Relativistic Quantum Phenomena in Circuit …...Master in Quantum Science and Technology Relativistic Quantum Phenomena in Circuit Quantum Electrodynamics Master Thesis by Julen S.

Master in Quantum Science and Technology

Relativistic Quantum Phenomenain Circuit Quantum Electrodynamics

Master Thesis

by

Julen S. Pedernales

Director:

Prof. Enrique Solano

Departamento de Quımica FısicaFacultad de Ciencia y Tecnologıa

Universidad del Paıs Vasco UPV/EHU

Leioa, September 12, 2012

Relativistic Quantum Phenomena

in Circuit Quantum Electrodynamics

Julen S. Pedernales

September 12, 2012

iii

Abstract

We present a method for simulating relativistic quantum physics in circuit quantumelectrodynamics. By using three classical driving fields, we show that a superconductingqubit strongly coupled to a resonator field mode can be used to simulate the dynamics ofthe 1+1 dimensional Dirac equation and Klein paradox in all parameter regimes. Usingthe same setup, we also propose the implementation of the Foldy-Wouthuysen canonicaltransformation, where the time derivative of the position operator becomes a constantof the motion. In our setup, the internal degrees of freedom of a superconducting qubitencode those of the simulated particle, while the quadratures of the field in the resonatorare associated with the mechanical ones. In this manner, we pave the way to the quantumsimulation of relativistic quantum phenomena in a nonrelativistic quantum system, asis the case of superconducting circuits.

v

Laburpena

Elektrodinamika kuantikoko zirkuituak erabiliz fisika erlatibista kuantikoa simu-latzeko prozedura aurkeztu egiten dugu. Qubit supereroale bat erresonadorearen modubatera indartsuki akoplatuz eta hiru eremu elektromagnetiko klasikoen bitartez, Dirac-en ekuazioa 1+1 dimentsiotan eta Klein-en paradoxa simulatzea proposatzen dugu. Aregehiago, egitura bera Foldy-Wouthuysen transformazioa inplementatzeko balio zaigu.Foldy-Wouthuysen transformazio kanonikoak posizio operadorearen deribatu denboralahigiduraren konstante batean bihurtzen du. Gure diseinuan qubit supereroalearen askata-sun gradu intrintsekoek simulatutako partikularenak kodifikatzen dituzte. Bestalde-tik, erresonadorearen barruko eremuaren koadraturak, partikularen askatasun gradumekanikoekin lotu egiten dira. Modu honetan, efektu kuantiko erlatibistak sistema ez-erlatibista batean simulatzeko aukera dugu.

vii

Resumen

Presentamos un esquema para la simulacion de fısica cuantica relativista en circuitosde electrodinamica cuantica. Demostramos que utilizando tres campos clasicos y unqubit supercondutor fuertemente acoplado a un modo del resonador puede simularsela dinamica de la ecuacion de Dirac en 1+1 dimensiones y la de la paradoja de Klein.El mismo sistema puede ser utilizado para implementar la transformacion canonica deFoldy-Wouthuysen, la cual convierte la derivada temporal del operador posicion en unaconstante del movimiento. Nuestro diseno codifica los grados de libertad internos dela partıcula simulada en los del qubit, mientras que los grados de libertad mecanicosse mapean en las cuadraturas del campo. De este modo, habilitamos la simulacion deefectos cuanticos relativistas en un sistema no relativista, como lo es el de los circuitossuperconductores.

Contents

Abstract iii

Laburpena v

Resumen vii

Introduction 1

1 Quantum Optics 51.1 Quantization of the electromagnetic field . . . . . . . . . . . . . . . . . . . 61.2 Field states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 The number state . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.3 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Quasi probability distributions . . . . . . . . . . . . . . . . . . . . . . . . 151.3.1 The P representation . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 The Q representation . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.3 The Wigner representation . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Light-matter interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.1 Two-level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.2 The quantum Rabi model . . . . . . . . . . . . . . . . . . . . . . . 221.4.3 The rotating-wave approximation . . . . . . . . . . . . . . . . . . . 231.4.4 The Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . 241.4.5 Ultrastrong and deep strong coupling regimes . . . . . . . . . . . . 25

2 Relativistic Quantum Mechanics 292.1 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.1 Derivation of the Dirac equation . . . . . . . . . . . . . . . . . . . 312.1.2 Representations of the Dirac equation . . . . . . . . . . . . . . . . 332.1.3 Fourier space and spectral subspaces of the Dirac operator . . . . 342.1.4 The Foldy-Wouthuysen transformation . . . . . . . . . . . . . . . . 362.1.5 1+1 Dirac equation, its nonrelativistic limit and squeezing . . . . . 37

2.2 Quantum relativistic e↵ects . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.1 Zitterbewegung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

x Contents

2.2.2 Klein paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Circuit quantum electrodynamics 453.1 Elements of circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . 463.1.2 Transmission line resonator . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Dual-path detection method . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Simulating Relativistic Quantum E↵ects in circuit QED 534.1 The method and proposal of implementation . . . . . . . . . . . . . . . . 554.2 Numerical simulations and discussion of the results . . . . . . . . . . . . . 57

4.2.1 The free Dirac particle and Zitterbewegung . . . . . . . . . . . . . 574.2.2 Dirac particle in an external potential and the Klein paradox . . . 62

5 Conclusions 67

Acknowledgments 69

Bibliography 71

Introduction

Analogies are undoubtedly an extremely powerful tool in the acquisition of knowledgeand in research. With analogies, one can understand the unknown based on what onedoes know. However, if the achieved equivalence is pushed too far, it naturally fails andyields misleading predictions and interpretations. It is precisely this intrinsic imperfec-tion in any analogy what turns to be its most interesting point, because it is there, atthe imperfections, where new knowledge is created. Quantum simulations, as they try toreproduce behaviours of certain physical systems in others, provide powerful analogiesbetween the simulated physics and the physics of the simulator system. Although this isnot the usual interest of quantum simulations, more focused on beating the capacities ofclassical computers, we see this aspect as a truly attractive one, for the sake of aesthetics,exchange and generation of scientific knowledge. In this work, we propose the simula-tion of quantum relativistic physics in the context of circuit quantum electrodynamics,two distinct and unconnected fields. The important concepts in relativistic quantumphysics, developed by Dirac and others in the 1920’s, predict interesting counterintu-itive phenomena which can hardly be observed in direct experimentation. However, bymeans of quantum simulations, we are able to create a small nonrelativistic realm whereeverything behaves as if it was ruled by quantum relativistic physics. More importantly,the proposed setup allows for the observation of phenomena that are demanding or evenof impossible access in other scenarios.

Relativistic quantum mechanics merges the physics of quantum mechanics with thatof special relativity in a theory that, unlike quantum field theories, still describes physicsat the level of single particles. The central equation in this theory is the Dirac equation.The main characteristic of the Dirac equation is that, for every positive-energy solution,it allows a negative-energy one. Moreover, the time evolution is given in such a way thatthese opposite energy states interact with each other. This gives rise to some peculiarphenomena, never observed in nature and with di�cult interpretation, which openedquestions that still nowadays generate debates. These e↵ects, as Zitterbewegung andthe Klein paradox, are somehow overcome in the formalism of quantum field theories.For this and because it becomes inconsistent out of its save validity regime, relativisticquantum mechanics is many times abandoned as an incorrect theory and interpretedas a mere step towards quantum field theories. However, Dirac equation has been anextremely successful relativistic wave equation, which has predicted the existence ofantiparticles, introduced the spin and the spin-orbit coupling from first principles and

2 Introduction

given a correct solution to the fine structure of the Hydrogen atom spectrum for example.Even nowadays, it has appeared in new scenarios as for example graphene where itdescribes quasi-electrons in 2+1 dimensions. For all this, we consider the Dirac equationextremely important in the history of physics. If treated carefully, in the correct rangeof validity and having in mind its limitations, the Dirac equation describes undoubtedlyvalid and interesting relativistic quantum dynamics.

On the other hand, quantum optics, the quantum theory of light, where the interac-tion between matter and light is modeled, has provided a very interesting platform forthe development of experiments with high controllability. Using techniques and knowl-edge stemming from quantum optics, one can simulate the physics of a wide range ofsystems. As the last exponent of these technologies, we have circuit quantum electro-dynamics, which can be also considered as a quantum simulation of quantum opticalcavity QED. This relatively new quantum platform allows for the implementation oflight-matter interactions that are easier to control and construct in on-chip circuits.For its high controllability and simple construction, circuit quantum electrodynamics isconsidered as a promising candidate for quantum information processing and quantuminformation protocols, thus becoming a contemporary hot topic and capturing muchinterest. Our proposal could be, in principle, implemented in quantum optical cavityQED, although with technical limitations. Therefore, we propose the implementation ofthe Dirac equation and relativistic quantum phenomena in circuit QED, a more flexiblequantum platform for the sake of implementation and quantum measurement. Thus allthe calculations presented are done using parameters consistent with this novel quantumtechnology.

Even if a quantum simulation of the Dirac equation has already been reproducedin other setups, as is the case of trapped ions, our scheme provides crucial physicaldi↵erences with respect to those, paving the way for quite di↵erent possibilities. Forinstance, the mechanical and internal degrees of freedom of our simulated particle, areheld by two di↵erent entities in the circuit QED setups. While in the simulation donein trapped ions, the ion encodes both the internal degree of freedom of the particle andthe mechanical ones, in our case these are delocalised. That is, we consider the spinorialdegree of freedom as held by the internal states of a superconducting qubit, while themechanical ones are mapped into the phase space quadratures of an electromagnetic field.Moreover, the introduction of a relativistic linear potential is easily done in our scheme,just using an external classical driving along the cavity axis, while in the simulation intrapped ions one needs to use a second trapped ion.

This thesis is divided in five chapters. The first one aims to give an introduction toquantum optics. The basics of this area are presented with the objective of providinga ground floor where we can later construct our theoretical and experimental proposal.The second chapter is devoted to quantum relativistic physics: we o↵er a short outlineof what it is and the implications it has, mainly focusing on the Dirac equation and thephenomena related to it. This will help us to interpret the physics we want to simulate.On the third chapter, we provide a presentation of circuit QED technologies, which isa possible physical implementation platform for our proposal. The fourth one describes

3

our proposal and the calculations done to validate it. Furthermore, we o↵er an analysisand interpretation of the obtained results. This thesis ends with a final chapter were weexpose the conclusions and the scope of our work.

Chapter 1

Quantum Optics

Quantum optics is the subject that deals with optical phenomena that can only beexplained describing light quantum mechanically.

The reader might be surprised to find that there are few phenomena that reallyrequire a fully quantum mechanical approach to be understood. Some of these could bethe spontaneous emission of a photon by an excited atom that decays to its ground state,some states of the electromagnetic field such as the one photon state, or the interactionbetween a two-level atom and a single mode electromagnetic field.

In the historical approach to the description of light we can identify three clearstages. First we have the classical approach, which begins in the late XVII century withthe contemporary and rival theories of Newton and Huygens. The dichotomy betweenthe discreteness and the continuum character of light already appears in these earlyyears, since Newton defended a corpuscular theory of light while Huygens proposed awave theory. The reputation of Newton helped to make his theory the dominant oneduring all the following XVIII century. In the early IXX century Fresnel developedhis own wave theory of light which included a wave interpretation of di↵raction, andYoung performed his famous double slit experiment showing clear evidence of the wavecharacter of light. Moreover, at the end of the same century Maxwell laid the foundationsof electromagnetism predicting the existence of electromagnetic waves. At this point thecorpuscular theory of Newton got relegated to mere historical interest.

The second stage of the theory of light is usually called the semi-classical approach.In 1901, Max Planck solved the ultraviolet problem proposing a quantization of the radi-ation emitted by a blackbody. Probably inspired by Planck’s work, four years later in his“Miracle year”, Einstein solved the photoelectric e↵ect problem, using the quantizationof light. However, none of these problems or their solutions give real evidence of thequantum character of light. Actually, both of them can be explained treating matter(atoms) quantum mechanically and light classically, hence the name of semi-classicaltheory. The first real attempt to confirm quantum character of light was tried and failedby G.I. Taylor in 1909. The idea was to replicate Young’s double slit experiment atan intensity low enough to ensure that just one photon was crossing the double slit ata time. The result was the same as that of the classical experiment. Now we know

6 Chapter 1 Quantum Optics

that the quantum theory predicts the same results the classical theory does. In orderto find the first quantum e↵ects, we have to involve higher order correlations, as theinterference of intensities (in Young’s experiment we deal with the interference of am-plitudes, probability amplitude in the quantum case and electric field amplitude in theclassical). In the 1920’s, quantum mechanics was developed and so a formal theory forlight quantization. In 1926 Gilbert Lewis coined the term ’photon’. However, in thefollowing years quantum mechanics itself developed more than the quantum theory oflight did, and it was not until 1956 that new and meaningful steps were given towardsa full quantum mechanical description of light.

In 1956, Hanbury Brown & Twiss performed an experiment involving interferencefrom starlight intensities, which is considered a landmark in the field (and usually pointedout as the birth of quantum optics), even if right away it was demonstrated that the re-sults could be explained applying quantum mechanics to the photodetection process onlyand not to the light itself. In 1963, Roy J. Glauber realized that there were some lightstates with statistical properties that could not be explained classically. One of thesenon-classical photon statistics is called antibunching and was experimentally confirmedby Kimble, Dagenais, and Mandel in 1977. Later, in 1985, Slusher et al. generated an-other nonclassical light state, the squeezed state. These two experiments for the first timerevealed the quantum nature of light, making of quantum optics a mature field. In thefollowing years quantum optics grew to incorporate light-matter interaction, quantumentanglement, quantum simulation and quantum information processing among otherfields. This fact is not surprising, since quantum optics provides an easy and “clean”way of performing many kind of experiments with high controllability, experiments thatgive access to physics that were unreachable prior to quantum optics. In the same way,quantum optics has enable direct testing of many quantum e↵ects that had just indi-rect proofs before. All these experimental possibilities that quantum optics o↵ers, haveincreased the interest of many di↵erent fields, making of quantum optics a very diverse,active and productive field.

This chapter aims to present some of the basic concepts of quantum optics uponwhich all this work stands up. In any case this chapter tries to be a complete or a fullreview on quantum optics, but a short summary of key concepts that will help the readerto understand the rest of the work. Nevertheless, this chapter will not be presented inthe usual way but in a more contemporary and appropriate one for the topic that isbeen covered here. In the same way, references to the latest advances in the field will begiven, which may help the reader to understand the context of our proposal. This willbe true specially for section 1.4.

1.1 Quantization of the electromagnetic field

In this section we aim to quantize the electromagnetic field so its description is given bya set of photons instead of the classical continuous electromagnetic waves. For such pur-pose we are going to follow Walls and Milburn [1]. Our starting point will be Maxwell’s

1.1 Quantization of the electromagnetic field 7

equations for the free electromagnetic field, this is,

r ·B = 0, (1.1a)

r⇥E = �@B@t

, (1.1b)

r ·D = 0, (1.1c)

r⇥H =@D

@t, (1.1d)

where B = µ0

H and D = ✏0

E, with µ0

the magnetic permeability, ✏0

the electricpermittivity, and µ

0

✏0

= 1/c2, c being the speed of light. On the other hand, we havethe scalar and vector potentials, � and A respectively, which transform like a four-vectorin the Minkowski space, Aµ = (�,A), and can describe both, the electric field, E, andthe magnetic field, B,

E = �r�� @A

@t, (1.2a)

B = r⇥A. (1.2b)

One of the key features of Maxwell’s equations is that they are gauge invariant,this means that under any transformation of the type Aµ ! Aµ + @µ where is anarbitrary scalar field, their solutions, the electric and magnetic fields, remain unchanged.This characteristic gives us freedom to choose the most convenient gauge depending onthe problem we are trying to solve. For problems in quantum optics a very appropriategauge is the so called Coulomb gauge, that we determine through the transversalitycondition,

r ·A = 0, (1.3)

and which results in the following condition for the scalar potential

r2� = 0. (1.4)

Thus, in this gauge, the scalar potential is no more a dynamical variable but a geometricalone. Consistent with Eq. (1.4) we can choose � = 0, letting expressions for the electricand magnetic fields only dependent on the vector potential, A,

E = �@A@t

(1.5a)

B = r⇥A. (1.5b)

Moreover, from expressions (1.1d) and (1.5a) we can show that the vector potential,which is a function of space and time, fulfills the wave equation

r2A(r, t) =1

c2@2A(r, t)

@t2. (1.6)


Solution to Eq. (1.6) can we written as

A(r, t) = A(+)(r, t) +A(�)(r, t) (1.7)

where the term A(+)(r, t) stands for all the amplitudes varying with e�i!t and A(�)(r, t)for those varying with ei!t, with ! > 0 and A(�)(r, t) = (A(+)(r, t))⇤.

