of molecules in a liquid averaging out inter molecular

dipole coupling as previously discussed. Decoherence time extension based

on this principal is investigated in chapter 7. The second approach is to ap-

ply a cyclic driving sequence such that the effect of the dephasing interaction

is equal and opposite during successive periods, thereby removing the con-

tribution of dephasing at the end of the cycle. This is investigated in detail

in chapter 6.

3.4 Liquid Phase NMR 61

3.4.1 Limitations of Liquid State NMR

The steady state population of nuclear spin states, in contact with an envi-

ronmental bath of temperature T is dominated by the first term in equation

3.5∑

k ωkZk. Since kT ≫ ~ωk the system is in a highly mixed state. The

population excess in the ground state that is utilised to perform NMR exper-

iments is typically very small. The Overhauser effect can be used to induce

larger population differences [41] however they require the ability to couple

the spins of interest to a highly polarised spin reservoir. These techniques

are not applicable to all nuclear spins and total spin polarisation is rarely

achievable.

Observables in NMR are traceless since it is the superposition states that

radiate perpendicular to the static magnetic field direction. Therefore any

excess diagonal terms due to thermal states do not contribute to the mea-

surement. It can therefore be argued that if the experiment is properly

constructed the excess diagonal thermal population is not detected and the

result of an experiment performed on a highly mixed state with a small pop-

ulation excess is identical to an experiment which used a pure initial state

[35]. Hence the term pseudopure state is often used when referring to NMR

experiments. There are two methods used to achieve this termed temporal

labelling and logical labelling [35].

Despite the result of a properly constructed experiment achieving math-

ematically the same result using a pseudopure state as using a pure state

the systems fails DiVincenzo’s second requirement: that the system must

be able to be initialised into a specific initial state. While this may seem

to be splitting hairs, it is contentious if manipulating the small population

excess used as the ensemble is categorically a quantum computation. This is

because there are an infinite number of expansions possible of a pseudopure

state and therefore the interpretation of the evolution and interactions of

such a state is not explicitly defined.

The most fundamental limitation to liquid state NMR providing a prac-

tically useful quantum computer is the number of interacting qubits that can

be achieved. While using molecular bonds as the interconnections between

qubits has obvious advantages for creating ready made quantum computing

circuits it introduces two principle limitations.

Firstly, the coupling between spins at opposite ends of a molecule becomes

very weak as the size of the molecule increases. This requires that very small

energy differences be resolvable, which in turn means long detection periods.

This poses problems for manipulating and observing the system well within

the decoherence time and severely limits the speed at which the computer


can operate. For the current state of the art 7 qubit implementation this

resulted in an energy difference of 0.25Hz being resolvable, resulting in a

detection period approaching 1 second. The typical signal to noise ratio

requires averaging a number of experiments making the total computation

period very long. Clearly this pushes detection limits and requires excessive

coherence times if the number of qubits is to be increased. To compound this

both logical and temporal labelling result in an exponential decrease in signal

strength as the number of bits is increased. This is because as the number of

molecules in any particular state decreases as the number of possible states

is increased if the available thermal energy exceeds the energy required to

change the molecular state.

Secondly, the coupling between qubits is not switchable since it is defined

by the molecular bonds. Many NMR Phase Cycling pulse techniques are

specifically designed to decouple spins within the same molecule, however

they are often optimised to preserve a specific state and therefore do not

necessarily preserve an arbitrary state [41]. Incorporating such a decoupling

scheme significantly increases the complexity of the required pulse sequences

to manipulate desired qubits. Consequently this increases the required deco-

herence time and fidelity with which control sequences are implemented. As

previously discussed, both of which are already challenging existing technol-

ogy.

Direct detection of RF photons makes detecting a conventional NMR

signal challenging. Detection takes place in the near field using a High Q

resonant circuit with the detection coil placed as close as possible to the

sample. Due to the low energy of RF photons current technology does not

allow efficient detection approaching the single photon limit. Consequently

conventional NMR experiments require a high concentration of spins, often

in combination with signal averaging to achieve a sufficient SNR.

3.5 The Case for Solids

Solid state quantum systems suitable for QIP are difficult to find. Each

quantum object that makes up a solid is typically strongly coupled to every

other quantum object in it’s locality due to bonding and close proximity.

There are many microstructures in solids that exhibit coherent oscillations,

such as quantum dots [116] or superconducting structures [115]. While these

structures are attractive since their properties can be tailored by fabrication

parameters, their decoherence time is typically sub nanosecond [115] due to

uncontrollable interactions with the environment and within the microstruc-

3.5 The Case for Solids 63

ture.

Using dopant ions as the quantum system to implement qubits has po-

tential to remove much of the fabrication and allows exploitation of known

spectroscopy of dopant ions. The maximum utilisation of available quantum

systems is for each dopant ion to be a physical qubit. This however comes

at the expense of easily addressing the qubits. The wavelength of the driv-

ing field is typically significantly longer than the inter qubit distance, which

removes the ability to utilise spatial selectivity to address qubits. Therefore

large field gradients [130] or exploitation of inhomogeneous broadening or

specifically introducing defects to produce known, unique detunings, as dis-

cussed in section 3.2, is required to individually address the qubits spectrally.

Optically detecting nuclear spins has several benefits when compared to

conventional NMR. Through the use of Raman heterodyne spin concentra-

tions orders of magnitude lower than conventional NMR can be used while

retaining high SNR. This allows the use of sufficiently dilute samples that

resonant relaxation between the spins of interest is negligible, often the lim-

iting factor on coherence in conventional solid state NMR [39]. Given the

efficiency of photodetectors and the ability to use interferometric techniques

single photon detection can be readily achieved [136]. It is therefore possi-

ble, in principal, to detect the nuclear spin state of a single optically active

nuclear spin system. Detecting RF photons, as in conventional NMR is a

significantly more difficult problem since detection takes place in the near

field regime and the energy of a single RF photon is significantly below the

thermal noise of detectors. Consequently conventional NMR is limited to

detecting ensembles for fundamental reasons.

There are two distinct benefits to ODNMR when compared to conven-

tional NMR. Directly detecting the RF photon, as in conventional NMR, is

difficult due to the low quantum efficiency, circumvented by detecting Ra-

man heterodyne beats on optical detectors. Conventional NMR achieves

population differences between nuclear spin states by a combination of ma-

nipulating the Boltzmann distribution and pulsed techniques. By applying

large magnetic fields (∼ 15T ) higher energy spin states become energetically

unfavourable and some spin polarisation due to the Boltzman distribution

arises. A variety of Overhauser based pulse sequences exist [41] to exchange

spin polarisation between different spin populations, typically used if a par-

ticular spin species of interest is difficult to polarise. The ability to prepare

nuclear spin population differences via optical excited states is a significant

advantage. Complete polarisation of a nuclear spin state can be achieved

using holeburning techniques [47].


Optical state manipulation also has a significant benefit for satisfying

Divincenzo’s initialisation requirement (section 1.2). The initialisation re-

quirement can be achieved using spontaneous emission from an optically

excited state to spin polarise the ions interacting with the laser beam [127].

Holeburning spectroscopy has exploited this ability extensively [47]. This is

in stark contrast to liquid phase NMR which is forced to utilise pseudopure

states afforded by applying large magnetic fields (∼ 15T ) such that some de-

gree of spin polarisation is induced by the Boltzmann distribution. A variety

of Overhauser based pulse sequences exist [41] to exchange spin polarisation

between different spin populations, typically used if a particular spin species

of interest is difficult to polarise. Complete polarisation is rarely achievable

through any method.

The rare-earth proposal like all others does however has a fly in it’s oint-

ment. This takes the form of low oscillator strength of the optical transitions,

which is due to the transitions being forbidden in high symmetry situations

and only weakly allowed for low symmetry hosts [47]. This limits the number

of photons able to be detected before spontaneous emission changes the spin

state by decaying to a different spin ground state. At present this will re-

strict us to an ensemble approach, whereby small clusters of qubits perform

identical computation, the results of which are read simultaneously. There

is also potential to couple the final answer to another colour centre or high

Q nanostructure can probed more easily. One such colour centre is the N-V

centre in diamond [137]. Nanostructuring a metallic film on the surface would

allow the formation of high Q microcavities, based on plasmon resonances,

which the ions to be read out could by stark shifting the ion to be read out

into resonance microcavity.

The density of qubits achievable with solid state systems is encouraging

for scaling systems toward a useful quantum computer. Scalability of solid

state systems has been amply demonstrated by conventional electronics.

3.6 Limitations of Rare Earth Quantum Com-

putation due to Hyperfine Decoherence

Rare- earth quantum computing schemes such as those discussed in the sec-

tion 3.2 as well as other proposals [3, 1, 138] use the hyperfine states as long

term storage for information contained on optical transitions. On commenc-

ing this work the T2 of the optical transitions was approximately equal to

that of hyperfine transitions [5, 102] at a few ms. While this is very close

to being lifetime limited for the optical transitions the hyperfine transition

3.6 Limitations of Rare Earth QC due to Hyperfine Decoherence 65

T1 can be several weeks [4]. This significantly longer upper limit to T2 has

led to the use of hyperfine states in the QIP proposals without any actual

experimental evidence that T2 can approach T1 for hyperfine transitions.

If the T2 of the hyperfine transitions are approximately the same as the

optical then hyperfine transitions are not useful for QIP. Transferring infor-

mation from an optical to a hyperfine transition will always introduce some

noise due to non-unit fidelity. Consequently, the decoherence of the hyperfine

transition and the decoherence induced by the swap operation must be less

than the optical transition over the same period to be beneficial. However for

the proposed QIP applications to be successful we require hyperfine transi-

tion decoherence times to significantly exceed that of optical transitions such

that decoherence of the hyperfine transition is negligible.

Research involving rare-earth ions has typically focused on the optical

properties of the ions for application to new lasing materials or holographic

memory and signal processing. Consequently development of rare-earth ma-

terials has focused on obtaining narrow homogeneous optical transitions.

This has resulted in rare-earth doped insulators possessing the narrowest

optical transitions observed in a solid [5]. Hyperfine transitions have typi-

cally only been used for their ability to store population differences created

by driving optical transitions for use in holographic signal processing. As

such hyperfine decoherence has not been a limiting factor in applications

and has not received the same attention as the optical transitions.

In this work we seek to increase the hyperfine T2 to such a point that rare-

earth ions are useful for QIP applications as described in current proposals

[3, 2, 1, 138]. To achieve this we investigate the dependence of hyperfine

transition T2 to both static and dynamic magnetic field techniques.

Chapter 4

Decoherence of Pr3+:Y2SiO5

Hyperfine Transitions with

Small Applied Magnetic Fields

“When we try to pick out anything by itself, we find it hitched to everything

else in the Universe.”

– John Muir

Understanding decoherence mechanisms is paramount to formulating strate-

gies to remove or minimise the effect of decoherence. As previously mentioned

in section 3.6 decoherence studies of rare-earth ions have focused on optical

transitions [139, 53, 54, 140, 141, 142, 7, 122, 5, 6, 143, 144, 88, 145, 4].

These studies showed that T2 for optical transitions are limited by magnetic

interactions [53, 54, 7, 5]. In this chapter the understanding gained from de-

coherence studies in rare-earth systems will be used to build up a description

of decoherence in Pr3+:Y2SiO5, culminating with experiments investigating

the described interactions.

4.1 Pr3+:Y2SiO5 Hyperfine Decoherence

The use of Optically Detected NMR (ODNMR) allows sufficiently dilute

dopant concentrations that dopant-host interactions dominate dopant-dopant

interactions. As discussed in section 3.5, this regime is not achievable in con-

ventional solid state NMR due to detection SNR limitations. Therefore, the

properties of the host are the most important for determining the decoher-

ence properties of the dopant ion.

High quality insulator hosts have no free charges and therefore time vary-

ing electric field perturbations typically contribute less to dephasing than

4.1 Pr3+:Y2SiO5 Hyperfine Decoherence 67

magnetic perturbations for both optical and hyperfine transitions [5, 7, 142].

Consequently the materials developed to achieve long T2 optical transitions

also help achieving long hyperfine transition T2. Magnetic fluctuations in in-

sulating hosts of interest originate from changes in the state of host nuclear

spins. These host nuclear spins exchange spin via cross relaxation as well

as spin flipping due to spin-lattice relaxation. Therefore the magnetic field

due to the array of host nuclear spin magnetic dipoles as seen by the Pr ion

fluctuates in time. This magnetic field Zeeman shifts the Pr ion’s hyperfine

states, causing the transition frequency to change. This results in the transi-

tion phase relative to a reference oscillator becoming uncertain, resulting in

decoherence.

During the early 1990s “low noise” Yttrium oxide based hosts became

available. While Yttrium had been used as a substitutional ion for some time

in common lasing materials such as YAG and YLF these materials contained

Fluorine or Aluminium which have magnetic moments orders of magnitude

larger than Yttrium. Consequently hosts such as Yttrium oxide (Y2O5) and

Yttrium orthosilicate (Y2SiO5), in which Yttrium is the only ion possessing

spin (assuming isotopically puse 28Si) have a significantly smaller fluctuating

magnetic field, hence their description as “low noise” host. This results in

significantly longer decoherence times with the optical transition linewidths

of 122Hz observed in Eu3+ : Y2SiO5 [5], the narrowest ever observed in a

solid. Approximately 20Hz of this linewidth was attributed to magnetic field

fluctuations due to host nuclear spins.

For a Pr3+:Y2SiO5 sample with a 0.05% concentration of Pr dopants

the resonant cross relaxation rate, given by equation 2.17, for two Pr ions

separated by the mean distance (|r| ≈ 30A) is maximally ∼ 60 Hz if both are

in the ±5/2 state and minimally ∼ 2 Hz if both are in the ±1/2 state. The

inhomogeneous broadening of hyperfine transitions in similar samples result

in linewidths of up to 30kHz [6]. Due to the inhomogeneous linewidth being

significantly larger than the dipole-dipole interaction the cross relaxation will

typically be non-resonant, detuned by several homogeneous linewidths and

therefore proceeds at a significantly reduced rate. Since T2 is 500µs [102],

decohering interactions with the host dominate and we can consider each Pr

ion as being surrounded by a continuous host which does not contain any

other Pr ion.

The Pr-Y interaction is significantly more complex than an ion of inter-

ested surrounded by a thermal spin bath. The magnetic moment of Pr ions is

over an order of magnitude larger than that of Y. The presence of the Pr ion

Zeeman shifts the nearby host nuclear spins and with no applied magnetic

68 Hyperfine Decoherence with Small Applied Mangetic Field

field also locally defines the quantization axis of Y spins. By Zeeman shifting

the nearby Y spins it lifts their hyperfine degeneracy and detunes them from

other nuclear spins in the host, hereafter referred to as the “bulk”. This

results in a drastically reduced cross relaxation rate for the spins that are

Zeeman shifted. This is known as a “frozen core” [53], since the spin state

of the ions within it are almost static compared to the bulk. Frozen cores

have been observed in many analogous systems [146, 147, 53, 88]. This effect

tends to increase the decoherence time of the Pr ion since the nearest Y spins

which induce the largest field at the Pr site spin flip at a reduced rate.

Yttrium ions for which the Pr-Y magnetic dipole-dipole interaction (HPr−YDD )

exceeds the Y-Y magnetic dipole-dipole interaction (HY−YDD ) are considered

as being part of the frozen core. In order to define this we first require to

know what the mean magnetic dipole-dipole interaction is for the bulk Y.

The distribution of magnetic fields due to the bulk Y was calculated

by using the Y2SiO5 crystal structure [99, 106] and summing the magnetic

dipole contributions after the spins had been randomly oriented. The code

used to calculate the Y positions, written by Jevon Longdell, is shown in

appendix A. After numerous randomisations of the spins had been performed

a histogram of magnetic field values at the Pr site was generated. It was found

that including more than 50 Y spins did not significantly alter the resulting

distribution of fields. The result of these calculations, shown in figure 4.1,

indicate that the most probable magnetic field experienced by the Pr ion

due to randomly oriented Y dipoles is ∼0.035 G. This results in a bulk Y

linewidth of 7.6 Hz. Calculating the Pr-Y magnetic dipole-dipole interaction

using equation 2.17 results in the nearest 81 Y ions satisfying HPr−YDD > HY−Y

DD

and are therefore considered as being part of the frozen core.

As previously mentioned the magnetic field due to the Pr defines the

quantisation axis of the nearby Y nuclear spins. Due to the Pr ion defining

the local Y quantisation axis a change in the Pr ion spin state will change the

quantisation axis of nearby Y spins. The change in quantisation axis mixes

the Y spin states and results in a high probability of a spin flip. This coupling

introduces the possibility that as a direct result of driving a Pr hyperfine

transition several Y spins also change state, known as the superhyperfine

interaction [7, 143, 148].

When transitions involving both Pr and Y spin flips are excited, it results

in many quantum pathways between the initial and final states that interfere.

The frequency of the quantum pathways will differ by integer multiples of

the Y Zeeman splitting. The echo will therefore be modulated by all quan-

tum pathways with the modulation intensity proportional to the transition


0 0.02 0.04 0.06 0.08 0.1 0.120

500

1000

1500

2000

2500

Magnetic Field (G)

Nu

mb

er

of

con

fig

ura

tio

ns

wit

hin

win

do

w

Figure 4.1: Histogram of the magnetic field magnitude at the Pr site due to randomlyoriented Y spins


Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

0= =

23= 2=

23=

OnResonance

Detuned

Resultant

Figure 4.2: Bloch sphere representation of a spin echo at the rephasing time 2τ whentwo ensembles, one detuned from resonance result in a modulated echo.

strength. This can be visualised by considering a Bloch sphere with several

sub ensembles, detuned by integer multiples of the Y Zeeman splitting as

shown in figure 4.2. As the echo delay is increased these sub ensembles will

rotate around the Bloch sphere according to their detuning, being initially

in phase with the sub ensemble on resonance, 180 out of phase at time

τ = nπ/∆ω and in phase at time τ = 2nπ/∆ω where n = 1, 2, ..k. It should

be noted that T2 cannot strictly be used to describe the loss of coherence

from the system since it is no longer a two level system and therefore does

not obey the Bloch equations.

The expected magnetic field strengths can be calculated created at the

Pr site due to the nearest neighbour Y ion (BY nn) and visa versa (BPrnn).

The magnetic field due to a dipole is given by the following equation:

B (r) = ~γ

[

I

|r|3 +3I · r|r|5 r

]

(4.1)

where γ is the gyromagnetic ratio, which for Pr is the effective Zeeman tensor,

M , I is the nuclear spin state and r is the distance from the dipole. For

Y2SiO5 the nearest neighbor Y sites are seperated by |r| = 3.4 A, γY = 209

Hz/G and the Pr effective Zeeman tensor is described in section 2.3.2. This

calculation results in a BY nn = 0.011 G and BPrnn = 1.4 G.

Since the magnetic field defines the spin quantisation axis the superhyper-

fine transition probability can be calculated by determining the projection of


the initial magnetic field onto the final magnetic field as described in equation

4.2.

PSHF =Bi · Bf

max (Bi · Bi,Bf · Bf)(4.2)

where Bi is the initial magnetic field, Bf is the final magnetic field and

normalised by dividing by the maximum projection of each field onto itself.

As an applied magnetic field is increased the spin quantisation axis is

increasingly defined by the applied field rather than the magnetic field due

to the Pr ion. Therefore mixing of Y spin states due to changes in the spin

state of the Pr ion is decreased and consequently the superhyperfine transi-

tion probability is also decreased. Increasing the applied magnetic field will

also increase the Y Zeeman splitting, and therefore the echo modulation fre-

quencies. The echo modulation frequencies will increase as integer multiples

of γY and since Y is a spin 1/2 system the Zeeman splitting will be linear

within the range of fields used. as the applied magnetic field is increased we

should therefore observe the superhyperfine transition frequency increase as

the transition strength decreases.

The superhyperfine transition probability as an external magnetic field is

applied was calculated using equation 4.2 and is shown in figure 4.3. This

predicts that the superhyperfine transitions will be significantly attenuated

for applied magnetic fields of the order of 10 G. The complex behaviour of the

superhyperfine transition probability when the applied field has a magnitude

of ∼1 G is due to the applied magnetic field and the magnetic field due

to the Pr ion being approximately equal. It is unlikely that the complex

behaviour can be observed experimentally due to residual magnetisation in

the equipment, such as the optical table, being of the order of a few Gauss

making it difficult to achieve true zero applied magnetic field.

