Applications of Quantum Optics- From the Quantum Internet to Anal

descrip-2.1 Quantum oscillators and photons The quantum harmonic oscillator Atthe most primitive level, the electromagnetic field can be thoughtof as aseaof masslessharmonic oscillators,

Louisiana State University

LSU Digital Commons

3-15-2021

Applications of Quantum Optics: From the Quantum Internet to Analogue Gravity

Anthony Brady

Louisiana State University and Agricultural and Mechanical College

Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations

Part of the Quantum Physics Commons

APPLICATIONS OF QUANTUM OPTICS: FROM THE

A DissertationSubmittedto the Graduate Faculty of theLouisiana State University andAgricultural and Mechanical College

in partialfulfillment of therequirementsfor the degree ofDoctorof Philosophy

inThe Department of Physics and Astronomy

byAnthony BradyB.S., Universityof North Georgia, 2016

May2021

I dedicatethis work to my family – to my beautiful daughters, Willow, Leigha, and ournewest addition, Clover; and to my better half, Autumn Especially to Autumn, who held

down the fort while I was in "thesismode" I love you all, dearly

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Foremost, I want totake this opportunity toacknowledge my motherwho passed away thispastFall(October2020) Underdifficultcircumstances,she setmy"life’s stage"upremark-ably and was the most supportive person I have ever known She persistently encouraged

my sister andI topursueour dreams,as ifthere wasnootherway togoabout life,and hadzero doubt that we would be successful in doing so Even if we ended up failing, she wasalways there tocatch usand heal our wounds I owe almost everything tomy mother

I also want toacknowledge my late advisor,Jonathan P Dowling, whopassed away thispast Summer (June 2020) I was extremely fortunate to know Jon and have him as myadvisor Hegaveme somuchsupport andfreedominmy researchandtrustedme,almostas

a colleague ratherthan a graduate student The amount of freedom and support that I feltwhileworking with Jonremindsmeof the freedomand supportthat I hadwith my mother,even though these two personalities were starkly different There is nothingcomparable tohavingthat kindof supportsystem inyourfamilylife aswellasinyourcareer I am forevergrateful toJon for illuminatingthe fact that the workplace can be so human

Iwanttothankmycurrentadvisor,IvanAgullo,forreachingouttomeafterJon’spassingand extendinganinvitation to"pickme up"asagraduate student,aswellasfor welcoming

my research style and approach to physics I can only hope for more smooth-sailing fromhere I want to thank my first collaborator, Sumeet Khatri My first real research projectwasincollaboration withSumeet, and byobserving andworking closely withhim, Ilearnedhow to properly and precisely do research, write papers, and work collaboratively Thislesson was invaluable I would also like to thank a current post-doc here at LSU and mygoodfriend, Lior Cohen,for beingpresent and extremelysupportiveduringthis last yearortwo It is also good to see another "family man" in the office! I would also like to thankStav Haldar for allowing me to play the role of mentor and letting me drag him along myresearch endeavors It has been fun,and hopefully, we can explore some more I would like

to thank some specific members of the Quantum Science and Technologies (QST) group,Eneet Kaur and Kunal Sharma, that I have befriended though did not get the pleasure towork with I would like to thank them both forbringing fun intothe office and outside theofficeaswell Iwould alsoliketothankall theprior andpresentmembersof theQSTgroupwhich havemet me and somehow tolerated my shenaniganswithout complaining toomuch.And last, but certainly not least, I would like to thank Siddharth Soni for not only being

my first friend at LSU but also for becoming a life-long best friend I hope that we get toexplore the world a bit in the upcoming years

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS iii

ABSTRACT v

1 INTRODUCTION 1

2 FUNDAMENTALS 3

2.1 Quantumoscillators and photons 3

2.2 Photondynamics 27

2.3 Photonsas information carriers 41

3 AN APPLICATION: SPACE-BASED ENTANGLEMENT DISTRIBUTION 49

3.1 Introduction 49

3.2 Network architecture 51

3.3 Overviewof simulations 52

3.4 Comparisonto ground-based entanglement distribution 64

3.5 Summaryand future work 66

4 ANOTHER APPLICATION: OPTICAL ANALOGUE-GRAVITY 69

4.1 Introduction 69

4.2 The model and basic formalism 72

4.3 In-out relations: a Gaussiananalysis 79

4.4 Quantumcorrelations 83

4.5 White-black hole circuitry 89

4.6 Summaryand future work 91

5 EPILOGUE 94

A ENTANGLEMENT DISTRIBUTION: SUPPLEMENTARY METHODS 96

A.1 Extendednoise model 96

A.2 Quantumrepeater rates 102

REFERENCES 103

VITA 117

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The aim of this thesis is to highlight applications of quantum optics in two very distinctfields: space-based quantum communicationand the Hawking effect in analogue gravity.Regarding the former: We simulate and analyze a constellation of satellites, equippedwithentangledphoton-pairsources,whichprovideon-demandentanglementdistributionser-vicestoterrestrialreceiverstations Satelliteservicesareespeciallyrelevantforlong-distancequantum-communication scenarios, as the lossin satellite-based schemes scales more favor-ablywithdistancethaninopticalfibersorinatmosphericlinks,thoughestablishingquantumresourcesinthespace-domainisexpensive Wethusdevelopanoptimizationtechniquewhichbalancesboththenumberofsatellitesintheconstellationandtheentanglement-distributionratesthattheyprovide Comparisonstoground-basedquantum-repeaterratesarealsomade.Overall, our results suggest that satellite-based quantum networks are a viable option forestablishing the backbone of future quantum internet