Now, we aim to expand the vector potential in normal modes. For that we have torestric our field to a volume V of the infinite free space. The expansion looks like

A(+)(r, t) =X

k

ckuk(r)e�i!kt (1.8)

where the normal mode functions, uk(r), are required to form a complete orthonormalset, Z

Vu⇤k(r)u

0k(r)dr = �kk0 . (1.9)

Thus, each of the normal mode functions has to satisfy both the wave equation,

(r2 +!2

k

c2)uk(r) = 0, (1.10)

and the transversality condition,

r · uk(r) = 0, (1.11)

since the vector potential does.In order to get an expression for the normal mode functions, uk(r), we have to impose

the boundary conditions. For instance, boundary conditions considering reflecting wallswould lead to solutions of the form of standing waves, while periodic boundary conditionsto propagating waves. We will impose periodic boundary conditions on a cube of sideL, which results in plane wave solutions such as

uk(r) =e(�)pVeik·r (1.12)

where V = L3 and e(�) is the unit polarization vector. Note that mode index k is notonly standing for wave vector k but also for polarization index �, which can take values� = 1 and � = 2.

From the transversality condition we know that the polarization vector has to bealways perpendicular to the propagation vector and from the boundary conditions weget that each component of the propagation vector is restricted to the following discretevalues

ki =2⇡ni

Lwhere ni = 0,±1,±2, ... and i = x, y, z. (1.13)

1.1 Quantization of the electromagnetic field 9

Now that we know the normal mode functions we can write the whole expansion forthe vector potential as

A(r, t) =X

k

✓~

2!k✏0

◆1/2

[akuk(r)e�i!kt + a⇤ku

⇤k(r)e

i!kt], (1.14)

where we have used the fact that A(�)(r, t) = (A(+)(r, t))⇤ and we have introduced the

normalization constant⇣

~2!k✏0

⌘1/2

so that the amplitudes, ak and a⇤k, are dimensionless.

Furthermore, with the selection of such normalization constant we guarantee the correctHamiltonian for the field, as we will see later.

So far, we have quantize nothing yet, we have just expanded the vector potentialin terms of its normal modes. In Eq. (1.14) the Fourier amplitudes, ak and a⇤k, arejust complex numbers. The quantization comes when we promote these dimensionlessamplitudes to mutually adjoint operators, this is,

ak ! ak (1.15a)

a⇤k ! a†k (1.15b)

Moreover, since we are dealing with photons, which are bosons, we are going to requirethis operators to satisfy the bosonic commutation relations,

[ak, a0k] = [a†k, a

0†k ] = 0, [ak, a

0†k ] = �kk0 . (1.16)

For simplicity and to match the usual notation in this field we will omit the hats,ˆ, inthe bosonic operators. The commutation relations in Eq. (1.16) arise from the attemptto preserve the canonical commutation relations when quantizing the field, as they dowhen we quantize the harmonic oscillator. Furthermore, for each mode k, this bosonicoperators are identical to the ladder operators in a quantum harmonic oscillator.

For instance, using Eq. (1.5a) the electric field can now be written as

E(r, t) = iX

k

✓~

2!k✏0

◆1/2

[akuk(r)e�i!kt � a†ku

⇤k(r)e

i!kt]. (1.17)

Now we will find an expression for the Hamiltonian in terms of these bosonic opera-tors. We know that for the electromagnetic field the Hamiltonian is given by

H =1

2

Z(✏

0

E2 + µ0

H2)dr. (1.18)

From Eq. (1.18), with the expression for the electric field given in Eq. (1.17), a similarone for the magnetic field that we can calculate from Eq. (1.5b), and making use of Eq.(1.9) it is not hard to reach the following expression for the Hamiltonian in terms of thebosonic operators, this is,

H =X

k

~!k

✓a†kak +

1

2

◆. (1.19)


For each k this is nothing but the Hamiltonian of the quantum harmonic oscillator.Remember from the basic physics of the quantum harmonic oscillator that eigenstatesof the Hamiltonian are all equally separated in energy, more precisely they are separatedby ~! energy units. Furthermore, we know that when applying operator a†(a) to aneigenstate of the Hamiltonian, we move the state to the eigenstate right above (below)in energy, thus increasing (decreasing) its energy by an amount ~!. This is why ladderoperators are also called raising and lowering operators, because they either raise orlower the energy of the state by a quanta of energy ~!. In quantum field theory thisquanta of energy is interpreted as a particle, and thus, the raising and lowering operatorsare called creation and annihilation operators. Going back to Hamiltonian in Eq. (1.19)

we observe operator a†kak. In the case of the quantum harmonic oscillator, for each kthis operator represents the energy level of an eigenstate, in other words the numberof energy quanta in the system. Thus, in our interpretation of the energy quanta asparticles, this operator represents the number of particles and it is called the numberoperator. So far, we can interpret Hamiltonian in Eq. (1.19) as the sum of the numberof particles in each mode times the energy of the mode, plus the term

Pk

1

2

~!k. Thislast term appears whatever it is the number of particles we have, even if we have noparticles, that is why it is interpreted as the energy of the vacuum.

Electric and magnetic fields are now described by a set of independent harmonicoscillators. Each mode of the field is related to a harmonic oscillator and representedby a wave function in the appropriate Hilbert space. The description of the whole fieldis given in a tensor product of Hilbert spaces each of them corresponding to a mode ofthe field.

From the quantum harmonic oscillator we know that position and momentum canbe written in terms of ladder operators as follows

x =

r~

2m!(a+ a†) (1.20a)

p = i

rm!~2

(a† � a). (1.20b)

The reader might check that commutation relations in Eq. (1.16) are chosen so that[x, p] = i~, as was pointed out before. In the context of quantum optics the dimensionlesspart of position and momentum are called the field quadratures,

X =a+ a†p

2(1.21a)

P = ia† � ap

2, (1.21b)

and obey commutation relation [X,P ] = i.They can be understood as the real and imaginary part of the complex operator a,

which remember represents the amplitudes of the normal modes when expanding thefield in terms of these.

1.2 Field states 11

Field quadratures satisfy Heisenberg’s uncertainty relation,

�X�P � 1/2, (1.22)

which is straightforward from the commutation rules.Field states can be represented in a phase diagram plotting its quadratures against

each other. These phase diagrams live in the so called optical phase space, and can showinteresting properties of the field that might be hidden in other kind of representation.

1.2 Field states

Now we want to introduce some of the most typical quantum states that represent theelectromagnetic field: the number state, the coherent state and the squeezed state. Fromthese three only the coherent state has a classical interpretation.

1.2.1 The number state

Looking at the Hamiltonian in Eq. (1.19) that rules the evolution of the field, the mostnatural field states seem its eigenstates, that is, eigenstates of the number operator.These states are called number states or Fock states, and their key feature is that theyhave a well defined number of particles (photons in our case). For each mode this statesare usually represented as | nki and have energy ~!k(nk +

1

2

). This is,

Nk|nki = nk|nki, (1.23a)

hnk|H|nki = ~!k(nk +1

2), (1.23b)

where nk is the number of particles in mode k, and Nk = a†kak. Creation and annihilationoperators act on the number states in the following way

a†k|nki =pn+ 1|nk + 1i, ak|nki =

pn|nk � 1i. (1.24)

The ground state is |0i and defined as

ak|0i = 0. (1.25)

It is interpreted as the vacuum of the field since it has no photons to be annihilated. Itsenergy is given by

h0|H|0i =X

k

1

2~!k. (1.26)

Notice that the sum in Eq. (1.26) is in general an infinite sum, since the allowed valuesfor k are discrete but infinite. Conceptually this is a hard di�culty because it means thatthe energy of the vacuum is infinite. Nevertheless, this is not a practical problem, becauseexperimentally we always deal with energy di↵erences, which cancel this constant term.


Both field quadratures of vacuum have 0 expectations value and equal uncertainty.Moreover, the product of their uncertainties gives the lowest value allowed by the un-certainty relation, that is, 1/2.

Number states can be generated by recursively applying the creation operator on thevacuum state and normalizing appropriately, this is,

|nki =(a†k)

nk

(nk!)1/2|0i, nk = 0, 1, 2... (1.27)

On the other hand, number states are orthogonal,

hnk|n0ki = �kk0 , (1.28)

and form a complete basis of the Hilbert space,

1X

nk=0

|nkihnk| = 1, (1.29)

which we call the Fock basis.In the most general case of a number state the field will be in a tensor product of

several single mode number states. This is called a multi-mode number state,

|n1

, n2

, ..., ns, ...i = |n1

i ⌦ |n2

i ⌦ ...⌦ |nsi ⌦ ... (1.30)

Number states are already quantum field representations that cannot be describedclassically, they are experimentally very hard to create, and just states with low n havebeen produced.

1.2.2 Coherent states

Coherent states unlike number states have an unfixed number of photons. Actually, theyare a coherent superposition of infinite number states. Fields described by a coherentstate in the quantum theory of light do not show any property or e↵ect that cannot bedescribed classically. As in the case of vacuum, for coherent states uncertainty in bothfield quadratures is the same and the minimum possible.

In order to give a more formal description of coherent states, let us first introducethe so called displacement operator,

D(↵) = e↵a†�↵⇤a, (1.31)

where ↵ is an arbitrary complex number, defined in the phase space as ↵ = X + iP ,with X and P the field quadratures.

Some of the more useful properties of the displacement operator are

D†(↵) = D�1(↵) = D(�↵), (1.32a)

D†(↵)aD(↵) = a+ ↵, (1.32b)

D†(↵)a†D(↵) = a† + ↵⇤, (1.32c)

D(↵+ �) = D(↵)D(�)e�iIm{↵�⇤}. (1.32d)

1.2.2 Coherent states 13

The displacement operator when applied over the vacuum state, generates a coherentstate which is completely defined by the complex number ↵,

|↵i = D(↵)|0i. (1.33)

Using Eq. (1.32b) is not hard to demonstrate that | ↵i is an eigenstate of operator awith eigenvalue ↵,

a|↵i = D(↵)D†(↵)aD(↵)|0i = D(↵)(a+ ↵)|0i = ↵|↵i. (1.34)

Let us now try to write the coherent state in the number state basis,

|↵i =X

n

|nihn|↵i, (1.35)

where we have used Eq. (1.29). The coe�cients of expansion in Eq. (1.35) are

hn|↵i = ↵n

(n!)1/2h0|↵i. (1.36)

This can be worked out from the recursive relation

(n+ 1)1/2hn+ 1|↵i = ↵hn|↵i (1.37)

which is straightforward from Eq. (1.34) multiplying at both sides with hn| from theleft. Imposing the normalization of the coherent state,

1 = h↵|↵i =X

n

h↵|nihn|↵i =X

n

↵2n

n!|h0|↵i|2 = e|↵|

2 |h0|↵i|2, (1.38)

where we have used Eq. (1.29) and Eq. (1.36), we have that

h0|↵i = e�|↵|2/2. (1.39)

So finally, the expansion of the coherent state in terms of number states can be writtenas

|↵i = e�|↵|2/2X

n

↵n

pn!|ni. (1.40)

Remember that ↵ is a complex number, this means that the superposition of thedi↵erent number states takes a relative phase. When dealing with pure number statesthere was no chance of introducing any phase, since this would have to be global, whichin the end makes it totally random. With coherent states instead, we can define thisrelative phase, that is not other thing than the phase of the complex number ↵. Theidea of having a defined phase is closer to the classical fields, which gives us the feelingof the classicality of coherent states that we pointed out before.


Even if in this text we are not going to enter the topic of the photon distributions andits relation to the quantum being or not of light, we note that the photon distributionfor a coherent state, that is, the probability of measuring certain number of photons, isa Poissonian distribution with a mean number of photons |↵|2,

P (n) = |hn|↵i|2 = |↵|2e�|↵|2n

n!. (1.41)

Poissonian distributions are very well understood from the classical point of view, whichagain brings us closer to the idea that coherent states have all the classical propertiesand not quantum ones.

Coherent states are not orthogonal,

h�|↵i = h0|D†(�)D(↵)|0i = e�(|↵|2+|�|2)/2+↵�⇤, (1.42)

where we have used Eq. (1.32a) and Eq. (1.32d). Although, in the limit where |↵��|� 1coherent states can be taken to be orthogonal.

Coherent states satisfy the following completeness relation

1

⇡

Z|↵ih↵|d2↵ = 1, (1.43)

which, actually, is telling us that coherent states form an over-complete two dimensionalcontinuum of states. Proof of Eq. (1.43) might be found for example in [1].

1.2.3 Squeezed states

As we did with coherent states, let us start introducing the operator that applied to thevacuum generates squeezed states, the Squeezing operator

S(✏) = e1/2(✏⇤a2�✏a†2), (1.44)

where ✏ = re2i�. Some of the more fundamental properties of this squeezing operatorare the following

S†(✏) = S�1(✏) = S(�✏), (1.45a)

S†aS(✏) = a cosh r � a†e�2i� sinh r, (1.45b)

S†a†S(✏) = a† cosh r � ae�2i� sinh r, (1.45c)

S†(✏)(Y1

+ iY2

)S(✏) = Y1

e�r + iY2

er, (1.45d)

where Y1

and Y2

are the rotated quadratures such that

Y1

+ iY2

= (X1

+ iX2

)e�i�. (1.46)

When we apply the squeezing operator to the vacuum the uncertainty of one of the ro-tated quadratures gets attenuated while that of the other one is amplified. The intensityof this attenuation and amplification is given by r = |✏|, the squeeze factor. In general

1.3 Quasi probability distributions 15

Figure 1.1: Phase diagrams of di↵erent states. a) Phase diagram of vacuum, expectedvalue for both quadratures is 0, the uncertainty is the same for both and the minimumpossible. b) Phase diagram of a coherent state |↵i, that is, a vacuum state displaced inthe complex plane by ↵ = |↵|ei'. c) Phase diagram of a displaced squeezed state, |✏,↵i.Squeezing is ✏ = re2i�. In the rotated quadratures {Y

1

, Y2

} the uncertainty is minimum.

squeezed states can be also displaced, by first applying the squeezing operator to thevacuum and then the displacement operator, which gives a displaced squeezed state,

|↵, ✏i = D(↵)S(✏)|0i. (1.47)

Expectation values and variances for a displaced squeezed state are the following

hX1

+ iX2

i = hY1

+ iY2

iei� = 2↵, (1.48a)

�Y1

= e�r/p2, �Y

2

= er/p2, (1.48b)

hNi = |↵2|+ sinh2 r, (1.48c)

(�N)2 = |↵ cosh r � ↵⇤e2i� sinh r|2 + 2 cosh2 r sinh2 r. (1.48d)

Notice how one of the rotated quadratures is attenuated by e�r while the other one isamplified by er. Thus, for this rotated quadratures the product of their uncertainties stillis the minimum allowed by the uncertainty relation, as for vacuum and coherent states.In one of the rotated quadratures the uncertainty is lower than that of the vacuum,we call this ’squeezing below the vacuum’, and it is a fully quantum characteristic ofsqueezed states that cannot be explained classically.

1.3 Quasi probability distributions

Among all the possible ways of representing the state of the electromagnetic field, prob-ably the most intuitive one is the density matrix expanded in terms of number states,

⇢ =X

Cnm|mihn|, (1.49)


where Cnm are complex coe�cients and ⇢ is the density matrix of the field state. Thetrouble with this representation is that the number of coe�cients is infinite, which makesit useless specially for problems where the phase plays an important role. We can still geta useful description of fields with a random or unknown phase, taking just the values ofthe diagonal terms in Cnm, this is, Pn = Cnn, the probability of having n photons in thefield. Obviously, this probability distribution gives no information about the coherence,for instance, we can not distinguish between a coherent state and a mixture of numberstates with the same probability distribution.

Thus, for many problems it is much more convenient to expand the field in termsof coherent states. Actually, as we said in section 1.2.2, these form an over-completeset of states which are not orthogonal to one another. For that reason the expansionof a state in terms of them shows some unusual properties. As we will show, it allowsto represent the field with several di↵erent quasi-probability distributions, which whereintroduced by R. J. Glauber in 1968 [2]. These quasi-probability distributions give acomplete description of the field state and can be used to reconstruct its density matrix.Moreover, they can be used to compute expectation values of a wide range of operatorsas integrals similar to those of the classical probability theory. The aim of this sectionis to present this quasi-probability distributions and some of their properties.

