Logical gates and quantum processors with trapped ions and cavities MIGUEL ORSZAG FACULTAD DE FISICA PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE Course at.

Logical gates and quantum processors with trapped ions and cavities

• MIGUEL ORSZAG• FACULTAD DE FISICA

PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE

Course at the Institut Fourier,Universite Joseph Fourier,

Grenoble, FranceMay 2006

QUANTUM COMPUTATION AND INFORMATION

•1982 Feyman: “It is possible to improve the computation using Quantum Mechanics.”

•1985 Deutsch: Describes quantum computer model similar to the TURING MACHINE.

•1994 Shor: Proposes a factorization algorithm.

Classical Information unit: Bit

Quantum Information Unit: Quantum Bit.QUBIT

Qubit: “Microscopic system limited by two quantum states”.

Qudit: “Microscopic system limited by N Quantum States”.

QUANTUM COMPUTING AND QUANTUM INFORMATION

Superposition

1 qubit gate

0011Control

Target

C-NOT

0011

target

2 qubit gate

control

QUANTUM GATESC_NOT GATE

QUANTUM GATESControlled Not (C-NOT)

Qudit

a

a

b

b

Toffoli

Control1

Target

0011

0011

Control2

3 qubit gate

QUANTUM GATESToffoli

a

b

c

The target Only changes if bothControls J=k=1

3 qubit gate

control1

control2

target

•NO CLONING THEOREM Wooters-Zurek (1982)

“ It is impossible to clone an arbitrary quantum state”.

•UNIVERSAL QUANTUM COPYING MACHINE UQCM Buzek-Hillery (1996)

“Analize copies restricted by the no-cloning theorem”.

Quantum Copying Machines

Fidelity: 2

Universal Quantum Copying Machine UQCM

• Ideal copying process:

• Analysis of the copies:

Quantum Copying Machines

1.

2.

3. An ideal copy should be written as

original

blanc

machine

Hilbert-Schmidt NORM

QUANTUM COPYING MACHINE

• Result:

WANTED STATE

UNWANTED STATE

Universal Quantum Copying MachineUQCM

IMPLEMENTED BY A CIRCUIT

QUANTUM COPYING MACHINE

Preparation phase

Quantum Copying Machine implemented by a circuit

QUANTUM COPYING MACHINES

UQCM:

Duplicator:

Triplicator:

QUANTUM COPYING MACHINES

Analysis of the copies

Universal, doesnot depend on the input state

“Universal”,but restricted

To real numbers(input coeff)

“Apparatus capable of developing any function within the DATA REGISTER, such

funtion being specified by a PROGRAM REGISTER”.

“It is not possible to build a deterministic quantum processor with a

finite number of resources, but it is possible to build a probabilistic one”.

CLASSICAL PROCESSOR

QUANTUM PROCESSOR

QUANTUM PROCESSORS

• “When implementing a set of inequivalent operations, the program

state apace must contain a set of mutually orthogonal states. This means that the dimension of the

Program Register must be as big as the number of unitary operators we want to implement.Since this number (of unitary operators) is infinite, it is not possible to build a processor with a

finite number of resources”.

• HOWEVER, WE CAN ALWAYS BUILD A PROBABILISTIC PROCESSOR

• (quantum measurements are implied)

QUANTUM PROCESSORSLimitations due to Quantum Mechanics.

Fixed Unitary Operator (processor)

State of the DATA

Program State

Residual State(independent of the data)

This Type of Processor is necessarily stochastic

U Unitary Operator ACTING ON THE DATA ONY

Stochastic Processor for a qubit

We want:

How to implement it???

A STOCHASTIC QUANTUM PROCESSOR

1.-G.Vidal,L.Masanes,J.I.Cirac,PRL,788,047905(2002)2.-M.A.Nielsen,I.L.Chuang,PRL,79,321(1997)

3.-M.Hillary,V.Buzek,M.Ziman,PARA,65,022301(2002)

Data qubit

Program 1 qubit

Program 2 qubit

C_NOT

GATEToffoli Gate

To understand the procedure, consider a single Program Register.First we define the program and the data

PROGRAM STATE

DATA STATE

)10(2

1

)10(2

1

)10(2

1

22P

i

P

i

PPP

ddd

ee

BAd

U

d

d

Pd

i

d

i

Pd

i

d

i

P

i

P

i

dd

Pd

dU

dU

BeAeBeAe

eeBANOTC

dNOTC

1)10(2

10)10(

2

1

)10)(10(2

1)(

)(

2222

22

+

In this case, a measurement on the program register will cause a collapse On the data qubit with the outcome .