Suppressing the multiple superhyperfine transitions results in an increase

in decoherence time. This has been known for some time and is often used

to reduce the magnetic linewidth contribution for optical transitions [5, 142].

These studies inferred the suppression of superhyperfine interactions from

increases in the homogeneous optical lifetime but did not directly observe

the beat frequency. More detailed studies were carried out on Pr3+:LaF3

by Pryde [148]. Superhyperfine interaction however has not been studied for

hyperfine transitions of Pr3+:Y2SiO5.

The previous discussion of superhyperfine interaction focuses on Y spin

flip process that create a different magnetic field configuration at the site

of the Pr ion. Changes in the Pr spin state can also induce two Y ions to

mutually spin flip, or cross relax, resulting in near degenerate initial and

final magnetic field configurations as investigated by Pryde in Pr3+:LaF3


0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Applied field (G)

Tra

nsi

tio

n p

rob

ab

ility

Figure 4.3: Calculation of the superhyperfine transition probability for changes in thePr spin state of mI = +1/2to + 3/2, mI = +1/2to − 3/2, mI = −1/2to + 3/2, mI =−1/2to− 3/2. The applied magnetic field is aligned to the z axis.

[148]. Pryde’s work showed that these interactions do not display the same

magnetic field dependence as the non-degenerate superhyperfine interactions

since they require a significantly smaller change in the spin quantisation axis.

Therefore, near degenerate superhyperfine interactions are unlikely to be sup-

pressed by the application of moderate magnetic fields. Pryde demonstrated

that this interaction resulted in transition frequency shifts peaked around 4

kHz with a transition probability of 0.0035. Comparing the gyromagnetic

ratios of Flourine (γF ) and Yttrium (γY ) we find that γF ≈ 20γY and conse-

quently expect only a ∼ 10 Hz contribution for Pr3+:Y2SiO5. The timescales

of coherent transients in Pr3+:Y2SiO5 with low applied magnetic field are of

the order of 500 µs [102] and as such the frequency shifts due to the degen-

erate Y spin flips will not be possible to measure experimentally. Precise

calculation of these parameters was therefore not performed. It is important

to note that these near degenerate superhyperfine interactions are errors in

state manipulation when performing quantum computing operations. The

estimated transition probability calculated above, while small is sufficient to

induce errors of the order of 10−5, the expected threshold for fault tolerant

quantum computing [35].

As previously mentioned we need to understand the decoherence mech-

anisms in order to develop strategies to suppress interactions or the effect

4.2 Experimental Setup 73

of interactions that cause decoherence. From previous work on rare-earth

doped insulators the dominant decohering interaction with low applied mag-

netic fields is the superhyperfine interaction [5, 6, 7] and consequently it is

the first interaction investigated. Studies at low field also provide a bench-

mark to assess whether our attempts to increase T2 beyond the original value

are beneficial.

4.2 Experimental Setup

Coherence is studied using two pulse spin echos detected via Raman het-

erodyne, as discussed in section 2.5.1, the pulse sequence of which is shown

in figure 4.6. The experiments, shown schematically in figure 4.5 were per-

formed using a Coherent 699 frequency stabilised tunable dye laser tuned to

the 3H4 − 1D2 transition at 605.977nm. This laser can be operated in “high”

resolution mode with a FWHM ∼1MHz or in “ultra-high” resolution mode

in which the linewidth is reduced to < 100 Hz. For these experiments it

is operated in high resolution mode. The laser power incident on the crys-

tal was 40mW, focused to ∼100 µm and could be gated using a 100MHz

acousto-optic modulator (AOM). The RF used to drive the AOM was sup-

plied by an 80 MHz oscillator switched by a Mini Circuits ZASW RF switch

before being amplified using an in house built amplifier to 1W maximum.

The ZASW 2 input 1 output bidirectional RF switch provides 100dB isola-

tion with a switching time of 10 ns [149]. This was the only RF switch used

in the experiments, hereafter simply referred to as an RF switch. The RF

used to drive the hyperfine transitions was supplied by the Pulse Blaster Tx

output amplified by an Amplifier Research 10W1000 RF amplifier. This was

applied to the sample using a non-resonant 6 turn 5 mm diameter RF coil

which resulted in a RF Rabi frequency of ΩRF = 50 kHz. The magnetic

field was applied using a solenoid in the z direction, defined by the crystal

properties as shown in figure 4.4.

The Pr3+:Y2SiO5 crystal was placed in a Cryo Industries gas exchange

cryostat with the temperature held at ∼ 4.2K for the duration of the ex-

periment. The laser prepared a population difference in the excited ions

for 1s before the pulse sequence and was scanned over 1.2GHz of the opti-

cal transition to avoid hole burning effects. The laser is off during the RF

pulse sequence to minimize coherence loss from optical pumping as shown

in figure 4.6. The laser and RF pulses were controlled using a Pulse Blaster

pattern generator with a timing resolution of 10ns. The Pulse Blaster is

computer controlled, allowing automation of the experiment. The RF pulses


x

y

z

3mm

5mm

4mmC2 axis

OpticalPolarisation

Figure 4.4: Relationship of laboratory axes to crystallographic axes: y is the C2 axis, zis the direction of polarisartion of the optical transition for site 1 and x is perpendicularto both.

applied were sufficiently hard (ΩRF = 50 kHz) that all transitions of interest

are within the Fourier width of the pulses. The required magnetic field are

supplied by a solenoid aligned with the z axis of the crystal.

The transmitted laser beam is incident on a photodetector with a 125MHz

bandwidth, the output of which is fed into a spectrum analyser. The spec-

trum analyser is operated in a non scanning mode such that when tuned

to the transition it detects the intensity of the Raman heterodyne beat fre-

quency. The input is band pass filtered using the internal resolution band-

width filter, set to 300 kHz. The RF applied to the hyperfine transition is

generated by the spectrum analyser’s tracking generator, thereby ensuring

the detection and excitation frequencies are locked. The intensity measured

by the spectrum analyser was recorded by a digital oscilloscope and down-

loaded onto a computer.

4.3 Results

The modulation of the echo intensity is clearly visible when a large number of

echo delays are studied as shown in figure 4.7. If this decay is to be fit to an

exponential decay the maximum peak heights should be chosen and aliasing

due to an inadequate number of samples will yield an incorrect measurement

4.3 Results 75

Co

he

ren

t6

99

RFAmp

GPIB

Trig

RFSwitch

Pulse TTLout

Tx Rx

BlasterRF out

Cryostat

RF CoilsDC CoilsAOM

DC CoilsSample

Detector

Computer

CRO

Spectrum

Analyzer

Tracking

Generator

RF in I out

Figure 4.5: Schematic diagram of the experimet steup. The laser is gated via an AOMbefore entering the cryostat and being incident on the sample. A static magnetic field canbe applied to the sample using pairs of coils, with the RF radiation applied via the Rfcoils. The transmitted laser beam is incident on a photodetector, the output of which isfed into a spectrum.

Laser

RF2

Figure 4.6: Two pulse Raman heterodyne spin echo pulse sequence.


0 1 2 30

0.2

0.4

1

0.6

0.8

Delay Time (ms)

15 shot running aveln

(E/E

)0

Figure 4.7: Two pulse echo series with 1000 single shot points with an applied field of7.5G. Solid line is a 15 shot running average used for the Fourier analysis. τ is the delaybetween the π/2 and π pulses, as shown in figure 4.6.

of the coherence in the system.

When the echo decay is viewed in the frequency domain it is clear that

there are a number of components of the ensemble with different detunings

contributing to the echo (figure 4.8). Given the π pulse width of 10µs all

frequency components are well within the Fourier width of the pulse for all

field values and therefore excitation bandwidth can be ignored. As an exter-

nal magnetic field is applied, the frequency of each superhyperfine transition

increases by integer multiples of the Y Zeeman splitting.

Figure 4.8 shows that at low applied magnetic fields superhyperfine tran-

sitions involving up to three Y spin flips have significant contribution to the

echo amplitude. The DC component of the FFT is proportional to the un-

modulated echo and consequently each FFT spectrum is normalised to the

DC component. The superhyperfine transitions are attenuated by the ap-

plied field at a rate proportional to how many Y spin flips are involved. This

is consistent with observation in analogous systems [150]. Fitting an expo-

nential decay to the Signal to Carrier Ratio for the spectral component due

to single Y spin flips yields a decay constant of 7G. This is consistent with

the calculations presented in figure 4.3. It is possible that the superhyperfine

transitions are mediated by an exchange interaction involving a shared bond-

ing electron, not simply magnetic dipole-dipole coupling. This would cause

the interaction to persist long after direct magnetic dipole-dipole coupling

4.3 Results 77

−10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

3

4

5

Frequency (kHz)

Am

plit

ud

e (a

.u)

Figure 4.8: Fourier transform of averaged echo sequences at different field values aslabelled on the trace. The FFT of each field value is normalised to the DC componentand offset for clarity.


was quenched by the external field. Since the superhyperfine transitions are

significantly attenuated when the external field is of the order of the Pr-Y

magnetic dipole-dipole coupling, an exchange interaction can be ruled out.

The Zeeman shift due to the Pr ion on local Y ions is the origin of the

frozen core since it detunes the local Y ions from the bulk Y ions. However, as

shown by this experiment, with low applied magnetic field it is premature to

consider the local Y ions as constituting a frozen core for coherent transient

techniques due to the superhyperfine interaction. The local Y ions can be

considered a frozen core when the applied field is much larger than the Pr-Y

dipole-dipole coupling, when changes in the Pr spin state only weakly perturb

the local Y quantization axis.

Although a simple exponential coherence decay is not a complete descrip-

tion of coherence decay at low field the stated value in the literature [144] of

T2 = 500µs is a good fit to the decay envelope. If the peak heights of the

echo are used to determine T2 it is clear that with an applied field of 7.5 G

as shown in figure 4.7 T2 is approximately 3ms.

The Pr-Y magnetic dipole-dipole coupling strength allows us to approxi-

mate the field strengths required for dynamic coherence time extension tech-

niques such as magic angle line narrowing, investigated in chapter 7. This

technique flips the dephasing spins fast, at the magic angle such that their

average contribution is zero. For it to be effective the RF field must exceed

the Pr-Y magnetic dipole-dipole coupling. Therefore an RF field of > 10 G

would be required.

The study by Pryde [148] of near degenerate superhyperfine transitions in

Pr3+:LaF3 indicates that near degenerate Pr-Y will persist. Therefore larger

applied magnetic fields than those used in this experiment should be used to

suppress the superhyperfine interactions such that the hyperfine transitions

behave more like a closed two level atom. Any deviation from closed two

level atom behaviour results in information being lost to the environment

directly due to driving the transition. The majority of previous studies of

Pr3+:Y2SiO5 were done at zero or low field [6, 141, 103, 151, 102]. This is gen-

erally because of the simpler energy level structure at zero field, of particular

benefit if the experiment requires an optical repump scheme [102]. This work

however demonstrates that at low applied magnetic field the system cannot

be considered to be a two level atom and as such QIP investigations should

be performed with larger applied magnetic fields. Larger applied magnetic

fields are investigated in the following chapter.

Chapter 5

Maximising Hyperfine T2 using

Moderate Static Magnetic

Fields

“You must be the change you want to see in the world.”

– Mahatma Gandhi

In the previous chapter it was demonstrated that the decoherence time

of Pr hyperfine transitions in Pr3+:Y2SiO5 is improved by the application of

a small field. It was established that the applied magnetic field minimises

the change in the local Y spin bath due to a change in the Pr spin state. In

this low field regime the sensitivity of the Pr hyperfine transitions to fluctu-

ating fields magnetic fields is largely unchanged. In this chapter I investigate

minimising the sensitivity of the Pr hyperfine transitions to fluctuating mag-

netic fields by application of a static magnetic field with precise direction

and magnitude.

In a spin 5/2 system with zero field splittings when the Zeeman split-

ting approaches the zero field splitting there are numerous level anticross-

ings which result in non-linear Zeeman shifts. The sensitivity of transition

frequency to magnetic field perturbations is given by the gradient of the

Zeeman shift. The non-linearities demonstrate that the sensitivity of the

transition frequency to magnetic field is dependant on the applied magnetic

field. Therefore it is possible that a particular magnetic field magnitude

and direction can be found for which the hyperfine transition sensitivity is

minimised and therefore T2 is maximised.

80 Maximising Hyperfine T2 using Static Magnetic Fields

5.1 Theory

Searching the magnetic field space experimentally for the global minimum of

transition sensitivity is impractical given 15 transitions and the complexity

of the Zeeman shift due to the low site symmetry. As such, knowledge of

the parameters of the reduced Hamiltonian for hyperfine ground state in-

teractions (equation 2.13) is required to enable computational searches of

the magnetic field space. The field rotation study undertaken by Longdell

et al. [105] determined the M and Q tensors for hyperfine interactions in

Pr3+:Y2SiO5, as discussed in section 2.3.3.

H = B · M · I + I · Q · I

The predicted hyperfine spectrum as the magnetic field is applied along

the x, y and z axis is shown in figure 5.1. The complexity of the Zeeman

shift non-linearities make finding a global minimum of transition sensitivity

prohibitively complex. While numerous turning points in hyperfine transi-

tion frequency are observed in one dimensional magnetic field sweeps (figure

5.1), representing one dimensional sensitivity minima, the minimum three

dimensional sensitivity was not known.

Transition sensitivity minima with zero first order Zeeman shifts in any

direction were searched for using a Nelder-Mead simplex (direct search)

method, implemented as part of Matlab Function Functions and ODE Solvers

toolbox. For each magnetic field value tried by the minimisation routine the

frequency shift due to a perturbing magnetic field along each axis was calcu-

lated as follows:

〈∆ω〉 =(

δωx2 + δωy

2 + δωz2)1/2

(5.1)

where 〈∆ω〉 is the geometric mean frequency perturbation and δωi is the

change in transition frequency due to the perturbing magnetic field aligned

with the i axis. The perturbing magnetic field strength was chosen to be

of the order expected due to the array of Y ions. This was calculated in

section 4.1 to have a mean value of 0.035G with the distribution previously

shown in figure 4.1. The minimisation routine was initiated from 125 points

within the ±2 kG region for all hyperfine transitions, extensively searching

the anticrossing region.

The minimisation routine found 61 magnetic field values that satisfied

the criterion zero first order Zeeman shift in any direction for a particular

hyperfine transitions. These points will hereafter be referred to as a critical

point with reference to the definition of a critical point on a curve.

5.1 Theory 81

0 1 2

Bz (kG)

0

10

20

30

40

Fre

qu

en

cy (M

Hz)

0

10

20

30

40

Bx (kG)0 1 2

Fre

qu

en

cy (M

Hz)

0 1 2

By (kG)

0

10

20

30

40

Fre

qu

en

cy (M

Hz)

Figure 5.1: Ground State hyperfine spectrum predicted by theory as the applied mag-netic field is varied from zero to 2kG in the x, y and z directions.


mI Bx By Bz f (MHz) δfδB2

+12↔ −3

2732 173 −219 8.6 102

-309 -1124 -425 3.5 119-451 361 -678 5.6 193

+12↔ −5

2-465 372 -698 7.7 192

−32↔ +3

2438 -351 659 2.2 386237 -1094 382 1.7 486

−52↔ −3

2-683 -357 -1000 7.5 241664 373 971 5.8 2441567 454 155 10.2 320-1086 93 489 10.2 345-890 487 -174 11.5 828212 55 -844 11.5 828

-1093 -253 -453 11.5 829-418 -793 561 11.5 830

+52↔ −3

2-518 566 494 8.5 132-256 1109 -411 5.8 243-1062 -857 602 10.2 320-1562 -457 -150 10.2 321-1289 -646 209 10.2 3451079 -87 -499 10.2 345-214 -54 841 11.5 829

Table 5.1: List of all critical points which have less than 1kHz/G2 second order frequencydependance.

Those critical points that had a second order sensitivity < 1kHz/G2 are

listed in table 5.1 with the full list presented in Appendix B. The critical

point with the lowest seconder order sensitivity was investigated on the mI =

−1/2 ↔ +3/2 transition with BCP = 732, 173,−219G and a transition

frequency of 8.6MHz.

In order to understand the dependence of the transition sensitivity on

the applied magnetic field in the vicinity of the critical point used in the

experiment the first order Zeeman shift was calculated for a large range of

field values around the optimal critical point, shown in figure 5.2. As figure

5.2 shows, there is a continuous reduction in first order transition sensitivity

as the critical point is approached.

The optimum critical point the magnetic field is required to be accurate

to within the magnetic field fluctuations inerrant to the material, which for

Y2SiO5 is of the order of 0.035G. Therefore, given the critical point field

magnitude is ∼780G, this requires an accuracy of one part in ∼2×104. Since

the fitting of the hyperfine parameters to the field rotation data was only

accurate to ∼4% [105] the predicted transition frequencies are not accurate


Bx (G)

∆ω

∆Β

Hz/

G

Figure 5.2: Theoretical calculation of the first order Zeeman shift due to a 0.035 Gperturbation in each spatial direction calculated as per equation 5.1.

enough. Further, the inhomogeneous linewidth of hyperfine transitions not

at the critical point is of the order of 10kHz [6]. Therefore, measuring the

spectrum alone is not sufficient to ensure an optimal field alignment has been

obtained. We can, however, predict the expected critical point magnetic

field with sufficient accuracy to optimise the field alignment experimentally.

Given the topology near the critical point, in particular the lack of nearby

local minima, we can be confident of achieving an optimised critical point

when T2 is optimised from an approximate initial magnetic field alignment.


The experiment configuration shown in figure 5.3 has many schematic sim-

ilarities with the previous configuration described in section 4.2. However,

in this experiment the Pr3+:Y2SiO5 crystal maintained at temperature of

∼1.5K in an Oxford Instruments MD10 liquid helium bath cryostat [152].

The applied magnetic fields to achieve the critical point field configuration

were supplied by two orthogonal superconducting magnets supplying a z

field, and x, y field. The axis used to describe the magnetic fields are related

to the crystal properties as shown in figure 4.4 and was discussed in section

2.3.3. The sample was rotated about the z axis to provide the correct ratio


Co

he

ren

t6

99

GPIB

Trig

Pulse TTLout

Tx Rx

BlasterRF out

Cryostat

RF CoilsDC Coils

DC CoilsSample

Detector

Computer

CRO

RF out

J850 DDS

RFAmp

RFAmp

USB

USB1 2 3 4

AOM

RFSwitch

RFSwitch

Figure 5.3: Schematic diagram of the experiment setup. The laser is gated via an AOMbefore entering the cryostat and being incident on the sample. A static magnetic field canbe applied to the sample using pairs of coils, with the RF radiation applied via the RFcoils. The transmitted laser beam is incident on a photodetector, the output of which isfed into a spectrum.

of fields along the x and y axes for the critical point in magnetic field space.

The x, y coil is a split pair magnet, incorporated into the cryostat, specif-

ically designed to produce very homogeneous fields for NMR experiments

and therefore used to provide a majority (∼71%) of the magnetic field. The

field in the x, y plane could also be adjusted using a small correction coil

mounted orthogonal to the MD10 split pair magnet. The inhomogeneity in

magnetic field across the sample was measured using a hall probe to be <2G,

dominated by inhomomgeneities in the z coil.

The Coherent 699 dye laser was operated in “ultra-high” resolution mode,

resulting in a sub kilohertz linewidth. The laser power incident on the crystal

was 40mW , focused to ∼100µm and could be gated using a 100MHz acousto-

optic modulator. The RF driving the AOM was upgraded to a purpose built

Direct Digital Synthesis (DDS) agile RF source combined with an array of

RF switches and amplifiers capable of producing 1W . This unit was built

to our specification by the RSPhysSE Electronics Unit, referred to by it’s

project number as the J850. This enabled implementation of the complex

burn back sequence described later.