1.3.1 The P representation

We will start with the P representation, which was introduced independently by Glauberand Sudarshan. The P representation is a diagonal representation of the density operatorin terms of coherent states,

⇢ =

Zd2↵P (↵)|↵ih↵| (1.50)

where d2↵ = dRe(↵)dIm(↵). In such representation the expectation value of an operatorA would be given by

hAi = Tr(⇢A) =X

n

hn|Z

d2↵P (↵)|↵ih↵|A|ni =Z

d2↵P (↵)X

n

h↵|A|nihn|↵i

=

Zd2↵P (↵)h↵|A|↵i =

Zd2↵P (↵)A(↵), (1.51)

where A(↵) = h↵|A|↵i. Notice how useful this representation is when operator A iswritten in normal order, this is, with all the annihilation operators to the right of thecreation ones. Since coherent states are eigenstates of annihilation operators, the expec-tation value for such kind of operators reduces to a simple c-number expression in thisrepresentation.

We can be tempted to interpret the P (↵) function as a probability distribution, how-ever, it is important to stress again that coherent states are not orthogonal. Although,as we have already noted, when |↵� ↵0|� 1, | ↵i and | ↵0i can be taken to be orthogo-nal. Thus, for P (↵) distributions that vary slowly over such large ranges of ↵ we can insome sense make a standard interpretation. Nevertheless, P (↵) is not positive definite,

1.3.2 The Q representation 17

which is a must for classical probability distributions. Moreover, fields for which theP (↵) distribution takes on negative values have no classical interpretation, which makesthe P representation a good candidate to detect quantumness. For all these unusualcharacteristics of the P (↵) distribution we cannot interpret it as a standard probabilitydistribution and we refer to it as a quasi-probability distribution.

At this point of the discussion may be interesting to introduce the characteristicfunction. In fact, there are three characteristic functions depending on how you defineit. On one hand, we have the symmetrically ordered characteristic function, which isnothing but the expectation value of the displacement operator,

�(⌘) = Tr[⇢e⌘a†�⌘⇤a]. (1.52)

And on the other hand, we have the normally and antinormally ordered characteristicfunctions, that are just variations of the symmetric one where the exponential is brokeninto a product of exponentials and the operators are either normally or anti-normallyordered,

�N (⌘) = Tr[⇢e⌘a†e�⌘⇤a], (1.53)

�A(⌘) = Tr[⇢e�⌘⇤ae⌘a†]. (1.54)

It is easy to see that the relation between all them is given by

�N (⌘) = �(⌘)e|⌘|2/2 = �A(⌘)e

|⌘|2 . (1.55)

If we now introduce the P representation of ⇢ in the expression for �N (↵) we get thefollowing

�N (⌘) =

Zh↵|e⌘a†e�⌘⇤a|↵iP (↵)d2↵ =

Ze⌘↵

⇤�⌘⇤↵P (↵)d2↵. (1.56)

This means that �N (⌘) is the two-dimensional Fourier transform of P (↵), and thus,P (↵) is the inverse Fourier transform of the normally ordered characteristic function,

P (↵) =1

⇡2

Ze↵⌘

⇤�↵⇤⌘�N (⌘)d2⌘. (1.57)

Therefore, the P representation will exist if and only if the Fourier transform of thenormally-ordered characteristic function exists.

1.3.2 The Q representation

We can define a quasi-probability distribution related to the antinormally ordered char-acteristic function, as the inverse Fourier transform of it,

Q(↵) =1

⇡2

Ze↵⌘

⇤�↵⇤⌘�A(⌘)d2⌘, (1.58)


which we call the Q representation. Thus, the antinormally ordered characteristic func-tion is written as the Fourier transform of the Q function,

�A(⌘) =

Ze⌘↵

⇤�⌘⇤↵Q(↵)d2↵. (1.59)

From Eq. (1.54) and Eq. (1.59) it is easy to see that the Q function can be written as

Q(↵) =h↵|⇢|↵i⇡

. (1.60)

Thus, unlike the P representation the Q representation is positive definite, that is,positive in all the space.

We can work out an expression of the Q distribution in terms of the P distribution,that turns out to be a Gaussian convolution of the P function,

Q(↵) =1

⇡2

ZP (�)e|↵��|2d2�. (1.61)

On the other hand, the Q representation can be very useful to compute expectationvalues of antinormally ordered operators. For instance,

ham(a†)ni = Tr[⇢am(a†)n] =X

n

hn|⇢am(a†)n|ni

=

Zd2↵

X

n

hn|⇢am|↵ih↵|(a†)n|ni

=

Zd2↵↵m↵⇤nQ(↵). (1.62)

where we have used the completeness relations in Eq. (1.29) and Eq. (1.43).

1.3.3 The Wigner representation

In the same way we can define another quasi-probability distribution called the Wignerfunction after physicist Eugene Wigner, who first introduced it into quantum mechanics.However, it was Glauber who put it into the context of quantum optics. The Wignerfunction is defined as the Fourier transform of the symmetrically ordered characteristicfunction, �(⌘),

W (↵) =1

⇡2

Ze⌘

⇤↵�⌘↵⇤�(⌘)d2⌘. (1.63)

Working out a little bit the relation between the P (↵) and W (↵) it can be shownthat

W (↵) =2

⇡

ZP (�)e�2|��↵|2d2�. (1.64)

That is, the Wigner function is a Gaussian convolution of the P function. Notice, howthe Gaussian in Eq. (1.61) is

p2 times wider than the Gaussian in this convolution,

1.4 Light-matter interaction 19

making the Q function positive definite. In the way it is defined the Wigner function isnormalized, Z

d2↵W (↵) = 1. (1.65)

In the same way that the P representation is useful to evaluate expectation valuesof normally ordered operators and the Q function for antinormally ordered ones, theWigner function can be used to evaluate expectation values of symmetrically orderedoperators. In this sense, the following expression holds,

Tr[⇢{(a†)nam}0

] =1

⇡

ZW (↵)(↵⇤)n↵md2↵, (1.66)

where {(a†)nam}0

represents a normally ordered operator with n creation operators andm annihilation ones. We will not give the demonstration of expression (1.66) in thistext, the interested reader might find it in Ref. [2]. In the same reference we can findalternative expressions for the Wigner function, for instance a very useful one is

W (↵) = 21X

n

(�1)nPn(�↵), (1.67)

where Pn(�↵) is the population of the photon field displaced in the complex amplitude(�↵),

Pn(�↵) = |Cn(�↵)|2, (1.68)

where

D(�↵)| i =1X

n

D(�↵)Cn|ni =1X

n

Cn(�↵)|ni, (1.69)

with | i any field state.As the P distribution, the Wigner function is not positive-definite and fields that

show negative values in this representation can be interpreted as quantum states oflight, since they have no classical counterpart. Unlike the P distribution, the Wignerfunction always exists and it is a uniformly continuous function of ↵.

1.4 Light-matter interaction

So far, we have presented a formalism in which the treatment of light quantum mechan-ically is natural. The next step is to describe the interaction of light with matter. Aswe already pointed in the introduction of this chapter, this interaction can be treatedin three di↵erent ways, classically, semi-classically and in a quantum manner, which bythe way is the only fully consistent one. In this section we aim to present this quantumversion of the interaction between light and matter. For that, as good physicist do, weare going to start with the most simple possible scenario. Matter will be represented bya two-level atom, and light by a single mode quantized optical field.


Figure 1.2: Up: Wigner function of vacuum state, | 0i. Center: Wigner function ofa displaced state, | ↵i, with ↵ = 5 + i5. Down: Wigner function of a squeezed satate,| ✏i, with ✏ = rei2� = 1

1.4.1 Two-level atom 21

1.4.1 Two-level atom

We usually refer to two level quantum systems as qubits, that is, a quantum system withjust two possible states, |0i or |1i. In the most general case a qubit state will be givenby

| i = c1

|0i+ c2

|1i, (1.70)

where of course|c1

|2 + |c2

|2 = 1. (1.71)

Actually, c1

and c2

can be parametrized without loss of generality as

c1

= cos(✓), (1.72a)

c2

= ei� sin(✓), (1.72b)

which preserves information on the phase between the amplitudes of the two states, andfulfills condition in Eq. (1.71). Parameters ✓ and � can also be understood as polarcoordinates. If so, they define a sphere of radius 1 in the R3 space. Any state of thequbit can be mapped into a point of this sphere. This geometrical representation of thestate of a qubit is called the Bloch representation and the unit sphere in which the statesare mapped is known as the Bloch sphere.

A two-level system can be the theoretical representation of many physical systems ora good approximation of them. Probably the most intuitive one is the internal degreesof freedom of a spin 1/2 particle. However, for our purposes we are interested in thetwo-level atoms. Of course, in practice these can be achieved only as an approximation,since real atoms have infinite possible levels which are impossible to fully cancel. In thisissue of light-matter interaction only those levels of the atom with an energy di↵erenceclose to the energy of the photons will take part in the interaction. Thus, if we useelectromagnetic fields with photon energies close to an atomic transition which di↵ers ina substantial manner from all the other possible transitions, then we can treat the atomas a two-level system, and thus as a qubit.

A two-level atom is mainly characterized by the energy di↵erence between its twoenergy levels, what we will call the splitting of the qubit. We shall call ground state,| gi, the state in which the system has lower energy (frequency) and excited state, | ei,the one with higher energy (frequency). It is not hard to see that the Hamiltonian for afree two-level atom is given by

Hq = ~!g|gihg|+ ~!e|eihe|, (1.73)

where !g and !e are the frequencies of the ground and excited states respectively.The physics of qubits is well described by the SU(2) algebra, of which Pauli matrices

are a representation. The reader might be familiar with these matrices from the physics ofspin 1/2, which in fact, is a two-level system. In terms of Pauli matrices the Hamiltonianmay now be written as

Hq =~!

0

2�z, (1.74)

where !0

= !e � !g and the Pauli matrix �z is diagonal in the {|gi, |ei} basis.


Figure 1.3: A generic Bloch Sphere

1.4.2 The quantum Rabi model

We now have all the pieces of the puzzle, we have quantized light and we have the two-level atom, which in fact is the simplest representation of quantized matter. We nowwant to see how these two interact. For that, we take a look at classical physics. Therethe coupling between light and charged particles, has its simplest version in the dipoleapproximation, which has a contribution in the Hamiltonian of the system of,

W = �~d · ~E. (1.75)

We now aim to model this for quantum optics. On one hand, we need an expressionfor an electric field of just one mode. Actually, in Eq. (1.17) we have already given anexpression of a quantized electric field, however this expression was in the Heisenbergpicture, if we instead move to the Schrodinger picture where operators are time inde-pendent and we select just one mode of the expansion, we are left with an expressionlike

E = C(a+ a†), (1.76)

where C is an appropriate constant with electric field dimensions. On the other hand,we need to represent the analog of the dipole momentum of a charge particle for a qubit.This will be given by expression

d = d(�+

+ ��) = d�x, (1.77)

where �+

= |eihg| and �� = |gihe|, are the raising and lowering operators of the qubit.Notice how this model of the dipole moment accounts for the two possible transitionsin the qubit. With these definitions we yield the interaction term of the Hamiltonianruling light-matter interaction,

HSint = ~g(�

+

+ ��)(a+ a†), (1.78)

1.4.3 The rotating-wave approximation 23

where S denotes Schrodinger picture and g is the coupling constant, which quantifiesthe strength of the coupling and has dimensions of frequency. The reader should notbe surprised of the structure of this interaction term, actually Eq. (1.78) is describingthe interaction between two harmonic oscillators, one of them of just two levels. Theelectric field stands for the harmonic oscillator while the qubit can be understood as atwo-level harmonic oscillator.

The complete Hamiltonian is given by

H = ~!a†a+ ~!0

�z + ~g(�+

+ ��)(a+ a†), (1.79)

where ! is the frequency of the field mode and !0

= !e � !g is the frequency di↵erencebetween the two energy levels of the qubit. The first two terms stand for the freeHamiltonians of the field and the qubit respectively, while the third one is the justintroduced interaction Hamiltonian.

This is known as the quantum Rabi model, after physicist Isidor Isaac Rabi, ande↵ectively describes light matter interaction for any coupling strength, g. However,although its simplicity analytical solutions have only been found recently (2011) by D.Braak.

1.4.3 The rotating-wave approximation

Let us now move to the interaction picture,

HIint = ei/~H0t(HS

int)e�i/~H0t

= ~g(�+

aei(!0�!)t + ��a†e�i(!0�!)t

+ �+

a†ei(!0+!)t + ��ae�i(!0+!)t), (1.80)

where superscript I denotes interaction picture. Remember that in the interaction picturethe Hamiltonian ruling the evolution of the system is the interaction Hamiltonian. TheHamiltonian in the interaction picture shows two clearly di↵erent type of terms, on onehand we have terms rotating with frequency (!

0

+ !), which are called the counter-rotating terms, and on the other hand we have terms rotating with frequency (!

0

� !),which we call the rotating terms. The rotating wave approximation (RWA) tells us thatwhen the coupling, g, is much lower than the rest of frequencies and if the qubit andthe field are in resonance or close to it, then we can neglect the counter-rotating terms.That is, if

g ⌧ !0

,!, (1.81a)

!0

� ! ⌧ !0

+ ! (1.81b)

thenHI

int = ~g(�+

aei(!0�!)t + ��a†e�i(!0�!)t), (1.82)

which back to the Schrodinger picture is

HS = ~!a†a+ ~!0

�z + ~g(�+

a+ ��a†). (1.83)


Intuitively one can understand the RWA thinking that in the Interaction picturethe counter-rotating terms rotate much more faster than the rotating ones, thus in acharacteristic time lapse these average to zero while the rotating terms change slowlyand need to be taken into account. More rigorously one can derive the Dyson series ofthe evolution operator of the Rabi model Hamiltonian and observe how counter-rotatingterms are negligible under conditions in Eqs. (1.81). In the same way numerical analysisjustifies the RWA approximation and also experimental results give validity to it.

1.4.4 The Jaynes-Cummings model

Equation (1.83) is known as the Jaynes-Cummings Hamiltonian, and unlike the Quan-tum Rabi model its analytical solutions are very well known and have an easy intuitiveinterpretation. One of its main characteristics is that it conserves the number of excita-tions of the system. For instance, if the qubit gets excited then the field losses a photonand vice-versa. This can be easily demonstrated first defining the excitation numberoperator as

Ne = a†a+ |eihe|, (1.84)

and then observing that it commutes with the Hamiltonian

[Ne, H] = 0. (1.85)

On the other hand, it can be shown that the Jaynes-Cummings Hamiltonian has eigen-states

|+, ni = sin ✓n|e, ni+ cos ✓n|g, n+ 1i (1.86a)

|�, ni = cos ✓n|e, ni � sin ✓n|g, n+ 1i, (1.86b)

where

sin ✓n =2gpn+ 1p

(�n ��)2 + 4g2(n+ 1), (1.87a)

cos ✓n =�n ��p

(�n ��)2 + 4g2(n+ 1), (1.87b)

�n =p�2 + 4g2(n+ 1), (1.87c)

with � = !0

� !, the detuning between the qubit and the field. In the same wayeigenvalues can be shown to be

E±n = ~!0

(n+ 1/2)± ~�n

2. (1.88)

These expressions are much simplified for the resonant case, � = 0,

|+, ni = 1p2(|e, ni+ |g, n+ 1i) (1.89a)

|�, ni = 1p2(|e, ni � |g, n+ 1i) (1.89b)

E± = ~!0

(n+ 1/2)± ~gpn+ 1. (1.89c)

1.4.5 Ultrastrong and deep strong coupling regimes 25

Using these we can calculate, for instance, how does state |e, ni evolve in the resonantcase,

|e, ni ! cos(gpn+ 1t)|e, ni+ sin(g

pn+ 1t)|g, n+ 1i. (1.90)

As we can see, the system oscillates between states |e, ni and |g, n + 1i with frequency⌦n = g

pn+ 1. This are known as Rabi oscillations, and frequency ⌦n is usually called

the quantum Rabi frequency. These oscillations can be understood as the field and thequbit interchanging an excitation every half period. Rabi oscillations can be representedplotting the probability of finding the qubit excited vs time. The probability of findingthe qubit excited if the evolution starts in state |e, ni is

Pe(t) = cos2(gpn+ 1t). (1.91)

The probability of finding the qubit in its initial state is 0 every half period, and it iswhat we call the collapse of the probability. On the other hand, every whole period theprobability reaches 1, what we call a revival. If the resonance is not perfect, then thisrevivals will not go up to 1 , that is, revivals are only perfect in the resonant case.