Regardingthelatter: TheHawkingeffectwasdiscussedintheastrophysicalcontextofthespontaneousdecayofblackholesintoblackbodyradiation,i.e Hawkingradiation However,this effect seems tobe universal, appearinganywhere that anevent horizon(aregion whichrestrictstheflowofinformationtoonedirection)forms Here,weanalyzetheHawkingeffect

in an optical-analogue gravity system, building on prior theoretical results regarding thiseffect indielectric media We provide a simplification of the process via the Bloch-Messiahreduction,which allowsustodecomposethe Hawkingeffect intoadiscretesetofelementaryprocesses With this simplification and leveraging the positivity of partial transpose (PPT)criteria, we examine the quantum correlations of the stimulated Hawking effect, explicitlyshowing that an environmental background temperature, along with backscattering, canlead to entanglement “sudden-death", even when the number of entangledHawking-pairs iscomparativelylarge We alsodiscussthe prospectofenhancingand“reviving" entanglementpre-mortem using single-mode, non-classical resources at the input Though much of thediscussion is phrased in terms of an optical-analogue model, the methods used and resultsobtained apply just as well to a variety of other systems supporting this effect Finally, weprovide Bloch-Messiah reductions of more exotic scenarios consisting of e.g a white-hole–black-hole pair which share an interiorregion

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CHAPTER 1 INTRODUCTION

Ithinkitisnottoouncommon(pardonthedoublenegative)inphysicstoconstrainoneselfto

aparticular(sub-)field,alongwithits(sub-)setofprinciplesanditstechnicalmachinery, andexplorethesurroundingworld(or universe)throughthis lens,contributingvaluableresearch

to one’s field along the way However, this is not the route I have taken I have chosen tofocus,uptothispoint,ononeparticularsysteminstead–thequantumelectromagneticfield,photons, flyingquantum oscillators,whateverone wantstocallthem(!) –and asked,"Whatcan one say about various fields or sub-fields in the language of photons? And how do theprinciplesofsuchfieldstranslate?"Thishastakenmedownvariousexploratorypaths–fromspace-based quantum communication (see [1]and Section 3) to linear-optical simulation ofquantum gravity [2] to Hawking radiation in optical analogue-gravity systems (see Section4) Perhaps this winding path of mine is due in part to the wandering history and modernmeandering of lightitself

Forinstance,itwasexperimentswithlightintheearly20thcenturywhiche.g (i)resolvedthe"ultra-violetcatastrophe"(theclassicalpredictionthatblack-bodiesareunstableathighfrequencies)andsparkedthequantumrevolutionthroughthepostulatedexistenceofphotons

byEinstein(followingPlanck’slead)[3],(ii) refutedtheexistenceoftheluminiferousaether,providing implicit support for Einstein’s special theory of relativity [4],1 and (iii) providedinitial support for Einstein’s theory of general relativity through early observations of thebending of light by the gravitational field of the sun [5, 6].2 In the latter half of the 20thcentury(andveryearly21stcentury),experimentswithphotonshave,asexamples,providedthe first experimental support for the intrinsic non-locality of quantum mechanics [7, 8]

as well as aspects of wave-particle duality at the level of individual quanta [9] In moremodern times, optical interferometers are measuring distortions in space-time induced bygravitationalwaves(aprediction ofthe generaltheory ofrelativity), even utilizing quantumstates of light to enhance the sensitivity of detection events [10]; networks of linear-opticalcomponents,togetherwithsingle-photon sourcesanddetectors, areactivelybeingdeveloped

toworkasquantum simulatorsand evenuniversalquantumcomputers[11, 12];andphotonsserveasanessentialingredientforlong-distancequantumcommunicationandarecrucialforbuilding large-scale, inter-connectedquantum networks[13, 14, 15, 16]

This meandering of light through time, subsequently translating into my own researchendeavors,hasmadewritingacoherentandcomprehensiblethesisabitchallenging Inorder

tofacilitatesomeorderofcoherency,I havestructuredthis thesisintoafewdigestibleparts:

• Thefirstpart,Chapter2,laysoutthemathematicalformalismusedtodescribephotonsandtheirdynamics,ingeneraltermsandinasimplifiedfashion,andservesasthebasisfor later chapters

• The second part, Chapter 3, is (with a few additional intricacies) an application ofthe formalism introduced in Chapter 2, to the domain of space-based entanglement-distribution It is based onmy published work [1]

1 It was Einstein’s theoretical investigations of light which brought him to the special theory of relativity Indeed, one of the postulates has to do with the invariance of the speed of light under changes of reference.

2 There is a lot of "Einstein" here, but of course, he was not the only one!

1

• Thethirdandfinalpart,Chapter 4,isanotherapplicationofthe formalismintroduced

in Chapter 2, now applied in the context of Hawking radiation in optical gravitysystems Thisworkisstillindevelopment,thoughnearingitscompletion,withmany of the main results appearingin this chapter forthe first time

analogue-In this thesis, one should view Chapters 3 and 4 simply as physical (though remarkablydistinct) applications ofphotons and the underlying formalismused to describethem, sincethis perspective adds a bit of coherencyto the document as awhole

As a final remark, I note that I have also completed other works, which fall underthis broad category of "applications of quantum optics", but which I have not included

in this thesis For example, I (and collaborators) have investigated the prospect of ulating/computing transition amplitudes in loop quantum-gravity (an exotic quantum de-scription ofgeneralrelativity)with alinear-optical quantumsimulator [2] Inanotherwork,

sim-I and a fellow graduate student investigated aspects of local, geometric quantum-optics incurved space-time, for the purpose of exploringpotential overlaps between classicalgeneralrelativity and seminal quantum-interference experiments in quantum optics (see reference[17]; currently underreview) Thesetopics couldhave justaswell servedasChapters 5 and

6of this thesis, butI did not includethem forthe sake of brevityand inhopes toavoid anymore meanderingthan necessary

2

CHAPTER 2 FUNDAMENTALS

The chapter serves as a pedagogical introduction to many of the concepts and techniquesused throughoutthe thesis The emphasis ison photons: whatto dowiththem and howtoanalyze them invarious scenarios, ina simplistic and, more orless, generalized framework

MyapproachistoprovidewhatIdeeminteresting, pedagogical,and/oressentialinorder

to comprehend the bulk of this thesis It is not my concern to dive into the history of thephoton nor provide philosophical insights into what a photon is, only to say that it is thequantum(bundleofenergy,particle,etc.) oftheelectromagneticfield, andIprovideonlyanoverly-simplistic mathematical description of what that means,at the level of the quantumharmonicoscillator, and howtoformally deal withit,atthelevel ofFockspaces,symplectictransformations,etc