If we measure 0 in the program, we get the good answer,If we measure 1, we get the wrong answer

dddorUdU With a probability 1/2+

Bad result is for 1in P1

To improve upon this scheme, we introduce a Toffoli Gate, as in the Figure.

The data line and the first program qubit are unchanged, however,If the output of the program register is

Indicating a failure , the Toffoli gate effectively acts like a C-Not Gate Between the data line and a second program qubit

11

P

22

P

There is again a probability 1/2 of getting this time

UUU 2

THEREFORE, WE HAVE INCREASED THE SUCCESS PROBABILITY to 3/4 (and so on…)

Bad result only in the 11 case

One can generalize the argument, including MORE GENERALIZEDTOFFOLI GATES, and having a success probability of

N)21(1

Where N is the number of Program qubits or generalized Toffoli Gates (also the number of measurements)

The price we pay is the increase of gates and number of measurements

PROCESADORES CUANTICOS

Rendimiento de ProcesadoresEfficiency of quantum processors

PROCESADORES CUANTICOSProcessors performance


Processors performance I

State of Data

Operation to implement:

QUANTUM PROCESSORS

Processor Performance I

Program State:

Program Register base:



State that inputs the machine

QUANTUM PROCESSORS


The output state is:

QUANTUM PROCESSORS


QUANTUM PROCESSORS

Processors performance

QUANTUM PROCESSORS

Processors performance II

Program state:

QUANTUM PROCESSORS


Input state

Output State:

QUANTUM PROCESSORS


QUANTUM PROCESSORS


Output State:

QUANTUM PROCESSORS


QUANTUM PROCESSORS


If we measure in:

Success probability

Error probability:

1.

2.

PROCESADORES CUANTICOSProcessors performance


Processors performance III

Program State:

Input State:

QUANTUM PROCESSORS


Output State:

QUANTUM PROCESSORS


Output State:

QUANTUM PROCESSORS


QUANTUM PROCESSORS

Hillery-Buzek (2002) Qubits

Base de Bell

P=1/4

QUANTUM PROCESSORS

Hillery-Buzek (2002) Qudits

QUANTUM PROCESSORS

Efficiency of processors

QUANTUM PROCESSORS

Efficiency of the Vidal-Cirac 2002 processor

Every time one gets an error, it is possible to correct it, provided one has the l-th program state

State of the Program:

This state can be used to implement the Unitary U with a probability

QUANTUM PROCESSORSEfficiency of the Vidal-Cirac 2002 processor

CONCLUSIONSQuantum Copying

Machines Every preparation Program is realized with 2 C-Not and 3 rotations, for each preparation state one can find 8 possible sets of rotation angles..

In spite of the No-Cloning Theorem, it is always possible to build a copying machine with a fidelity different from one.(5/6 is the optimum)

The machine described here can produce 2 or 3 copies according to the preparation.

The copies obtained here have a fidelity factor of F =variable(max 5/6)

CONCLUSIONSProcessors

In spite of the fact that is impossible to build a deterministic processor (with finite resources), it is feasible to build a probabilistic one.

The Buzek proposal for a processor , for qubits, can be also used to implement an unknown operation U, with a certain probability p.

It is possible to generalize this processor and work with qudits of dimension N.

The probability of implementation of the Cirac proposal can be incremented by a feedback procedure, approaching one.

There is an effort to implement the probability with a single step, together with the factibility of implementing it with known gates….

CONCLUSIONSCopying machines,

processors

The cloning machine uses the same set of operators P31 P21 P13 P12 as compared to the processor.