Instead of starting with the field in a number state if we start in a superpositionof number states each of them will interact with the qubit in the way described above,that is, each of the number states will interchange an excitation with the qubit withfrequency ⌦n = g

pn+ 1. Notice how the frequency of each level depends on the square

root of the level number and thus are incommensurable. This means that no matter howmuch time we let the system evolve it will never come back to the initial state unlessthere is only one level of the field interacting with the qubit. In general, for the resonantcase but starting in an initial state such as

Pn cn | e, ni, that is, with the qubit excited

and the field in a superposition of number states, the probability of finding the qubitexcited is given by

Pe(t) = 1/2 + 1/2X

n

|cn|2 cos(2gpn+ 1t). (1.92)

This probability also has some collapse and revivals as shown in Fig. (1.4).The Jaynes-Cummings model is very interesting not only for the simplicity of its

solutions and the easy interpretation it has, but also because it correctly describes thephysics of many processes. For instance, atoms in nature couple to light very weakly,thus allowing us to describe them with the Jaynes-Cummings model when interactingwith light close to resonance. Also the dynamics of cavity QED, trapped ions and severalsetups in mesoscopic physics are accurately described by this model.

1.4.5 Ultrastrong and deep strong coupling regimes

The Jaynes-Cummings model works in the so called strong coupling regime, where thecoupling constant, g, is comparable to all decoherence rates, but still small compared tothe rest of frequencies, so the RWA is yet applicable.

For long time this regime of the coupling was enough to describe all known physicsrelated to light-matter interactions. However, nowadays, semiconductor microcavities


Figure 1.4: Up: Probability of finding the qubit excited for the Jaynes-Cummingsmodel in the resonant case, starting with the qubit excited and the field in a numberstate n. Down: Probability of finding the qubit excited for the Jaynes-Cummingsmodel in the resonant case, starting the evolution with the qubit excited and the fieldin a coherent state with a mean number of photons of 5.

1.4.5 Ultrastrong and deep strong coupling regimes 27

[3] and circuit QED systems [4, 5] have achieved coupling regimes where g/! & 0.1,what we call the ultrastrong coupling (USC) regime. In the USC regime, the RWA isno longer applicable since the coupling constant is not small enough compared to therest of frequencies, therefore the full quantum Rabi model is needed to describe thephysics. Moreover, proposals of quantum simulations that allow for simulated couplingconstants of g/! & 1 already exist [30]. This regime of the coupling is called thedeep strong coupling (DSC) regime and shows physics di↵erent to those of the strongcoupling regime [6]. For instance, in the DSC regime the Hilbert space is broken intotwo subspaces, one of them formed by states of positive parity (p=+1) and the otherone by states of negative one (p=-1). States in each subspace can be ordered in twoseparate chains,

|g, 0i $ |e, 1i $ |g, 2i $ |e, 3i $ ... (p = +1),

|e, 0i $ |g, 1i $ |e, 2i $ |g, 3i $ ... (p = �1), (1.93)

where each state is related to the state right ahead or behind by either rotating orcounter-rotating terms of the Rabi Hamiltonian. For example, state |e, 2i is related to|g, 3i via the rotating term ��a†, while the transition |g, 1i |e, 2i is induced by thecounter-rotating term ��a. Notice how transitions from states of one chain to statesof the other are impossible and how, within a chain, each state has only two possibletransitions, except states |g, 0i and |e, 0i which allow only for one transition. When wego back to the strong coupling regime, where the RWA is applicable, the transitionsinduced by the rotating terms of the Hamiltonian are prohibited, breaking the chainsinto the well known JC doublets {|g, n+ 1i, |e, ni}.

What is important to understand is that in both of these regimes, USC and DSC,the RWA breaks down and the JC model is not valid. Thus, in these regimes physics isagain ruled by the whole quantum Rabi Hamiltonian in Eq. (1.79).

In order to prepare the reader for the incoming chapters, we point out here thesimilarity of the quantum Rabi Hamiltonian with the Dirac equation in the appropriaterepresentation,

HD = mc2�z + cp�y, (1.94)

which rules the dynamics of a free quantum relativistic particle. The reader should keepin mind that p = i(a� a†)/

p2 and �y = i(�� +)/2. Thus,

HD = mc2�z +cp2(a�+ + a†�� a†�+ � a��), (1.95)

which up to the coe�cients and a phase in the counter-rotating terms is the Rabi Hamil-tonian except for the field energy term, ~!a†a, of which we can get rid o↵ in the ap-propriate rotating frame. One of the central aims of this work is to propose a quantumsimulation of the Rabi Hamiltonian where these coe�cients and phases are tunable andthus where the Dirac Hamiltonian can be simulated.

Chapter 2

Relativistic Quantum Mechanics

Historically, special relativity and quantum mechanics have been developed indepen-dently and are related to di↵erent physical phenomena. The two theories being suc-cessful in their domains, it would be intellectually unsatisfactory not to find a theorythat is consistent with both of them. This has occurred plenty of times in the historyof physics, say electricity and magnetism, thermodynamics and statistical mechanics orinertial and gravitational forces. More importantly, the existence of phenomena whichcannot be understood within a purely nonrelativistic quantum mechanical framework (e.g. the fine structure of the Hydrogen atom) or the solid theoretical reasons to expectnew phenomena to occur at relativistic velocities, are enough motivation to try a fusionbetween quantum mechanics and relativity.

In 1926, Erwin Schrodinger finds the very well known equation that takes his name,which is a nonrelativistic equation and the basis of nonrelativistic quantum mechanics.In the same year, Klein proposed a relativistic correction of this equation, based onthe quantization of the relativistic energy-momentum invariant, E2 = p2c2 + m2c4.The quantization procedure, consisting in substitutions E ! i@t and p⌫ ! i@⌫ with⌫ = x, y, z, results in the Klein-Gordon equation � @2

@t2 = �r2 +m2 . Apparently

Schrodinger himself tried this equation before the Schrodinger equation, but discardedit because of the various unphysical consequences that seems to have. On one hand, asthe equation is second order in time, we need initial conditions not only for (t = 0),but also for @t (t = 0), adding an extra constraint absent in the Schrodinger equation.On the other hand, the density ⇢ = †@t � @t †, is not a positive definite quantityand thus it cannot represent a probability. Moreover, the equation has negative energysolutions which had no physical interpretation at that time, and even if we discard them,the evolution in time for any of the not discarded states could have projections on theunwanted eigenstates.

All these uncomfortable consequences of the Klein-Gordon equation took Dirac tokeep looking for a better relativistic version of the Schrodinger equation. In 1928, hefound the now known as Dirac equation to describe fermions [14]. This equation, unlikethe Schrodinger equation, is first order in time, however, still has negative energy eigen-states. Dirac himself gave a solution to this problem. He proposed that all the negative

30 Chapter 2 Relativistic Quantum Mechanics

energy states were already filled with electrons. Since electrons are fermions, two ofthem cannot be in the same state, thus the negative energy states being occupied areinaccessible for electrons with positive energy. This idea of having all negative energystates filled with electrons is usually called the Dirac sea of electrons. Moreover, onecould excite one of the electrons in the Dirac sea making it have positive energy andleaving a hole in the sea. This hole which is merely the absence of a negative charge,could be interpreted as a positively charged particle. At the time this was proposedelectrons and protons were the only known subatomic particles, so Dirac interpretedthis hole as a proton. However, this particle should have the mass of an electron andwe know that protons are much heavier. In 1932, Carl D. Anderson found the positronwhile he was looking for cosmic rays in a cloud chamber, a particle of positive chargeand the mass of an electron. Dirac’s equation had predicted the existence of the firstknown antiparticle. In 1933 he won the Nobel prize.

Although the Dirac equation anticipated the existence of antimatter, a simultaneousdescription of particles and antiparticles needs an extension of quantummechanics knownas quantum field theories. The Dirac equation interpreted as a single particle evolutionequation still is an equation with no clear range of validity and with not very wellunderstood physics at its limits. However, many of the problems posed by the Klein-Gordon equation or by the Dirac equation are successfully solved when interpretingthese equations as field equations instead of as particles. Nowadays, the entire realmof quantum theories includes nonrelativistic quantum mechanics (NRQM), reativisticquantum mechanics (RQM) and quantum field theories (QFT). NRQM and RQM areparticle theories, that is, the solutions to their equations are interpreted as particles,while QFT is a field theory. On the other hand, only the last two are relativisticallycovariant.

In this chapter we mainly focus on the Dirac equation as single particle evolutionequation in the field of RQM and the various phenomena related to it. The Diracequation, despite is many times rejected as a single particle wave equation and consideredmerely as a milestone towards quantum field theories, is undoubtedly a cornerstone inthe history of physics. In a safe range of validity, at velocities high enough to appreciaterelativistic kinematical e↵ects but where the energies are su�ciently small to ensure thatpair creation is impossible, the Dirac equation is still a very interesting description forthe evolution of a single particle. Indeed, it is for example the first equation to introducefrom first principles the spin and the spin-orbit coupling, the equation that predictedthe existence of antimatter and the equation that accounted for the fine structure of theHydrogen atom spectrum. Even nowadays still happens to appear in new scenarios asfor example in graphene, where the two dimensional version of it accurately describesthe physics of quasi-electrons behaving as relativistic particles in two dimensions.

2.1 The Dirac equation

In this section we are going to talk about the Dirac equation itself, we are going to deriveit and to show some of the di↵erent representations it has.

2.1.1 Derivation of the Dirac equation 31

2.1.1 Derivation of the Dirac equation

To start we will derive the Dirac equation in the same way Dirac did it in 1928. Forthat we are going to follow Ref. [7]. As was pointed out in the introduction to thischapter Klein derived his relativistic equation by replacing the classical quantities in therelativistic energy-momentum invariant,

E2 = p2c2 +m2c4, (2.1)

by the appropriate operators. However, this led him to an equation that was secondorder in time derivative. To avoid this Dirac tried to linearize the expression in Eq.(2.1) before quantizing it. He wrote an expression like

E = c3X

i=1

↵ipi + �mc2 = c↵ · p+ �mc2 (2.2)

where ↵ = (↵1

,↵2

,↵3

) and p = (p1

, p2

, p3

). Our goal now is to determine ↵ and �.Comparing the square of Eq.(2.2) with Eq. (2.1), we immediately find the followingrelations

↵i↵k + ↵k↵i = 2�ik1, i, k = 1, 2, 3, (2.3a)

↵i� + �↵i = 0, i = 1, 2, 3, (2.3b)

�2 = 1, (2.3c)

where �ik denotes the Kronecker delta, that is, �ik = 1 if i = k and �ik = 0 if i 6= k. Thisrelations are telling us that ↵ and � have to be anticommuting quantities, which fromordinary quantum mechanics we know are more naturally represented by n⇥n matrices.Actually, expressions 1 and 0 in Eqs. (2.3) are the n-dimensional unit and zero matrices.On the other hand, ↵ and � have to be Hermitian because E is self-adjoint. The readermight use Eqs. (2.3) to verify the following

tr ↵i = tr �2↵i = �tr �↵i� = �tr ↵i�� = �tr ↵i ! tr↵i = 0, (2.4)

where tr denotes the trace of a matrix. Moreover, from Eq. (2.3a) its easy to see that↵2

i = 1, which makes its eigenvalues either 1 or -1. If the eigenvalues have to be 1 or -1and the trace 0, then the dimension of the matrix, n, has to be an even number. Thisis clear in the diagonal representation of the matrix where the elements in the diagonalare the eigenvalues of it. For n=2 there is no enough room for 4 anticommuting linearlyindependent matrices. This can be seen if for example we take the Pauli matrices,

�1

=

✓0 11 0

◆, �

2

=

✓0 �ii 0

◆, �

3

=

✓1 00 �1

◆, (2.5)

which are anticommuting and together with the unit matrix, 1, form a basis of the 2⇥ 2Hermitian matrices space. For n=4 the following matrices satisfy all the requirementsof Eqs. (2.3),

� =

✓1 00 �1

◆, ↵i =

✓0 �i�i 0

◆, i = 1, 2, 3. (2.6)


This representation of the matrices was introduced by Dirac and it is known as the”Standard representation”.

So, we now have a linear expression for the energy-momentum invariant, the pricewe have paid is that the expression is now described in a 4⇥4 dimensional space insteadof the c-number expression of Eq. (2.1). We can proceed now to the quantization ofthis new representation of the energy-momentum invariant. As usual the quantizationis made by the following substitutions

E ! i~ @@t

, p! �i~r. (2.7)

This takes Eq. (2.2) to the Dirac equation,

i~ @@t (t,x) = H

0

(t,x), (2.8)

where

H0

= �i~c↵ · r+ �mc2 =

✓mc21 �i~c� · r

�i~c� · r �mc21

◆, (2.9)

with ↵ = (↵1

,↵2

,↵3

) and � = (�1

,�2

,�3

) triplets of matrices. On the other hand, H0

acts on vector-valued wavefunctions

(t,x) =

0

B@ 1

(t,x)...

4

(t,x)

1

CA , (2.10)

called spinors and which are elements of a 4 dimensional complex vector space, C4,not be confused with regular vectors since they do not behave the same under spatialtransformations.

Notice how, for a massless particle, the Dirac equation reduces to

i~ @@t (t,x) = �i~c↵ · r (t,x), (2.11)

In this case we need just 3 linearly independent anticommuting matrices, a conditionthat can be fulfilled in a 2⇥ 2 Hermitian matrices space by the Pauli matrices,

i~ @@t (t,x) = �i~c� · r (t,x). (2.12)

Equation (2.12) is known as the Weyl equation and obviously describes massless rela-tivistic quantum particles.

In the same way, when dealing with relativistic particles of 1 or 2 dimensions we areleft with 3 or less linearly independent anticommuting matrices which can be representedin a 2 ⇥ 2 dimensional space, where the most common representation are the Paulimatrices.

2.1.2 Representations of the Dirac equation 33

2.1.2 Representations of the Dirac equation

In general, one can work with Dirac matrices without referring to any particular rep-resentation, just relaying on their properties. However, many times it is convenient torepresent these matrices to facilitate the calculations. Actually, there is plenty of waysof doing it, and depending on the problem we are working on we might prefer to useone or another. Here we show some of the most commonly used representations of Diracmatrices. From now on, 1 represents a 2 ⇥ 2 unit matrix, i = i1, and the 0’s representall the needed 0’s to make any matrix a square matrix.

On one hand, we have the already presented Standard representation some timesalso called the Dirac-Pauli representation,

� =

✓1 00 �1

◆, ↵ =

✓0 �

� 0

◆. (2.13)

On the other hand, we have the Supersymmetric representation where Dirac matricestake the form

�s =

✓0 �ii 0

◆, ↵s =

✓0 �

� 0

◆. (2.14)

The unitary matrix

Ts =1p2

✓1 ii 1

◆(2.15)

relates the Standard and the Supersymmetric representations, via the relation �s =Ts�T�1

s .We also define the Weyl (or spinor) representation as

�w =

✓0 11 0

◆, ↵w =

✓� 00 ��

◆(2.16)

which is related to the Standard representation by

Tw =1p2

✓1 11 �1

◆. (2.17)

And finally we have the Majorana representation,

�m =

✓0 i�i 0

◆, (↵m)

1,3 =

✓�1,3 00 ��

1,3

◆, (↵m)

2

=

✓0 11 0

◆. (2.18)

In this case the relation to the Standard representation is given by

Tm =1p2

✓1+ i�

2

1� i�2

�i+ �2

i+ �2

◆. (2.19)

The same ↵-matrices, together with

�m =

✓�2

00 �

2

◆(2.20)


is also known as the Majorana representation and corresponds to Majorana’s originalchoice.