Thischapterisbrokenintothreeparts Section2.1introducesphotonsthroughtionofthesimpleharmonicoscillator Afterathoroughdiscussionoftheharmonicoscillator,

quantiza-we swiftly transition tofields, providing a more satisfactory and “closer to reality"tion of the quantized electromagnetic field This section is meant only to develop somefamiliarity with the structure of quantum fields and set notation Section 2.2 introducesphotondynamics, restrictingto quadraticinteractions (the “Gaussian sector")and posed inthe form ofscattering-like processes Though this focus seems quiterestrictive and perhapstrivial attimes, itis richenough toencompass avariety of phenomena inmarkedlydistinctscenarios – from e.g photon scattering in the atmosphere in quantum-optical communica-tion tolinear-optical quantum-computationtothe spontaneousdecayof astrophysicalblackholes, etc Gaussian states/systems and the Gaussian formalism is also introduced in thissection Finally, in section 2.3, I introduce some basic concepts from quantum informationtheory with a focus on photonic encoding Quantum entanglement is also discussed, withfocus on the positivity of partial transpose (PPT) criteria for the separability of quantumstates Since much of what I write in this chapter is “textbook material", I will limit thereferences to textbooks for the most part, a listof which can be found at the beginning ofeachsubsection, as needed

descrip-2.1 Quantum oscillators and photons

The quantum harmonic oscillator

Atthe most primitive level, the electromagnetic field can be thoughtof as aseaof masslessharmonic oscillators, with an oscillator positioned at each point in space vibrating at somefrequency Thus, to understand the physics and quantum propertiesof the electromagneticfield, it is sufficient to grasp the corresponding properties of a single, point-like quantumharmonic oscillator We do this by first characterizing aclassical oscillator, which we doso

byderivingtheequationsofmotionintheLagrangianformalismandbyalsointroducingthecanonicalvariablesfortheoscillator Thelatterprovidesaneasyroutetoquantization Aftersolving the equations of motion, we proceed to quantize the oscillator modes via canonicalquantization, which will naturallylead us toextend these notions tofields

Basic notions regarding the Lagrangian and Hamiltonian formalisms can be found in

3

Goldstein’s classic book [18] Discussions on the method of canonical quantization can befound in Dirac’s classic book[19].

The classical oscillator

I firstprovidegeneralmethodsofanalyzingphysicalsystemsviathe Lagrangianand tonian formalisms We then apply such toa point-like, simple harmonic oscillator

Hamil-Considertheactionfunctionalfora(non-relativistic)point-likeparticleinonedimension,

S[x]=

Z

dtL(x,x˙), (2.1)whereL isthe Lagrangian forthe system, xisthe positionof theparticleinspace (relative

to some origin, taken at x = 0), and the overdot represents a derivative with respect totime Theequation ofmotion isthen found through Hamilton’s principle,which states thatthe evolution of the system isgoverned bythe path, x(t),which extremizesthe action (e.g.,

δS = 0) For a general Lagrangian, the extremization of the action (assuming vanishingboundary conditions) impliesthe Euler-Lagrange equations of motion,

Given a Lagrangian, L, defined in terms of configuration variables (x,x˙), the nian, H, defined in terms of the canonical variables (x,p), can be found via the Legendretransform of the Lagrangian –i.e.,

Hamilto-H(x,p)=xp˙ −L, (2.3)

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Settingp= ∂L/∂x˙ completes the proof.

Some immediatecorollaries followfrom this Byvirtue ofthe Euler-Lagrangeequations,equation (2.2), and by definition of the momentum, equation (2.4), it follows from thepreceding proof that,

re-f(x,p),1 which can be used to describe any given property of our system under question.Using equation (2.5), we have,

where f and g are functions of the canonical variables, (x,p)

1 Ignoring explicit time dependence for brevity.

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The Poisson bracketis important,asitwill giveusa directroutetothe quantum theoryvia the correspondence principle For later convenience, we list the most pertinent bracketrelations,

{f,H}PB=f˙ (2.8){x,p}PB=1 (2.9){x,x}PB={p,p}PB=0 (2.10)The second equation – also known as the (classical) canonical commutation relation – isperhapsthe mostprominent,asitsquantumcounterpartformsthebasis ofquantumtheory

We nowapply the precedingformalismtosolvethe simpleharmonic oscillator Considerthe Lagrangian for a simpleharmonic oscillator,

p=mx,˙ (2.12)which, unsurprisingly, isjustthe massof the oscillatortimes itsvelocity Fromhere and theLegendre transform, equation(2.3), the Hamiltonian is derived,

xt=x0cosωt+ p0

mωsinωt (2.15)

pt=p0cosωt−mωx0sinωt, (2.16)where x0 is the initial position of the oscillator and p0 = mx˙(0) is the initial momentum

As an aside, let me remark that: If we regard (x0,p0) as canonical variables and define thePoisson bracket with respect to these variables, such that canonical commutation relationsfor them hold [equations (2.9)-(2.10)] by definition, Hamiltonian dynamics then preservesthese relations for any and all times, t, i.e {xt,pt}PB = {x0,p0}PB = 1 with the Poissonbracket taken with respect to(x0,p0).2

Another way to write this solution is in terms of (complex) plane-wave solutions, ut =

Nexp(−iωt) (also calledthe mode functions) Here, N ∈R+ isa to-be-determined ization constant In these terms,

where α and α∗ are dimensionless, complex coefficients These coefficients play a nent role in the quantum theory, where they take on the form of creation and annihilationoperators, but more onthis later.

promi-We can extract the coefficients, α and α∗, directly by introducing the so-called Gordon inner product, which we define just below and which will be useful later when wediscuss fields and quantizationof such Consider two complex solutions to the equations ofmotion, f and g, and define πf = mdf/dt (and similarly for g) Then, the Klein-Gordoninner product between f and g is defined as,

Klein-(f,g)KG

def

= i f∗πg−gπf∗

Some genericobservations are inorder:

• The Klein-Gordon inner productis not aninner product inthe strict sense, since itisnot positive definite: if (f,g)KG >0, then (f∗,g∗)KG =−(f,g)KG <0.3 (Note that allreal solutions have vanishingKlein-Gordon inner product.)