The same machine has two applications

Quantum Copying

Quantum Processor

QUANTUM PROCESSORSEffiviency of the processors

Hillery-Buzek 2004

The processor is given by the unitary operator

Ajk is an operator in Hd, {|j, |k are the basis for Hp

The required operation is

The data and program states are

It is useful to take c1= z c0

QUANTUM PROCESSORS

Efficiemcy of Processors

Hillery-Buzek 2004

1. Case with only one program state, the operator A is

Summ módulo 2

2. To improve the probability, one uses a vector program given by

ck+1= z ck

In the two program case, the operator A

Summ módulo 4

3. To generalize this process, an N dimensional vector program is used

QUANTUM PROCESSORS

Rendimiento de Procesadores

Hillery-Buzek 2004

In the case of N program states, A is given by

Summ módulo N

QUANTUM PROCESSORS

Rendimiento de Procesadores

Hillery-Buzek 2004

Therefore, the processor is given by the operator

Summ módulo N

The success probability approaches to 1

PROPOSAL OF A UNIVERSAL QUANTUM COPYING MACHINE IN CAVITY QED

• Joanna Gonzalez• Miguel Orszag• Sergio Dagach• Facultad de Física• Pontificia Universidad Católica de

Chile

The Jaynes Cummings model

The Jaynes-Cummings Model describing the interactionOf a single two level atom with a single quantized cavity mode

Of the radiation field plays a central role in quantum optics.In spite of the mathematical simplicity of this model, is physically

Realistic.It describes purely quantum mechanichal phenomena like

Rabi oscillations, collapses and revivals of the atomic inversion,subPoissonian statistics and squeezing of the cavity field.

Also, experiments with highly excited Rydberg atoms in high QCavities have allowed to investigate experimentally

the interaction of a single atom with a single cavity modeThus proving experimentally the predictions of the JC

Model.

))((2

aagaaH z

ab

The Hamiltonian in the dipole approximation can be written as

a

a

a

a Creation photon and destr atomic unit of en

destr photon and creation atomic unit of en

destr photon and destr atomic unit of en

Creation photon and creation atomic unit of en

JAYNES CUMMINGS MDELTWO LEVEL ATOM AND DIPOLE APPROXIMATIONS

Normally in QO this last two terma are neglected in theSo-called RWA

a

b Two level atom

)1cos()( ngttCan

It is simple to prove that under the JC dynamics, and ifThe atom is initially in the a state, and the field with n photons

(Fock state)

Rabi oscillation and was originally studied in NMR

If initially we donot have a Fock state but a ratherA combination of Fock states, a coherent state

nn

en

n

02

!

2

Then a curious phenomena takes place,Called

COLLAPSE AND REVIVAL

1gtcPa-Pb

Constructive and destructive interference between the variousRabi oscillations.Apparently the Poisson distribution has the correct

Factors for total collapse. If we try a thermal state, will observeONLY PARTIAL COLLAPSES

OSCILLATION VERSUS EXPONENTIAL DECAY

For a long time the Jaynes-Cummings Model appeared to be A highly academic model.

It described an atom and a field that periodically pass the energy From one to the other like two coupled pendulum or two

coupled oscillators.On the other hand, people saw in the LAB a different reality.

ATOMS DECAYED FROM EXCITED TO GROUND STATES.Of course there was the Wigner Weiskopf theory of spontaneousEmission (Also Einstein phenomenological theory) that explained

Such a decay if one took

AN INFINITE NUMBER OF OSCILLATORSCOUPLED TO THE ATOM.

RYDBERG ATOMS

However, the situation has changed over the last two decades.The introduction of highly tunable Dye Lasers, which can excite

Large population of highly excited atomic states with a highMain quantum number n.These atoms are referred as

Rydberg Atoms.Such excited atoms are very suitable for atom-radiation experimentsBecause they are very strongly coupled to the radiation field, since

The transition rates between neighbouring levels scale as n^4.Also transitions are in the microwave region where photons

live longer, thus allowing longer interaction times.Finally, Rydber atoms have long lifetimes with respect to

spontaneous decay.THE STRONG COUPLING OF THE RYDBERG ATOMS TO THE FIELDCAN BE UNDERSTOOD SINCE THE DIPOLE MOMENT SCALES WITH

n^2, (typical n~70 )

MICROMASER

MICROMASER

A one atom maser is described in the previous figure.A collimated beam of Rubidium atoms is passed through

A velocity selector.Before entering a high Q superconducting microwave cavity

The atom is excited to a high n-level and convertedIn a Rydberg atom.