The hyperfine transition was excited using a six turn coil with a diameter

of 5mm, driven by a 10W RF amplifier resulting in a Rabi frequency ΩRF =


π2π

ττ x10

x500

x10

ω 3

ω CP

ωp

ωr

ω r scan

ω1 ω2,

State Preperation

Figure 5.4: Pulse sequence used in the experiment, showing the repump scheme, with thenumber iterations for wach section indicated below, followed by the 2 pulse echo sequence

91kHz. The RF pulse and digital control sequences were generated using

a direct digital synthesis system referenced to an oven controlled crystal

oscillator. The pulse sequences used in the experiment are illustrated in

figure 5.4 The Raman heterodyne signal, seen as a beat on the optical beam,

was detected by a 125MHz photodiode. This signal was analysed using a

mixer and a phase controlled local oscillator referenced to the RF driving

field.

Prior to applying each Raman heterodyne pulse sequence the sample was

prepared with the optical/RF repump scheme as shown in figure 5.5. The re-

pump frequencies were ωr−ωp = 18.2MHz, ω1 = 12.2, ω2 = 15.35MHz and

ω3 = 16.3MHz. Optical inhomogeneous broadening is significantly larger

than the excited state Hyperfine splittings. Therefore, since the repump

scheme is only designed to select for ground state hyperfine levels there will

be 6 optical subgroups for which the optical transition frequency is satis-

fied. Therefore the common excited state is not labelled. The repump RF

was pulsed with a duty cycle of 10% to reduced sample heating, while the

readout laser frequency ωr was scanned 200kHz to hole burn a trench in the

inhomogeneous optical line where detection would take place. This was im-

plemented by applying ω1 and ω2 pulses alternately while ωr was scanned,


- 1 2/

- 3 2/

+ 3 2/

- 52/

+ 52/

+ 1 2/

3 2/+-5 2/+-

+- 1 2/

r

1

23

p

Figure 5.5: All ground state hyperfine levels other than mI = −1/2 interact via RFfrequencies ω1−3 with laser radiation either at the read frequency ωr or pump frequencyωp. Through spontaneous emission from the electronic excited state the group of ionsinteracting with both ωr and ωp via any common excited state will be holeburned into themI = −1/2 state. Bold states are the critical point transition.

then applying ω3 pulses while ωp was applied and iterating this 500 times as

shown in figure 5.4. This repump scheme ensures that all Pr ions interacting

with the laser radiation are forced into the mI = −1/2 state, creating a pure

state ensemble. It also ensures there is no initial population near the laser

frequency used for Raman heterodyne detection. The use of a sub kilohertz

linewidth laser and the repump scheme resulted in a significant improvement

in the signal to noise compared to work performed earlier [152].

5.2.1 Finding a Critical Point

Sweeping the magnetic field from 0G to 2BCP results in transition frequen-

cies changing as shown in figure 5.6. As can be seen, the transition of interest

goes through a very gradual critical point. Site 1 consists of two magnetically

inequivalent sites as discussed in section 2.3.1. Consequently the y compo-

nent of the magnetic field experienced by “site 1a” ions is the opposite of

“site 1b” ions, resulting in the more complex level structure seen in figure 5.6.

As the sites are crystallographically equivalent it is not important which site

is used and consequently we will refer to “a” as the site that is used. However

since both “a” and “b” ions will interact with the laser radiation they will

both contribute to Raman heterodyne signals and are therefore important

experimentally.

While there are many distinct spectral features, such as other turning

points and points where transition frequencies cross, most are not close

enough to the critical point to be of use. The main spectral feature of use is


200 400 600 800 1000 1200 1400

2

4

6

8

10

12

14

16

18

20

B field (G)

Fre

qu

en

cy (

MH

z)

Figure 5.6: Ground State hyperfine transition frequencies as the applied magnetic fieldis varied from zero to twice the Critical Point Field. Site 1a ions are shown as a solid linewith site 1b ions shown as dashed lines. The critical point transition is bold.


that the critical point transition is crossed near the optimum magnetic field

value by the mI = −1/2 ↔ +3/2 transition of site 1b as shown in figure

5.6. As discussed in section 5.1, measuring the spectrum is not sufficient to

achieve an optimal critical point magnetic field configuration.

Coarse magnetic field alignment focused initially on the x, y field by ap-

plying the predicted field magnitude and rotating the sample rod until the

hyperfine spectrum matched the predicted spectrum shown in figure 5.7.

Sample rod rotation has the least resolution of any magnetic field adjustment

due to seals producing hysterisis in the adjustment and not having a microm-

eter actuated rotation mechanism. Consequently this was only adjusted if

there was insufficient field strength from the perturbing coil to achieve an

optimised critical point field alignment. In practice the sample rod needed

to be adjusted several times due to the inhomogeneous linewidth masking the

exact transition crossing points and a lack of other distinct spectral features.

The z axis magnetic field is then applied and the magnetic field is swept

by a small amount to verfy that the frequency of the critical point transi-

tion changes very little compared to surrounding transitions. The predicted

response to a small field sweep on each axis is shown in figure 5.8.

Once the approximate position of the critical point in magnetic field space

has been obtained the decoherence time of the transition is the best guide

for improving the magnetic field alignment. Fitting an entire echo sequence

is time consuming whereas maximising the amplitude of an echo with a long

delay is sufficient. The echo delay should be adjusted so that the echo is eas-

ily distinguishable from the background but there is still plenty of dynamic

range left in the detection system. The magnetic field can then be iteratively

adjusted until the amplitude of the echo is optimised using the longest delay

possible. The lack of further gains will be due to one of three things: the

magnetic field is more accurate than the magnetic field fluctuations in the

material; the magnetic field alignment is limited by inhomogeneity of the ap-

plied field or there is insufficient resolution in the adjustment of the magnetic

field.

5.3 Results

Initial experiments to verify the concept resulted in a significant increase

in decoherence time as shown in figure 5.9. The typical gains due to the

application of a magnetic field were observed, increasing the T2 of transitions

not at the critical point from 0.5ms to 9.9ms for the site 1b mI = −1/2 ↔+3/2 transition and 5.9ms for the site 1a mI = −3/2 ↔ +3/2 transition.

5.3 Results 89

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40

Rotation angle (degrees)

Fre

qu

en

cy (

MH

z)

Figure 5.7: Hyperfine spectrum as the sample rod is rotated with the x, y field magnitude(752 G) applied. Site 1 ’b’ ions are dotted lines and the critical point transition is bold


−10 −8 −6 −4 −2 0 2 4 6 8 108.642

8.643

8.644

8.645

8.646

8.647

8.64810G field sweep around Critical Point

Detuning from optimal Critical Point field (G)

Tra

nsiti

on F

requ

ency

(M

Hz)

Figure 5.8: Frequency change of the mI = +1/2 ↔ +3/2 transition due to a change of10G in the x (solid),y (dashed) and z (dotted)

5.3 Results 91

0 0.02 0.04 0.06 0.08 0.1−4

−3

−2

−1

0

delay (s)

ln(E(2 )/E ) 0

τ

Figure 5.9: A comparison of two pulse echos at zero field (×) and three transitions withthe for the mI = +1/2 ↔ +3/2 at site 1a (). The mI = +1/2 ↔ +3/2 site 1b transition(∗); the mI = −3/2 ↔ +3/2 transition (+) and the low field.

The application of the critical point field has a further significant increase

on these values to yield T2 = 82ms. These experiments were limited by only

having control of the applied magnetic field in two dimensions, requiring the

ratio of the x and y field being adjusted via rotation of the sample rod. Given

the low resolution and hysteresis of such an adjustment we were only able to

attain a magnetic field alignment within ∼5G of the ideal critical point field.

Despite these limitations the critical point transition displayed an order of

magnitude longer T2 than transitions not near a critical point [152].

With the addition of the y axis perturbation coil, allowing 3D control of

the applied magnetic field, greater timing resolution and improved RF control

the experiment was significantly improved. This is clearly demonstrated in

the two pulse echo data presented in figure 5.11, in which T2 was increased to

860 ms for an optimised critical point. A summary of the T2 for the different

magnetic field configurations is shown in table 5.3.

The initial critical point results were fit to the exp[−2τ/TM ] model de-

rived by Mims [110] which describes the echo decay due to a dephasing pro-

cess long compared to the echo time scale. Besides being longer the decay


ln(E(2

)/

E ) 0

τ

τ

0

-0.04

0 20 40 60

(ms)

Figure 5.10: Two pulse echo decays with improved experimental apparatus showing theshort time decay of the optimised critical point field alignment.

Mangetic Field Configuration (G) T2 (ms)zero field ∼0.5

7 ∼312 ∼5780 5.9780 9.9

Initial CP 82Optimised CP 860

Detuned 2G (z) from CP 320Detuned 5G (z) from CP 100

Table 5.2: List of the observed T2 for the static magnetic field values experimentedwith. CP is an abbreviation of critical point and all ∼ are due to the superhyperfineinteraction resulting in T2 not being a good description of the system for the magneticfield configuration. 780G is the field magnitude of the critical point field alignment withechos performed on transitions not at a critical point.

5.3 Results 93

0 0.2 0.4 0.6 0.8 1 1.2

0

delay (s)

ln(E

(2

)/E

) 0τ

Figure 5.11: Two pulse echo decays with improved experimental apparatus for anoptimised critical point (⋄), ∼2G detuned from the optimised critical point (·) and ∼5Gdetuned in the z direction to yield a T2 = 100ms decay ().


can no longer be described by standard echo decay function with a single

time constant [110]. There are three distinct regions for the optimised criti-

cal point field, as seen in figures 5.10 and 5.11. As shown in figure 5.10, for

pulse separations less than 20 ms the decay rate is less than 1/4s−1. When

the pulse seperation is of the order of 30 ms there is a distinct shoulder with

the decay rate increasing to 1/0.4s−1 as the pulse separation reaches 60 ms.

From 150 ms onwards the decay rate asymptotically approaches a value of

1/0.86s−1. This asymptotic decrease in the decay rate was only observed

for magnetic fields within 0.5 G of the optimal field. When the field was

more than 0.5 G away from the critical point a simple exponential decay was

observed for delays longer than 50 ms.

5.4 Discussion

During the initial stage of the decay (τ = 0 → 10ms) there is almost no

decay, indicating that there is very little reconfiguration of the local Yttrium

ions on this time scale. The shoulder in the decay at τ = 30 ms is interpreted

as indicating that the majority of the dephasing is due to perturbations that

occur on time scales between 10 and 100ms. This is consistent with an

exp [−(τ/TM )2] decay when the dephasing time scale is longer than the echo

time scale as discussed in section 2.5.3.

For delays longer than 100mswe would normally expect to see an exp [−τ/T2]

decay emerge, corresponding to a dephasing process acting faster than the

echo time scale (see section 2.5.3). The asymptotic behaviour of the decay

is attributed to a variation in the T2 within the ensemble resulting primar-

ily from inhomogeneity in the applied magnetic field. Ions experiencing a

field closer to the critical point condition will have a longer T2 and conse-

quently their contribution to the echo intensity will dominate for long pulse

separations. Quadrupole and pseudoquadrupole inhomogeneities will also

contribute to producing a range of ideal critical point field values within an

ensemble. However, given the measured magnetic field inhomogeneity of 2G

it is expected to be the current limit rather than the quadrupole interactions.

Therefore, the asymptotic decay rate gives an upper limit for the contribu-

tion to the decoherence due to second order magnetic interactions of 1.16

Hz.

The decay affords insight as to which Y ions are contributing to the

decoherence. The kinds of magnetic field perturbations experienced by the Pr

ion can be divided into two groups. Y ions near the Pr ion are part of a frozen

core and therefore spin flip slowly, causing large but infrequent magnetic

5.4 Discussion 95

field perturbations. Nearest neighbour Y ions resonantly exchange spin at

approximately 10 Hz, however in the bulk given the vast number of spins that

can resonantly exchange spin the actual cross relaxation rate is significantly

faster. This results in rapid but small magnetic field perturbations.

The critical point technique results in only second and higher order con-

tributions to the Zeeman shift. Consequently the rapid, small magnetic field

perturbations originating from the bulk Y ions have a vanishingly small ef-

fect on the hyperfine transition frequency. This leaves only the large, slow

magnetic field perturbations of the frozen core Y able to significantly change

the hyperfine transition frequency.

Removing transition frequency perturbations due to bulk Y is fundamen-

tally important for QIP applications. Any change in transition frequency

that is not controlled within a quantum computer results in a phase error

accumulating. The vast number of Y ions in the bulk means that on any time

scale it would be reasonable to expect some change in the bulk Y spin con-

figuration. This in turn means that Dynamic Decoherence Control (DDC,

or error correction protocols) could not be applied fast enough to correct

all errors. Further, DDC assumes that the transition frequency is constant

during the application of the pulses required to implement the particular

DDC scheme. This assumption is not valid if the bulk Y can perturb the

transition frequency. Therefore, despite the optimisation of the critical point

only showing significant gains at long delay times it is still fundamentally

important to error correction of the system.

QIP operations have to be performed in a time short compared to decay

of coherence from the system. Therefore the most interesting timescale for

QIP is within the first ∼ 10ms, shown in figure 5.10 during which decoher-

ence is negligable. On this timescale decoherence for each member of the

interacting Pr ensemble is different due to different local Y environments

with individual perturbation magnitudes and dynamics. For long delays, as

previously discussed, each Pr ion has experienced a represantitive set of local

Y configurations and only the ions experiencing an optimised critical point

significantly contribute. Therefore the T2 at long delays remains the most

usefull measure of the Pr ion sensitivity to magnetic perturbations for QIP

applications despite operating on a significantly shorter timescale.

A subtle point is that if there is no change in Pr hyperfine transition

frequency to first order as the magnetic field is perturbed it follows than the

magnetic moment dipole of each hyperfine state must be the same to first

order. Since the magnetic dipole moment of each state is the same to first

order there is no change to the spin quantisation axis to first order. This


property of a critical point magnetic field alignment makes it the optimal

field alignment for suppressing near degenerate superhyperfine interactions.

Consequently, changes in the Pr spin state when the transition being driven

is at a critical point no longer result in back action from the bath on the Pr

ion. We have therefore created a situation in which it is accurate to consider

the Pr ion be surrounded by a decohering thermal spin bath. As there will

always be second order contributions to the Zeeman shift it is considered

that the critical point with the lowest second order Zeeman shift represents

the global minimum of hyperfine transition sensitivity and therefore is the

global maximum of T2, excluding spin polarisation of the bulk Y.

Minimising the back action of the bath on the Pr ion is fundamentally

important for QIP applications. If there is any back action from the bath in

response to changes in the state of a qubit it cannot be considered as a two

level atom and therefore driving the system increases the probability of an

error in QIP applications. Dynamic Decoherence Control (error correction

schemes), a requirement of fault tolerant quantum computing, are derived

under the universal assumption is that the interactions used to perform error

correction do not induce errors. This is assumption is not valid if there is

any back action from the bath. It is an open question as to wether any error

correction can be succesfully implemented if there is appreciable back action

from the bath.

The main requirements for the critical point technique are the existence

of a zero field splitting in the spin states and the ability to apply a sufficiently

strong magnetic field so that the Zeeman splitting is comparable to the zero

field splittings. Zero field splitting are observed for sites with axial or lower

symmetry. Axial symmetry sites, due to the higher symmetry than investi-

gated in this work will result in a circle in magnetic field space of minimised

decoherence instead of a single point. Anticrossings are also required and

hence the system should have a nuclear spin I > 1.

An interesting aside is that the critical point technique is unlikely to have

been discovered using conventional NMR. Limits imposed by resonant dipole-

dipole cross relaxation between the spins of interest is removed by using dilute

samples in ODNMR (sections 2.5.1 and 3.1), compared to molecular tumbling

or Magic Angle Spinning for liquid and solid state NMR respectively. Both

molecular tumbling and sample spinning render the critical point concept

inapplicable due to the continuous reorientation of molecules or crystal with

respect to the applied field. Further, the applied fields used to tune the

nuclear spin transition frequency into resonance with the detection coil are

typically many Tesla and is significantly beyond the anticrossing region of

5.4 Discussion 97

most nuclear spin systems. Therefore, ODNMR experiments allow studying

spin systems which have a common, static reference frame for all spins of

interest, with the ability to apply an arbitrary magnetic field magnitude and

direction, required for the critical point technique.

5.4.1 Future Improvements

Improvements to the magnetic field homogeneity over the sample can be

made by either using a thinner sample or improving the homogeneity of the

applied field. The solenoid in the Oxford MD10 bath cryostat is specifically

designed to provide a homogeneous field along the optical axis with the field

inhomogeneity contributing less than 0.1G variation accross the laser beam

volume, determined from the calibration data supplied with the cryostat by

Oxford. Therefore it was used to provide a majority of the applied magnetic

field (x and y components), the ratio of which was adjusted by sample rod

rotation. The MD10 cryostat used in these experiments has a small sample

space and as such limits the coil designs in both the z and y axis. With

modifications to the sample mount the z field could also be produced using

the Oxford solenoid, thereby making both the z and y coils perturbation

coils. A superconducting switch is implemented on the Oxford solenoid and

should also be implemented on the y perturbation and z coils to increase

current stability. Using a thinner sample reduces the volume over which the

field is required to be homogeneous. The sample used in these experiments

is 3 × 4 × 5 mm and the group has recently acquired some 1 × 4 × 5 mm

samples. Signal strength is currently sufficiently large to allow single shot

measurements. When using thinner samples if SNR becomes a problem the

measurements can be averaged, the sample can be placed in a cavity, or

interferometric detection can be used.

With no applied field the hyperfine transition linewidths in Pr3+:Y2SiO5

are ∼70kHz, dominated by magnetic broadening. For an optimised critical

point field alignment the linewidth was measured as 4kHz, which did not

change as the magnetic field was detuned to the point where T2 = 100ms.

Since there is almost an order of magnitude change in the magnetic field sensi-

tivity, indicated by T2, with no corresponding change in the inhomogeneous

linewidth of the transition it was concluded the 4kHz of inhomogeneous

broadening of the transition at the critical point field is not due to magnetic

interactions.

The inhomogeneous broadening at the critical point field is most likely

dominated by strain broadening within the crystal, which is not intrinsic to

the site and can be reduced by refining standard crystal growing techniques.


The strain broadening couples to the hyperfine transition via the crystal field

interacting the quadrupole and pseudo quadrupole moment [47]. Techniques

for reducing strain broadening are discussed in section 6.4.1

It is possible, in principal, to spin polarise the bath and thereby stop bath

spin flips using large magnetic fields and low temperatures. This is achieved

by making one of the spin states energetically unfavourable due to the Boltz-

mann distribution, such that kT ≪ µB. This has been used to increase T2

in electron spin systems [153] and T2 increases in nuclear spin systems have

been attributed to this mechanism despite lack of complete spin polarisation

[154]. For Y this would require extreme experimental conditions of B > 10T

and T ≈ 1mK. While technically possible, this regime is very challenging

and, in general, is not feasible for nuclear spin systems. The hyperfine transi-

tions would remain sensitive, with potential enhancement of their sensitivity

due to non linearity of Zeeman shifts at high fields and therefore magnetic

field fluctuations generated by experimental equipment will strongly deco-

here the transitions. Due to the very low temperatures it would also be a

fragile thermal state and performing frequent state manipulations is likely

to heat the sample, if only the spin temperature, beyond the required tem-

perature. Therefore this regime is considered impractical. Consequently we

consider the critical point to provide the optimum practical magnetic field

configuration for achieving a long hyperfine T2 in Pr3+:Y2SiO5.

Chapter 6

Dynamic Decoherence Control

“If everything seems under control, you’re just not going fast enough.”

– Mario Andretti

In this chapter we investigate the possibility of increasing T2 using time

varying magnetic fields to periodically drive transitions of the quantum sys-

tem of interest. We can describe a quantum system of interest, HS , inter-

acting with a driving field HRF and an environment HE using the following

Hamiltonian:

H′ = HS + HSS + HE + HSE + HRF (6.1)

where HSE is the system - environment interaction and HSS in the interaction

within the quantum system of interest. As discussed in section 4.1, the first

step is to understand what the dominant decohering interaction is. Once this

has been determined strategies can be formulated to use HRF to minimise

the effect of this interaction on HS . The strategy used in this chapter is to

apply a cyclic driving sequence to HS such that the effect of the dephasing

interaction is equal and opposite during successive periods, thereby removing

the contribution of the dephasing interaction at the end of the cycle.