• The Klein-Gordoninner productisconserved forsolutions tothe equationsofmotion,i.e d(f,g)KG/dt =0, which one can quicklycheck using the formalismabove and thedefinition of the Klein-Gordon inner product

• (f,f∗)KG =0.Thelastpropertyisimmediatelyusefulasitgivesusawaytodistinguishthemodefunctions,

ut and u∗t, since (ut,u∗t)KG = 0 We can also use this property to extract the coefficient αfrom equation (2.17) First, however, let us normalize the mode functions with respect tothis inner product such that,

Proof We prove the preceding statement Observe that πu = mdut/dt = −imωut bydefinition of ut=Nexp(−iωt), where ω>0 Then,

The final definition then holds for N =1/√2mω

3 Although, it will allow us to define a basis of complex solutions, which we will useful only when we discuss fields.

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With thenormalization ofthemode functionsinhandand usingequation(2.17),wefindthe complex coefficients, α and α∗, in termsof the Klein-Gordon inner product,

α=(ut,xt)KG and α∗ =−(u∗t,xt)KG (2.20)Thus, theoscillatorsystemhas beenformally solved,but wecontinuethediscussiontomakemore connections with the quantum theory

For curiosity’s sake, let us compute the Poisson bracket between α and α∗ by usingequation (2.20) We find that,4

{α,α∗}PB=−i(ut,ut)KG =−i (2.21)wherethe lastequality followsfromthenormalization ofthe modefunctions withrespecttothe Klein-Gordon inner product A similar relation will carry over to the quantum theorywhere α and α∗ will be replaced bythe creation and annihilationoperators

Proof We prove the preceding equations Consider the following setof equalities,

One can also extract the real coefficients x0 and p0, introduced in equations (2.15) and(2.16), byusing the Klein-Gordon inner product First we find the relations,

(ut,cosωt)KG =

rmω

2 and (ut,sinωt)KG =i

rmω

2 (2.22)fromwhich we compute,

4 Note that, with this normalization, α has dimensions of √

action (square root of length times momentum) since the Poisson bracket itself has dimensions of one over action.

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ω|α|2 = p

2 0

2m +

1

2mω

2x20, (2.26)which is justthe energy of the oscillator With quantum theory in mind,we seethat, upto

a factor of ℏ,the number |α|2 is the classical analogue tothe numberof particles (photons)

inthe system

Recap: We took a somewhat odd approach to solving the classical harmonic oscillator,but this will allow us to ‘go over’ to the quantum theory with relative ease (especially inthe case of fields) To summarize, letme distill the approach we took down to a handful ofsteps:

• Definea Lagrangian, L,and find the canonical variables,(x,p),from the Lagrangian

aswell as the corresponding Hamiltonian,H

• Establish the canonical commutation relations and dynamics (Hamilton’s equations)via the Poisson bracket

• Solve Hamilton’s equations interms of the mode functions,u

• Define the Klein-Gordon inner product and find the complex coefficients, α, therebysolving Hamilton’s equations ingeneral

These are almostthe exact steps that wewill take in orderto describeand quantize a field;

so itis beneficialto keep them inmind

The quantum oscillator

Wenowsetout tosolvethequantum harmonicoscillator,butabulkofthe workhasalreadybeen done! We need only make correspondence between the classical theory and quantumtheory and then transition fully into quantum mechanics by describing the quantum states

of the oscillator This will allow us to ease our way into a quantum field and photons inlater sections But first, allow me tolist some basic elements of quantum theory,which wewill be implicitly orexplicitly understood and appliedin this thesis

• Quantumstates: Thestateofaquantumsystemisgivenbyavector,|ψ⟩,whichresides

ina Hilbertspace, H (a complexvector space, equipped with aninner product,⟨·|·⟩;the “space of states") and which has unit norm, i.e ⟨ψ|ψ⟩ = 1 The vector spacestructure implies that a linear combination (a superposition) of quantum states isagainaquantumstate Thatis,giventwoquantumstates,|ψ⟩,|φ⟩∈H,andcomplexcoefficients, α and β, then α|ψ⟩+β|φ⟩∈H , with anextra algebraic condition on αandβ inordertopreserveunitnormforthesuperposedstate Forexample,assumingψandφareorthogonalwithrespecttotheinnerproduct,⟨ψ|φ⟩=0,then|α|2+|β|2 =1

• Density matrix: More generically, we can forma density matrix, ρ, which isa convexcombination (a probabilistic mixture) of pure quantum states That is, given a set of

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unit vectors, {|ψi⟩}, which are not necessarily orthogonal, one can define the densitymatrix,

ρ=X

i

pi|ψi⟩⟨ψi|, (2.27)where Trρ=P

ipi =1and pi ≥0 ∀i

• Composite Systems: The quantum state for a compositesystem, consisting of tems A and B with respective Hilbert spaces HA and HB, is a vector in the tensorproductspace,i.e |Ψ⟩∈HA⊗HB Notethat, for|Ψ⟩∈HA⊗HB,itisnotgenerallytrue that |Ψ⟩=|ψ⟩⊗|φ⟩ for some|ψ⟩∈HA and φ∈HB, which onlyholds when thesubsystemsare separable (independent, nocorrelations)

subsys-• Observables: Every observable, O, in the classical theory corresponds to a tian (‘real’) operator, Oˆ, in the quantum theory, which act on quantum states in theHilbert space and which has mean value ⟨ψ|Oˆ|ψ⟩, with respect to the quantum state