The microwave cavity is made of niobium and cooled downTo a low temperature.

The Rydberg atoms are detected in the upper or lower levelBy two field ionization detectors with their fields

Adjusted so that in the first detector, only atoms in theUpper state are ionized.

MICROMASER

MASER OPERATION (Walther et al) WAS DEMONSTRATED BY TUNING THE CAVITY TO THE MASER TRANSITION AND

RECORDING SIMULTANEOUSLY, THE FLUX OF ATOMS IN THE EXCITED STATE.

AS SHOWN IN THE FIGURE, OM RESONANCE, A REDUCTION OF THE SIGNAL WAS OBSERVED FOR RELATIVELY SMALL

ATOMIC FLUXES (1750 ATS^-1)

HIGHER FLUXES PRODUCE POWER BROADENINGAND A SMALL FRQUENCY SHIFT.

ALSO THE TWO PHOTON MICROMASER WAS DEMONSTRATED(HAROCHE ET AL)

Meschede,Walther,Muller,PRL,54,551(1984),Rb85 63p3/2….. 61d3/2, Qcavity~10^8, Tcav~2K, nth~2

The No-Cloning Theorem(Wooters and Zurek,Nature299,802(1982)) showed that it is not possible to construct a device that will produce an exact copy of an arbitrary quantum state.This Theorem is an unexpected quantum effect due to the linearity of Quantum Mechanics, as opposed to Classical Physics, where the copying Process presents no difficulties, and this represents the most significant difference between Classical and Quantum Information.Thus, an operation like:

Thus, an operation like:

0 ´a b x a b x

Q Q

Is not possible, with:

a

0b

xQ

´x

Q

=INPUT QUBIT

=initial state of cloner

=Blank copy=final stateof cloner

Because of this Theorem, scientist ignored the subject up to 1996 when Buzek and Hillery (V.Buzek,M.Hillery,Phys.Rev.A,54,1844(1996) proposed the Universal Quantum Copying Machine(UQCM)-that produced two imperfect copies from an original qubit, the quality of which was independent of the input state.

UNIVERSAL QUANTUM COPYING MACHINE BASIS

2 1

3 3

2 1

3 3

,

1

2

5

6

I

I

I

B A A

B A A

B BlankCopy

A A AUXILLIARY QUBITS

F

�

The quality of the copy is measured through the FIDELITY

copy ideal

ideal input

F

In the present work, we propose a protocol that produces 2 copies from an input state, with Fidelity

5

6F

In the context of Cavity QED, in which the information is encoded in the electronic levels of Rb atoms, that interact with two Nb high Q cavities.SOME PREVIOUS BACKGROUND TO THE PROPOSALConsider a two level atom that is prepared in a superposition state , using the Microwave pulses in a Ramsey Zone, with frequency r

Near the e(excited)-g(ground) transition. It generates superpositions

cos sin

sin cos

( )

i

i

r eg

e e e g

g e e g

where

t

Depends on the interaction timeIs prop. detuning

On the other hand, the atom-field interaction is described by the Jaynes Cummings Hamiltonian

0

0

1( ) ( ),.