Dynamic Decoherence Control is a label quantum information theorists

give to decoupling techniques suitable for QIP, many of which are existing

techniques which NMR spectroscopists broadly refer to as Phase Cycling or

Coherent Averaging techniques. Only a subset of existing NMR techniques

are useful for QIP because many decoupling techniques are designed to pro-

tect a specific state from decohering and consequently do not protect an

arbitrary state well.

The simplest DDC technique used in NMR is the Carr-Purcell Mieboom-

Gill (CPMG) pulse sequence [80, 81]. This sequence is also the most com-

monly discussed DDC method for QIP applications [8, 9, 10, 11, 12, 13, 14,

100 Dynamic Decoherence Control

15]. In QIP discussions the CPMG technique was renamed Bang Bang de-

coupling [8], in reference to an analogous classical error correction protocol.

The Bang Bang decoupling method is a direct extension of a spin echo

and consequently has a long history in NMR. As discussed in section 2.5.2

spin echos “refocus” the inhomogeneous broadening of an ensemble. If the

reconfiguration of the dephasing bath is longer than the time scale which

pulses can be applied to the system then the frequency perturbations due to

the dephasing bath remain constant during each echo period and are therefore

also rephased. Consequently if π pulses can be applied sufficiently quickly

the driven T2 will exceed the undriven T2.

This was noted very early in the development of NMR techniques by Carr

and Purcell [80]. It was also noted by Mieboom and Gill [81] that to max-

imise the measured T2 the applied RF power had to be rigorously optimised.

They deduced that this was due to systematic errors in Rabi frequency ac-

cumulating as shown in figure 6.1. Mieboom et. al. then modified the pulse

sequence with the second π pulse having a 180 phase shift, to cancel any

systematic error in Rabi frequency. This modified sequence is commonly

referred to as the CPMG sequence and is also shown in figure 6.1.

Detailed analysis of the evolution of an ensemble interacting with a pulsed

driving field have been conducted initially by liquid NMR spectroscopists

[80, 41, 42, 82, 48] and more recently by quantum information theorists [8,

10, 11, 13]. All of these analyses lead to an equivalent statement of the simple

time scale argument [13] that:

ωbτc . 1 (6.2)

where ωb is the cut-off frequency of the dephasing bath modes and τc is the

delay between the pulses, or cycling time as shown in figure 6.1. Therefore if

the decoupling sequence is to be effective we need the ability to apply pulses

faster than the rate at which the bath reconfigures. This provides a simple

test for whether application CPMG like sequences are expected to increase

T2 of the system based directly on experimental measurements.

6.1 Application to Pr3+:Y2SiO5

In the previous chapter it was discussed that the magnetic field perturbations

due to the Y ions can be divided into two groups: frozen core Y ions near

the Pr ion producing large but infrequent magnetic field perturbations; and

Y ions in the bulk which result in small, rapid magnetic field perturbations.

6.1 Application to Pr3+:Y2SiO5 101

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

π π

Z

X

Y

−π

π

Z

X

Y

π

Z

X

Y

π/2

ππ/2

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

1

Ensemble Coherence

Carr Purcell RF τ2τ τ4 τ6 τ8 τ10 τ12 τ14 τ16 τ18

ππ/2

Time

Ensemble Coherence

Mieboom - Gill RF

τ2τ τ4 τ6 τ8 τ10 τ12 τ14 τ16 τ18

τ

3τ 5τ

Time t0

CarrPurcell

Common

MieboomGill

Figure 6.1: Bloch sphere visualisation of the Carr Purcell and Carr-Purcell Mieboom-Gill(CPMG) pulse sequence following a subgroup of an inhomogeneously broadened ensemble.This illustrates the effect of the π phase shift in cancelling systematic Rabi frequency errors


Z

X

Y

π/2Z

X

Y

Z

X

Y

π/2

Z

X

Y

π

π

Z

X

Y

Z

X

Y

Z

X

Y

−π

−π

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

τ

2τ 3τ 4τ

Time t0

Figure 6.2: Bloch sphere visualisation of the action of the Bang Bang pulse sequenceon an inhomogeneously broadened ensemble with two different initial coherent states: inphase and shifted by π/2.

6.1 Application to Pr3+:Y2SiO5 103

Perturbation Timescale

Fre

qu

en

cy S

hif

t

1/τ c

Figure 6.3: conceptual representation of the effect of DDC and critical point techniqueson the expected decohering magnetic field perturbations. The critical point techniqueminimises all transition frequency perturbations at all frequencies and is particularly ef-fective for the small perturbations. This is indicated by the difference between the zerofield distribution (solid line) and the critical point (dashed line). DDC techniques are onlyeffective at removing the perturbations occurring slower than τc, indicated by the blueshading.

DDC techniques and the critical point technique are complimentary tech-

niques. While the critical point technique minimises the decohering effect of

magnetic field perturbations at any frequency, it is best at rephasing small

perturbations since the second order contribution remains negligible as shown

in figure 6.3. Therefore it is most effective at removing the decoherence due

to the bulk Y. Larger magnetic field perturbations due to nearby Y ions

have a larger second order contribution and therefore still contribute to de-

coherence. We can consider the effect of the DDC as a decoherence filter,

in which low frequency decoherence within the bandwidth of the “filter” is

corrected. Consequently DDC will be effective at removing the decoherence

due to nearby Y ions but poor at negating the effect of the bulk Y. Without

application of the critical point technique the required τc will be significantly

smaller and therefore the schemes are complementary and DDC will be in-

vestigated in addition to the critical point technique.

Successful application of DDC techniques rests upon the whether the

required τc can be achieved and if so, can sufficiently hard pulses be achieved.

We therefore need to estimate the characteristic reconfiguration time of the

dephasing environment. For this we revisit the two pulse spin echo decay

for short delays, initially presented in figure 5.10, reproduced in figure 6.4.

As discussed in section 5.4 for echo delays less than 10ms there is almost

no decoherence due to reconfiguration of the spin bath. The duration of a


delay (ms)0 20 40 60

0

-0.2

-0.4

ln(E

(2

)/E

) 0τ

Figure 6.4: Two pulse spin echo amplitude as a function of delay for short delays. Thevery low decoherence during the first 10ms indicates that dynamic decoupling should beable to significantly extend T2, ideally as a tangent to the echo decay.

π pulse in the current experimental configuration is ∼10µs controlled with

10ns timing resolution by the PulseBlaster. Therefore inequality 6.2 can be

satisfied and Bang Bang DDC techniques should be effective at decoupling

the a Pr hyperfine transition at a critical point. This should result in the

hyperfine coherence evolving at a tangent to the decay curve, as shown in

figure 6.4, dramatically extending T2.


The experimental configuration remained largely unchanged from the critical

point experiment configuration. All of the DDC experiments use either an op-

timised critical point magnetic field configuration or a with the field detuned

by 5G in the z direction. By detuning the magnetic field the sensitivity of

the Pr hyperfine transition to magnetic field fluctuations is increased thereby

requiring the decoupling sequence to rephase more decoherence. The critical

point was optimised using long delay echos, as outlined in section 5.2.1 and

from this optimised configuration the field was detuned if desired.

The major change to the experimental apparatus was the addition of

phase sensitive detection of the Raman heterodyne signal as shown in figure


Coherent

699

GPIB

Trig

Pulse TTLout

Tx Rx

BlasterRF out

Cryostat

RF CoilsDC Coils

DC CoilsSample

Detector

Computer

CRO

RF out

J850 DDS

RFAmp

RFAmp

USB

USB1 2 3 4

AOM

RFSwitch

RFSwitch

Figure 6.5: Schematic diagram of the experiment setup. The laser is gated via an AOMbefore entering the cryostat and being incident on the sample. A static magnetic field canbe applied to the sample using pairs of coils, with the RF radiation applied via the RFcoils. The transmitted laser beam is incident on a photodetector, the output of which isfed into a spectrum.

6.5. Phase sensitive detection is required for performing quantum process

tomography [35] which, as described in section 1.3.3, is used to determine

the process operator O of the DDC technique. This involved down converting

the RF Raman Heterodyne beat note using two mixers with a 90 phase shift

on the Local Oscillator input such that both the in phase and quadrature

components were detected. In order to perform phase sensitive measurements

over T1 time scale of ∼ 100s at 8.65MHz requires a reference oscillator

stability of ∼10−9. All of the RF equipment was referenced to an Oven

Controlled Crystal Oscillator (OCXO) contained within a Stanford Research

Systems DS345 which has an Allen variance of < 5× 10−11 over a 1s period.

Therefore, our measurements should not be limited by the phase stability of

the reference oscillator.

The pulse sequences used in this experiment are shown in figure 6.6. As

before, two pulse spin echos were used to measure the undriven T2. The

inversion recovery sequence is used to measure T1, the upper limit that could

be expected from DDC techniques. The decoupling pulse sequence is also

shown in figure 6.6, with τ1 denoting the period between the initial pulse and

the start of the decoupling sequence and τc is the cycling time, the period

between the pulses used in the decoupling sequence.


π x10

x500

x10

ω 3

ω CP

ωp

ωr

ω r scan

ω1 ω2,

State Preperation

1τ1τ

π

2π

1τ

a)

2τ2τ2π/

−π

π π1τ cτcτ 1τ2π/

b)

c) N

Figure 6.6: Pulse sequence used in the experiment, showing the repump scheme, withthe number iterations for each section indicated below, followed by the a) 2 pulse echosequence b) Inversion recovery pulse sequence and c) decoupling pulse sequence. Thedecoupling pulse sequence is enclosed by the grey box and is iterated N times.

In the current investigation τ1 = 1.2ms and the cycling time τc was varied

from 0.05ms to 20ms. The choice of τ1 was made such that it could remain

constant for all choices of τc without being a simple integer multiple of τc.

This avoids echo artefacts that result from the DDC sequence rather than

the initial coherence by temporally separating the signals. Therefore we can

be certain that the measured echo amplitude is due to coherence generated

by the initial pulse.

6.3 Results

The decoupling pulse sequence was investigated varying the cycling time τc

from 0.05ms to 20ms, with several echo sequences shown in figure 6.7. In

order to differentiate the decoherence times for the driven and undriven cases

in the following discussion they will be referred to as T2D and T2 respectively.

Also shown in figure 6.7 is the result of an inversion recovery measurement

used to determine the T1 of the transition. The inversion recovery measure-

ments were performed using the pulse sequence described in figure 6.6(b).

As can be seen in figure 6.7 the decoupling sequence significantly increases

the Pr hyperfine coherence time though T2D/T1 < 1/2, even for the longest

value of τc.

6.4 Discussion 107

0 10 20 30 40 50−4

−3

−2

−1

0

Total cycling time (s)

ln(E(2 )/E ) 0

τ

Figure 6.7: Decoupled echo decays, varying τc 7.5ms(×), 10ms(∇), 15ms(⋄), 20ms(⋆)corresponding to T2D = 27.9, 21.1, 15.2, 10.9s. Inversion recovery measurements () yieldT1 = 145s.

T2D was studied as a function of τc for both an optimised critical point

(T2 = 860ms) and with the magnetic field detuned by ∼5G in the z direction

such that the two pulse echo decay was reduced to T2 = 100ms. As shown in

figure 6.8, there are three regions: for τc > 5ms, there are significant gains

in T2D made by reducing τc. T2D then plateaus at 33s for 0.2 < τc < 5ms

before again rising to a maximum of 70s for an optimised critical point.

It was also noted during the experimental work that there was a significant

drop in the echo decay amplitude during the first iterations of the decoupling

pulse sequence. As seen in figure 6.9 this initial fast decay was short lived

and all T2D measurements were fit to the latter portion of the decay.

6.4 Discussion

For τc > 5ms there is a significant reduction in the T2D observed using

the decoupling sequence for the measurements made with the detuned field.

As τc is decreased throughout this region T2D increases exponentially at a

similar rate for both data sets. The offset between the optimal critical point

and the detuned data sets is due to the increased sensitivity to magnetic

field fluctuations. Away from the critical point smaller, faster bulk processes


10-2

10-1

100

101

102

10

20

30

40

50

60

70

80T 2

(s)

Bang Bang cycling time τc (ms)

Figure 6.8: Dependence of decoherence time on the Bang Bang cycling time τc both atthe critical point () and with the magnetic field misaligned to give a coherence time ofT2D = 100ms (×). Trend lines do not represent a physical model.

0 5 10 15 20 25 300.5

0.6

0.7

0.8

0.9

1

Iterations

ln(E

(2

)/E

) 0τ

Figure 6.9: Echo decay during the initial iterations of a Bang Bang decoupling sequenceshowing an initial fast decay.

6.4 Discussion 109

contribute an equivalent amount of dephasing and consequently τc must be

shorter to leave the same residual dephasing contribution.

During the plateau region of 0.2 < τc < 5ms both data sets average

to ∼30s. This implies that T2D in this region is not limited by the ion’s

sensitivity to magnetic field fluctuations. As τc is reduced further (0.05 <

τc < 0.2ms) there is another dramatic increase in T2D. The most likely source

of this limitation was determined to be the driving RF waveform. Changes in

RF amplitude and phase that differed between the π and −π pulses are not

cancelled by the pulse sequence and hence accumulate, causing decoherence

relative to the reference oscillator. This second increase in T2D is due to the

driving pulses being applied to the ensemble faster than phase errors occur in

the driving field. Therefore, the phase of the ensemble tracks the phase of the

driving field, observed in numerous spin locking experiments [41]. Reducing

τc further was not possible due to the RF coil heating the liquid helium bath

past the superfluid lambda point. An interesting feature of the data set

for a detuned critical point is that T2D is actually shorter for τc = 0.05ms

compared to τc = 0.1ms. This was not due to noise in the experimental data

and was verified on different days that T2D decreased from ∼60s to ∼53s

for these delays. We consider this to be due to the combination of phase

noise and increased sensitivity to magnetic field fluctuations. While this is

somewhat unresolved it is considered to be an issue dominated by noise of

the driving RF and consequently should be revisited one the RF control is

improved.

The phase noise of the Pulse Blaster was characterised using a mixer and

the Stanford Research Systems DS345, reference oscillator. A continuous

stream of pulses, differing by 180 was mixed down to DC using the reference

oscillator. Over long time scales (¿1s) the phase noise was 1.15. As the time

scale was reduced to the range of 100ms → 0.5ms, transients in the phase

error became resolvable. It was not until the pulse width was reduced to

∼100µs that the number of phase noise glitches per pulse were noticeably

reduced by the time scale of the measurement. Pulse widths of 20µs typically

had no phase difference between the pulses. This indicated that the phase

noise was primarily glitches in the Pulse Blaster DDS circuitry, unrelated

to specifics of the pulse duration. This observation was consistent with the

decoupling sequence data, which demonstrated an increase in T2D in the

region which phase noise glitches were minimised. If the T2D increased at

the same rate as prior to the plateau the for a cycling time of τc = 1ms a

T2D of the order of 100s is expected. This could allow significant progress

toward the T1 = T2 limit.


-1

0

1

|0 0|

|0 1|

|1 0|

|1 1| |0 0|

|0 1|

|1 0|

|1 1|

-1

0

1

|0 0|

|0 1|

|1 0|

|1 1| |0 0|

|0 1|

|1 0|

|1 1|

-1

0

1

|0 0|

|0 1|

|1 0|

|1 1| |0 0|

|0 1|

|1 0|

|1 1|

Real Imaginary

-1

0

1

|0 0|

|0 1|

|1 0|

|1 1| |0 0|

|0 1|

|1 0|

|1 1|

Ideal

WorstCase

Figure 6.10: The λ matrix for any DDC scheme is to leave an arbitrary state unchangedand therefore is the identity matrix. The worst case is that the DDC process takes anarbitrary state and maps it to an incoherent state loosing all quantum information

6.4.1 Bang Bang Process Tomography

In order to assess how well the decoupling sequence preserved an arbitrary

quantum state process tomography was performed to determine the process

superoperator O, as described in section 1.3.3. Using the quamtum process

tomography method described in section 1.3.3 we determine λ, which can be

used to construct O using equation 1.24. Process tomography was performed

on the input state and 1, 10, 100 and 1000 iterations of the decoupling

sequence. The initial delay was τ1 = 1.2ms, with a cycling time of τc =

2ms. The total period over which the tomography was performed was 4ms,

40ms, 400ms and 4s respectively. Figure 6.10 shows the best and worst

case scenarios for a decoupling sequence. Ideally the decoupling sequence

preserves the input state without manipulating it, resulting in λ being the

identity matrix. The representation of an incoherent state, also shown in

figure 6.10 is the worst case, since any input state will be turned into an

incoherent state by the application of the pulse sequence.

The imaginary component of the process tomography, as shown in figure

6.11 is only shown for the experimental data since the modelling results were

always zero. Ideally the decoupling sequence process operator is the identity

matrix, leaving the state unchanged. It was observed that the component

6.4 Discussion 111

−1

0

1

−1

0

1

−1

0

1

− 1

0

1

− 1

0

1

− 1

0

1

− 1

0

1

− 1

0

1

− 1

0

1

− 1

0

1

|0 0|

|0 1|

|1 0|

|1 1| |0 0|

|0 1|

|1 0|

|1 1|

|0 0|

|0 1|

|1 0|

|1 1| |0 0|

|0 1|

|1 0|

|1 1|

|0 0|

|0 1|

|1 0|

|1 1| |0 0|

|0 1|

|1 0|

|1 1|

0

1

100

1000

10

ImaginaryReal Realπ,−πpairs

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

Experiment Theory

Figure 6.11: Plots of λ constructed from process tomography of the Bang Bang pulsesequence for 1, 10, 100 and 1000 iterations for experiment and theoretical modelling.Imaginary components of theory are omitted for clarity.


of the Bloch vector in the coherence plane for a given state was preserved

well, while the population component of the Bloch vector rapidly decayed.

The fidelity of the decoupling sequence for 1, 10, 100 and 1000 iterations was

99%, 65%, 54% and 43% respectively.

The evolution of the ensemble was modelled using the Bloch equations

assuming an infinite T1 and T2, with an inhomogeneous linewidth of 4kHz

(FWHM) and a Rabi frequency of 100kHz. The results from this mod-

elling are shown in figure 6.11, along side the experimental data. Despite the

model not including any homogeneous dephasing the rapid decay of popula-

tion terms compared to coherence terms of the simulated process tomography

(figure 6.11) match the experimental data. Simulations indicate that the de-

cay rate of the population terms can be reduced by increasing the ratio of the

Rabi frequency to the linewidth. A suitable criteria for when the application

of the decoupling sequence is useful for preserving arbitrary quantum states

is when the decay rate of the population terms in the presence of the pulse

sequence is slower than that of the coherence terms in the absence of the

decoupling sequence. For the present case, where T2 = 0.86s the simulation

indicates that to meet this criteria it will be necessary to achieve a ratio of

Rabi frequency to linewidth of ΩRF /ωinh ≈ 100. There is limited capacity to

increase the Rabi frequency of the driving field without the possible excitation

of off-resonant transitions. Therefore for the application of the decoupling

sequence to the investigated critical point on the mI = −1/2 ↔ +3/2 hyper-

fine transition to be useful it will be necessary to reduce the inhomogeneous

broadening of the transition by a factor of ∼10.

The simulations also exhibit the fast decay during the initial few iterations

of the decoupling sequence. Shown in figure 6.12 is the predicted projection

on the Bloch sphere axes as a function of decoupling sequence iteration for

the experimentally measured parameters. While the magnitude of the effect

differs from the experiment the general behaviour is the same.

Further simulations indicated that the decay rate of the population terms

can be reduced by increasing the ratio of the Rabi frequency to the linewidth.

A suitable criteria for when the application of the decoupling pulse sequence

is useful for preserving arbitrary quantum states is when the decay rate of

the population terms in the presence of the pulse sequence is slower than that

of the coherence terms in the absence of the sequence. For the present case

where T2 = 0.86s the simulation indicates that to meet this criteria it will be

necessary to achieve a ratio of Rabi frequency to linewidth of ΩRF/ωinh≈100.

In order to achieve a high fidelity (1 −mathcaF < 10−5) coherence time

extension using the decoupling pulse sequence the modelling predicted that

6.4 Discussion 113

0 100 200

0

1

Ech

o a

mp

litu

de

Decoupling Sequence iterations

0.5

Figure 6.12: Modelling prediction of the Bloch sphere projection during the initialiterations of a Bang Bang decoupling sequence for a coherent state (dash) and a populationstate (solid) with the experimentally measured parameters.