The reader might also find in the wide literature on the Dirac equation di↵erent waysof writing it. For instance, the Dirac equation in natural units (c = ~ = 1) might alsobe written with the �-matrices (also known as Dirac matrices) which are a matricialrepresentation of Cli↵ord algebra,

(i�µ@µ �m) = 0, (2.21)

where �µ = (�0, �1, �2, �3), represents the set of four �-matrices, not to be confused witha vector. Here we have used Einstein’s notation where �µ@µ = ��0@t+�1@x+�2@y+�[email protected] (2.21) is easily achieved if one takes Eq. (2.8) multiplies at both sides with �and defines the �-matrices as

�µ = �(1,↵), (2.22)

where 1 is a four dimensional unit matrix. With this definition we can write the �-matrices in any of the representations given above. For instance, in Dirac’s representationthe �-matrices look like

�0 =

✓1 00 �1

◆, �k =

✓0 �k��k 0

◆, k = 1, 2, 3. (2.23)

2.1.3 Fourier space and spectral subspaces of the Dirac operator

As we have already pointed out, the Dirac operator H0

in Eq. (2.9), acts on C4-valuedfunctions of x 2 R3, hence it lives in the Hilbert space

H = L2(R3)� L2(R3)� L2(R3)� L2(R3) = L2(R3)4 = L2(R3)⌦ C4. (2.24)

The Dirac operator in the way it is written in Eq. (2.9) is a matrix di↵erentialoperator. Its Fourier transform is represented by a matrix multiplication operator definedin the momentum Hilbert space, L2(R3, d3p)4, as

(FH0

F�1)(p) = h(p) =

✓mc21 c� · pc� · p �mc21

◆, (2.25)

where the Fourier transformation is defined as

(F k)(p) =1

(2⇡)3/2

Z

R3e�ip·x k(x)d

3x, k = 1, 2, 3, 4. (2.26)

From its representation in the momentum space the Dirac operator is diagonalizedvia the unitary transformation

u(p) =(mc2 + �(p))1+ �c↵ · pp

2�(p)(mc2 + �(p))= a

+

(p)1+ a�(p)�↵ · pp

, (2.27)

2.1.3 Fourier space and spectral subspaces of the Dirac operator 35

where

a±(p) =1p2

p1±mc2/�(p) and �(p) =

pc2p2 +m2c4 > 0 (2.28)

with p =| p |. So one can write

u(p)h(p)u(p)�1 = ��(p), (2.29)

and confirm that the matrix gets diagonalized with eigenvalues

�1

(p) = �2

(p) = ��3

(p) = ��4

(p) = �(p). (2.30)

Hence, we define the unitary transformation

W = uF (2.31)

as the transformation that converts the Dirac operator H0

into a diagonal matrix inmomentum space,

(WH0

W�1)(p) = ��(p). (2.32)

On the other hand, the Hilbert spaceWL2(R3)4 where the Dirac operator is diagonal,can be decomposed into two orthogonal subspaces, Hpos composed by all the spinors ofpositive energies and Hneg by the spinors of negative energy. Because these subspacesare orthogonal one can write

H = Hpos � Hneg. (2.33)

Moreover, from Eq. (2.32) we see that the two upper components of wavefunctionscorrespond to the positive energies while the two lower components belong to the negativeones. Thus, vectors of the Hpos subspace are of the type

pos = W�1

1

2(1+ �)W , 2 L2(R3, d3x), (2.34)

while vectors

neg = W�1

1

2(1� �)W , 2 L2(R3, d3x) (2.35)

build up the Hneg subspace. These last two expressions are easily interpreted. First webring the wavefunction to a picture in which the Dirac operator is diagonal, by thetransformationW , then we apply 1

2

(1+�) (12

(1��)), which selects the two upper (lower)components of the vector and finally we take the wavefunction back to the original picturevia the inverse transformation W�1. Thus, the positive/negative projection operatorsare

P posneg

= W�1

1

2(1+ �)W =

1

2(1+H

0

| H0

|�1), (2.36)

where

| H0

|=qH

0

2 =p�c2�+m2c41 (2.37)

with � = @2/@x1

2 + @2/@x2

2 + @2/@x3

2. The Fourier transform of the square rootoperator, | H

0

|, is the multiplication operatorpc2p2 +m2c4 defined in space L2(R3, d3p)


One can see how any wavefunction projected with the positive (negative) energyprojection operator has positive (negative) energy, for instance

( pos, H0

pos)(p) = (W�1�+

,W�1�(p)�+

) = (�+

,�(p)�+

) > 0 (2.38)

where �± = 1

2

(1± �)W .

2.1.4 The Foldy-Wouthuysen transformation

The Foldy-Wouthuysen (FW) transformation (after Lesley L. Foldy and Siegfried A.Wouthuysen) [31] consists in the following transformation

UFW = e�↵·p✓ = cos ✓ + �↵ · p sin ✓, (2.39)

which depends on the parameter ✓. When we apply it to the Dirac operator

H 00

= UFWH0

UFW�1 = (cos ✓ + �↵ · p)(c↵ · p+ �mc2)(cos ✓ � �↵ · p). (2.40)

From the commutation relations of the Dirac matrices one can write

H 00

= (↵ · p+ �m)(cos ✓ � �↵ · p sin ✓)2 = (↵ · p+ �m)e�2�(↵·p+�m)✓ (2.41)

= (c↵ · p+ �mc2) = (c↵ · p+ �mc2)(cos 2✓ � �↵ · p sin 2✓) (2.42)

which can we factor out as

H 00

= ↵ · p (cos 2✓ � m

psin 2✓) + �(mc2 cos 2✓ + pc sin 2✓). (2.43)

This is not diagonal in general, but only certain values of the parameter ✓ result in adiagonal transformation. If we assume that we are in the Standard representation, where� is diagonal and ↵-matrices are not, we can choose tan ✓ = p/m so that the term with↵ · p in Eq. (2.43) vanishes. This is called Newton-Wigner representation.TheodoreDuddell Newton and Eugene Wigner found a diagonal representation of the Dirac op-erator in 1949. The FW transformation in the specific case of tan ✓ = p/m coincideswith transformation W in Eq. (2.31), thus being the transformation that diagonalizesthe Dirac operator.

Using elementary trigonometric relations, one will easily work out that

H 00

= �pm2c4 + p2c2, (2.44)

which is the same as Eq. (2.32)

Notice how the diagonal representation of the Dirac equation is equivalent to twodouble-component square-root Klein-Gordon equations.

2.1.5 1+1 Dirac equation, its nonrelativistic limit and squeezing 37

2.1.5 1+1 Dirac equation, its nonrelativistic limit and squeezing

The 1+1 dimensional Dirac equation can be defined in a 2⇥2 dimensional Hilbert spaceof which the Pauli matrices are a basis. The equation looks like

i~d dt

= (mc2�z + cp�y) , (2.45)

with �i for i = x, y, z Pauli matrices. Using arguments of quantum optics one can easilyfind the nonrelativistic limit of this equation. We will use the commutator theorem.This theorem assumes that one can write in some interaction picture a time-dependentHamiltonian as

HI = ~X

j

[Aj†ei�jt +Aje

�i�jt], (2.46)

where Aj† is a time-independent function of system operators, say Aj

† = gjaj2bj† + ...

with gj coupling strengths. Then it states that under conditions | �j |� gj , 8j and| �j ± �k |, 8{j 6= k} the Hamiltonian in Eq. (2.46) is e↵ectively equivalent to

Heff = ~X

j

[Aj†, Aj ]

�j. (2.47)

This can be demonstrated using the Dyson series of the evolution operator related toHamiltonian in Eq. (2.46), which can be approximated as the Dyson series of the evo-lution operator of H

e↵

.Back to our 1+1 dimensional Dirac equation we can identify

H0

= mc2�z, (2.48)

to write the Dirac operator in the interaction picture for H0

,

HID = eiH0t(HD �H

0

)e�iH0t. (2.49)

We recall here that

�y = �i|eihg|+ i|gihe|, (2.50)

�z = |eihe|� |gihg|, (2.51)

where |ei and |gi are the eigenvalues of �z. With this in mind the reader will easilycheck out that

HID = c�yp(�iei2mc2t|eihg|+ ie�i2mc2t|gihe|). (2.52)

Now following the commutator theorem, under condition 2mc2 � chpi we can e↵ectivelyapproximate the Hamiltonian as

He↵

= c2p21

2mc2[�i�+, i��] = p2

2m�z, (2.53)


where �+ = |eihg| and �� = |gihe|. Equation (2.53) gives the e↵ective Hamiltonian forthe nonrelativistic limit of the Dirac operator. That is, the free Schrodinger Hamiltonianwith the spin operator �z, which is reminiscent of the original Dirac operator.

The evolution of such Hamiltonian is given by

U = eip2�zt/2m = ei(�a2�(a†)2+aa†+a†a)�zt/2m, (2.54)

where we have used p = i(a† � a). Although it is not exactly the same because ithas some more components, the evolution operator looks very similar to the squeezingoperator introduced in Eq.(1.44) with squeezing factor ✏ = it/m. One can draw theuncertainties of momentum and position in a phase-space representation, in a similarway we did in chapter 1 with the quadratures of the field states. If the particle is in aneigenstate of �z then one will see a ’cloud’ of uncertainties squeeze as time increases. Ifthe particle instead is in an eigenstate of �y then one will see a squeezed Schrodingercat, since eigenstates of �y are superpositions of eigenstates of �z which will squeeze indirections perpendicular to each other. Notice that because we are in the limit where2mc2 � chpi, the squeezing will not be very significant.

2.2 Quantum relativistic e↵ects

In this section we are going to present two characteristic quantum relativistic e↵ects,the Zitterbewegung and the Klein paradox. Both of them have no nonrelativistic coun-terpart, and were very anti-intuitive at the time they were discovered and even now,generating much debate around them.

2.2.1 Zitterbewegung

Zitterbewegung is a German word meaning trembling motion. It makes reference to thepeculiar oscillatory motion of free particles described by the Dirac equation. The e↵ectwas first noticed by Schrodinger in 1930 [15], and consists in a fluctuation of the positionof the particle around its mean value. This fluctuation occurs with a frequency of 2mc2/~and is usually interpreted as an interference between the positive and negative energystates. Zitterbewegung has never been observed in real particles, however, quantumsimulations with trapped ions have been able to simulate this behaviour [8].

The velocity operator

One of the most delicate parts of a quantum theory, but necessary for its interpretation,is the selection of operators representing observables. Here we are interested in the oper-ator representing the position observable. By analogy with the nonrelativistic quantummechanics, we select the multiplication operator x = (x

1

, x2

, x3

), that we will call the”standard position operator”. The expected value of each coordinate of the position isgiven by

2.2.1 Zitterbewegung 39

D(xi) =

Z4X

k=1

| (xi k)(x) |2 d3x, i = 1, 2, 3, (2.55)

where

(xi )(x) =

0

B@xi 1

(x)...

xi 4

(x)

1

CA . (2.56)

In a similar fashion we define the ”standard velocity operator” as the time derivativeof the standard position operator,

d

dtx(t) = i[H

0

,x(t)] = eiH0ti[H0

,x]e�iH0t = eiH0tc↵e�iH0t = c↵(t). (2.57)

Notice how eigenvalues of c↵(t), which is unitarily equivalent to c↵, are c and �c. Thismeans that, the velocity operator has a discrete spectrum, with the surprising resultthat any measurement at any time gives only two possible values, c or -c.

Inspired by the classical relativistic kinematic relation v = c2p/E, we define the”classical velocity operator” as c2p/H

0

. This operator commutes with H0

, which meansthat is a constant of motion, and has a continuous spectrum in the interval [�c, c].

Back to the standard velocity operator, one can calculate its time derivative,

d

dtc↵(t) = eiH0ti[H

0

, c↵]e�iH0t = 2iH0

F(t), (2.58)

whereF = c↵� c2pH�1

0

. (2.59)

Because F anticommutes with H0

,

FH0

= �H0

F, (2.60)

it is straightforward to check that the time dependence of F(t) is given by

F(t) = e2iH0tF,

Z t

0

F(t)dt =1

2iH0

e2iH0tF. (2.61)

One can now integrate Eq. (2.58) and get an expression for the time dependence ofthe standard velocity operator,

c↵(t) = c2pH0

�1 + F(t), (2.62)

where we have used c↵(0) = c↵. Hence, the standard velocity oscillates around theconserved mean value given by the classical velocity , c2pH

0

�1. We call this phenomenonZitterbewegung.


Time Evolution of the Standard Position operator

The time evolution of the standard position operator is given by the integral of Eq.(2.57),

x(t) = x

Z t

0

c↵(t)dt, (2.63)

where we have chosen x(0) = x. Using Eq. (2.61) and Eq. (2.62) it is easy to yield

x(t) = x+ c2pH�1

0

t+ Z(t), (2.64)

with

Z(t) =1

2iH0

(e2iH0 � 1)F. (2.65)

In this expression the first two terms represent a linear evolution of the position, as innonrelativistic quantum mechanics or in classical physics for a free particle. Howeverthe third term is an oscillating term, with no counterpart in nonrelativistic quantummechanics. This term appears from the fact that the standard position operator and theDirac operator do not commute, and it is manifestation of the Zitterbewegung.

The reader can check that operator F in this third term of the position operatoranticommutes with the positive/negative projection operators introduced in the previoussection, P pos

neg, i. e. FP pos

neg= Pneg

posF . This means that F maps the positive (negative)

energy subspace into the negative (positive) one. As a consequence, PposFPpos = 0.This is particularly interesting because it can be shown that the expectation value ofthe third term of the position operator is 0 for positive or negative energy states,

( pos(t),x(t) pos(t)) = (Ppos ,x(t)Ppos ) = ( , Pposx(t)Ppos ) (2.66)

= ( , {PposxPpos + Pposc2pH�1

0

tPpos + Ppos1

2iH0

(e2iH0 � 1)FPpos} ) (2.67)

= ( , {PposxPpos + Pposc2pH�1

0

tPpos} ), (2.68)

which obviously shows no Zitterbewegung. The same holds for a negative energy state.Hence, because Zitterbewegung appears only on states of particles with negative andpositive components and not in those with just one of them, it is usually interpreted asan interference phenomenon between the negative and positive energy components of aparticle state.

We will now list some of the more characteristic properties of Zitterbewegung. Wewill not give here theoretical proofs for this properties, but these are easily checked outwith numerical simulations.

1. It disappears as time tends to infinity.

2. The frequency of Zitterbewegung grows linearly with increasing mass.

3. Amplitude of Zitterbewegung decreases as mass is increased.

2.2.1 Zitterbewegung 41

4. Positive (negative) energy states do not show Zitterbewegung.

5. For particles with a higher average momentum Zitterbewegung disappears faster.

Concepts of frequency and amplitude are just approximate for Zitterbewegung. Fromproperties 2 and 3 it is clear that for both limits of the mass, massless particles and veryheavy ones, Zitterbewegung disappears.

Alternative Position operators

Although it is mathematically very well understood the Zitterbewegung is a very un-comfortable feature of a quantum relativistic particle. There is no reason for a freeparticle to violate Newton’s second law. Since it is an interpretation problem it couldbe that the election of the standard position operator as the position of the particle wasincorrect. Actually, it exists another position operator which is also canonical conjugateof the momentum, but unlike the standard position operator does not mix the positiveand negative energy subspaces, thus showing no Zitterbewegung. Moreover, the timederivative of this position operator is the classical and well behaved velocity operator,c2pH

0

�1. This is called the ”Newton-Wigner position operator” and it is nothing butthe standard position operator under the Foldy-Wouthuysen transformation,

xNW = UFW�1xUFW . (2.69)

It is easy to see how this position operator commutes with the positive/negativeenergy projector and thus does not mix subspaces Hpos and Hneg,

[xNW , P posneg

] = UFW�1[x, 1/2(1± �)]UFW = 0. (2.70)

In a similar fashion, the time dependence of this position operator can be shown tobe

xNW (t) = xNW + c2pH0

�1t, (2.71)

where

xNW = x� �

2i�(c↵� c2

�(�+mc2)c(↵ · p)p)� c2

�(�+mc2)S ^ p. (2.72)

The Newton-Wigner operator is sometimes also called the Foldy-Wouthuysen op-erator. Another way of thinking of it, is that the standard position operator showsno Zitterbewegung in the FW picture, that is, if we transform the Dirac operator tothe FW picture and measure the untransformed standard position operator we will seeno Zitterbewegung. Or in other words, the FW transformation of the Newton-Wigneroperator is the standard position operator.


2.2.2 Klein paradox

The Klein paradox is a peculiar result of the scattering problem for particles evolvingaccording to Dirac’s equation. It was found in 1929 by physicist Oscar Klein [17],and consists in the fact that quantum relativistic particles ruled by the Dirac equationtunnel potential barriers even if their energy is lower than that of the barrier. Moreoveras the energy of the barrier increases the transmission is more probable. Indeed, forinfinite energy barriers the particle is always transmitted. This is very surprising, sincein nonrelativistic physics the tunneling of particles into a barrier is accompanied by anexponential damping, and of course for infinite barriers there is no tunneling at all.

We will study the case for the 1+1 dimensional Dirac equation with an electrostaticstep potential of the type

�el(x) =

⇢�0

for x > 00 for x < 0

. (2.73)

The whole Dirac operator is now given by

H = �i�y @@x

+mc2�z + �el(x)1. (2.74)

Solutions of the free Dirac operator, H0

, for positive and negative energies are respec-tively

+

= N+

(✏)

✓ikc

✏�mc2

◆eik·x�i✏t (2.75)

� = N�(✏)

✓ikc

�✏�mc2

◆eik·x+i✏t, (2.76)

where ✏ = |E| = pk2c2 +m2c4 and k = |k| with k the wavevector. N+

and N� are theappropriate normalization constants for each solution and E the energy of the particle.