2 2 2

. ,2

egJ C zH a a a a

Coupling constant

The atom-field state evolves like

0 0

0 0

,0 ( ) cos( ) ,0 sin( ) ,12 2

,1 ( ) cos( ) ,1 sin( ) ,02 2

e

g

e t t e t g

g t t g t e

For example, for

0

0

1 1,0 ,0 ,1 , , , , , , ,1 ,0 ,1

2 2 2

,0 ,1 , , , , , , , , , , ,1 ,0

t e e g g e g

t e g g e

Now, consider an external Classical pulse, interacting with the atom

( )ext gH t

We use the dressed state basis that diagonalizes the J-C Hamiltonian:

2 2

, ) sin , 1 cos ,

, ) cos , 1 sin ,

cos 24 ( 1)

n n

n n

n

n g n e n

n g n e n

n

eg ,

The Energies of the dressed states are

E

)1(42

)2

1( 22 nnE

In the limit0

, , ............ .. .. ......... ,1 ,1)

, , 1 .......... .. .. ....... ,0 ,1

n e n in this case e

n g n in this case g

Consider the external field in resonance with the (+,1)-(-,0)Transition, that is

)cos()( ttf sg Where f(t) is some smooth function of time to represent the pulse shape, with(in the dispersive case)

201 3

eg

nns EE

The above Hamiltonian has been studied by several authors (Domokos et al;Giovannetti et al) and arrive to the conclusion thatFor a suitable pulse, a C-NOT gate can be achieved, where the photonNumber (0 or 1) is the control and the atom the target

The mechanism of the above C-NOT gate that forbids, for example the (g,0>--(e,0> transition is the Stark Effect, caused by one photon in the cavity. In order to resolve these two transitions, we have to make sure that

int

1

t

g,0or -,-1g,0or -,-1

e,0or ,0e,0or ,0

e,1or ,1g,1or ,0

g,1or -,0e,1or ,1

og, 1,g

0,e 1,e

s s

Where Is the frequency difference between these two transitions.

The exchange

,1 ,1g e

IS POSSIBLE

C-NOT GATE

N=0 ATOMIC STATE IS NOT CHANGEDN=1 ATOMIC STATE IS EXCHANGED

CONTROL TARGET

UQCM

PROPOSED PROTOCOL

ATOM 1

A1, initially at is prepared in a superposition, via a Ramsey Field

1g

1 1 1

2 1

3 3

1cos

3

g g e

ar

A1 interacts with the cavity Ca(initially in )through a Rotation, so

0a

,0 ,1

,1 ,0

e e

g e

State swapping.The excitation of atom 1 is transferred to the cavity a

1 1 1

2 1 2 1( ) 0 ( 0 1 )

3 3 3 3a a ag e g

g

ATOM 2

IT CONTAINS THE INFORMATION TO BE CLONED

2 2 2e g

This state can be prepared in the same fashion as the atom 1, for example with a Ramsey Field.Then we apply a Classical pulse, as described before, generating a C-NOT gate ,nothing happens with 0 photons

,1 ,1g e

2 2

2 2 2 2

2 1( )( 0 1 )

3 3

2 10 1

3 3

a a

a a

e g

e g g e

C-not

A3 and A4 are the atoms carrying the two

copies(IDENTICAL)

FINAL STATE

3 4 3,4

3 4 3,4

1 1 3 4

1

1 2

3,4 3 3 4

2 1

3 3

2 1

3 3

, ,

2 1( )

3 3

0 0

0 1

1( )

2

I

I

a b

I a b

e e A A

g g A A

where

e g

B g e g g

A g g

A g g

e g g e

DISCUSSION

Experimental numbers(Haroche et al)

KHzKHzKHz 502

,502

,1002

An interaction time of ss 50 Marginally satisfies the earlier requirement.

With the flight time of 100 s The whole scheme should

Take about 700 s Which is reasonable in a cavity with a

Relaxation time of 16ms.They achieved a resolution required toDistinguish between 1 or 0 photons

Discussion of the dispersive C-NOT Gate

We have solved numerically the Hamiltonian

)()(exp)cos(

)(exp

2

2

ssext

extJC

ttfH

with

Ht

HH

We introduced the exponentials to simulate numerically the flight time and duration of the pulse s