ΩRF /ωinh≥103. This can be addressed in part by using a resonant RF coil

such that all RF power is applied to the sample rather than a significant frac-

tion being dumped into the 50Ω load. The maximum transition specific Rabi

frequency is ∼1MHz which would result in ∆ωinh/ΩRF = 250. Therefore

inhomogeneous broadening of the hyperfine levels should ideally be reduced

by a factor of ∼10 for Bang Bang decoupling sequences to be effective.

It was argued in section 5.4 that the remaining inhomogeneous broad-

ening at the critical point field was due to strain broadening. The strain

broadening couples to the hyperfine transition via the crystal field interact-

ing the quadrupole and pseudo quadrupole moment [47]. A reduction in

strain broadening by over an order of magnitude has been achieved in anal-

ogous materials [143] through reducing the dopant concentration and using

isotopically pure materials. In the available Pr3+:Y2SiO5 crystals Si29, a spin

1/2 isotope of silicon, exists in natural abundance of ∼4%. The magnetic

moment of Si29 is a factor of 4 larger than that of Y, thereby contributing to

dephasing via magnetic field fluctuations and increasing disorder in the crys-

tal lattice. Repeated annealing of the crystal has demonstrated a reduction

in optical linewidth from 2.4 GHz to 520 MHz for Eu3+:Y2SiO5 [88]. Both

of these inhomogeneous linewidth observations were for optical transitions


and the hyperfine linewidth was not measured during these studies. If the

assertion in section 5.4 that the remaining inhomogeneous broadening on the

hyperfine at the critical point is due to inhomogeneities in the quadrupole

and pseudo quadrupole is correct then it is the same broadening mechanism

for both the optical and hyperfine transitions. Therefore we could expect a

similar reduction in hyperfine linewidth at the critical point as observed in

the optical transitions.

Concluding the discussion above, by using isotopically pure materials with

very low dopant concentrations and repeated annealing of the crystal we ex-

pect a reduction of inhomogeneous linewidth by a factor of approximately 50.

Other materials also warrant investigation and will be discussed in chapter

8.

Holeburning techniques can also be used to modify the hyperfine inhomo-

geneous linewidth. Sinc pulses have a “top hat” frequency response, thereby

allowing a narrow window of detunings to be resonant with the pulse. By

applying 2π sinc pulses to the hyperfine transition only those within the top

hat bandwidth of the sinc pulse will be returned to the initial state. Us-

ing holeburning techniques the remainder of the inhomogeneous line can be

transferred to other spin states that will not be involved in the experiment.

Therefore we can narrow the inhomogeneous linewidth of the ensemble used

to the required ratio even if refining the crystal growth techniques does not

provide sufficient inhomogeneous line narrowing.

Using a combination if Rabi frequency increase and inhomogeneous linewidth

reduction there is potential to increase ∆ωinh/ΩRF such that it is no longer a

limitation to the performance of the decoupling pulse sequence. In combina-

tion with actively modifying the inhomogeneous linewidth using holeburning

techniques this creates a real possibility of performing high fidelity DDC

experiments using Pr3+:Y2SiO5 in the near future.

Composite pulses are often used in NMR to minimise undesired effects due

to detuning and Rabi frequency errors [41]. By applying several consecutive

pulses the resulting operation, or composite rotation, can be made less depen-

dent on Rabi frequency and detuning errors [42, 41, 58, 155, 156, 157, 158].

Again we were restricted to composite pulses that can act on an arbitrary

state, such as those designed by Levitt [158], Wimpres [58] and Tycko [157].

Application of the decoupling sequence utilising the Wimpres BB1, BB2 [58]

and Levitt composite pulses resulted in a shorter T2D. Modelling confirmed

that the ratio ΩRF/ωinh was too low for the composite pulses to be effec-

tive. Composite pulses become effective when ΩRF /ωinh ≈ 103, the same

condition for the decoupling sequence itself. This is not surprising since this

6.4 Discussion 115

represents the regime where the pulses can be considered hard. While the

simulations typically showed only minor improvement due to using compos-

ite pulses there was no modelling of noise in the driving field and hence, in

practice, the improvement should be more significant.

To be certain that the detected spin echo was only due to the initial pulse

and not an artefact of the decoupling sequence, τ1 and τc were chosen such

that τc would not be an integer multiple of τ1. The optimal condition is

actually τc = 2τ1 [81] such that the inhomogeneous broadening is refocused

in between the π pulses. In the τc = 2τ1 case coherence generated by the

decoupling sequence will contribute to a spin echo at the same time as the

echo due to the input pulse. As this is the first test of Bang Bang DDC in

a solid the the experiment was designed such that the interpretation was as

clear as possible. While T2 can be expected to be longer for this optimised

condition more detailed investigation would be required to determine the

artefact contribution.

While T2 has been dramatically extended by the application of DDC, it

has also imposed a periodic condition for QIP operations. If we consider

the optimised case of τc = 2τ1 the inhomogeneous broadening is rephased at

the points half way between the π pulses. Consequently it is only at these

times that QIP operations can be performed without introducing phase shifts

relative to ions not involved in the QIP operation. To predict the phase shifts

incurred performing QIP operations not at τc/2 the detuning of the ions

involved must be precisely known. Since DDC techniques are employed to

remove the effect of time dependant inhomogeneities in transition frequency

this is not information that is possible to determine prior to executing the

operation. It therefore appears that as the cycling time is reduced, T2 is

increased and there are more opportunities per unit time to operate on the

system. This of course comes at the expense of applying more DDC pulses,

therefore causing errors due to pulse imperfections accumulate faster.

Careful attention to the reference oscillator and the phase distortions

inherent in RF synthesizing equipment will become increasingly important

in future experiments. In current experiments phase sensitive measurements

over a period of 100s at 8.65MHz requires a reference oscillator stability of

∼10−9. The phase stability of the driving field should ideally be an order

of magnitude greater than the measurement requirements such that it is not

the limiting noise source. State of the art OCXOs that provide stabilities of

∼10−12 over a 10s period are available from oscillator manufacturers [159].

Chapter 7

Extending T2 Through Driving

the Environment

“In theory, there is no difference between theory and practice. But, in prac-

tice, there is.”

– Jan L.A. van de Snepscheut

In the previous chapter we investigated a DDC technique that drove the

quantum system of interest, HS , to minimise the decoherence of that system.

In this chapter we investigate driving the environment, HE to decrease the

decoherence of HS . Magic Angle Line Narrowing is an established NMR

technique to drive HE to decouple the environment from the system, thereby

minimise the effect of system-environment interaction HSE [41] and increase

T2.

Magic Angle Line Narrowing (MALN) is used to decouple our spin of

interest, A, from the B spin bath undergoing dipole-dipole interactions by

driving the bath such that the B spins are rapidly flipped. Rapidly flipping

the B spins averages out dipolar interactions, resulting in a similar effect

to the tumbling of molecules in liquid NMR [41]. In order to be effective

the interaction of B spins with the driving field must dominate the dipolar

interaction between the A and B spins. When the driving field is applied at a

particular detuning relative to the transition linewidth to provide the “Magic

Angle” all orientations experience the same Rabi frequency. Therefore, all of

the B spins are decoupled effectively, resulting in optimal decoupling when

applied to complex systems or powdered samples.

The primary criterion for successful application of MALN is achieving the

following ratio of magnetic fields [54]:

B0 ≫ B1 > BAB (7.1)

7.1 Experiment Setup 117

where B0 is the static magnetic field, B1 is the magnetic field of the RF

driving the spin transition and BAB is the magnetic field due to the A spin

as seen by the nearest neighbour B spin. This can be interpreted as B0 must

significantly exceed B1 in order for there to be a defined quantisation axis

for spins and B1 > BAB such that the interaction with the driving field is

stronger than the interaction between the spins. The Magic Angle condition

for the detuning is given by [54]:

θM =ωd∆ω

= 3cos2β (7.2)

where β = 54.75 is the Magic Angle, ωd is the driving frequency and ∆ω is

the detuning from line centre.

MALN has been applied to many solid state systems with the first demon-

stration in solid state ODNMR spectroscopy performed on Pr3+:LaF3 by

Rand et. al. [53] and MacFarlane et. al. [54]. In systems that have large

nuclear spins in close proximity this can often result in impractically large

RF fields. This often occurs in solid state systems limiting the applicability

of this technique.

7.1 Experiment Setup

The experimental setup underwent minor modifications to the magnetic field

control from the previous experiments, described in section 6.2. Given the

magnetic field applied to reach the critical point the Y nuclear spin transi-

tion frequency, ωY , is 162khz. In order to get a sufficiently large RF field,

perpendicular to the dominant field direction ωY was applied to the sample

using the z coil as shown in figure 7.1. This required an RF choke to isolate

the RF from the DC power supplies. Since ωY is very low frequency in terms

of RF electronics, the choke could not provide very high isolation with prac-

tical inductances. Therefore we limited the RF power applied to minimise

the risk of damaging the DC power supply. The RF power supply delivered

1.4W into the z coil, resulting in a field of B1 = 6.4G, which produced a

Rabi frequency of ΩY = 1.3kHz.

T2 was measured using Raman heterodyne two pulse echos using the same

repump scheme and pulse sequence as the critical point experiments shown

in figure 7.1. The only addition to this was the RF driving field ωY was

applied during both the state preparation and echo sequence.

To determine how well we expect MALN to work we need to examine these

experimental conditions in light of the inequality 7.1 and the Pr-Y interaction

118 Extending T2 Through Driving the Environment

Coherent

699

GPIB

Trig

Pulse TTLout

Tx Rx

BlasterRF out

Detector

Computer

CRO

RF out

J850 DDS

RFAmp

RFAmp

USB

USB1 2 3 4

AOM

RFSwitch

RFSwitch

DC Coils

DC CoilsRF Coil

Sample

Figure 7.1: Experimental setup for the MALN experiments showning the DC and RFdrive to the z coil.

strength measured in chapter 4 as 9G. The experimental configuration fails

to provide an RF field B1 larger than BAB. We were unable to generate a

larger RF field with the available equipment due to the low RF frequency

limiting our ability to isolate the DC power supply.

Although the experiment did not satisfy inequality 7.1 it was close and

consequently we persisted. Despite not being an optimal configuration if

significant gains in T2 are achievable through optimised MALN we should

be able to achieve minimal gains with the current configuration. Since we

have not achieved B1 > BAB, the “Magic Angle” is not meaningful and

therefore we will refer to the current experiments as Driven Environment

Line Narrowing (DELN).

7.2 Results

The frequency of the DELN driving field was optimised using a Raman het-

erodyne two pulse spin echo with a long delay as per critical point optimi-

sation (section 5.2). When the DELN driving field was applied at a fixed

frequency in the range 140kHz < ωY < 180kHz the increase of the echo

amplitude not significantly above the small shot to shot variation.

It became clear that due to not satisfying the inequality 7.1 only a sub-

7.3 Discussion 119

group of the Y ions were interacting with the DELN field. The requirement

in inequality 7.1 that B1 > BAB can be interpreted that the Rabi frequency,

and therefore power broadening, must exceed the inhomogeneous broadening

of the B spins due magnetic field of the A spin. Therefore, if the inequality

7.1 is satisfied all Y ions interact with driving field.

Scanning the driving field was tried such that all of the Y ions would

interact with the driving field. This was the only experimental configuration

that resulted in a gain in the long delay spin echo amplitude. The spin echo

amplitude was optimised when ωY was centred at 162kHz and scanned over

600Hz with a 10Hz scan rate. The decoupling field was applied during both

the state preparation and echo sequence. It should be noted that given the

field of 9G magnetic field due to the Pr as seen by the nearest neighbour Y

and γY = 209Hz/G we expect the Y inhomogeneous linewidth to be 1.8kHz.

The Rabi frequency of 1.3kHz combined with scanning 600Hz matches the

Y inhomogeneous broadening.

The two pulse echo decay with and without the DELN decoupling field

are shown in figure 7.2. It is clear that the decay is significantly longer

when the DELN decoupling field is applied. In this experimental run the

critical point T2 was optimised to 820ms which DELN increased to 1.12s,

an increase of 36%. The overall characteristics of the decay are unchanged

with the three decay regions evident, as discussed in section 5.4. The two

pulse echo decay becomes linear at different times due to the DELN field

(0.7s and 1s ) but at the same echo amplitude, i.e. ln(E(2τ)/E0) ≈ −3.4.

Therefore, the proportion of ions experiencing an optimised critical point is

not changed by DELN. At short time there is no discernible difference caused

by the DELN driving field as shown in figure 7.3.

7.3 Discussion

Magic Angle Line Narrowing represents the optimal experimental conditions

for removing decoherence due to dipole-dipole interactions by driving the

bath spins. Since the MALN conditions were not satisfied these experiments

should be regarded as a feasibility of increasing T2 through driving the bath.

This experiment has demonstrated a 36% increase in T2 and as such an

optimised MALN experiment should perform significantly better.

The increase in T2 due to the DELN was most prominent in the second re-

gion of the two pulse echo decay (see section 5.4) where ions not experiencing

an optimised critical point create non-linear time response in the decay. The

increase in coherence in this region of the echo decay indicates that driving


0

0

0.4 0.8 1.2 1.6

Delay (s)

ln(E

(2

)/E

) 0τ

Figure 7.2: Spin echo decay amplitude as a function of pulse delay with (·) and without() the DELN decoupling field.

7.3 Discussion 121

0 0.1 0.2 0.3 0.4 0.5

−3

−2

−1

0

Delay (s)

ln(E(2 )/E ) 0

τ

Figure 7.3: Spin echo decay amplitude as a function of pulse delay with no Y drivingfield(×), fixed frequency DELN () and the DELN frequency scanned over 600Hz (·).


the bath is effective at extending the T2 of ions both at and near the critical

point.

Inhomogeneous broadening of Y hyperfine transitions is dominated by

magnetic interactions with the Pr ion. Since spin 1/2 nuclei, such as Y

have no electric dipole moment and are therefore not broadened by crystal

field inhomogeneities. Therefore, the Y hyperfine transition frequency is

proportional to the proximity of Pr ions, resulting in the frozen core and

bulk delineation of the Y spin bath, previously discussed in section 4.1. By

not scanning ωY either the bulk Y and the outer frozen core are driven or

the inner frozen core is driven. This results in many bulk Y or a few inner

frozen core Y still contributing as normal to dephasing the Pr ions. As such

either the bulk contributes many weak dephasing sources or the inner frozen

core provides a few strong dephasing sources. Some Y ions will be strongly

coupled to the driven subset despite not being driven themselves, such as the

ions on the edge of the frozen core when either the bulk Y or inner frozen

core Y are driven. The ions strongly coupled to the driven Y ions will have

an increased spin flip rate due to the rapid flipping of nearby Y ions with

a small detuning. This will tend to increase the decoherence contribution

from the undriven coupled ions as their spin flip rate is increased, but not

sufficiently increased to decouple the dipole - dipole interactions. This results

in the minimal gain in T2 due to the fixed frequency DELN driving field.

An increase in RF power will allow the MALN magnetic field condition

(inequality 7.1) to be satisfied. Higher RF powers were not available without

risk to equipment and hence reserved for future investigations. Further gains

in T2 are expected and, in the opinion of the author, achieving T2 ≈ 2s is

reasonable for an optimised MALN experiment.

In the previous chapter problems associated with imperfect driving fields

acting on HS became apparent. While much of this can be avoided through

the improvement of the RF circuitry the fidelity will always be less than

unity. By driving HE rather than HS to extend T2, errors in the driving field

are coupled to HS non-resonantly and as a result accumulate differently.

The MALN driving field introduces a time dependant Zeeman shift on the

hyperfine transitions of the Pr ion at the driving frequency. This is often

considered to be a phase modulation by NMR spectroscopists due to the

appearance of sidebands on the transition frequency, separated by integer

multiples of the driving frequency. Pulsed techniques have been developed

to counter the phase modulation in a similar manner to the refocusing in spin

echo experiments. The most commonly used technique is TOtal Sideband

Suppression (TOSS) [83] which involves a minimum of 4 π pulses, precisely

7.3 Discussion 123

timed relative to the period of the MALN driving frequency. It should be

noted that TOSS is identical to a CPMG or Bang Bang decoupling scheme

which rephases time dependant frequency shifts of the transition due to the

MALN field.

Consequently, as with the Bang Bang DDC we have introduced a periodic

condition for performing operations on the system if phase shifts between

qubits due to operations are to be avoided. While TOSS can be implemented

with a minimum of 4 π pulses, significantly more can be used such that QIP

operations can be performed more often. This however highlights that there

is no consequence free DDC method to extend T2 for QIP applications.

Implementing a TOSS scheme is very similar to the CPMG or Bang Bang

DDC schemes investigated in the previous chapter, with the addition of the

MALN field. The main importance of the DELN experiment is to show

that the technique is compatible with the critical point technique, allowing

for concatenation of QIP error correction schemes. Concatenation of these

schemes could be usefull in situations, such as removing residual first order

Zeeman contributions if it is not possible to achieve an optimised critical

point for all qubits.

There are a series of experiments will only be worthwhile if the Magic

Angle conditions are met, in particular studying the effectiveness of TOSS

at and near the critical point. These experiments require an optimised MALN

driving field, thereby removing the requirement to scan the frequency of the

driving field, and is therefore left to future work.

Chapter 8

Future Decoherence Challenges

“A bend in the road is not the end of the road... unless you fail to make the

turn.”

–Anon

The investigation of the decoherence mechanisms and suppression thereof

in Pr3+:Y2SiO5 has been very successful, however other materials are of

greater interest for long term rare-earth QIP development. Currently ma-

terials using Europium appear most attractive and stoichiometric crystals,

rather than lightly doped materials also appear promising. Decoherence ben-

efits and challenges of these future directions are discussed in this chapter.

8.1 Exchanging Praseodymium for Europium

In examining possible replacement ions for Pr we have strict criteria. The

ion must be an optically active non-Kramers ion, ie. it has a nuclear spin,

but not an electron spin. Rare-earth ions that are non-Kramers ions in the

3+ oxidation state are Pr, Pm, Eu, Tb, Ho and Tm, however Pm can be

excluded as being impractical due to having a short nuclear decay half life.

We require the ability to use the critical point technique, and as such the

nuclear spin to be I > 1 5.4, such that in a low symmetry host there is a

zero field splitting and an anticrossing region. This allows us to exclude Tm

since it is a spin 1/2 nucleus. Holmium is a spin 7/2 system with a much

larger dipole moment than Pr, reflected in the significantly larger linewidths

when used in the same hosts as Pr [88]. This indicates that inhomogeneous

broadening issues that arose in this work would be exacerbated by using Ho.

Terbium is still a candidate system, exhibiting equivalent linewidths to Pr

8.2 Stoichiometric Materials 125

[88]. However, the lack of interest in Tb for both holographic research and

achieving narrow optical transitions suggests there are experimental issues

with the ion. The most promising replacement ion appears to be Europium,

as will be discussed in the remainder of this section.

The greater promise of materials containing Eu is due to differences in

hyperfine transition T1, which for Pr3+:Y2SiO5 is 145s compared to 23 days

for Eu3+:Y2SiO5 [4]. Europium therefore provides a significantly longer up-

per bound on T2. This is due to negligible cross relaxation rates in low

concentration samples and in general cross relaxation is only significant in

stoichiometric samples. The smaller magnetic moment of Eu (γ ≈ 3kHz/G,

measured in Eu3+:Y2SiO5 [160]) implicitly reduces the hyperfine transition

sensitivity to magnetic field fluctuations.

The larger nuclear quadrupole of Eu results in zero field hyperfine split-

tings of the order of 100MHz in Eu3+:Y2SiO5 [140, 7, 6]. This larger splitting

allows higher Rabi frequencies to be used and remain transition specific. The

smaller γ of Eu when compared to Pr does however have two minor draw-

backs. Higher power driving fields are, however, required to obtain the same

Rabi frequency and larger magnetic fields are required to reach the hyperfine

anticrossing region to achieve critical point field configurations. Both are

technical issues.

Europium materials were not used in these initial investigations primar-

ily due to the difficulty of obtaining good Raman heterodyne signals in non

stoichiometric samples and the lack of field rotation data for Eu:Y2SiO5. Re-

cent developments within our research group [160, 128] have demonstrated

that using a suitable optical pumping scheme high SNR Raman heterodyne

signals can be obtained. Therefore investigating critical point field configura-

tions and dynamic decoupling methods in lightly doped Eu materials warrant

investigation after field rotation studies are performed.