Considering the presence of the Klein step and a particle incident from the left thesolution is

=

8>><

>>:

✓ikc

E �mc2

◆eikx +B

✓ �ikcE �mc2

◆e�ikx for x < 0

✓ �ipc�0

� E �mc2

◆e�ipx for x > 0

. (2.77)

Moreover, continuity condition at x = 0 implies

ik(1�B) = �ipF (2.78)

(E �mc2)(1 +B) = (�0

� E �m)F (2.79)

resulting in relation

1�B

1 +B=

p(E �mc2)

k(�0

� E �mc2)=

1

. (2.80)

2.2.2 Klein paradox 43

Reflection coe�cient is given by

R = |B|2 =✓1� 1 +

◆2

(2.81)

while the transmission coe�cient is

T = 1�R =4

(1 + )2, (2.82)

where is defined by Eq. (2.80) as

=p(E +mc2)

k(E +mc2 � �0

). (2.83)

These are very surprising results, notice how when the energy of the step is higherthan the energy of the particle plus the rest mass of the particle, V > E + mc2, isnegative. Negative values of parameter imply reflection coe�cients greater than oneand negative transmission coe�cients. This results are very hard to interpret since theytell that more than the incident particles are reflected. This is many times called theKlein paradox. What Klein pointed out however was not that.

Klein tells that in x > 0 the momentum of a particle is given by p2 = (�0

�E)2�m2c4,while the group velocity is

vg = dE/dp = p/(E � �0

) (2.84)

This implies that for the group velocity to be positive when �0

> E, p has to be negative,indeed

p = �p(�� E)2 �m2c4. (2.85)

As a consequence

=

s(�

0

� E +mc2)(E +mc2)

(�0

� E �mc2)(E �mc2). (2.86)

If mc2 E V � mc2, then is positive and greater or equal to one, and thus Rand T are positive or zero, and sum up to one. However, in the limit where V !1, tends to 1 and thus the transmission is complete, while there is no reflection.The same happens for massless particles. This tunneling of high potential barriers hasobviously no classical counterpart and it is hard to interpret, hence it is usually describedas paradoxical. Although these results still generate much debate, in the context ofquantum field theories these di�culties are in some sense overcame with arguments ofpair-creation. A vague idea is that so abrupt potential barriers are able to create a pair ofparticle and antiparticle. The antiparticle is attracted to the potential while the particleis repealed. The repealed particle accounts for the extra reflected particles while theantiparticle explains the tunneling. Notice how, the paradoxical results happen whenthe barrier has enough energy to create a pair of particles of mass m.

Chapter 3

Circuit quantum electrodynamics

Circuit quantum electrodynamics is the technology that allows to experimentally imple-ment light-matter interaction in a circuit at a macroscopic level and in the microwaveregime. It is a quite novel subject (2004) that arises from the mixing of quantum opticsand solid-state physics. Until the appearance of circuit QED, light-matter interactionwas always investigated at a microscopic level, say in quantum dots, trapped ions orsemiconductor spin qubits. With the new scenario of circuit QED however, light-matterinteraction is implemented at a macroscopic size for the first time. This technologyprofits from the physics of superconductors, which is one of the few areas that showsmacroscopic quantum e↵ects.

The physics of circuit QED is analog to that of cavity QED. Here the atom is repre-sented by a superconducting qubit and the cavity by a transmission line resonator. Bothare on-chip implementations working at cryogenic temperatures, that makes of circuitQED a technology of pretty easy construction compared to for example cavity QED.Moreover, the macroscopic size of its elements makes them easier to couple with eachother. However, this is at the same time a problem, because also interactions with theenvironment are easily allowed, making very di�cult to isolate the setup. Nevertheless,circuit QED has become a very promising quantum platform, with good perspectives inboth scalability and coherence. As a consequence, it is a prominent example of quantuminformation processing and a great candidate for possible quantum computation.

Circuit QED technology can serve as one of the possible experimental implementa-tions of our proposal. In this chapter we present some of the basic knowledge on thistopic. We discuss the basics of circuit QED in order to understand how it can be usedfor our proposal. One important aspect of circuit QED is that it works in the microwaveregime were there are no photodetectors. For that reason we also include an explana-tion of the dual-path method for measuring field quadratures of propagating quantummicrowaves in the absence of photodetectors.

46 Chapter 3 Circuit quantum electrodynamics

3.1 Elements of circuit QED

Circuit QED works mainly with two components, the superconducting qubit playing therole of the atom, and the transmission line resonator as the cavity. In this section wetalk about these two elements.

3.1.1 Superconducting qubits

Superconducting (SC) qubits are SC circuits which take advantage of the anharmonicenergy distribution of the quantum levels of a device called Josephson Junction (JJ). Dueto the anharmonicity of the energy distribution, one can approximate the system to atwo level system truncating the Hilbert space to dimension 2. SC qubits are macroscopicobjects which can be made to interact with electromagnetic fields. Thus we can use themalong with on-chip one dimensional microwave resonators of circuit QED to investigatelight-matter interaction. Moreover, they provide high controllability since both thesplitting frequency and the coupling strength can be easily tuned with the help of externalelectromagnetic fields. However, the coherence time of SC qubits is not very long dueto their large size.

Josephson junctions consist of two superconducting electrodes separated by a verythin nonsuperconducting barrier. The barrier is thin enough to allow for quantum tun-neling of the Cooper pairs. These are pairs of electrons. It was shown by Leon Cooperthat in superconducting materials electrons can have a lower energy when paired thanindividually. It was predicted by Josephson that this Cooper pairs would tunnel the thinbarrier of the JJ. This tunneling creates a supercurrent (that is a current in the absenceof any applied voltage and which lasts indefinitely in time) across the device. This isknown as the Josephson e↵ect.

It turns out that these Josephson junctions can be described by a quantum version ofthe classical circuit theory. More precisely the JJ behaves as a nonlinear inductance inparallel with a capacitor plate, which corresponds to the two coupled superconductingelectrodes. As we know, the physics of an LC circuit is analogous to that of a harmonicoscillator. In our case, the position operator would be associated to the gauge invariantphase di↵erence, ', while the momentum is represented by operator N = �i @

@' , whichrepresents the number of Cooper pairs and thus is proportional to the total charge Q.These operators of course are canonically conjugated variables and fulfill commutationrelation [N,'] = �i. The Hamiltonian of the system is that of a nonlinear oscillator,where the capacitance is given by the Josephson capacitance CJ and the nonlinear in-ductance by the Josephson inductance LJ = �0

2⇡IC cos' , where �0

= h2e is the SC flux

quantum, and Ic is the critical current, which depends on the material and the size ofthe JJ.

Josephson junctions are characterized by two energies, the Cooper pair charging

energy Ec = (2e)2

2CJ, the energy used to store a Cooper pair in the capacitor, and the

Josephson energy EJ = �02⇡ IC , which is the required energy for storing the flux quantity

�0

in the Josephson inductance. On the other hand, SC qubits are also described by

3.1.1 Superconducting qubits 47

two characteristic times, T1

and T2

. The first one is the time that takes the qubit to getunexcited, that is, to go back from the excited to the ground state. T

2

is the coherencetime of the qubit, the time the qubit can last isolated.

About the physical implementation of the SC qubit, we distinguish among threedi↵erent regimes of the ratio EJ/EC . Varying the size of the junction one can achievedi↵erent values of the Josephson capacitance CJ and thus of EJ . We call charge qubitsthose operating in the charge regime EC � EJ where the number of Cooper pairs is welldefined while the phase ' fluctuates strongly. Operating in the flux regime Ec ⌧ EJ

we have the oposite situation with a well defined phase and strong fluctuation of thenumber of Cooper pairs. The flux, phase and trasmon qubits operate in this regime.

Now, we will expose some of the most usual SC qubits.

The charge qubit

A JJ connected in parallel with a capacitor leaves a piece of superconducting materialsomewhat isolated between the gap of the capacitor and the barrier of the JJ. Thissuperconducting island is called Cooper pair box. The capacitor is then connected to avoltage source. We are now interested in the number of Cooper pairs in the island. Itcan be shown that in the charge regime due to the high anharmonicity of the states onecan use the lowest two states, |0i and |1i, corresponding to no o↵set Cooper pairs in theisland and 1 pair in the island, as a qubit.

As we have just said, theoretically the charge qubit works in the charge regime,however actual experimental implementations are done in a regime where EJ ⇠ EC ,with the same predicted physics. The main disadvantage of these qubits is that they arevery sensitive to the charge noise, reducing notably the coherence time of the qubit.

The main characteristics of it are:

• EJ ⇡ EC

• Size: 0.01µm2

• Energy splitting of the two lowest states: 10 GHz

• Coherence time: T2

= 0.1� 1µs

This qubit has been experimentally built and tested in several groups all aroundthe world as for example groups of Yale, Saclay, Chalmers and the Nippon ElectronicsCorporations.

The flux qubit

In this case qubits exploit the flux degree of freedom. The qubit states are given bycirculating currents. The two states correspond to clockwise and anticlockwise currentson a SC loop with Josephson junctions. These currents appear induced by an LC circuitinductively coupled to the SC loop and controlled with an external current. Nowadays,due to its high sensitivity to external magnetic fields flux qubits are built with three JJs.


In general they are very sensitive to flux noise and fluctuations of the critical current.We list here the key features of this qubits:

• EJ ⇡ 102Ec

• Size: 0.1� 1µm2



= 1� 10µs

Experimental groups of Delft, Berkeley, IBM laboratory and Walther-Meissner In-stitute have already built and tested this kind of SC qubits.

The phase qubit

The macroscopic phase of the JJ is use to implement this kind of qubit. The JJ is biasedwith a very large dc current, close to the critical current, producing high nonlinearityon the phase states. Flux qubits operate in the real superconducting regime, whereEJ/EC � 1. Increasing the critical-current noise flux qubits avoid the sensitivity tocharge noise. The main characteristics are:

• EJ ⇡ 104Ec

• Size: 10� 100µm2



= 0.1� 1µs

Universities of Santa Barbara (UCSB), Maryland and the National Institute of Stan-dards and Technologies (NIST) work on these qubits.

The transmon qubit

The design of the transmon qubit is very similar to that of the charge qubit, but nowwe have an additional large capacitance, CB, shunted in parallel to the JJ. This reducesthe charging energy, EC , which results in two e↵ects, on one hand, the charge dispersionreduces exponentially with EJ/EC , and on the other hand the anharmonicity decayswith an small power of EJ/EC . The sensitivity of the Cooper pair box to charge noiseis directly related to the charge dispersion, so in this way we have reduced the chargenoise sensitivity with just an small loss of anharmonicity. The main consequence is thatthe coherence time is drastically augmented.

The transmon qubit is characterized by:

• EJ ⇡ 102Ec


3.1.2 Transmission line resonator 49

Figure 3.1: Schematic representation of a transmission line resonator. The center con-ductor of an on-chip coplanar waveguide is cut at two points. These two gaps act asmirrors for the propagating waves of voltage and current and thus the setup behaves asa resonator. The field is created between the lateral ground planes and the central line.The circuit is made of superconducting materials and works at cryogenic temperatures.


⇡ 35µs

This qubit has been developed in Yale.

3.1.2 Transmission line resonator

In circuit quantum electrodynamics the role of the cavity is played by an on-chip coplanarwaveguide resonator, which stores photons in the microwave regime. The structure ofthis resonator is depicted in Fig. (3.1). As any coplanar waveguide it consists of twolateral ground planes and a central superconductor, which carries the signal along thewaveguide. In order to generate a resonator this central conductor is broken at twopoints. Now the central line consists of three parts, what we will keep calling the centralconductor or central line in the middle and the input and output lines at both sides of it,which are capacitively coupled to the central conductor via these gaps that we have justcreated. These capacitance points have very large impedance and thus act as mirrorsfor the voltage and current waves propagating along the central line. Hence, the centralconductor behaves as a resonator, where standing waves of the voltage and the currentwith the appropriate wavelength are generated. Due to these new boundary conditions,the continuum of modes for the electromagnetic field in free space is broken into a welldefined discrete set of allowed modes. The electric field is confined between the centralguideline and the lateral ground planes, with antinodes at the end of the resonator. Thisresults in a very small volume per mode and thus the electric field happens to be reallyintense, which allows for strong coupling with a superconducting qubit. The SC qubitwould be built in the space between the central line and the ground planes, where the


electric field is generated. The input and output lines can be used to introduce rf pulsesor dc currents into the resonator to manipulate the qubit. In the same way this linesare used for measurements of amplitudes and phases of the field.

The system can be described as a harmonic oscillator close to resonance, modeledby an LC circuit. The frequency thus is !r = 1/

pLC. The Hamiltonian is that of a

harmonic oscillator

Hr = ~!r(a†a+ 1/2) (3.1)

The average number of photons is given by ha†ai and the rate of dissipation is = !r/Q where Q is the quality factor of the resonator.

We now list some of the typical values of a TLR:

• Resonator length: dres ⇡ 20 mm

• Capacitance per length: c = Cdres⇡ 2 pF

• Inductance per length: l = Ldres⇡ 4 nH

• Impedance: Z =q

LC ⇡ 50⌦

As an example, the Yale group has achieved n < 0.06 photons inside the resonatorat a temperature T < 100mK, with the typical frequencies between 2 and 10 GHz.

As for any chip, for the construction of these devices lithographic techniques are used,which are very well dominate d nowadays. Superconductor materials as for examplealuminum (Al) or niobium (Nb) are placed on dielectric substrates like silicon (Si) orsapphire (Al

2

O3

).The derivation of the interaction Hamiltonian between the superconducting qubit

and the resonator varies depending on the type of superconducting qubit we use. Ingeneral the whole Hamiltonian will be given by a JC Hamiltonian with the interactionterm taking a negative sign,

H = ~!ra†a+ ~!scq�z � ~gscq(a†�� + a�+). (3.2)

Summing up, what the reader should keep in mind is that TLRs are one dimensionalresonators working in the microwave regime, with the advantages that they are quiteeasy to construct and generate very high electric fields facilitating the coupling with aqubit.

3.2 Dual-path detection method

In the theory of quantum optical measurement and experiments one of the most basicaspects is the reconstruction of the Wigner function or the density matrix of a field state.For such a reconstruction it is crucial to have access to all the quadrature moments ofthe field we want to reconstruct. In the optical domain this is done, for example, with a

3.2 Dual-path detection method 51

Figure 3.2: Schema of the dual-path method. The signal, S, enters a 4-port 50-50 beamsplitter, at the other port we have an ancilla state, V. The two outputs, S+V and -S+V,are passed through detection chains which consist of a linear amplifier that adds noise(�

1

and �2

) to the signal and a linear detector able to measure the voltage, C1

and C2

.

technique called homodyne tomography. This technique uses what we call photodectors,which are just devices capable to detect single photons. In circuit QED, however, wework on the microwave regime, where photodetectors exist only as theoretical proposals.This fact forces us to use linear amplifiers. These are devices that amplify the signalso it can be detected. The detection is made by a linear detector which instead ofdetecting photons (intensity), measures voltage (field quadrature). The drawback isthat the amplifiers obscure the signal we want to reconstruct by the introduction of 10to 20 photons of random noise per photon.

The dual-path Method [27] is a last generation method (2010) capable to overcomethese problems and have access to all the quadrature moments of the field using onlybeam splitters, linear amplifiers and linear detectors, that is, without photodectors.

For this method, we use a four-port 50-50 microwave beam splitter. This device hastwo input ports and two output ones. The relation between the input and the output is

✓c1

c2

◆=

1p2

✓1 1�1 1

◆✓SV

◆, (3.3)

where c1

and c2

are bosonic creation operators of the output fields, while S and V arethose of the input fields. If one of the input ports is left empty vacuum is assumed asinput. We will discuss the procedure to detect classical signals, but this can be easilygeneralized to the quantum case.

The procedure of the dual-path is the following. In one of the input ports we enterthe signal we want to reconstruct, S, while in the other port we put a well knownancilla state, as for example vacuum, V. The output signals are then passed through thedetection chain where they are amplified by a linear amplifier, which introduces noise


contributions, and then recorded by linear detectors. These record time traces like

C1

= G(S + V + ⇠1

) (3.4)

C2

= G(�S + V + ⇠2

) (3.5)

where ⇠1

and ⇠2

are the noise contributions of each amplifier, and G is the amplificationgain, which is equal for both detection chains.