Scrodingers Eq was solved for the state

gbeagbeat ,1,1,0,0)( 1100

int

62.8 ,2

35.8 ,2

31.42

28.6422 54

2 27s

KHz

KHz

KHz

fKHz

t s

s

gbeagbeat ,1,1,0,0)( 1100

BIBLIOGRAPHY1.-W.K.Wooters and W.H.Zurek,Nature,London,299,802(1982)2.-V.Buzek,M.Hillery,Phys.Rev.A 54,1844(1996)3.-D.Bruss et al, Phys.Rev.A 57,2368(1998)4.-N.Gisin,S.Massar, Phys.Rev.Lett,794,153(1997)5.-D.Bruss et al, Phys.Rev.Lett,81,2598(1998)6.-V.Buzek,S.L.Braunstein,M.Hillery,D.Bruss, Phys.Rev.A,56,3446(1998)7.-C.Simon,G.Weihs,A.Zeilinger, Phys.Rev.Lett,84,2993(2000)9.-P.Milman,H.Olivier,J.M.Raimond, Phys.Rev.A,67,012314(20003)10.-M.Paternostro,M.S.Kim,G.M.Palma,J.of Mod.Opt,50,2075(2003)11.-M.Brune et alPhys.Rev.A,78,1800(1995)12.-V.Giovannetti,D.Vitali,P.Tombesi,A.Eckert,Phys.Rev.A,52,3554(1995)13.-M.Orszag,J.Gonzalez,S.Dagach,sub Phys.Rev.A14.- M.Orszag,J.Gonzalez,Open Sys and Info Dyn,11,1(2004)

Pontificia Universidad Católica de Chile

A SINGLE ION STOCHASTIC QUANTUM PROCESSOR

MIGUEL ORSZAGPAUL BLACKBURN

OUTLINE OF THE TALK

1.-INTRODUCTION

2.-QUANTUM GATES C-NOT ,TOFFOLI

3.- A STOCHASTIC QUANTUMPROGRAMMABLE PROCESSOR (General, using quantum gates)

4.-IMPLEMENTING THE GATES VIA TRAPPED ION

5.-DISCUSSIONActual implementation of the processor with a trapped ion, decoherence and measurements

We first show how to realize the rotation of a qubitvia a quantum processor, using two and three qubit gates.

Next we discuss these C-NOT and TOFFOLI gates, implemented by making use the two and three dimensional

Center of mass vibrational qubits using a single three level ion.

Control and coupling of the ion´s internal electronic states is achieved via far detuned lasers exciting a Raman

transition scheme.

Finally we put things together and come upwith the proposal

Stochastic Processor for a qubit

We want:

How to implement it???

A STOCHASTIC QUANTUM PROCESSOR

1.-G.Vidal,L.Masanes,J.I.Cirac,PRL,788,047905(2002)2.-M.A.Nielsen,I.L.Chuang,PRL,79,321(1997)

3.-M.Hillary,V.Buzek,M.Ziman,PARA,65,022301(2002)

Data qubit

Program 1 qubit

Program 2 qubit

C_NOT

GATEToffoli Gate

To understand the procedure, consider a single Program Register.First we define the program and the data

)10(2

1

)10(2

1 22

ddd

P

i

P

i

P

BAd

ee

PROGRAM STATE

DATA STATE

d

d

Pd

i

d

i

Pd

i

d

i

P

i

P

i

dd

Pd

dU

dU

BeAeBeAe

eeBANOTC

dNOTC

1)10(2

10)10(

2

1

)10)(10(2

1)(

)(

2222

22

+

In this case, a measurement on the program register will cause a collapse On the data qubit with the outcome .

If we measure 0 in the program, we get the good answer,If we measure 1, wqe get the wrong answer

dddorUdU With a probability 1/2+

Bad result is for 1in P1

To improve upon this scheme, we introduce a Toffoli Gate, as in the Figure.

The data line and the first program qubit are unchanged, however,If the output of the program register is

Indicating a failure , the Toffoli gate effectively acts like a C-Not Gate Between the data line and a second program qubit

11

P

22

P

There is again a probability 1/2 of getting this time

UUU 2

THEREFORE, WE HAVE INCREASED THE SUCCESS PROBABILITY to 3/4 (and so on…)

Bad result only in the 11 case

One can generalize the argument, including MORE GENERALIZEDTOFFOLI GATES, and having a success probability of

N)21(1

Where N is the number of Program qubits or generalized Toffoli Gates (also the number of measurements)