Hermi-ψ Independentrealizations/measurementsoftheobservableO takeonvaluesfromthespectrum(theeigenvalues)ofOˆ andoccurwithafrequency,determinedbyprobability

of occurrence for a given quantum states These probabilities formally correspond tothe squared coefficients of the quantum state ψ written inthe eigen-basis of Oˆ.With some basic elements of quantum mechanicsin hand, we now‘go over’to quantumtheoryfromtheclassicaltheoryinonestrokebythe methodofcanonicalquantization That

is, we make the physical correspondence between Poisson brackets, taken between ables inthe classical Hamiltoniantheory, and commutators,taken between the observable’sHermitian-operator-counterparts in the quantum theory Given two classical functions ofcanonical variables, A and B, and their operator counterparts, Aˆ and Bˆ, we assume thecorrespondence(up tooperator-ordering ambiguities),

observ-{A,B}PB→ [A,ˆ Bˆ]

where [A,ˆ Bˆ] =AˆBˆ−BˆAˆ is the commutator and ℏ is Planck’s reduced constant We nowapply this correspondence to our classical system discussed previously: given the canonicalvariables (x,p) and Hamiltonian H, define the Hermitian operators x → xˆ, p → pˆ, and

H → Hˆ Applying the correspondence to the Poisson bracket relations, equations (2.10), wehave,

(2.8)-d ˆf

dt =

1

iℏ[ ˆf,Hˆ] (2.29)[ˆx,pˆ]=iℏI (2.30)[ˆx,xˆ]=[ˆp,pˆ]=0, (2.31)where ˆis some function of the canonical operators, xˆ and pˆ, and I is the identity operator(which we will drop for brevity)

Astoourprimaryexample,considertheHamiltonianforaquantum harmonicoscillator,

We find Hamilton’s equationsin operatorformbythe commutation relations (2.29)-(2.31),

dˆxt

dt =pˆt/m and dˆpt

dt =−mω

2xˆt, (2.33)which is, as in the classical case, a coupled set of first-order differential equations for thecanonical operators xˆ and pˆ These equations are easilysolved,

ˆ

xt=xˆcosωt+ pˆ

mωsinωt (2.34)ˆ

pt=pˆcosωt−mωxˆsinωt (2.35)One can check that evolution under the Hamiltonian operator, Hˆ, preserves the canonicalcommutator, i.e [ˆxt,pˆt]=iℏ

One can rewritethese operator equationsin terms of complexmode functions,

ˆ

xt=ˆaut+ˆa†u∗t, (2.36)where ut = Nexp(−iωt) is the mode function and aˆ and aˆ† are non-Hermitian operators(generalized from complex numbers, α and α∗), called the annihilation creation operators,respectively We extract these operator coefficients via the Klein-Gordon inner product, asbefore,butbeforedoingso,were-scaleour previousdefinition, equation(2.18),bythe factor

ℏ inorder tomakes the creation and annihilationoperators dimensionless

Given two complex solutions to the classical equations of motion,f and g, and defining

πf = mdf/dt (and likewise for g), the re-defined Klein-Gordon inner product between fand g reads,

[ˆa,ˆa†]=1 (2.39)Proof We prove the last equality Given the relations (2.38), consider the following set ofequalities,

In the first equality, we used equation (2.38) In the second and third equalities, we pandedthe Klein-Gordon innerproductand thenused the linearityof the commutator Fortheremainingequalities,weusedtheanti-symmetryofthe commutatortogatherliketerms,assumed the canonical commutation relations (Hamiltonian evolution preserves these rela-tions),andintroducedtheKlein-Gordoninnerproductforthemodefunction,u Normalizingthe mode functionthen establishes the equalitywhich wasto be shown.

ex-As before,onecan findarelationbetweentheannihilationand creationoperators,aˆandˆ

a†,and thecanonical operators,xˆand pˆviathe Klein-Gordon innerproduct Isimply writedown the result,

ˆ

a=

rmω

2ℏ

ˆ

x+i pˆmω

ℏω/2,whichisknownasthe vacuumenergy Thevacuuminthiscasebeingthegroundstate

ofthe aboveHamiltonian,with energyℏω/2 Thequantum vacuumisstarklydifferentfromthe “classical vacuum”,which has trivial properties(zero energy, etc.)

At this stage, we have some clear definitions for the observables of our theory: theyare given as general functions of the canonical operators, xˆ and pˆ The quantum oscillatorsystem is only half-solved at this point, however, because we have not yet determined thequantum states to which these observables are measured with respect to This is now ourgoal

From the quantum vacuum, all other quantum states originate This is because we canbuild abasis of quantum states (the Fock basis),which are eigenvectors of the Hamiltonianoperator,bysuccessiveapplicationsofthe creationoperatortothe vacuumstate This basisspans an infinitedimensional Hilbert space (the Fock space), which any quantum state canthenbewrittenintermsofbysuperposition Toshowthis,letusfirstarguethatthereexists

a groundstate – the vacuum state, |0⟩ – with energy ℏω/2 and which satisfies ˆa|0⟩=0.Physicist’s proof We prove the preceding statement Define the number operator, nˆ def= aˆ†ˆa,such that

ˆ

H =ℏω(ˆn+1/2), (2.42)Now, define the vacuum state, |0⟩, to be the minimum energy state of the Hamiltonianoperator, and thusthe minimumeigenvector of the number operator; i.e.,

ˆ

n|0⟩=µ|0⟩, (2.43)12

with µ being the smallest eigenvalue of nˆ We argue that µ=0 onthe groundsof stabilityforthephysicalsystem(ontheassumptionforthe existenceofaminimumenergystate)andthat ˆa|0⟩=0 We argue bycontradiction.

Consider the unnormalized state,

|ψa⟩=ˆa|0⟩ (2.44)Then,

ˆ

aˆn|0⟩=µ|ψa⟩ (2.45)Now,

ˆ

anˆ|0⟩=aˆˆa†ˆa|0⟩ (2.46)

=(ˆn+I)|ψa⟩, (2.47)which, from equation(2.45), impliesthat,

(ˆn+I)|ψa⟩=µ|ψa⟩ =⇒ nˆ|ψa⟩=(µ−1)|ψa⟩ (2.48)But this contradicts our assumption that there exists aground state, unless µ=0 to beginwith Furthermore, for our assumption to generally hold, it must also be so that the state

ψa isthe trivial state, i.e ˆa|0⟩=0

We now define the Fock basis, which serve as a basis for the Hilbert/Fock space, F,fromwhichallquantum statescan bebuiltfrombysuperposition Ilistthe mostimportantproperties and definitions and then prove various claims:

• Thesetoforthonormalbasisvectors{|n⟩},forn ∈N,defineabasisfortheFockspace,

F They satisfy,

⟨m|n⟩=δmn, (2.49)with δmn being the Kronecker delta-function, such that δmn = 1 for m = n and 0otherwise

• The basis states (also called Fock states or number states), |n⟩, can be constructedfromthe vacuumby successive applications of the creation operator, ˆa†, such that

• The setof quantum statesare then definedas,

n

|ψ⟩ |ψ⟩=

We now prove the first threestatements.