8.2 Stoichiometric Materials

Stochiometric materials containing Eu are also of interest. Stoichiomet-

ric samples produce the lowest possible strain and therefore results in the

narrowest inhomogeneous linewidth. Reducing inhomogeneous linewidth is

beneficial for DDC techniques, as discussed in section 6.4.1. Investigat-

ing crystals that have inherently narrow inhomogeneous linewidths such as

EuCl6H2O will be interesting for further Dynamic Decoherence Control ex-

periments. Optical linewidths of the order of 100MHz have been observed

for EuCl6H2O [128], significantly narrower than the several GHz observed

126 Future Decoherence Challenges

in Eu3+:Y2SiO5.

The drawback with these materials is that resonant cross relaxation be-

tween the spins of interest dramatically increases the relaxation rate, with

the resulting decrease of T1 and T2. Stoichiometric materials lightly doped

with magnetic defects, such as Er3+:Eu2SiO5, discussed in detail in section

3.2, will hopefully provide a solution to this by only using the Eu ions within

the “frozen core”, and therefore allow low strain crystals which also have long

T2. Performing QIP in such a situation has potential complications above

and beyond simple T2 considerations and will be discussed in section 8.3.

8.3 Considerations for Implementing QIP in

Stoichiometric Materials

Performing QIP in stoichiometric samples as described in quantum computer

architecture (section 3.2) creates some problems due to the close proximity

of the ions used as qubits. The following discussion considers these issues

in relation to the Er3+:Eu2SiO5 system as proposed in section 3.2. It is

assumed that the bulk Eu ion can be optically spin polarised via holeburn-

ing techniques and therefore decohereing magnetic perturbations originating

from the bulk are vanishingly small. The following discussion also assumes

that a critical point for Eu can be found with similar properties to the critical

point investigated in this work.

Both the electric and magnetic dipoles change when the ion is optically

excited. Consequently there is a change in magnetic field on the order of a

Gauss seen by nearest neighbour ions. The use of the critical point technique

will be helpful to minimise the effect of the field perturbation, however there

are two primary differences between this situation and the Pr-Y critical point

investigated in this work. Firstly the Eu ions near the Er ion have a range

of hyperfine detunings due to the frozen core generated by the Er ion, and

therefore result in a range of critical point field configurations. Secondly,

the magnetic field change due to optically exciting Eu is significantly larger

than for a Y spin flip and therefore second order Zeeman shifts will be more

important. This is another benefit of the smaller magnetic moment of Eu

with respect to Pr.

The back action of the Y bath due to a change in the Pr spin state, as

investigated in chapter 4, also needs to be reconsidered in the context of

Er3+:Eu2SiO5. Changing the magnetic dipole moment of an Eu ion will

change the local spin quantisation axis and therefore can induce superhy-

perfine and near degenerate mutual spin flip transitions amongst nearby Eu

8.4 Minimising Decoherence in Stoichiometric Defect QIP Systems 127

ions. Superhyperfine transitions should be well suppressed by the applica-

tion of the critical point field assuming the magnetic field magnitude for a

critical point in Eu is, like that for Pr, many orders of magnitude larger than

the magnetic dipole moment of the Eu ion. Therefore, the near degener-

ate superhyperfine transitions that represent the biggest problem. The near

degenerate superhyperfine spin flips will most likely be other qubits rather

than members of the bath due to their proximity. This represents a very se-

rious potential for errors to occur as a direct consequence of QIP operations.

Quantum Error Correction Codes can be employed to correct these kinds of

errors at the expense of using several physical qubits for one logical qubit

[72, 65, 63, 161, 74, 68, 162, 62, 70, 66].

There is also potential for the change in electric dipole moment of an ex-

cited ion coupling to the hyperfine state of a neighbour through the quadrupole

and pseudo quadrupole moments. This needs to be carefully investigated to

determine the limits it creates for particular applications. The Er3+:Eu2SiO5

system discussed in the previous section provides a very good system to in-

vestigate these effects and how they scale with distance.

In light of these considerations the previous discussion in section 3.2 of the

optical qubit as being interacting and the hyperfine qubit as being isolated is

not completely true. Assessing the limits these interactions impose on QIP

is a particularly important direction for future research in the area. Char-

acterisation of these interactions will be required to formulate appropriate

quantum error correction strategies.

8.4 Minimising Decoherence in Stoichiomet-

ric Defect QIP Systems

The quantum computing architecture discussed in section 3.2 creates uniquely

addressable qubits by introducing defects. All QIP therefore occurs inside

the frozen core of the defect site. While it was previously stated that devices

possessing greater than 10 qubits could be produced, the frozen cores can be

very extensive. If we consider the Er3+:Eu2SiO5 system a rough calculation

shows that γEr/γEu ≈ 25γPr/γY and using the size of the Pr-Y frozen core

therefore we can expect the Er-Eu frozen core to contain of the order of 104

Eu ions. This clearly has the potential to create far more than 10 individ-

ually addressable qubits. The size of the frozen core also has the distinct

advantage for decoherence since the bulk is a long way from the inner frozen

core used for QIP. Therefore, the fluctuating magnetic field due to the bulk

as seen by the inner frozen core will be small and tend to reflect the mean

128 Future Decoherence Challenges

field value.

This work has demonstrated the dramatic increase in T2 afforded by the

critical point technique and that DDC techniques can be used to further

increase T2. The introduction of defects to the system provide two primary

challenges the the methods followed in this work to increase T2. Firstly the

state of the defect must be controlled. In the case previously discussed of

Er3+:Eu2SiO5 we require spin polarisation of the Er since any change of state

of the Er will introduce catastrophic errors via the superhyperfine interaction.

Therefore magnetic field configurations will be dictated by achieving this

result with the benefits of a critical point configuration being secondary.

The second challenge is that we introduced the defect to create repeat-

able inhomogeneities, however inhomogeneities limit the effectiveness of the

critical point technique. The magnetic field configuration required to spin po-

larise the defects may preclude the use of the critical point technique. There

is, however, also the possibility of satisfying a critical point field configura-

tion for a particular set of detunings such that many of the other rare-earth

ions experience partial benefit. Operation of the quantum computer would

need to reflect the difference in T2 and response to field perturbations among

the available qubits in this case.

The defect related inhomogeneities will also have implications for DDC

techniques. As was shown in section 6.4.1 the inhomogeneities were the

reason that an arbitrary state was not well preserved. Consequently this will

require harder RF pulses or a qubit specific Raman optical implementations

of DDC techniques.

Chapter 9

Conclusions and Future Work

“Any sufficiently advanced technology is indistinguishable from magic.”

–Arthur C. Clarke

This work has demonstrated that hyperfine decoherence times sufficiently

long for QIP and quantum optics applications are achievable in rare-earth

doped insulators. Prior to this work there were several QIP proposals using

rare earth hyperfine states for long term coherent storage of optical inter-

actions [1, 2, 3]. The very long T1 (∼ 23 days [4]) observed for rare-earth

hyperfine transitions appears promising but the hyperfine transition T2 was

typically only a few ms [144], comparable to rare-earth optical transitions.

Therefore transferring information from an optical to a hyperfine transition,

with the associated time and fidelity considerations for only a minimal in-

crease in T2 made the usefulness of such proposals dubious. This work demon-

strated an increase in hyperfine T2 by a factor of ∼ 7× 104 compared to the

previously reported hyperfine T2 for Pr3+:Y2SiO5 through the application of

static and dynamic magnetic field techniques. This increase in T2 makes pre-

vious QIP proposals useful and provides the first solid state optically active

Λ system with very long hyperfine T2 for quantum optics applications.

The first technique employed the conventional wisdom of applying a small

static magnetic field to minimise superhyperfine interaction [5, 6, 7], as stud-

ied in chapter 4. This resulted in hyperfine transition T2 an order of mag-

nitude larger than the T2 of optical transitions, ranging from 3 to 10 ms.

Estimates of the required T2 to the time taken to perform a quantum oper-

ation, τop, such that the system is useful for complex QIP yield T2/τop > 105

[35]. In rare-earth systems the operation time is expected to be ∼10 µs given

Rabi frequencies of approximately 1 MHz [127] and swapping the information

130 Conclusions and Future Work

to and from the optical transition. Therefore minimising the superhyperfine

interaction resulted in T2/τop ≈ 103, which falls too short of the 105 estimate.

In addition to this requirement, most of the proposals involving rare-earth

ions used the hyperfine states as the long term storage [1, 2, 3]. In analogy

to classical computing the optical transitions are used as a RAM/processor

combination, while the hyperfine transitions are used as a hard drive. This

implies that hyperfine T2 is required to be longer, such that the decoherence

is negligible over the desired storage times.

Development of the critical point technique during this work was crucial

to achieving further gains in T2. The critical point technique is the applica-

tion of a static magnetic field such that the Zeeman shift of the hyperfine

transition of interest has no first order component, thereby nulling decohering

magnetic interactions to first order. This technique also represents a global

minimum for back action of the Y spin bath due to a change in the Pr spin

state, as discussed in section 5.4, allowing the assumption that the Pr ion

is surrounded by a thermal bath. The critical point technique resulted in a

dramatic increase of the hyperfine transition T2 from ∼10 ms to 860 ms. The

critical point method should be applicable to any spin system with I ≥ 1 and

zero field splitting, as discussed in section 5.4. This technique is anticipated

to be of general interest. Using the critical point technique allowed us to

realise a system with T2/τop ≈ 105, thereby experimentally demonstrating

the suitability of rare-earth doped systems for QIP applications.

Satisfied that the optimal static magnetic field configuration for increas-

ing T2 had been achieved, dynamic magnetic field techniques, driving ei-

ther the system of interest or spin bath were investigated. These tech-

niques are broadly classed as Dynamic Decoherence Control (DDC) in the

QIP community. The first DDC technique investigated was driving the

Pr ion using CPMG or Bang Bang decoupling pulse sequence. This sig-

nificantly extended T2 from 0.86 s to 70 s. This decoupling strategy has

been extensively discussed for correcting phase errors in quantum computers

[8, 9, 10, 11, 12, 13, 14, 15], with this work being the first application to

solid state systems. The time scales for pulsed decoupling are easily satisfied

for the Pr3+:Y2SiO5 system unlike expectations of other solid state quantum

computing candidate systems [13].

Magic Angle Line Narrowing was used to investigate driving the spin

bath to increase T2. The experiment demonstrated the applicability of this

technique despite not reaching the desired field regime, as discussed in section

7.3. This experiment resulted in T2 increasing from 0.84 s to 1.12 s. Both

DDC techniques introduce a periodic condition on when QIP operation can

Conclusions and Future Work 131

be performed without the qubits participating in the operation accumulating

phase errors relative to the qubits not involved in the operation.

Without using the critical point technique Dynamic Decoherence Control

techniques such as the Bang Bang decoupling sequence and MALN are not

useful due to the sensitivity to magnetic field fluctuations. Critical point

and DDC techniques are mutually beneficial since the critical point is most

effective at removing high frequency perturbations while DDC techniques

remove the low frequency perturbations as discussed in section 6.1. A further

benefit of using the critical point technique is it allows changing the coupling

to the spin bath without changing the spin bath dynamics. This was useful

for discerning whether the limits are inherent to the DDC technique or are

due to experimental limitations.

Solid state systems exhibiting long T2 are typically very specialised sys-

tems, such as 29Si dopants in an isotopically pure 28Si and therefore spin free

host lattice [16]. These systems rely on on the purity of their environment

to achieve long T2. Despite possessing a long T2, the spin system remain

inherently sensitive to local magnetic field fluctuations. Sources of magnetic

field fluctuations, from local defects in the crystal, noise in the controlling

fields and noise from lab equipment can all strongly couple to the spin of in-

terest and therefore place practical limits on achievable T2. Nanostructuring

always introduces some level of disorder at the interface between different

materials, creating excess charge or trapped spins. Therefore problems asso-

ciated with the qubit’s sensitivity scale proportionally to the complexity of

their environment. Complex, nanostructured systems for realising quantum

computing architectures in such a system therefore face significant scaling

challenges.

In contrast, this work has demonstrated that decoherence times, suffi-

ciently long to rival any solid state system [16], are achievable when the spin

of interest is surrounded by a concentrated spin bath. Using the critical

point method results in a hyperfine state that is inherently insensitive to

small magnetic field perturbations and therefore more robust for QIP appli-

cations.

In the present work we have achieved a hyperfine transition with a res-

onator quality factor, Q of 6 × 108, which is expected to increase with up-

grades to RF equipment, more sophisticated DDC techniques and investigat-

ing other promising materials. Previously discussed approaches to achieve

these goals are summarised in the remainder of this chapter as well as inter-

esting quantum optics applications.


9.1 Strategies for Further Increases in Deco-

herence Time

While the 70s T2 achieved dwarfs the previously achieved few ms T2 limit

for optically active nuclear spin system there is great promise to increase

this further. The discussion will initially focus on improvements that would

benefit experiments on any system before addressing material specific issues.

9.1.1 Improved RF Control

The current limit on the T2 achievable using DDC techniques, such as Bang

Bang decoupling, appears to be due to the driving RF, as discussed in section

6.4. The PulseBlaster [163] DDS circuitry was identified as the dominant RF

noise source in section 6.4 and consequently using higher purity RF source is

critical to removing the plateau observed in the Bang Bang dynamic decou-

pling experiments. If T2 increased at the same rate as prior to the plateau

the for a cycling time of τc = 1ms a T2 of the order of 100s is expected. This

could allow us to approach the T1 = T2 limit.

9.1.2 Rabi Frequency and Inhomogeneous Broadening

As discussed in section 6.4.1 a combination of increasing in Rabi frequency

and a decrease in linewidth are required if Bang Bang decoupling sequences

can preserve an arbitrary state. If work is continued in Pr3+:Y2SiO5 repeat-

edly annealed isotopically pure crystals are desirable due to the expected re-

duction in hyperfine inhomogeneous linewidths, as discussed in section 6.4.1.

If such crystals are not available then modifying the inhomogeneous linewidth

via holeburning techniques is feasible for further DDC investigations.

Europium is the other rare-earth ion with particular application to this

work. Doping Eu into Y2SiO5 or analogous crystals such as Y2O5 allows

directly analogous systems to be investigated with potential for higher Rabi

frequencies due to the increased hyperfine splittings, as discussed in section

8.1. Other materials such as Eu6H2O have shown optical inhomogeneous

linewidths of the order of 10MHz [128] and consequently are expected to

have narrow hyperfine inhomogeneous widths. As such DDC investigations

could benefit from this or analogous materials.

9.2 Other Applications For Long T2 Optically Active Solids 133

9.1.3 Eulerian Decoupling

While in Pr3+:Y2SiO5 it is possible to achieve the required cycling time,

τc, for the Bang Bang pulse sequence to be effective the underlying theory

still requires unbounded Rabi frequencies during the applied pulses. This is

clearly not physically reasonable and as such the more sophisticated method

of Eulerian decoupling has been proposed to relax the pulse amplitude re-

quirement [11, 13]. Eularian decoupling is gaining favour with the QIP theory

community since it brings theory and experiment a step closer. I believe that

given how successful the application of the Bang Bang decoupling scheme was

rare earth ion systems are the best solid state system to investigate these new

dynamic decoherence control schemes.

9.2 Other Applications For Long T2 Optically

Active Solids

Quantum optics experiments are typically performed using dilute gas sys-

tems due to the availability of optically active Λ systems with long coherence

times. In atomic vapour systems the atoms will always have a non-zero ve-

locity resulting in a finite time for an atom to drift such that it experiences

a different phase or amplitude of the optical driving field and eventually

completely leave the interaction region. Utilising solid state systems removes

these problems, at the cost of introducing inhomogeneous broadening via the

crystal field, which as discussed earlier, can be engineered. The static rela-

tionship of ions in solids has been exploited for classical optics experiments,

using holograms for high bandwidth RF signal processing [91, 92, 93] and

optical routing [94]. Utilising the static, ultra coherent optically active Λ

systems afforded by this work has direct application to two experiments of

particular interest in quantum optics.

9.2.1 Slow and Stopped Light

Electromagnetically Induced Transparency (EIT), commonly referred to as

slow or stopped light, has been proposed as a potential quantum memory

[164, 165]. While there have been a number of EIT demonstrations in dilute

gas systems there has only been one demonstrations in a solid state system

prior to this work [102]. Recently the critical point technique, in combina-

tion with the Bang Bang DDC technique has been used to stop light in a

solid for 6s [166], dwarfing the longest previous storage time of 1ms [167].


The efficiency of the EIT feature was also an order of magnitude larger that

that of dilute gas experiments [43] and can be increased further by modifying

the dopant concentration or crystal dimensions to increase the optical depth.

Utilising a solid also allows a larger number of beam geometries to be consid-

ered, in particular the probe and coupling beams can counter propagate and

as such simplify the detection of a weak probe in the presence of a strong

coupling beam [43].

9.2.2 Stark Echo Quantum Memory

EIT memories have inherently small bandwidth due to using a narrow spec-

tral feature. A higher bandwidth quantum memory scheme was recently

developed in our group by Sellars and Alexander [128, 160]. This scheme

uses electric field gradients to create a controlled inhomogeneous broadening

on an optical transition that can be reversed, resulting in a photon echo [87].

Using Raman techniques to store the information on the optical transitions

will allow a both critical point and DDC techniques to extend the storage

time.