The price we pay is the increase of gates and number of measurements

3 level Trapped ion (harmonic trap) interacting with two laser fields highly detuned from the upper level

(RAMAN SCHEME)

1

FOR LARGE DETUNING WE PROCEED TO PERFORMAN ADIABATIC ELIMINATION OF THE UPPER LEVEL 2

,31

,21

13

12

Defining

Go to the Heisenberg picture

)(

)()(

),(

)()(

)()(

23

~)(

12*

12

~)(

2313

~

1313

~

13

~)(23

*

22

~

11

~)(

12*

12

~

1212

~

1212

~

12

~

12

23

~

23

23

~

23

12

~

12

tki

tki

tki

tki

xe

yedt

di

ye

xedt

di

tUtU

eoperatorsDefiningth

13

~)(

231213

23

~

23

12

~

12

2312

23

12 ,

ti

ti

ti

e

e

e

Using a second trasnformation to eliminate The fast rotating terms

)(

)()(

),(

)(

23)(

12*

12)(

2313231213

13)(

13*

22

~

11

~)(

12*

121212

12

23

23

12

xki

yki

yki

xki

e

edt

di

e

edt

di

We get the following equations in the “second picture”

Please notice that all the fast terms are gone

Under the assumption of large detuningsWe obtain a solution for

By setting

])2233([1

],)2211([1

0

31

~)(

12*

~~)(

23*

23

32

~

13

~)(

23*

~~)(

12*

12

12

~

12

12

12

~

1223

~

23

23

~

2312

~

12

txkityki

tykitxki

ee

ee

dt

d

By setting

And similarly for the 2-3transition

Upon inserting these adiabaticSolutions for the atomic operators

13

~

23

~

12

~

An d

In terms of

And replacing them in the HamiltonianWe get the effective Hamiltonian after the

Adiabatic elimination of level 2

ccbbaaH cba 3311 310

Now we go to the interaction picture, with

And transform to the new Hamiltonian

tiHtiH

I eHHeH00

)( 0

And replace the x, y, z operators by

,2

,2

,2

)(

),(

),(

0

0

0

0

0

0

c

b

a

mZ

mY

mX

with

ccZz

bbYy

aaXx

Width of groundState of oscillators

Called usually the Lamb Dicke parameter.The square of this quantity represents the ratio Between the recoil energy and the vibrational

Energy in the i direction .Experimentally it ranges between 0.1 and 0.2.

iii mk

2

Interaction Picture

Laser Frequencies

Integers(small)

+hc

Stationary Terms

We look for terms such that

)()(

0)()()(

1

~

3

~

231213

13

lnm cba

Laserfrequencies

Stark shiftedAtomic levels

Please notice that by careful tuning of the lasers we can select a given Hamiltonian

C-NOT

Toffoli

The Hamiltonians

To get the different effective Hamiltonians, we have to get the right laser frequencies

The Temporal Evolution

C-NOT

The Temporal Evolution

Toffoli

-

+

-

+

C-NOT

Control target

X,Y vibrations

bus

Toffoli

Control 1 Control 2 target bus

X,Y,Z vibrations

-

The actual operation of the processor consists in three phases1)Preparation,2)Processing and 3)Measurement

In the first phase, the ion is Prepared in the ground state and is loaded with the data and program states.

In the Processing phase, the lasers are switched on with the detunings and spatial orientations required for the required

Pulse periods.In the detection phase, the y and z vibrational state of the ion is

Measured, thus indicating whether the desired operationwas applied successfully on the data state.

The actual loading of the vibrational state could be done ina separate trap on an auxilliary ion. The vibrational

state of this auxilliary ion can be transferred to our ion,following for example, the proposal of Paternostro et al

M.Paternostro,M.S.Kim,P.L.Knight,PRA,71,022311(2005)

The feasibility of the proposal depends strongly on the Decoherence time which is of the order of 1-10ms.

The characteristic time required for an operation like H3T,For a Lamb Dicke parameter =0.3 and

Corresponds, for (full cycle)a time of 0.53 ms.