Proofs We first show that the single-particle state (defined below) is an eigenstate of thenumber operator, with eigenvalue equal to1 We then show, byinduction, that the vector,

|n⟩, constructed by n-successive applications of ˆa† to the vacuum state, is an eigenstate ofthe number operator, with eigenvalue equal to n Orthonormality of the vectors {|n⟩} isthen proven

Consider the single-particle state,

|1⟩=ˆa†|0⟩, (2.53)and observe that,

weused the relation[ˆa,ˆa†]=1and the definition ofthe single-particle state Thus, one seesthat the single-particle state isan eigenstate of the number operator, with eigenvalue equal

toone

We now prove a similar relation for the two-particle state The generalization to the

n-particle state will follow byinduction Consider the two-particle state,

|2⟩=N2ˆa†2|0⟩, (2.54)where N2 issome normalization constantsuch that ⟨2|2⟩=1 Observethat,

to 2 The generalization to the n-particle state, |n⟩ = Nnˆa† n|0⟩, follows by a repeated

14

applicationoftheabovesetofequalities Thus,then-partidleFockstate,|n⟩,isaneigenstate

of the number (Hamiltonian) operatorwith eigenvalue equal ton [En =ℏω(n+1/2)]

We nowproveorthonormalityof theFockstatesand providethe normalizationconstant,

Nn, while doing so Consider the Fock states |n⟩ and |m⟩ and take n > m without loss ofgenerality Then, considerthe following setof equalities,

⟨m|n⟩ = δmn, as was to be shown As an aside, note that there is no upper-bound for thevalueof n It thusfollows that the Fock space isan infinitedimensional Hilbertspace.With the dynamicsof the observablesthussolved and the quantum states of the system

inhand, our job is complete

Recap: Wewereabletosolvethequantumharmonicoscillatorwithswiftnessviacanonicalquantization,whicheffectivelyallowedustotakemostofourresultsfromtheclassicaltheoryandapplythemdirectlytothequantumsystem Theonlymissingingredient(andaltogetherabsent in our classical discussion) was to find the quantum states of our system I lay outthe procedure we tookfor quick reference:

• Given the classical, Hamiltonian description of the system, ‘go over’ to the quantumtheory bycanonical quantization, i.e

{A,B}PB→ [A,ˆ Bˆ]

iℏ Here, A and B are classical observables, which are general functions of the canonicalvariables, x and p, and Aˆ and Bˆ are their Hermitian operator-counterparts and areoperator-functionsof the canonical operators,xˆ and pˆ

• Solve the operator formof Hamilton’s equations for the canonical variables,

thereby determining the time evolution for any observable, Aˆ(ˆx,pˆ) Note that, thesolutions of these equations are, more or less, equivalent to the classical equations ofmotion.

• Determine the quantum states of the system For the quantum harmonic oscillator,this was found byconstructing the Fock basis, {|n⟩}, which was formed by successiveapplications of the creation operator tothe quantum vacuum,|0⟩ – i.e.,

|n⟩= ˆa

† n

√

n!|0⟩. (2.56)Thus,foroscillator-likesystems,constructtheFockbasisfromthevacuum,fromwhichany quantum state can written interms of by linearity

Photons

Weare almostinthepositiontodiscussthequantumofelectromagneticfield–i.e., photons.Formally,wemusttransitionfrompoint-likesystems tosystemsdescribedbyfields–entitieswhich takeonavalueateachpointinspace andtime Thiswillrequire someslightrewiring

ofoutthoughtprocesseswhendiscussingtheformalismbutwillnonethelessfollowanalogoussteps we took earlier indescribing point-like systems

Quantizing the electromagnetic field also presents some complexities and mathematicalmaneuvering, owing to complications when dealing with the polarization of the field Wewillbypassthese difficulties,however,byconsideringasimplemodelfor theelectromagneticfield: a massless scalar field This approximation will be sufficient for our purposes, andalmosteverything we havetosay laterabout photonscan be thoughtof directlyinterms ofthe formalism and concepts presented therefrom

We start bypresentingthe Lagrangian andHamiltonian formalismsfor aclassicalscalarfield in one dimension We will apply this formalismto solve the equations of motion for afree,masslessscalar-field(themasslessKlein-Gordon field),andthen moveontoaquantumdescription of the field via canonical quantization We will see many similarities with theoscillator system alongthe way, and hopefully, these similarities will makethe presentationseem familiar and make the quantum description of the field seem like a natural extension.For background on (relativistic) quantum field theories, in general, see David Tong’slectures[20] With respect tothe quantized electromagnetic field and its place“in the lab",see the many books on quantum optics, e.g [21, 22] We will use the calculusof variations

at the level of, e.g., functional derivatives when discussing fields For background on this,seeGelfand’s book [23]

The classical fieldFor the point-like oscillator, one thought of x as both a canonical variable describing thephysical system and a coordinate For fields however, the field, ϕ(x), itself is a canonicalvariableandxisjustacoordinate (whichplaysnopivotalrole)atwhichpointthefieldtakes

on a particular value Hence fields are the primary objects, not coordinates, and we mustlearn how to handle them in the context of the Lagrangian and Hamiltonian formalisms

16

Recall that,in ourprior endeavours, we tookvariations withrespect tothe variables x,x,˙ p,etc In our current situation, we will be taking variations with respect to the field, ϕ, andits conjugate momentum, π, which are themselves functions of coordinates To make thetransitionfrompointparticles tofieldsmoreefficient,wetakeashortdetourintofunctionalsand their variation.