With the linear detector measuring quadratures of C1

and C2

we have access tocorrelations of the form hC

1

lC2

mi where m, l 2 N0

. The first moment of both noises isobviously zero since these are random noises. For the vacuum state also we have a firstmomentum equal to zero, thus it is easy to see that the first momentum for the signalis hSi = hC

1

i/G = �hC2

i/G. Any other moment of the signal or the noise is calculatedby induction:

hSni = � hCn�1

1

C2

i/Gn

�n�1X

k=1

kX

j=0

✓n� 1k

◆✓kj

◆hSn�kihV jih�k�j

1

i

+n�1X

k=0

kX

j=0

✓n� 1k

◆✓kj

◆hSn�k�1ihV j+1ih�k�j

1

i, (3.6a)

h�n1

i = + hCn1

i/Gn

�nX

k=1

kX

j=0

✓nk

◆✓kj

◆h�n�k

1

ihSk�jihV ji, (3.6b)

h�n2

i = + hCn2

i/Gn

�nX

k=1

kX

j=0

✓nk

◆✓kj

◆(�1)k�jh�n�k

2

ihSk�jihV ji. (3.6c)

For the derivation of this formulas it is crucial that that S, V, �1

and �2

aremutually statistically independent, so the following can be fulfilled, hS�V ��

1

�✏2

i =hS�ihV �ih��

1

ih�✏2

i. We point out here that the assumption of equal gain at both de-tection chains is not an experimental restriction.

The dual-path method can be used in any implementation in circuit QED where weare interested in measurements of the quadrature moments of the field. For that, onecan let the field leak out of the resonator through the output line, this will behave as apropagating quantum microwave to which one can then apply the dual-path method.

Chapter 4

Simulating Relativistic QuantumE↵ects in circuit QED

When analytical techniques are not powerful enough, computational simulation of phys-ical phenomena is undoubtedly a very powerful tool in research. However, it is wellknown that many interesting calculations that scientists would like to perform are out ofour computational reach. This is specially true for the simulation of quantum systems.Due to the exponential growth of the Hilbert space with the size of the system, the sim-ulation of quantum systems rapidly becomes intractable with classical computers. Thebest known algorithms require physical resources which also grow exponentially with thenumber of particles been simulated. Hence, the simulation of quantum systems needsan implementation distinct to that of classical computers. The big idea came with R. P.Feynman, who said:

Nature isn’t classical, dammit, and if you want to make a simulation of nature,you’d better make it quantum mechanical, and by golly it’s a wonderful problem, becauseit doesn’t look so easy

This was in 1981, when Feynman gave his Keynote speech during the 1st Conferenceon Physics and Computers. For these talks he is credited as the introducer of the ideaof a quantum simulator [9]. And the idea is pretty clear, to simulate a quantum systemuse another more controllable one which is also quantum.

However, his contribution needed more than ten years to be fully appreciated. Proba-bly, the work of Deutsch [10] helped to hinder Feynman’s contributions. Deutsch showedthe existence of a universal quantum computer, a quantum mechanical generalization ofthe classical Turing machine using quantum logic gates and algorithms. This quantumTuring machine should not be confused with a quantum simulator, although it can alsobe used for the simulation of quantum systems, it is a more general idea. The quantumsimulation is just a physical process capable to simulate another one and from which datacan be extracted. Even though a quantum simulator is not necessarily more powerfulas a computational tool than a universal quantum Turing machine, for many physical

54 Chapter 4 Simulating Relativistic Quantum E↵ects in circuit QED

problems a quantum simulator can beat classical machines for lower number of qubitsthan a quantum Turing machine would do. This was put in clear by Seth Lloyd, whoargued [11] that whereas the simulation of a general 40 spin�1/2 particle system wouldbe enough to outperform existing classical computers, the factorization of a 100-digitnumber with Shor’s algorithm would need of thousands of qubits to become nontrivial.

Lloyd brought the rebirth of the concept of quantum simulator when in 1996 hepresented a general proof for Feynman’s conjecture [11]. Since then, many quantumsimulations have been performed and proposed in very diverse platforms as for example,trapped ions, cavity QED, circuit QED, quantum dots or nuclear spins using NMRmethodology. It is likely that the first quantum device to beat classical computers willbe a quantum simulator.Therefore, quantum simulators are attracting much attentionlately. But there is another reason why quantum simulators are becoming so interesting.They provide powerful analogies to understand the physics of the most varied typeof systems one can think of. Including condensed-matter physics, high-energy physics,cosmology and quantum chemistry [12, 13]. These analogies can be very powerful, givingaccess to properties that might be hidden in the original system. In every quantumsimulation two fields are mixed, that of the system that is being simulated and that ofthe platform that simulates. The simulation as a common point between them providesa natural communication bridge for both fields, which we strongly believe improves bothareas. Moreover, when compared to the original systems, quantum simulators show anunprecedented degree of controllability over all physical parameter regimes.

In this chapter we present a proposal for the simulation of quantum relativisticphysics in the platform of circuit quantum electrodynamics. However, the proposalcan be easily extended for implementation in cavity quantum electrodynamics. Moreprecisely we propose the implementation of the Dirac equation, with controllability ofparameters like the mass of the simulated particle and access to phenomena like Zit-terbewegung and Klein paradox. This is motivated by two reasons. First, since 2004[25, 26] cQED has become the quantum platform with the most promising perspectivesin terms of scalability and coherence. Second, there are crucial physical di↵erences withrespect to any previous implementation of the Dirac equation and Klein paradox. In ourproposal the physics of a relativistic spin�1/2 particle is simulated by the interaction oftwo degrees of freedom of di↵erent physical systems, i.e. a standing wave in a supercon-ducting resonator and the two lowest levels of a superconducting qubit, where none ofthem move at all. The position and momentum of the Dirac particle are then encodedin its phase-space or field quadrature representation. This opens up new possibilities forcombining intracavity fields with propagating quantum microwaves [27, 28] in scalablequantum network architectures with delocalised and/or sequential interactions.

This is the main chapter of this master thesis, where original new ideas are shownand the results of our investigations are presented.

4.1 The method and proposal of implementation 55

4.1 The method and proposal of implementation

In this section we are going to show how we can implement a simulation of the Diracequation in circuit quantum electrodynamics. We will show which are the magnitudesthat will map the degrees of freedom of the simulated particle and we will exhibit thecontrollability of our model.

For our protocol we require a two-level superconducting qubit, strongly coupled toa single electromagnetic field mode of the resonator. This interaction will be describedby the Jaynes-Cummings model (JCM) [29, 25, 26]. In addition, we introduce threeexternal classical drivings, these are just classical electromagnetic fields which can befocused very accurately. In this way we can make these external drivings couple onlyto the qubit if we point them directly to it and transversal to the resonator. Or wecan also make them couple to the single mode field if we place them longitudinal to theresonator. For our model we will place two of them transversal to the resonator [30],coupling only to the qubit, and one longitudinal that only sees the resonator mode. Thetime dependent Hamiltonian of the complete system is given by

H =~!q

2�z + ~!a†a� ~g

⇣�+a+ ��a†

⌘� ~⌦

⇣ei(!t+')�� + e�i(!t+')�+

⌘

�~�⇣ei(⌫t+')�� + e�i(⌫t+')�+

⌘+ ~⇠

⇣ei!ta+ e�i!ta†

⌘. (4.1)

with �y = i�� i�+ = i |gihe| � i |eihg| and �z = |eihe| � |gihg|, |gi and |ei being theground and excited states of the qubit. Here ~! and ~!q stand for the photon energy andqubit energy splitting, whereas g is the coupling constant. The two orthogonal drivingshave real amplitude ⌦, �, phase ', and frequencies !, ⌫. Finally the longitudinal drivingis characterised by an amplitude ⇠ and a frequency !. We have made the transversaldrivings to couple negatively while the longitudinal one does it with positive sign. Thiscan be done since we have control on the relative phase of the external fields. Noticethat while we have chosen two of the drivings to be resonant with the resonator fieldmode, the other parameters will be set later on. For simplicity in our presentation, wewill also assume that !q = !, i.e. qubit and resonator field interact resonantly.

Our derivation consists of two straightforward transformations. First, Hamiltonian(4.1) can be simplified using the reference frame rotating with the resonator frequency!,

HL1 = �~g⇣�+a+ ��a†

⌘� ~⌦

�ei'�� + e�i'�+

�+ ~⇠

⇣a+ a†

⌘

�~�⇣ei[(⌫�!)t+']�� + e�i[(⌫�!)t+']�+

⌘. (4.2)

Second, we will write this Hamiltonian in another interaction picture with respect to the


time independent term HL10

= �~⌦ �ei'�� + e�i'�+�. The resulting expression reads

HI = �~g2

��|+ih+|� |�ih�|+ e�i2⌦t |+ih�|� ei2⌦t |�ih+| ei'a+H.c.

�

� ~�2

��|+ih+|� |�ih�|� e�i2⌦t |+ih�|

+ei2⌦t |�ih+| ei(⌫�!)t +H.c.⌘+ ~⇠

⇣a+ a†

⌘, (4.3)

where we have introduced the rotated spin basis |±i = �|gi± e�i' |ei� /p2. To simplifythis Hamiltonian expression further, we can now set ! � ⌫ = 2⌦, and assume the am-plitude of the first driving ⌦ to be large as compared to the rest of frequencies in (4.3).This implies we can apply the RWA to yield the e↵ective Hamiltonian

He↵

=~�2�z +

~gp2�yp+ ~⇠

p2x, (4.4)

where we have also put ' = ⇡/2. Here we have used definitions in Eq.(1.21) for thestandard quadratures of the electromagnetic field, i.e. x = (a + a†)/

p2, p = �i(a �

a†)/p2, with the commutation relation [x, p] = i.

Making the following identifications

~�/2! mc2 (4.5)

~g/p2! c (4.6)

Hamiltonian in Eq. (4.4) is the Dirac operator in 1+1 dimensions with the additionof an external linear potential � = ~⇠

p2x, that depends linearly on the position of

the particle. Clearly in the dynamics simulated one can cover a wide range of physicalparameters. The controllability of our setup is obvious, while the simulated mass isproportional to the amplitude of the weak orthogonal driving �, the strength of thelinear potential can be tuned through the amplitude ⇠ of the longitudinal driving. Thisis an interesting advantage over the implementation in ion traps, where a second ion isneeded to simulate the external potential [22, 23].

The study of relativistic quantum e↵ects, such as Zitterbewegung for a free particle,or the Klein tunneling, should be done in the phase-space representation of the electro-magnetic field in the superconducting resonator. In all numerical cases studied in thefollowing section, the initial state of the bosonic degree of freedom of the simulated Diracparticle is assumed to be a wavepacket with position hx

0

i and momentum hp0

i

(x) = ⇡�1/4 exp {ihp0

ix} exp

⇢�(x� hx

0

i)22

�. (4.7)

This coincides with the x�quadrature representation of a coherent state of the elec-

tromagnetic field�� hx0i+ihp0ip

2

E= D

⇣hx0i+ihp0ip

2

⌘|0i, |0i being the vacuum state, and

4.2 Numerical simulations and discussion of the results 57

D(↵) = exp�↵a† � ↵⇤a

the coherent displacement operator. Summing up, field quadra-

tures of the electromagnetic field inside the resonator simulate position and momentumof the particle, while the internal degrees of freedom of the particle are mapped intothe superconducting qubit. Notice how powerful this analogy is, the simulated particlehas its mechanical and internal degrees o↵ freedom simulated by two distinct and in-dependent physical entities. Moreover, the field in the resonator can be let go out ofthe resonator delocalising both degrees of freedom, this is, the mechanical degrees offreedom of the particle, its position, can be far away of its internal degrees of freedom,its spin.

4.2 Numerical simulations and discussion of the results

In this section we show computational simulations of our proposal and try to give inter-pretation to the results in order to validate the model. We first analize the behaviour ofthe free Dirac particle and the phenomena related to it as the Zitterbewegung and thenwe investigate the particle in an external linear potential, the Klein paradox. For thenumerical simulations discussed in this section we have chosen a value of the supercon-ducting qubit and resonator frequencies of !q = ! = 2⇡ ⇥ 9 GHz, a qubit-field couplingof g = 2⇡ ⇥ 10 MHz and a strong transverse driving amplitude of ⌦ = 2⇡ ⇥ 200 MHz.All this values are consistent with an implementation on cQED. The interaction timeneeded is of about 60 nsec, which is well below standard decoherence times in cQEDexperiments.

4.2.1 The free Dirac particle and Zitterbewegung

Phase space analysis

In the absence of external potential, � = 0, the Dirac equation for a free particle (2.45)does not couple the di↵erent spinor components. The Dirac Hamiltonian

HD

=~�2�z +

~gp2�yp (4.8)

has two important limits depending on the value of the mass of the particle. First, in themassless case the Hamiltonian reduces to H

D

= ~g/p2�yp allowing for straightforward

interpretation of its dynamics in terms of phase-space representation. Fig. 4.1(a) showsthe evolution of a massless particle with initial state |+, 0i. Its spin being in an eigenstateof �y, the e↵ect that the interaction has is just to generate a coherent field displacementfrom its original position, the vacuum state |0i, to the final one in which the state ofthe field is |gt/2i. This simulates the evolution of a massless free Dirac particle movingat the simulated speed of light, ~g/

p2. As we expect, the dynamics of this massless

particle would be essentially unchanged if one assumes the initial state to have a non-zerovalue of the expectation value of momentum. This can be seen in Fig. 4.1(d), where theparticle starts its trajectory with hp

0

i = 2. The particle covers exactly the same amountof space in the x�quadrature, ending in the coherent state

��gt/2 +p2i↵.


Figure 4.1: Wigner function W (x, p) representation of the state of the electromagneticfield mode inside the resonator. The initial state evolves under the Hamiltonian (4.2)for a time period of 60 nsec. The physical parameters used are g = 2⇡ ⇥ 10 MHz,⌦ = 2⇡ ⇥ 200 MHz, ⇠ = 0, together with: (a) � = 0 being the initial state |+, 0i; (b)� =

p2g being the initial state |+, 0i; (c) � = 4

p2g being the initial state |e, 0i; (d)

� = 0 being the initial state��+,p2i↵; (e) � =

p2g being the initial state

��+,p2i↵; (f)

� = 4p2g being the initial state

��e,p2i↵.

Second, when the mass of the particle becomes large enough, the dynamics shouldtend towards that of a nonrelativistic one evolving under a free Schrodinger equation.As was explained in section 2.1.5 for �/2 � g/

p2hpi, i.e. mc2 � chpi, one can derive

a second order e↵ective Hamiltonian [21] for Eq. (4.4) to yield HNRel

= ~�zp2/�. InFig. 4.1(c) we depict the evolution of the initial state |e, 0i under the time dependentHamiltonian (4.2) for the case in which the mass of the particle is �/2 = 4⇥ g/

p2. This

generates a squeezed vacuum state with degree of squeezing increasing linearly in time.This behavior is in agreement with the aforementioned nonrelativistic approximation forthe Hamiltonian, proportional to p2. Such a simulation realizes a standard example inall quantum mechanics textbooks. Namely, the free evolution under the nonrelativisticSchrodinger equation of an initial wavepacket with hp

0

i = 0 produces a particle thatremains centered at the same initial position, with a wavepacket spreading over timein the x�quadrature. However, the e↵ect of the operator p2 is not simply squeezing.

4.2.1 The free Dirac particle and Zitterbewegung 59

Figure 4.2: Expectation value of the x�quadrature of the electromagnetic field modeinside the resonator. The interaction time is set to 30 nsec. The physical parametersused are g = 2⇡ ⇥ 10 MHz, ⌦ = 2⇡ ⇥ 200 MHz, ⇠ = 0, together with: � = 0 beingthe initial state |+, 0i (dashed line); � =

p2g being the initial state |+, 0i (dotted line);

� = 4p2g being the initial state |e, 0i (dash-dot); � = 4

p2g being the initial state |e, 0i

using the FW Hamiltonian (solid line). Whereas in (a) the calculations involve timedependent Hamiltonians (4.2), in (b) e↵ective ones (4.4) are used.

This can be clearly seen in Fig. 4.1(f) where we have taken a particle with initial state��e,p2i↵, that is a wavepacket with non-zero kinetic energy such that hp

0

i = 2. Now thewavepacket not only spreads over time, but its centre moves linearly in time to the rightas the expectation value of the position of a nonrelativistic particle would do.