In principle this time could be decreased since

As long as the laser power is not beyond Watts/cm2which would photoionize the atom

23 tT

KHz500)2(13

2113 EE1410

Alternatively,one may attempt to increase the motional stateDecoherence time of the trap.

C.Monroe et al have recently done experiments where the heating rate is very low (Cd+ ions)where they report

sec0248.0 mquanta

dt

dn

Which means that an n=1 state can have a life of40ms!!!!!!!!!!!GREAT……

L.Deslauriers,P.Haljan,J.P.Lee,K.A.Brickman,B.B.Blinov,M.J.Madsen,C.Monroe,PRA,70,043408(2004)

Measurement or reconstruction of the quantum mechanical state of a trapped ionWhere the information on the vibrational CM motion of a trapped ion can be Transferred to it´s electronis dynamic by irradiating a long living electronic

transition by laser light and probing a strong transition for resonance Fluorescence.Was first suggested by S.Wallentowitz,W.Vogel, PRL,75,2932(1995)

Other reconstruction schemes, applying coherent displacements of different Magnitudes was suggested both theoretically and experimentally by

D.Liebfried, D.M.Meekhof,B.E.King,C.Monroe,W.M.Itano,D.J.Wineland,PRL,77,4281,(1996)

QND measurements of vibrational populations in ionic traps . This scheme allowsThe production of of Fock states, associated with the CM motion and it is based

On the fact that the Rabi Frequency between two internal states of the ionInduced by a resonant carrier field depends on the vibronic number.

L.Davidovich,M.Orszag,N.Zagury,PRA,54,5118(1996)R.L.Matos Filho,W.Vogel,PRL,76,4520(1996)

Direct Measurement of the Wigner FunctionL.G.Lutterbach,L.Davidovich,PRL 78,2547(1997)

A generalization of the QND measurement of the vibronic states (which was originally suggested for one dimensional vibrations,

was extended to higher dimensions, quiteappropriate in the present scheme, was done by

W.Kaige et al“Quantum Non-Demolition Measurements and Quantum State

Manipulation in two dimensional Trapped Ion”

In “Modern Challenges in Quantum optics”,

M.Orszag,J.C.Retamal, Edts, Springer Verlag,2001

Quantum non-demolition measurements are design to avoidThe back-action of the measurement on the detected

Observable.For example, in the optical domain, we find experiments

Using Kerr effect in a solid or liquid medium.The signal field to be measured interacts non-linearly

With a probe field, whose phase changes by a quantityWhich depends linearly on the number of photons in the

signal beam.In the Haroche group, they developed a QND method

To measure the number of photons stored in a high Q cavityWhich is sensitive to a very small number of photons.

The method is based on the detection of a dispersive phase shift produced by the field on the wave function of

Non-resonant atoms which cross the cavity.This shift which is proportional to the photon number in the

Cavity,is measured by atomic interferometry, usingRamsey fields.

Since the atoms are non-resonant with the cavity,No photon is exchanged between them and the

Cavity and the measurement is indeed a QND one.However, the information aquired by detecting

A sequence of atoms modifies the field step by step ,untilIt eventually collapses into a Fock state, which a priory is

Unpredictable.A repetition of the measurement, for the same initial state

Of the field will yield a distribution of Fock states, which reproduces the initial distribution of the field.

In a similar way, it is possible to realize a QND Measurement of the vibrational population distribution

For an ion in a Paul trap.As in the cavity QED case, a Fock state is generated in

the process.

Miguel Orszag

We are proposing a scheme for implementing a single qubit Stochastic Quantum Processor using a single cold

Trapped ion.The Processor implements an arbitrary rotation around the

z-axis of the Bloch sphere of the data qubit,GIVEN TWO PROGRAM QUBITS.

The operation is applied succesfully with a probabilityP=3/4

We analize the preparation process, discuss decoherence and also propose various possible measurement schemes

on the program qubit space.

CONCLUSIONS

Logical gates and quantum processors with trapped ions and cavities MIGUEL ORSZAG FACULTAD DE FISICA PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE Course at.

Documents

qubit gate slide

quantum computation

quantum mechanics

quantum computing

phase slide

case slide

p1 slide

arbitrary quantum state