• Functionals: In simple terms, a functional is a function of a function We define afunctional, F, of the “field variable”, ϕ, which is itself a function of coordinates, x, inthe following manner,

F[ϕ]=

Z

dxf(ϕ(x),∂xϕ), (2.57)where f(ϕ,∂xϕ) is typically a polynomial in ϕ and its first derivatives This can beextended tofunctions of higher-orderderivatives, but itwill not be useful for us

• Variation of a functional: The variation of a functional, F, with respect to a field, ϕ,assuming vanishing boundary conditions, isgiven by,

where coordinate dependence onx is understood

We now take on fields Consider the action functional, S[ϕ], for a scalar field, ϕ(x,t),which we write interms of the Lagrangiandensity, L,

tothefieldϕ Generically,theextremizationoftheactionwithrespecttothefield(assumingfixedendpoints) implies the Euler-Lagrange equationsof motion,

Proof We prove the preceding implicationassuming First, note that,

δS =

Z

dtdxδS

δϕδϕ(x,t). (2.61)

by definition of the variation of a functional Hamilton’s principle, δS def= 0, then implies

δS/δϕ =0 fromthe previous equation Now, assuming that the Lagrangian density isonly

afunction of ϕ and itsfirst derivatives and assuming vanishingboundary conditions for thefield variations,we have that the functionalderivative of the action isgiven by,

H[ϕ(x),π(x)]=ϕπ˙ −L, (2.63)with,

wherethe Poisson bracket betweentwo functionals F and Ghas beenimplicitlydefinedas,

Proof We showthat δϕ(y)/δϕ(x) =δ(x−y) Byextension,asimilarrelationfollowsfor π.Consider that,

ϕ(x)=

Z

dy δ(x−y)ϕ(y), (2.76)and thusδ(x−y)ϕ(y)definesa functionaldensityforthe functionalϕ But,bydefinition ofthe variation of afunctional,

δϕ(x)def=

Z

dyδϕ(x)

δϕ(y)δϕ(y). (2.77)After taking a variation of the former equation and equating the result with the latter, weare led to conclude δϕ(y)/δϕ(x) = δ(x−y) The canonical commutation relations can befound by using this result and applyingthe definition of the Poisson bracket

We now apply this formalism tofind and solve equations of motionfor afree, massless,scalar field We take the speed of light to be one in what follows The Lagrangian densityfor afree, massless,scalar field is given by,

with H being the integrandin the equation beforelast It then follows that

˙

ϕ=π and π˙ =∂x2ϕ, (2.82)which are a coupled set of partial differential equations By taking a time derivative ofHamilton’s equation for ϕ, we are led to one second-orderpartial differential equation,

∂t2ϕ−∂x2ϕ=0, (2.83)alsoknownasthe(massless, free)Klein-Gordon equation Thisisawaveequation,whichwecan readily solve in Fourier space I write a generic solutionin terms of the complex modefunctions, uk(x,t),

ϕ(x,t)=

Z

dk(αkuk(x,t)+α∗ku∗k(x,t)) (2.84)

π(x,t)=−iZ dkωk(αkuk(x,t)−α∗ku∗k(x,t)) (2.85)

whereαk arecomplex coefficientsand uk=Nkexp[−i(ωkt−kx)]isthe mode function,with

Nk ato-be-determined normalization constant Here, ωk =|k|is theoscillation frequencyofthe Fourier modes and k ∈ R is the wave-number This is similar to the classical oscillatorsolution, however, in this case, we have an infinite set of oscillators, one for every point x,which have arange of possible oscillation frequencies, ωk

We now define the Klein-Gordon inner product for genericsolutions to the equations ofmotion This will allow us to determine the coefficients αk from the field ϕ and the modefunctions{uk} Consider twogeneric(complex)solutions totheequationsof motion,f(x,t)and g(x,t), and define pf =f˙ (and similarly for g) Then, the Klein-Gordon inner productbetween f and g is definedas,

Proof We prove the first equation and show Nk = 1/√4πωk The other equations can befound using similar methods Let

uk(x,t)=Nke− i ( ω k t − k x ), (2.90)

20

such that,

pu k =∂tuk =−iωkuk (2.91)Then, computing the Klein-Gordon inner product between the mode functions uk and ul,the following set of equalitiesare found,

The first and second equalities follow from equations (2.89) and by expanding the Gordon inner product The third, fourth, and fifth equalities follow from linearity of thePoisson bracket and by using the canonical commutation relations, equations (2.74) and(2.75) The finaltwo equalitiesfollowfromthe definitionofthe Klein-Gordon innerproductand the normalization of the mode functions.

Klein-As a final calculation, let uswrite the Hamiltonian in terms of the complex coefficients,

α andα∗ Using thedefinition ofthe Hamiltonian,equation(2.80),the classicalsolutionstoHamilton’s equations, equations(2.84) and (2.85), and aftera few integrals,one finds,

The quantum field

We now “go over" to the quantum theory in a similar manner as we did for the quantumharmonic oscillator We need to take some care, however, when dealing with the quantumstates, due to the continuous nature of fields and formally divergent quantities etc I will

dowhatever isconvenientand whatever gets ustothe answerswe needin themost efficientand sensible fashion (in my perspective)

Recall our procedure for producinga quantum theory fromthe classicaltheory, which Ireproducehere for convenience:

• Given the classical, Hamiltonian description of the system, ‘go over’ to the quantumtheory bycanonical quantization, i.e

{A,B}PB→ [A,ˆ Bˆ]

iℏ Here, A and B are classical observables, which are general functions of the canonicalvariables, ϕ and π, and Aˆ and Bˆ are their Hermitian operator-counterparts and areoperator-functionsof the canonical operators,ϕˆand πˆ

• Solve the operator formof Hamilton’s equations for the canonical variables,

22

• Determine the quantum states of the system For the quantum harmonic oscillator,this was found byconstructing the Fock basis, {|n⟩}, which was formed by successiveapplications of the creation operator tothe quantum vacuum,|0⟩ – i.e.,

|n⟩= ˆa

† n

√

n!|0⟩. (2.102)Thus,foroscillator-likesystems,constructtheFockbasisfromthevacuum,fromwhichany quantum state can written interms of by linearity