While for the limits of zero and large mass, the state of the field remains Gaussianduring the evolution, this is no longer true when the mass of the particle mc2 is compa-rable to chpi. This is the case of Figs. 4.1(b) and (e), where �/2 = g/

p2. Starting from

initial states |+, 0i and ��+,p2i↵, respectively, the Gaussian states of the field evolve into

nonclassical ones with negative Wigner function through a highly nonlinear interaction.The e↵ect is more evident for the case of an initial state |+, 0i in Fig. 4.1(b). Initialstates with non-zero kinetic energy, such as

��+,p2i↵in Fig. 4.1(e) make the qubit-field

coupling term in Eq. (4.4) dominant.

Zitterbewegung

With our scheme we can also study the appearance of Zitterbewegung for a free spin�1/2Dirac particle. In Fig. 4.2 we plot the time evolution of hxi for a particle prepared in aninitial state |+, 0i and with three di↵erent values of the mass. For better comparison,plots (a) and (b) correspond to calculations done using the time dependent Hamiltonian(4.2) and the e↵ective one (4.4). The good agreement found indicates that choosing a


value of the strong driving amplitude ⌦ = 2⇡ ⇥ 200 MHz is enough for our purpose.The dashed line in Fig. 4.2 shows the evolution of a massless particle, � = 0, whoseexpectation value of the position increases monotonically at the speed of light, ~g/

p2. If

we assume a mass of �/2 = g/p2, then we observe that the evolution of the hxi followed

by dotted line departs considerably from the constant rectilinear behaviour one wouldexpect for a free particle. Likewise, for a mass even larger, of about �/2 = 4 ⇥ g/

p2,

the e↵ect seen in the dash-dotted line is even more significant. Now the particle hardlymoves forward. Instead it develops an almost stationary oscillation around its initialposition.

To gain in physical insight, we are going to derive an analytical expansion for theevolution operator of a free Dirac particle. We start decomposing it as a sum of sineand cosine,

UD = e�i(�2 �z+

gp2p�y)t = cos(

�

2�z +

gp2p�y)t� i sin(

�

2�z +

gp2p�y)t. (4.9)

We now expand sine and cosine functions and use the properties of the Pauli matrices,this is, that their square is the unit matrix and that they anticommute, to yield

UD =1X

n=0

(�1)n[(�

2

�z +gp2

p�y)t]2n

2n!� i

1X

n=0

(�1)n[(�

2

�z +gp2

p�y)t]2n+1

(2n+ 1)!

=1X

n=0

(�1)n(q

�2

4

+ g2

2

p2t)2n

2n!(4.10)

� i(�2

�z +gp2

p�y)tq

�2

4

+ g2

2

p2t

1X

n=0

(�1)n[q

�2

4

+ g2

2

p2t]2n+1

(2n+ 1)!

= cos(

r�2

4+

g2

2p2t)

� i(�

2�z +

gp2p�y)t sinc(

r�2

4+

g2

2p2t), (4.11)

where sinc(x) = sin(x)/x. We now expand these terms over � = 0,

UD = cos ctp�sinc( gp

2

tp)

2

�2t2

4

� i(�

2�z +

gp2p�y)[sinc(

gp2pt) +

cos gp2

pt� sinc( gp2

pt)

2g2

2

p2t2�4t2

4] (4.12)

(4.13)

4.2.1 The free Dirac particle and Zitterbewegung 61

which are finally reorganized as

UD = exp

�it�y

r�2

4+

g2

2p2

!� i�z

�t

2sinc

r�2t2

4+

g2t2

2p2

!

� i�y

0

@ gtp/p2q

�2t2

4

+ g2t2

2

p2� 1

1

A sin

r�2t2

4+

g2t2

2p2

!

= exp

✓�i gtp

2�yp

◆� i

�t

2�zsinc(gtp/

p2)

+�2t2

4

�sinc(gtp/

p2)

2� i�y

cos(gtp/p2)� sinc(gtp/

p2)

gtpp2

!

� i�3t3

8�z

cos(gtp/p2)� sinc(gtp/

p2)

g2t2p2+ . . . , (4.14)

The first term corresponds to the evolution of a massless particle, which shows no Zitter-bewegung. Here one can easily see that all the other terms decay with time, taking theevolution operator closer to that of the massles particle, which explains why Zitterbewe-gung decays in time. Moreover, one can also see how a larger initial average momentumgthp(0)i/2 will lead to a faster collapse of the trembling motion for a particle preparedin a wavepacket [18, 19].

As already discussed, Zitterbewegung stems from the fact that the time derivative ofthe position operator is not a constant of the motion for the Dirac equation. This canbe seen if one writes down the solution for the evolution of this operator (2.64). TheZitterbewegung term Z(t) will be exactly zero if the initial state of the particle containspurely positive or negative components of the spinor. However, for a completely arbitraryinitial wavepacket Z(t) does not necessarily vanish. In addition, the preparation of apurely positive spinor for a massive Dirac particle is not straightforward. In Ref. [8] aheuristic method for preparing a state with only negative components was used. Thisshows that even for a massive Dirac particle, evolutions without Zitterbewegung arepossible. Another alternative to study the existence/absence of Zitterbewegung wouldbe to implement a measurement of the Foldy-Wouthuysen or Newton-Wigner positionoperator [16, 31], that was introduced in section 2.2.1, and which for our one dimensionalcase takes now the form

xNW

(t) = x0

+ c2pH�1

D

t+�, (4.15)

where � is the time independent operator

� =~c�x2E� ~c3�xp2

2E2(E +mc2), (4.16)

�x = i�z�y, E =pc2p2 +m2c4. As already said, this position operator in (4.15) has

the property of being also canonically conjugate of the momentum operator. The main


drawback of xNW

(t) from an experiential point of view is that is not diagonal in thecoordinate space. This means that measuring such an operator might require a fullquantum tomography of the evolved state of the system.

Here we follow a di↵erent route to the problem. As we already know, Foldy andWouthuysen are also credited for having introduced a unitary transformation [31] thatallows for the direct diagonalization of the spin degree of freedom in the Dirac Hamilto-nian. Using the parameters of our simulation, this new Hamiltonian is given by

HFW

= SFW

HD

S†FW

= �z

r~2�24

+~2g22

p2 (4.17)

where

SFW

= exp

⇢�i atan

✓g

�p2�xp

◆�. (4.18)

If one were able to follow the evolution of the system in the Foldy-Wouthuysen represen-tation, but without transforming the measurement apparatus, then no Zitterbewegungwould be seen. That is precisely, what we have done in the solid line of Fig. 4.2(b).This shows the evolution of hxi for a massive particle with �/2 = 4⇥ g/

p2, just like the

dash-dotted one, but using the evolution operator

UFW

(t) = exp (�iHFW

t/~) = SFW

exp (�iHD

t/~) S†FW

(4.19)

instead of UD

(t) = exp (�iHD

t/~). Now the particle remains centered at hxi = 0 duringthe evolution (solid line). More importantly, for large masses we can propose a physicalimplementation of the Foldy-Wouthuysen transformation using the same scheme we havedeveloped. Clearly, when g/�

p2 is small enough, we can linearized the interaction in

the unitary operator SFW

. This means that

UFW

(t) ⇡ exp

✓�i g

�p2�xp

◆exp (�iH

D

t/~) exp

✓i

g

�p2�xp

◆, (4.20)

where the first and last step involve the realization of a Dirac equation for a masslessparticle with an evolution time t = ��1. Such an implementation would require someadditional local rotations and Ramsey-like driving pulses [30]. It would be particularlystraightforward if the setup allows for a variable coupling from �g to +g, being g instrong-coupling regime. This is the case for instance, of a gradiometer type of couplingseen in flux qubits [32]. The solid line in Fig. 4.2(a) depicts the evolution of hxi,computed under these assumptions, for the same particle of large mass �/2 = 4⇥ g/

p2.

Even for such moderate value of the mass, it is evident how oscillations seen for theDirac evolution become significantly reduced.

4.2.2 Dirac particle in an external potential and the Klein paradox

The addition of an external potential has the e↵ect of coupling the di↵erent energycomponents of the spinor. As before, in the evolution under an external potential we

4.2.2 Dirac particle in an external potential and the Klein paradox 63

Figure 4.3: Wigner function W (x, p) representation of the state of the electromagneticfield mode inside the resonator. The initial state evolves under the Hamiltonian (4.2)for a time 60 nsec. The physical parameters used are g = 2⇡ ⇥ 10 MHz, ⌦ = 2⇡ ⇥ 200MHz, ⇠ = g/2 and together with: (a) � = 0 being the initial state |+, 0i; (b) � =

p2g

being the initial state |+, 0i; (c) � = 4p2g being the initial state |e, 0i; (d) � = 0 being

the initial state��+,p2i↵; (e) � =

p2g being the initial state

��+,p2i↵; (f) � = 4

p2g

being the initial state��e,p2i↵.

can identify two limiting cases. A massless Dirac particle is described by the HamiltonianH

K

= ~g/p2�yp + ~⇠

p2x. In Figs. 4.3 (a) and (d) the evolution of the initial state

|+, 0i and ��+,p2i↵, respectively, is shown. We use the same set of parameters as in

Fig. 4.1, with the only addition of a linear potential of strength ⇠ = g/2. Again startingwith the qubit state |+i, allows for a simple interpretation. The initial state of thefield remains coherent being subject to two independent displacements. Namely, as forthe free-particle case there is one in the x�quadrature proportional to g/

p2, and the

second one induced by the potential that drags it down in the p�quadrature. Hence, theexistence of such an external potential cannot alter the rectilinear movement in positionrepresentation at the speed of light. This is precisely the e↵ect of the Klein paradox.A massless Dirac particle tunnels through the potential barrier with unit probability,regardless of the potential strength. The tunneling is accompanied by the ’rotation’ ofthe positive energy components of the initial spinor. In our simple case, this means that


the components with positive momentum components in the Gaussian wavepacket willbe pushed downwards by the potential becoming negative. And the rate at which thishappens is given by ⇠

p2. This explains why the only noticeable di↵erence between the

plots in Figs. 4.1 and 4.3 is in the value of hp(t)i.If the mass of the particle is large enough, then the nonrelativistic Schrodinger equa-

tion should be a good approximation. Its Hamiltonian can be written as HNRel

=~�zp2/� + ~⇠

p2x, which guarantees that any initial Gaussian state should remain

Gaussian during the evolution. This is indeed what we see in Figs. 4.3(c) and (f)where we have prepared initial states |e, 0i and ��e,p2i↵, respectively, with a mass of

�/2 = 4 ⇥ g/p2. In the first of them the particle is being pushed backwards by the

potential. The same can be said for the second. Although having an initial positivekinetic energy has allowed it to enter the barrier further, after 60 nsec is already movingbackwards.

While for these two limiting cases we see complete transmission or reflection, a par-ticle with an intermediate mass will present only partial transmission/reflection. This isshown in Fig. 4.3(b) which corresponds to the initial state |+, 0i for a mass �/2 = g/

p2.

The wave packet has now broken up into spinor components of di↵erent sign which moveaway from the center of the potential. If the wavepacket has some initial kinetic energyas in Fig. 4.3(e), where we prepare

��+,p2i↵, then the particle tunnels slightly to get

stopped and eventually breaks up as well.As we did in Eq. (4.14), one can obtain an analytical expansion for the evolution

operator of a Dirac particle under the e↵ect of a linear potential. Using formula

e�it(X+Y ) = e�itXe�itY e�t2/2[X,Y ] +O�t3�

(4.21)

and with the following definitions

X = �/2�z +gp2p�y (4.22)

Y = ⇠p2x (4.23)

one easily yields

exp

⇢�i✓�t

2�z +

gt

2�yp+ ⇠

p2 t x

◆�

= exp

✓�iHD

~ t

◆exp

⇣�i⇠p2 t x

⌘exp

✓ig⇠t2

2�y

◆+O

�t3�, (4.24)

where we have used, [X,Y ] = g⇠�y. Notice how the free term of the expansion hasthree exponentials, the first one corresponds to the evolution of a free Dirac particle, thesecond one represents a coherent displacement in the P quadrature, and the third oneis a rotation of the spin degree of freedom. Hence, for short time intervals the evolutioncorresponds to that of the free Dirac particle (4.14), but with an initial state with rotatedspin and displaced initial momentum, with respect to the prepared wavepacket.

4.2.2 Dirac particle in an external potential and the Klein paradox 65

The reconstruction of the Wigner function will allow to characterise what type ofdynamics is being simulated. This can be done either using intracavity techniques [33]or allowing the quantum field to leak the superconducting resonator for its subsequentmeasurement. This will require the use of methods for reconstructing the moments ofa propagating quantum microwave signal [27, 28], already available in cQED, as forexample the dual-path method introduced in section 3.2

Chapter 5

Conclusions

We have shown that circuit QED provides a powerful quantum platform for simulatingrelativistic quantum physics in 1+1 dimensions. In a setup with a superconductingqubit strongly coupled to a resonator and three external classical driving fields, we mapthe internal degrees of freedom of a particle on those of the superconducting qubit andthe mechanical ones on two conjugate field quadratures. With a suitable setting of laserphases and amplitudes, we implement the Dirac Hamiltonian, which describes relativisticquantum physics in a nonrelativistic setup. Tuning the parameters of three classicaldriving fields, the system is made to evolve according to the Dirac equation for a freeparticle and under the e↵ect of a linear potential. Either with intracavity techniquesor measuring the outgoing cavity fields, we are able to access the field quadratures.In this way, one can observe phenomena as Zitterbewegung and the Klein paradox,which are di�cult to observe in true relativistic particles in nature. The degree ofcontrollability o↵ered by the proposed method would allow to study interesting physicalphenomena beyond any direct experiment ever achieved. A relevant example is thepossibility of implementing the Foldy-Wouthuysen transformation, which for decadeshas remained a purely abstract transformation. On the other hand, the simulation ofthe Dirac equation with a linear potential is easy to implement in our scheme compared toany previous simulation of the Dirac equation. Furthermore, having the di↵erent degreesof freedom of the simulated relativistic particle encoded in di↵erent circuit QED physicalentities, superconducting qubit and intracavity field, is an attractive feature of thisproposal. The simulation of the nonrelativistic limit of the Dirac equation correspondsto the evolution of a Hamiltonian proportional to the square of the momentum. Inquantum optics, this represents an interesting feature since the quadratic terms of aand a† in the evolution operator are di�cult to implement. These terms appearing inthe squeezing operator produce squeezing of the field. In this way, a future possibleapplication of this model could be the generation of squeezed states, which is di�cult inother kind of schemes since it requires nonlinear interactions. One of the limitations ofthe proposed model is the apparent lack of direct extension to higher dimensions. Onthe other hand, a source of error is the qubit dephasing and relaxation rates, as wellas the resonator decay rate. However, all numerical simulations show that the relevant

68 Chapter 5 Conclusions

physics associated to our method happen well below usual decoherence times. We havecombined for the first time the originally unconnected fields of relativistic quantumphysics and superconducting circuits, opening the door to an attractive exchange andflood of knowledge between them. In this newly merged field, we have proposed theimplementation of quantum relativistic physics in a nonrelativistic quantum platform.Beyond the natural unpredictable consequences, mainly for the sake of applications, webelieve our results are of importance when considering the fundamentals of relativisticand nonrelativistic quantum physics in superconducting circuits.

Acknowledgments

I would like to thank first all my classmates for helping to create a wonderful atmosphereof passion for physics, hard work, good humour and friendship, where work loses all itsnegative connotations. On the other hand, I thank all the members of the QUTIS groupof Prof. Enrique Solano, where I have developed this work, for accepting me in thegroup and making very easy my integration, always ready to help and to share in a goodmood the working days. I specially thank Jorge Casanova and Roberto Di Candia, forhelping me in this project, with insightful discussions and plenty of comments on mywork which have helped me a lot. Undoubtedly, I want to dedicate special words toEnrique Solano, the leader of the group, and the director of this master thesis. Knowingme so little, he put his trust on me and o↵ered me the amazing project that is writtenin these pages. I have to thank not only that enormous generosity, but also the greatworking atmosphere that he has created. He, as the snake charmer he is, has knownhow to transmit passion for knowledge, hard work and an always winning mentality,never losing the good humour. Although not always in agreement with him, speciallywhen it comes to football, I acknowledge his valuable opinions. And last but not least, Ishow my gratitude to Daniel Ballester, with whom I have worked side by side and whohas shown me many tricks of this business. I thank him for being always available andready to help, for dedicating me uncountable hours, for his good humour and for the”coke-walks” debating philosophical transcendental issues, which I admit have changedthe way I think on physics.

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Relativistic Quantum Phenomena in Circuit …...Master in Quantum Science and Technology Relativistic Quantum Phenomena in Circuit Quantum Electrodynamics Master Thesis by Julen S.

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