Appendices

Appendix A

Y2SiO5 site position calculation

This appendix lists the code used to calculate the Y site positions for Y2SiO5The

files lsted are pairs.cpp, rot.c, rot.h, assign.h, clapack.h, parameters.h and

makeJ.h

----- pairs.cpp -----

#include <iostream.h>

#include <complex>

#include <math.h>

#include <unistd.h>

#include <tnt/tnt.h>

#include <tnt/vec.h>

#include <tnt/fmat.h>

#include <tnt/cmat.h>

using namespace std;

using namespace TNT;

double a = 10.410,b=6.726,c=12.495; //unit cell lengths in Angstroms

double beta = (102.65)/180.0*M_PI; //beta = 102.65deg

Y2SiO5 site position calculation 137

Fortran_Matrix<double> xtal_to_cart(3,3);

double jmod(double x,double y)

double temp = fmod(x,y);

if (temp<0) temp+=y;

return temp;

double norm(Vector<double> v)

double norm=0;

for(int k=1;k<=v.size();k++)

norm+=v(k)*v(k);

norm=sqrt(norm);

return norm;

Vector<Vector<double> > Y_positions(Vector<double>atom_coords)

//return array of yttrium positions within unit cell

// in crystalographic coordinates

// space groups I 2/a origin same as int tables

Vector<Vector<double> > Y;

int k,j;

Y.newsize(8);

for(k=1;k<=8;k++)

Y(k).newsize(3);

Y(1) = atom_coords;

// if(n==1) //return site one positions

138 Y2SiO5 site position calculation

// Y(1)(1) = 0.30657;

// Y(1)(2) = 0.37701;

// Y(1)(3) = 0.14154;

// else

// Y(1)(1) = 0.42839;

// Y(1)(2) = 1.0-0.25506;

// Y(1)(3) = 1.0-0.03701;

//

// if(n==-99)

// Y(1)(1) = 0.1;

// Y(1)(2) = 0.2;

// Y(1)(3) = 0.3;

//

//do inversion operation

for(k=1;k<=3;k++)

Y(2)(k) = jmod(-Y(1)(k),1);

//do rotation operator

for(j=1;j<=2;j++)

Y(j+2)(1) = jmod(0.5-Y(j)(1),1);

Y(j+2)(2) = jmod(Y(j)(2),1);

Y(j+2)(3) = jmod(-Y(j)(3),1);

//do translation by 1/2 1/2 1/2

for(j=1;j<=4;j++)

for(k=1;k<=3;k++)

Y(j+4)(k) = jmod(Y(j)(k)+0.5,1);

return Y;

Vector <Vector<double> > Y_dipoles(Vector<double> dipole_of_Y1)


//return array of yttrium dipoles within unit cell corresponding to

//

// cartesian coords (see notebook for axes)

// space groups I 2/a origin same as int tables

Vector<Vector<double> > Y;

int k,j;

assert(dipole_of_Y1.dim() == 3);

Y.newsize(8);

for(k=1;k<=8;k++)

Y(k).newsize(3);

Y(1) = dipole_of_Y1;

//do inversion operation

for(k=1;k<=3;k++)

Y(2)(k) = -Y(1)(k);

//do rotation operator

for(j=1;j<=2;j++)

Y(j+2)(1) = -Y(j)(1);

Y(j+2)(2) = Y(j)(2);

Y(j+2)(3) = -Y(j)(3);

//do translation by 1/2 1/2 1/2

for(j=1;j<=4;j++)

for(k=1;k<=3;k++)

Y(j+4)(k) = Y(j)(k);

return Y;

Vector<Vector<Vector<double> > > all_within(Vector<double> r,


double distance,

Vector<double> atom_coords,

Vector<double> dipole)

//returns a vector length 2, element one is a vector of position vectors

// for all the atoms with coordinates atom_coords

// within distance angstroms of the position r

// element two are the corresponding dipole moments

// dipole = the dipole moment of the atom with position vector

// equal to the atomic coords

// r, atom_coords,output(1) in xtalographic units

// dipole,output(2) in cartesian

//

// r is assumed to be with the 0-1,0-1,0-1 unit cell

Vector<Vector<Vector<double> > > output_before_thining;

Vector<Vector<Vector<double> > > output;

Vector<Vector<double> > one_unit_cell_posns;

Vector<Vector<double> > one_unit_cell_dipoles;

int h,k,l,j,count;

int n = (int)(distance/c)+1;

one_unit_cell_posns = Y_positions(atom_coords);

one_unit_cell_dipoles = Y_dipoles(dipole);

output_before_thining.newsize(2);

output.newsize(2);

for(k=1;k<=2;k++)

output_before_thining(k).newsize((2*n+1)*(2*n+1)*(2*n+1)*8);

output(k).newsize((2*n+1)*(2*n+1)*(2*n+1)*8);

for(k=0;k<output_before_thining(1).dim();k++)

output_before_thining(1)[k].newsize(3);


count=1;

for(h=-n;h<=n;h++)

for(k=-n;k<=n;k++)

for(l=-n;l<=n;l++)

for(j=1;j<=8;j++)

output_before_thining(1)(count)(1)=one_unit_cell_posns(j)(1)+h;

output_before_thining(1)(count)(2)=one_unit_cell_posns(j)(2)+k;

output_before_thining(1)(count)(3)=one_unit_cell_posns(j)(3)+l;

output_before_thining(2)(count)=one_unit_cell_dipoles(j);

count++;

count=0;

for(k=1;k<=output_before_thining(1).dim();k++)

if (norm(xtal_to_cart*(output_before_thining(1)(k)-r))<distance)

count++;

output(1)(count) = output_before_thining(1)(k);


output(1).newsize(count);

output(2).newsize(count);

count=0;

for(k=1;k<=output_before_thining(1).dim();k++)

if (norm(xtal_to_cart*(output_before_thining(1)(k)-r))<distance)

count++;



return output;


main()

int k,l,j;

double temp;

double re;

int nj;

xtal_to_cart(1,1) = a;

xtal_to_cart(1,3) = c*cos(beta);

xtal_to_cart(2,2) = b;

xtal_to_cart(3,3) = c*sin(beta);

Vector<double> site_one_coords(3,"0.30657 0.37701 0.14154");

Vector<double> site_two_coords(3,"0.42839 0.74494 0.96299");

Vector<double> site_Si_coords(3,"0.6270 0.9070 0.1810");

Vector<Vector<Vector<double> > > posns_and_dipoles1 ;




Vector<Vector<Vector<double> > > posns_and_dipolesSi ;

Vector<Vector<double> > posns,dipoles;

Vector<double> r(1501);

Vector<double> n1(1501);




re= 200;//angstroms

/* cout<<"###site ones\n";

posns_and_dipoles1 = all_within(site_one_coords,re,site_one_coords,site_one


nj = posns_and_dipoles1(1).dim();

for(k=1;k<=nj;k++)

posns_and_dipoles1(1)(k) = xtal_to_cart*posns_and_dipoles1(1)(k);

for(j=1;j<=3;j++)

cout<<posns_and_dipoles1(1)(k)(j)<<" ";

cout<<endl;

/*posns_and_dipoles1(2)(k) = xtal_to_cart*posns_and_dipoles1(2)(k);

for(j=1;j<=3;j++)


cout<<endl;* /

;

cout<<"###site twos\n";

posns_and_dipoles2 = all_within(site_one_coords,re,site_two_coords,site_one_coord

nj = posns_and_dipoles2(1).dim();

for(k=1;k<=nj;k++)

posns_and_dipoles2(1)(k) = xtal_to_cart*posns_and_dipoles2(1)(k);

// cout<<"len="<<posns_and_dipoles2(1).size()<<endl;

for(j=1;j<=3;j++)


cout<<endl;

/* posns_and_dipoles2(2)(k) = xtal_to_cart*posns_and_dipoles2(2)(k);

for(j=1;j<=3;j++)


cout<<endl; * /

;*/

cout<<"###site Si\n";

posns_and_dipolesSi = all_within(site_Si_coords,re,site_Si_coords,site_Si_coords);

nj = posns_and_dipolesSi(1).dim();

for(k=1;k<=nj;k++)

posns_and_dipolesSi(1)(k) = xtal_to_cart*posns_and_dipolesSi(1)(k);

for(j=1;j<=3;j++)

cout<<posns_and_dipolesSi(1)(k)(j)<<" ";

cout<<endl;

;


// temp=0.0000001;

// for(k=0;k<1501;k++)

// r[k]=temp;

// temp+=0.005;

// posns_and_dipoles1 = all_within(site_one_coords, r[k],site_one_coords,site_

// posns_and_dipoles2 = all_within(site_one_coords, r[k],site_two_coords,site_

// posns_and_dipoles3 = all_within(site_two_coords, r[k],site_one_coords,site_

// posns_and_dipoles4 = all_within(site_two_coords, r[k],site_two_coords,site_

// n1[k] = posns_and_dipoles1(1).dim()-1;

// n2[k] = posns_and_dipoles2(1).dim();

// n3[k] = posns_and_dipoles3(1).dim();

// n4[k] = posns_and_dipoles4(1).dim()-1;

// cout<<r[k]<<"\t"<<n1[k]<<"\t"<<n2[k]<<"\t"<<n3[k]<<"\t"<<n4[k]<<endl;

//

// posns = posns_and_dipoles(1);

// dipoles = posns_and_dipoles(2);

// for(k=0;k<posns.dim();k++)

// cout<<k<<"\t"<<posns[k]<<endl;

//

---- end pairs.cpp ----

---- rot.c ----

#include <math.h>


//#include <f2c.h>

/* rotation matrix describing a rotation size |axis| around the vector axis */

#define EPS(i,j,k) ((int)((i<j)&&(j<k))-(int)((i>j)&&(j>k))+(int)((j<k)&&(k<i))-(int)

void axis2so3(double * axis,double * R)

double theta;

double unit_vec[3];

double col[3];

int k,l,m;

for(k=0;k<9;k++)

R[k]=0;

/* normalise */

theta = sqrt(axis[0]*axis[0]+axis[1]*axis[1]+axis[2]*axis[2]);

for(k=0;k<3;k++)

unit_vec[k]=axis[k]/theta;

//x’ = x*cos(t) + sin(t) (unit_vec X x) + (1-cos(t))u.(u.x)

for(k=0;k<3;k++)

R[4*k] = cos(theta);

for(k=0;k<3;k++)

for(l=0;l<3;l++)

for(m=0;m<3;m++)

R[k+3*l]+=EPS(k,m,l)*unit_vec[m]*sin(theta);

R[k+3*l]+=unit_vec[k]*unit_vec[l]*(1-cos(theta));


void euler2so3(double alpha, double beta, double gamma, double * R)

int k;

char side = ’l’;

int three = 3;

char trans = ’n’;

double naada = 0;

double unity = 1;

double sa = sin(alpha);

double ca = cos(alpha);

double sb = sin(beta);

double cb = cos(beta);

double sg = sin(gamma);

double cg = cos(gamma);

double rx[9] = ca,sa,0,-sa,ca,0,0,0,1;

double ry[9] = cb,0,-sb,0,1,0,sb,0,cb;

double rz[9] = cg,sg,0,-sg,cg,0,0,0,1;

double hold[9] = 0,0,0,0,0,0,0,0,0;

/* hold = ry*rz */

dgemm_(&trans,&trans,&three,&three,&three,&unity,ry,&three,rz,&three,

&naada,hold,&three);

dgemm_(&trans,&trans,&three,&three,&three,&unity,rx,&three,hold,&three,

&naada,R,&three);

/* for(k=0;k<9;k++) */

/* R[k] = hold[k]; */

/* */


---- end rot.c ----

---- rot.h ----

#include <math.h>

/* rotation matrix describing a rotation size |axis| around the vector axis */

void axis2so3(double * axis,double * R);

void euler2so3(double alpha, double beta, double gamma, double * R);

void apply_rot(double *R, double *A, double *RARt);

---- end rot.h ----

---- parameters.h ----

typedef struct

double alf;

double bet;

double gam;

double gx;

double gy;

double gz;

double az;

double el;

double alf2;

double bet2;

double gam2;

double E;

double D;

double B_0x;

double B_0y;


double B_0z;

Parameters;

---- end parameters.h ----

---- clapack.h ----

#ifndef __CLAPACK_H

#define __CLAPACK_H

typedef struct double r, i; doublecomplex;

/* Subroutine */ int zheev_(char *jobz, char *uplo, int *n, doublecomplex

*a, int *lda, double *w, doublecomplex *work, int *lwork,

double *rwork, int *info);

void zgemm_(char*,char*,int*,int*,int*,doublecomplex*,doublecomplex*,

int*,doublecomplex*,int*,doublecomplex*,doublecomplex*,int*);

#endif /* __CLAPACK_H */

---- end clapack.h ----

---- makeJ.h ----

void makeJ(doublecomplex** J, doublecomplex** JJ,const int hsdim);

void makeHeff(const double* const M,

const double* const Q,

doublecomplex** J,

doublecomplex** JJ,

doublecomplex* Heff,

const int nsq,

const double *B,

int t,

double *temp);

void revert_params(Parameters *params,double old_value,int kj);


void choose_new_params(Parameters *params,

Parameters *del,

double alpha,

int kj,

double *old_value);

---- end makeJ.h ----

Full Critical Point List 151

Appendix B

Full Critical Point List

Transition Bx By Bz δfδB2 f(MHz)

−12↔ +1

2-101 -369 -139 1769 18.27

−12↔ +3

2-101 -369 -139 1790 18.74

+12↔ −3

2-732 -172 219 102 8.64

-309 -1123 -424 118 3.54

-451 361 -677 193 5.57

+12↔ −5

2-464 371 -697 191 7.72

-38 -9 11 2207 10.18

+12↔ +5

2-492 391 -734 16028 23.88

496 -399 750 16119 23.88

-384 -89 112 17584 27.05

−32↔ +3

2-103 -376 -142 86 4.77

438 -350 658 385 2.16

236 -1093 382 486 1.73

−52↔ +3

294 713 312 1002 9.86

-312 -765 -247 1009 9.86

-102 -369 -139 1819 16.24

-203 -738 -279 277156 9.90

−52↔ −3

2-683 -356 -999 240 7.55

664 372 971 243 5.82

1566 453 155 320 10.21

-1085 92 488 345 10.20

-889 486 -173 827.95 11.55

211 55 -843 828 11.55

-1093 -252 -453 828 11.55

-417 -793 561 829 11.55

402 -252 -287 1049 11.39

99 370 138 1823 11.01

576 -464 871 18102 16.03

152 Full Critical Point List

Transition Bx By Bz δfδB2 f(MHz)

−52↔ −3

2317 -704 489 417327 12.49

-203 -738 -279 419271 9.90

-520 -35 -767 488098 12.49

+52↔ −3

2-517 566 493 131 8.47

-256 1109 -411 243 5.82

-1061 -857 601 320 10.21

-1562 -456 -149 320 10.21

-1289 -646 209 345 10.20

1079 -87 -499 345 10.20

-213 -53 840 828 11.55

-95 -713 -312 1005 9.86

-103 -368 -141 1823 11.01

+52↔ +3

2311 765 247 1013 9.86

101 369 139 1819 16.24

-408 316 -594 16350 18.40

373 -309 579 16532 18.40

203 738 279 250783 9.90

-317 705 -490 330466 12.49

-520 -33 -769 330550 12.49

Appendix C

Published Papers

This appendix contains papers published during the course of my PhD work.

Not all of the work is specifically discussed in this thesis, in particular the

paper on spectral features of Europium pair sites.

154 Published Papers

Published Papers 155




















Bibliography

[1] M. S. Shahriar et al., Solid-state quantum computing using spectral

holes, Phys. Rev. A. 66, 032301 (2002).

[2] N. Ohlsson, R. K. Mohan, and S. Kroll, Quantum computer hardware

based on rare-earth-ion-doped inorganic crystals, Optics Comm. 201,

71 (2002).

[3] K. Ichimura, A simple frequency-domain quantum computer with ions

in a crystal coupled to a cavity mode, Opt. Commun. 196, 119 (2001).

[4] F. Konz et al., Temperature and concentration dependence of optical

dephasing, spectral-hole lifetime, and anisotropic absorption in

Eu3+ : Y2SiO5, Phys. Rev. B. 68, 085109 (2003).

[5] R. W. Equall, Y. Sun, and R. L. Cone, Ultraslow Optical Dephasing

in Eu3+:Y2SiO5, Phys. Rev. Lett. 72, 2179 (1994).

[6] R. W. Equall, R. L. Cone, and R. M. Macfarlane, Homogeneous

broadening and hyperfine structure of optical transitions in

Pr3+:Y2SiO5, Phys. Rev. B. 52, 3963 (1995).

[7] R. M. Macfarlane, A. Cassanho, and R. S. Meltzer, Inhomogeneous

Broadening by Nuclear Spin Fields: A New Limit For Optical

Transitions in Solids, Phys. Rev. Lett. 69, 542 (1992).

[8] L. Viola and S. Lloyd, Dynamical supression of decoherence in

two-state systems, Phys. Rev. A. 58, 2733 (1998).

[9] D. Vitali and P. Tombesi, Using parity kicks for decoherence control,

Phys. Rev. A 59, 4178 (1999).

[10] L. Viola, E. Knill, and S. Lloyd, Dynamical Decoupling of Open

Quantum Systems, Phys. Rev. Lett. 82, 2417 (1999).

[11] L. Viola and E. Knill, Robust Dynamic Deoupling of Quantum

Systems with Bounded Controls, Phys. Rev. Lett. 90, 037901 (2003).

176 BIBLIOGRAPHY

[12] H. Gutmann, F. K. Wilhelm, W. M. Kaminsky, and S. Lloyd,

Compensation of decoherence from telegraph noise by means of bang

bang control, 2003. arXiv:cond-mat/0308107

[13] L. Viola, Advances in Decoherence Control, J. Mod. Opt. 51, 2357

(2004).

[14] P. Facchi, D. A. Lidar, and S. Pascazio, Unification of Dynamical

Decoupling and the Qunatum Zeno Effect, 2004.

arXiv:quant-ph/0303132

[15] W. M. K. H. Gutmann, F. K. Wilhelm and S. Lloyd, Bang-Bang

Refocusing of a Qubit Exposed to Telegraph Noise, Quant. Inf.

Process. 3, 247 (2004).

[16] T. D. Ladd, D. Maryenko, and Y. Yamamoto, Coherence time of a

Solid-State Nucelar Qubit, 2003. arXiv:quant-ph/0309164

[17] C. J. Myatt et al., Decoherence of quantum superpositions through

coupling to engineered reservoirs, Nature 403, 269 (2000).

[18] W. H. Zurek, Pointer basis of quantum apparatus: Into what mixture

does the wave packet collapse?, Phys. Rev. D 24, 1862 (1981).

[19] A. M. Turing, On computable numbers, with an application to the

Entscheindungs problem, Proc. Lond. Math. Soc. 42, 230 (1936).

[20] C. P. Williams and S. H. Clearwater, Explorations in Quantum

Computing, 1 ed. (TELOS, Springer-Verlag, JPL Caltech, 1998).

[21] J. Gill, Computational Complexity of Probabilistic Turing Machines,

SIAM J. Comp. 6, 675 (1977).

[22] G. E. Moore, Cramming more components onto Integrated Circuits,

1965.

[23] C. Bennett, Logical Reversibility of Computation, IBM Journal of

Research and Development 17, 525 (1973).

[24] R. P. Feynmann, Simulating Physics with Computers, Intl. J. Theo.

Phys. 21, 467 (1982).

[25] D. Deutsch, Qunantum theory, the Church- Turing principle and the

universal quantum computer, Proc. R. Soc. Lond. 400, 97 (1985).

http://www.arXiv.org/abs/cond-mat/0308107

http://www.arXiv.org/abs/quant-ph/0303132


BIBLIOGRAPHY 177

[26] D. S. Abrams and S. Lloyd, Simulation of Many-Body Fermi Systems

on a Universal Quantum Computer, Phys. Rev. Lett. 79, 553 (1999).

[27] D. S. Abrams and S. Lloyd, Quantum algorithm for providing

exponential speed increase for finding eigenvalues and eigenvectors,

Phys. Rev. Lett. 83, 5162 (1999).

[28] D. P. DiVincenzo, The Physical Implementation of Quantum

Computation, 2000. arXiv:quant-ph/0002077

[29] A. Ishijima and T. Yanagida, Single molecule nanobioscience, Trends

in Biochemical Sciences 26, 438 (2001).

[30] A. V. Balatsky and I. Martin, Theory of Single Spin Detection with

STM, Quantum Inf. Proc. 1, 355 (2002).

[31] L. Bulaevskii et al., Nondemolition Measurements of a Single

Quantum Spin using Josephson Oscillations, Phys. Rev. Lett. 92,

177001 (2004).

[32] J. Preskill, Reliable Quantum Computers, Proc. Roy. Soc. Lon. A.

454, 385 (1996).

[33] D. P. DiVincenzo, Two-bit gates are universal for quantum

computation, Phys. Rev. A. 51, 1015 (1995).

[34] S. Lloyd, Almost any Quantum Logic Gate is Universal, Phys. Rev.

Lett. 75, 346 (1995).

[35] M. A. Nielsen and I. L. Chuang, Quantum Computation and

Quantum Information, 1 ed. (Cambridge University Press,

Cambridge, UK, 2000).

[36] A. Muthukrishnan and C. R. Stroud, Multivalued logic gates for

quantum computation, Phys. Rev. A. 62, 052309 (2000).

[37] F. Bloch, Nucear Induction, Phys. Rev. 70, 460 (1946).

[38] D. Wolf, Spin Temperature and Nuclear Spin Relaxation in Matter, 1

ed. (Oxford University Press, Walton St, Oxford OX2 6DP, 1979).

[39] B. C. Gerstein and C. R. Dybowski, Transient Techniques in NMR of

Solids, 1 ed. (Academic Press, Orlando, Florida, 1985).

[40] I. to Nonlinear Laser Spectroscopy, M. D. Levenson and S. S. Kano,

1 ed. (Academic Press, 1250 Sixth Avenue, San Diego, CA, 1988).


178 BIBLIOGRAPHY

[41] P. T. Callaghan, Principals of Nuclear Magnetic Resonance

Microscopy (Oxford University Press, Oxford, 1991).

[42] R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear

Magnetic Resonance in One and Two Dimensions, 2 ed. (Oxford

University Press, Walton St, Oxford OX2 6DP, 1991).

[43] Personal communication, Jevon J. Longdell, 2005.

[44] I. L. Chuang and M. A. Nielsen, Prescription for experimental

determination of the dynamics of a quantum black box, J. Mod. Opt.

44, 2455 (1997).

[45] S. M. Tan, Inverse Problems, Physics 707 course notes, 2005.

[46] W. H. Zurek, Environment-induced superselection rules, Phys. Rev. D

26, 1516 (1982).

[47] V. M. Agranovich and A. A. Maradudin, Spectroscopy of solids

containing Rare Earth ions, 1 ed. (North-Holland Physics Publishing,

Amsterdam, 1987).

[48] M. D. Hurlimann, Carr Purcell Sequences with Composite Pulses, J.

Magn. Res. 152, 109 (2001).

[49] M. D. Hurlimann and D. D. Griffin, Spin Dynamics of Carr Purcell

Meiboom Gill-like Sequences in Grossly Inhomogeneous B0 and B1

Fields and Application to NMR Well Logging, J. Magn. Res. 143, 120

(2000).

[50] E. L. Hahn, Spin Echos, PhysRev 80, 580 (1950).

[51] Y. Yu and B. M. Fung, An Efficient Broadband Decoupling Sequence

for Liquid Crystals, J. Magn. Res. 130, 317 (1998).

[52] A. E. Bennett, C. M. Rienstra, M. le Auger, and K. V. Lakshmi,

Heteronuclear decoupling in rotating solids, J. Chem. Phys. 103, 6951

(1995).

[53] S. C. Rand, A. Wokaun, R. G. DeVoe, and R. G. Brewer,

Magic-Angle Line Narrowing in Optical Spectroscopy, Phys. Rev.

Lett. 43, 1868 (1979).

[54] R. M. MacFarlane, C. S. Yannoni, and R. M. Shelby, Optical line

narrowing by nuclear spin decoupling in Pr3+:LaF3, Opt. Comm. 32,

101 (1980).

BIBLIOGRAPHY 179

[55] M. S. Byrd and D. A. Lidar, Combined Error Correction Techniques

for Quantum Computing Architectures, J. Mod. Opt. 50, 1285 (2003).

[56] D. G. Cory et al., Experimental Quantum Error Correction, Phys.

Rev. Lett. 81, 2153 (1998).

[57] N. Boulnant et al., Experimental Concatenation of Quantum Error

Correction with Decoupling, Quantum Information Processing 1, 135

(2002).

[58] S. Wimperes, Broadband, Narrowband and Passband Composite pulses

for use in Advanced NMR Experiments, J. Magn. Res. A. 109, 221

(1994).

[59] G. J. Pryde, M. J. Sellars, and N. Manson, Optical non-Bloch

behaviour observed using an optical Carr-Purcell Meiboom- Gill pulse

sequence, J. Lumin. 86, 279 (2000).

[60] P. W. Shor, Scheme for reducing decoherence in quantum computer

memory, Phys. Rev. A 52, 2493 (1995).

[61] P. G. Silvestrov, H. Schomerus, and C. W. J. Beenakker, Limits to

Error Correction in Quantum Chaos, Phys. Rev. Lett. 86, 5192

(2001).

[62] M. S. Byrd and D. A. Lidar, Comprehensive Encoding and Decoupling

Solution to Problems of Decoherence and Design in Solid-State

Quantum Computing, Phys. Rev. Lett. 89, 047901 (2002).

[63] E. Knill, R. Laflamme, and W. H. Zurek, Resilient Quantum

Computation, Science 279, 342 (1998).

[64] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect

Quantum Error Correcting Code, Phys. Rev. Lett. 77, 198 (1996).

[65] D. Gottesman, Class of quantum error-correcting codes saturating the

quantum Hamming bound, Phys. Rev. A 54, 1862 (1996).

[66] D. A. Lidar and L. A. Wu, Encoded recoupling and decoupling: An

alternative to quantum error-correcting codes applied to trapped-ion

quantum computation, Phys. Rev. A. 67, 032313 (2003).

[67] P. Zanardi and M. Rasetti, Noiseless Quantum Codes, Phys. Rev.

Lett. 79, 3306 (1997).

180 BIBLIOGRAPHY

[68] D. A. Lidar, D. Bacon, J. Kempe, and K. B. Whaley,

Decoherence-free subspaces for multiple-qubit errors. I.

Characterization, Phys. Rev. A. 63, 022306 (2001).

[69] D. A. Lidar, D. Bacon, J. Kempe, and K. B. Whaley,

Decoherence-free subspaces for multiple-qubit errors. II. Universla,

fault-tolerant quantum computation, Phys. Rev. A. 63, 022307 (2001).

[70] D. A. Lidar and K. B. Whaley, Decoherence-free Subspaces and

Subsystems, Phys. Rev. A. 63, 022307 (2003).

[71] P. Wocjan, Efficient decoupling schemes with bounded controls based

on ”Eulerian” ornthogonal arrays, 2004. arXiv:quant-ph/0410107

[72] I. L. Chuang and R. Laflamme, Quantum Error Correction by

Coding, 1995. arXiv:quant-ph/9511003

[73] J. E. Ollerenshaw, D. A. Lidar, and L. E. Kay, Magnetic Resonance

Realization of Decoherence-Free Quantum Computation, Phys. Rev.

Lett. 91, 217904 (2003).

[74] D. A. Lidar, D. Bacon, and K. B. Whaley, Concatenating

Decoherence-Free Subspaces with Quantum Error Correcting Codes,

Phys. Rev. Lett. 82, 4556 (1999).

[75] L. Viola et al., Experimental Realization of Noiseless Subsystems for

Quantum Information Processing, Science 293, 2059 (2001).

[76] P. B. Kingsley, Product Operators, Coherence Pathways, and Phase

Cycling Part I: Product Operators, Spin-Spin Coupling and Coherence

Pathways, Concepts Mag. Res. 7, 29 (1995).


Cycling Part II: Coherence Pathways in Multipulse Sequances: Spin

Echos, Stimulated Echos and Multiple-Quantum Coherences,

Concepts Mag. Res. 7, 115 (1995).


Cycling Part III: Phase Cycling, Concepts Mag. Res. 7, 167 (1995).

[79] C. P. Slichter, Principals of Magnetic Resonance, 2 ed.

(Springer-Verlag, Dept. Phys, Univ Illinois, Urbanna, IL 61801, 1980).



BIBLIOGRAPHY 181

[80] H. Y. Carr and E. M. Purcell, Effects of Diffusion on Free Precession

in Nuclear Magnetic Resonance Experiments, PhysRev 94, 630

(1954).

[81] S. Meiboom and D. Gill, Modified spin-echo method for measuring

nuclear relaxation times, Rev. Sci. Instrum. 29, 688 (1958).

[82] J. Simbrunner and R. Stollberger, Analysis of Carr Purcell Sequences

with Nonideal Pulses, J. Magn. Res. B. 109, 301 (1995).

[83] W. T. Dixon, Spinning-sideband-free and spinning-sideband-only

NMR spectra in spinning samples, J. Chem. Phys. 77, 1800 (1982).

[84] M. E. Smith and J. H. Strange, NMR techniques in materials physics:

a review, Meas. Sci. Technol. 7, 449 (1996).

[85] M. S. Byrd and D. A. Lidar, Empirical determination of dynamical

decoupling operations, Phys. Rev. A 67, 012324 (2003).

[86] P. Shor, in Algorithms for quantum computation: discrete logarithms

and factoring, Foundations of Computer Science, 1994 Proc, 35th

Annual Symposium on (PUBLISHER, ADDRESS, 1994).

[87] A. L. Alexander, J. J. Longdell, M. J. Sellars, and N. B. Manson,

Photon Echoes Produced via Controlled Inhomogeneous Broadening,

2005. arXiv:quant-ph/0506232

[88] R. M. MacFarlane, High-resolution laser spectroscopy of rare-earth

doped insulators: a personal perspective, J. Lumin. 100, 1 (2002).

[89] M. Mitsunaga, Time-domain optical data storage by photon echo,

Opt. and Quant. Elec. 24, 1137 (1992).

[90] K. K. Rebane, Purely electronic zero-phonon lines in optical data

storage and processing, Phys. Chem. 7, 723 (2004).

[91] K. D. Merkel et al., Accumulated programming of a complex spectral

grating, Opt. Lett. 25, 1627 (2000).

[92] T. W. Mossberg, Planar holographic optical processing devices, Opt.

Lett. 26, 414 (2001).

[93] V. Lavielle et al., Wideband versatile radio-frequency spectrum

analyzer, Opt. Lett. 28, 384 (2003).


182 BIBLIOGRAPHY

[94] W. R. Babbitt and T. W. Mossberg, Spatial routing of optical beams

through time-domain spatial-spectral filtering, Opt. Lett. 20, 910

(1995).

[95] M. Attrep and P. K. Koroda, J. Inorg. Nuclear Chen. 30, 669 (1968).

[96] A. J. Freeman and R. E. Watson, Theoretical investigation of Some

Magnetic and Spectroscopic Properties of Rare-Earth Ions, Phys. Rev.

127, 2058 (1962).

[97] G. H. Dieke and H. M. Crosswhite, The spectra of the doubly and

triply ionized rare earths, Appl. Opt. 2, 675 (1963).

[98] W. T. Carnall, H. Crosswhite, and H. M. Crosswhite, Technical

report, Technical Report N0. 60439, Argonne Natl. Lab.

(unpublished).

[99] B. A. Maksimov, Y. A. Kharitonov, V. V. Ilyukhin, and N. V. Belov,

Crystal Structure of the Y-Orthosilicate Y2(SiO4)O, Dokl 13, 1188

(1969).

[100] R. Yano, M. Mitsunaga, and N. Uesugi, Ultralong Optical Dephasing

time in Eu3+:Y2SiO5, Opt. Lett. 16, 1884 (1991).

[101] J. M. Baker and B. Bleaney, Proc. R. Soc. London Ser. A 245, 156

(1958).

[102] A. V. Turukin et al., Observation of Ultraslow and Stored Light

Pulses in a Solid, Phys. Rev. Lett. 88, 023602 (2002).

[103] B. S. Ham, M. S. Shahriar, and P. R. Hemmer,

Radio-frequency-induced optical gain in Pr3+:Y2SiO5, J. Opt. Soc. B

15, 1541 (1998).

[104] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by

Simulated Annealing, Science 220, 671 (1983).

[105] J. J. Longdell, M. J. Sellars, and N. B. Manson, Hyperfine interaction

in ground and excited states of praseodymium-doped yttrium

orthosilicate, Phys. Rev. B 66, 035101 (2002).

[106] W. Y. Ching, L. Ouyang, and Y.-N. Xu, Electronic and optical

properties of Y2SiO5 and Y2Si2O7 with comparisons to α-SiO2 and

Y2O3, Phys. Rev. B. 67, 245108 (2003).

BIBLIOGRAPHY 183

[107] J. J. Longdell, Qunatum Information Processing in Rare Earth Ion

Doped Insulators, Ph.D. thesis, RSPhySSE, ANU, 2003.

[108] J. R. Klauder and P. W. Anderson, Spectral Diffusion in Spin

Resonance Experiments, Phys. Rev. 125, 912 (1962).

[109] B. Herzog and E. L. Hahn, Transient Nuclear Induction and Double

Nuclear Resonance in Solids, Phys. Rev. 103, 148 (1956).

[110] W. B. Mims, Phase Memory in Electron Spin Echos, Lattice

Relaxation Effects in CaWO4 : Er, Ce,Mn, Phys. Rev. 168, 370

(1968).

[111] I. L. Chuang et al., Experimental realization of a quantum algorithm,

Nature 393, 143 (1998).

[112] L. M. K. Vandersypen et al., Experimental Realization of an Order

Finding Algorithm with and NMR Quantum Computer, Phys. Rev.

Lett. 85, 5452 (2000).

[113] J. I. Cirac and P. Zoller, Quantum Computation with Cold Trapped

Ions, Phys. Rev. Lett. 74, 4091 (1995).

[114] C. Monroe et al., Demonstration of a Fundamental Quantum Logic

Gate, Phys. Rev. Lett. 75, 4714 (1995).

[115] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Coherent control of

macroscopic quantum states in a single-Cooper-pair box, Nature 398,

786 (1999).

[116] Y. A. Pashkin et al., Quantum oscillations in two coupled charge

qubits, Nature 421, 823 (2003).

[117] E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient

quantum computation with linear optics, Nature 409, 46 (2001).

[118] T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, Simple

scheme for efficient linear optics quantum gates, Phys. Rev. A. 65,

012314 (2001).

[119] S. R. Hartmann, H-3–Photon, Spin and Raman Echos, IEEE j.

Quant. Elec 4, 802 (1968).

[120] P. Hu, S. Geschwind, and T. M. Jedju, Spin-Flip Raman Echo in

n-Type CdS, Phys. Rev. Lett. 37, 1357 (1976).

184 BIBLIOGRAPHY

[121] G. K. Liu and R. L. Cone, Laser-induced instantaneous spectral

diffusion in Tb3+ compounds as observed in photon-echo experiments,

Phys. Rev. B 41, 6193 (1990).

[122] J. Zhang and T. W. Mossberg, Optically induced shifts in the

frequency of the 3H4 − 1D2 transition of Pr3+ in LaF3, Phys. Rev. B

48, 7668 (1993).

[123] N. M. Strickland and R. L. Cone, Persistent spectral hole burning in

deuterated CaF2 : Tm3+, Phys. Rev. B 59, 14328 (1999).

[124] G. J. Pryde, M. J. Sellars, and N. Manson, Solid state coherent

transient measurements using hard optical pulses, Phys. Rev. Lett.

84, 1152 (2000).

[125] M. Mitsunaga, T. Takagahara, R. Yano, and N. Uesugi,

Excitation-induced frequency shift probed by stimulated photon echoes,

Phys. Rev. Lett. 68, 3216 (1992).

[126] S. B. Altner, M. Mitsunaga, G. Zumofen, and U. P. Wild,

Dephasing-Rephasing Balancing in Photon Echoes by Excitation

Induced Frequency Shifts, Phys. Rev. Lett. 76, 1747 (1996).

[127] J. J. Longdell and M. J. Sellars, Experimental demonstration of

quantum-state tomography and qubit-qubit interactions for

rare-earth-metal-ion-based solid-state qubits, Phys. Rev. A 69, 032307

(2004).

[128] Personal communication, Matthew Sellars, 2005.

[129] D. P. DiVincenzo et al., Universal quantum computation with the

exchange interaction, Nature 408, 339 (2000).

[130] T. D. Ladd et al., All-Silicon Quantum Computer, Phys. Rev. Lett.

89, 017901 (2002).

[131] B. Kane, A silicon-based nuclear spin quantum computer, Nature 393,

133 (1998).

[132] M. E. J. Newman, Models of the Small World, J. Stat. Phys. 101, 819

(2000).

[133] E. M. Fortunato et al., Implementation of universal control on a

decoherence-free qubit, New. J. Phys. 4, 5.1 (2002).

BIBLIOGRAPHY 185

[134] I. L. Chuang, N. Gershenfeld, and M. Kubinec, Experimental

Implementation of Fast Quantum Searching, Phys. Rev. Lett. 80,

3408 (1998).

[135] L. M. K. Vandersypen et al., Experimental realization of Shor’s

quantum factoring algorithm using nuclear magnetic resonance,

Nature 414, 883 (2001).

[136] S. Cova et al., Avalanche photodiodes and quenching circuits for

single-photon detection, App. Opt. 35, 1956 (1996).

[137] M. D. Lukin and P. R. Hemmer, Quantum Entanglement via Optical

Control of Atom-Atom Interactions, Phys. Rev. Lett. 84, 2818 (2000).

[138] J. Wesenberg and K. Molmer, Robust quantum gates and a bus

architecture for quantum computing with rare-earth-ion-doped

crystals, Phys. Rev. A. 68, 012320 (2003).

[139] R. G. DeVoe, A. Szabo, S. C. Rand, and R. G. Brewer, Ultraslow

Optical Dephasing of LaF3 : Pr3+, Phys. Rev. Lett. 42, 1560 (1979).

[140] R. M. Macfarlane and R. M. Shelby, Sub-kilohertz optical linewidths

of the 7F0 ↔ 5D0 transition in Y2O3 : Eu3+, Opt. Commun 39, 1691

(1981).

[141] M. Lukac, F. W. Otto, and E. L. Hahn, Spin-spin cross relaxation

and spin-Hamiltonian spectroscopy by optical pumping of Pr3+:LaF3,

Phys. Rev. A. 39, 1123 (1989).

[142] J. Ganem et al., Nonexponential Photon-Echo Decays of Paramegnetic

Ions in the Superhyperfine Limit, Phys. Rev. Lett. 66, 695 (1991).

[143] R. M. Macfarlane, R. S. Meltzer, and B. Z. Malkin, Optical

measurement of the isotope shifts and hyperfine and superhyperfine

interactions of Nd in the solid state, Phys. Rev. B. 58, 5692 (1998).

[144] B. S. Ham, M. S. Shahriar, and P. R. Hemmer, Spin coherence and

rephasing with optically shelved atoms, Phys. Rev. B 58, R11825

(1998).

[145] R. M. MacFarlane, High-resolution laser spectroscopy of rare earth

materials for hole burning and coherent transient applications, J.

Lumin. 98, 281 (2002).

186 BIBLIOGRAPHY

[146] L. L. Wald, E. L. Hahn, and M. Lunac, Flourine spin frozen core in

Pr3+LaF3 observed by cross relaxation, J. Opt. Soc. Am. B 9, 789

(1992).

[147] A. Szabo, Spin dependance of optical dephasing in ruby: the frozen

core, Opt. Lett. 8, 486 (1983).

[148] G. J. Pryde, Ultra-High Resolution Spectroscopic Studies of Oprical

Dephasing in Solids, Ph.D. thesis, RSPhySSE, ANU, 1999.

[149] Product Descriptions,

http://www.minicircuits.com/dg03-216.pdf, 2005.

[150] N. Manson, M. J. Sellars, P. T. H. Fisk, and R. S. Meltzer, Hole

burning of rare earth ions with kHz resolution, J. Lumin. 64, 19

(1995).

[151] K. Yamamoto, K. Ichimura, and N. Gemma, Enhanced and reduced

absorbtions via quantum interference: Solid system driven by an RF

field, Phys. Rev. A 58, 2460 (1998).

[152] E. Fraval, M. Sellars, and J. J. Longdell, Method of extending

hyperfine coherence times in Pr3+ : Y2SiO5, Phys. Rev. Lett 92,

077601 (2004).

[153] in Proceedings of the Conference on Lasers and Electro-Optics

(CLEO), edited by T. bottger, C. W. Thiel, Y. Sun, and R. L. Cone

(PUBLISHER, ADDRESS, 2004).

[154] J. Ganem, Y. Wang, R. S. Meltzer, and W. M. Yen, Magnetic-filed

dependance of photon-echo decays in ruby, Phys. Rev. B. 43, 8599

(1991).

[155] H. K. Cummins, G. Llewellyn, and J. A. Jones, Tackling systematic

errors in quantum logic gates with composite rotations, Phys. Rev. A.

67, 042308 (2003).

[156] H. K. Cummins, G. Llewellyn, and J. A. Jones, Use of composite

rotations to correct systematic errors in NMR quantum computation,

New. J. Phys. 2, 1 (2000).

[157] R. Tycko, Broadband Population Inversion, Phys. Rev. Lett. 51, 775

(1983).

http://www.minicircuits.com/dg03-216.pdf

BIBLIOGRAPHY 187

[158] M. H. Levitt, Composite Pulses, Progress in NMR Spectroscopy 18,

61 (1986).

[159] Product Descriptions, http://www.mti-milliren.com/ocxo 2.html,

2005.

[160] Personal communication, Anabell L. Alexander, 2005.

[161] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Decoherence-Free

Subspaces for Quantum Computation, phys. Rev. Lett. 81, 2594

(1998).

[162] E. Knill et al., Introduction to Quantum Error Correction, 2002.

arXiv:quant-ph/0207170

[163] Product Descriptions,

http://www.spincore.com/pbplus index.html, 2005.

[164] E. A. Cornell, Stopping light in its tracks, Nature 409, 461 (2001).

[165] D. F. Phillips et al., Storage of Light in Atomic Vapor, Phys. Rev.

Lett. 86, 783 (2001).

[166] J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, Stopped

light with storage times greater than one second using EIT in a solid,

Phys. Rev. Lett. 95, 063601 (2005).

[167] C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Observation of

coherent optical information storage in an atomic medium using

halted light pulses, Nature 409, 490 (02001).

http://www.mti-milliren.com/ocxo_2.html


http://www.spincore.com/pbplus_index.html

Minimising the Decoherence of Rare Earth Ion Solid State ......Minimising the Decoherence of Rare Earth Ion Solid State Spin Qubits Elliot Fraval A thesis submitted for the degree

Documents