We follow this procedure to a tee We start by writing out the canonical commutationrelations Let ϕˆ and ˆπ = ∂tϕˆbe the canonical field operators for the massless, scalar field.Then,

dFˆ

dt =

1

iℏ[Fˆ,Hˆ] (2.103)[ ˆϕ(x),πˆ(y)]=iℏδ(x−y) (2.104)[ ˆϕ(x),ϕˆ(y)]=[ˆπ(x),πˆ(y)]=0, (2.105)where Fˆ is some operator functional of the canonical variables, ϕ and π The Hamiltonianoperator isgiven as,

∂tϕˆ=πˆ and ∂tπˆ =∂x2ϕ.ˆ (2.107)Towardssolvingtheseequations,thestoryisthesameasintheclassicalfieldcase SoIsimplywritedown thesolution,interms ofthecomplex mode functionuk(x,t)=Nkexp[−i(ωkt−kx)],

ˆ

ϕ(x,t)=

Z

dkˆakuk(x,t)+ˆa†ku∗k(x,t) (2.108)ˆ

π(x,t)=−iZ dkωk

ˆ

akuk(x,t)−aˆ†ku∗k(x,t) (2.109)

where ˆa and ˆa† are the annihilation and creation operators, respectively We can extractthese operators from the field via the Klein-Gordon inner product per usual, but first aredefinition,

(f,g)KG

def

= iℏ

23

only changes the normalization factor to Nk = ℏ/(4πωk) From this definition and thefield operator, equation(2.108), we have,

ˆ

ak=(uk,ϕˆ)KG and aˆ†k=−(u∗k,ϕˆ)KG (2.111)Using this relationand the canonical commutation relations, one can prove that,

[ˆak,ˆa†l]=δ(k−l) (2.112)[ˆak,ˆal]=[ˆa†k,ˆa†l]=0 (2.113)With all this in hand, we re-write the Hamiltonian operator in terms of the creation andannihilation operators After some algebra, a few integrals, and an application of the com-mutator (2.112),one finds,

sothat the fieldis justacontinuum approximationfor, e.g., alargebut finite setofcoupledoscillators, inwhich case the vacuum energy reducesto afinite sum (alarge term, perhaps,but finite),oritcould besothatour fieldtheory isonlyappropriateuptosomehigh energyscale (up to an upper value of ω), at which point a more sound theory steps in and savesthe day In any case, we will not think too much into it Instead, we will do what anyrespectable physicist does and simply sweep this infinity under the rug It will not bother

First, we build a Fock space, with suitably normalized basis states, by restricting to asingle-modewave-packet WewillgeneralizethistoM modes later Considerawave-packet,

f,which isa solutionto the equations of motionsatisfies,

(f,f)KG =1 (2.115)Since {uk} form abasis of solutions tothe equationsof motion,we may expand f as,

f =

Z

dk(αkuk+βku∗k), (2.116)24

af def

= (f,ϕˆ)KG (2.118)From which it follows,

[ˆaf,ˆa†f]=(f,f)KG =1 (2.119)From equation (2.118)and the expansion of the field operator(2.108), wecan expand ˆaf interms of the mode operators, ˆak and ˆa†k,

ˆ

af =

Z

dkα∗kˆak−βk∗ˆa†k (2.120)Proof We prove the preceding equation Consider the following setof equalities:

KG =(g,f)KG, andfor the final equality, we have used the fact that αk=(uk,f)KG and βk =−(u∗

k,f)KG.Now define the vacuum state, |0⟩, to be the ground state of the Hamiltonian operator,equation (2.114), satisfying ˆak|0⟩= 0 ∀ k and ˆaf|0⟩ =0 With this choice, it follows that

• The setof orthonormal basis vectors, {|nf⟩}, for n ∈N, define a basis for the mode Fock space, Ff, satisfying

single-⟨mf|nf⟩=δmn, (2.126)where δmn is the Kroneckerdelta

25

• Thebasis states(Fockstates), |nf⟩,can beconstructedfromthe vacuumbysuccessiveapplications of the creation operator, suchthat,

|nf⟩= ˆa

† n f

√

n!|0⟩. (2.127)From this construction, it follows that the Fock states are eigenstates of the numberoperator,nˆf =ˆa†fˆaf with eigenvalue n

• The time dependent annihilationoperator for the wave-packet is given as

ˆ

af(t)=

Z

dkα∗kˆake− i ω k t, (2.128)fromwhich allother dynamicalquantities of interest can be found

• BycomputingthematrixelementsoftheHamiltonianinequation(2.114)inthemode Fock basis of f,it can be shown that,

single-⟨mf|Hˆ|nf⟩=ℏωfnδmn, (2.129)where

ωf =

Z

dkωk|αk|2 and |αk|2 =|(uk,f)KG|2, (2.130)andthedivergentvacuumenergyhas beendiscarded Hence,althoughthestate|nf⟩isnot an eigenstate of the Hamiltonian, its energy nevertheless takes ondiscrete values,withanaveragegivenbyℏωfn Forexample,thestate|1f⟩correspondstothequantumstateofasinglephotonoccupyingthewave-packetf,whichcarriesaquantumofenergy

ℏωf onaverage (averaging over k) Iff issharply peaked around some mode k0, then

ℏωf ≈ℏωk 0,which isjust the Einstein-Planckrelationforthe energy ofa photonwithfrequencyωk 0

We now generalize the single-mode construction to an M-mode construction, by ducing a finite set of functions, {fi}M

intro-i =1, which are orthonormal in the Klein-Gordon innerproduct, i.e (fi,fj)KG =δij The Fock space is then given as a tensor product over all Mmodes, F{f } =N

iFf i Of physicalinterest is a setof wave-packets which are dynamicallydecoupledunderfreeHamiltonianevolution Forinstance, leteachfi havesupportonlyover

afinitesectorink-space,andleteachofthesesectorsnotoverlap(therearethusM dentsectors) Thatis,if fi =R

indepen-dkαi,kuk,then wesupposethat supp(αi,k) isnon-trivialonly

infinitedomainand that supp(αi,kαj,k)=∅ ifi̸=j Therefore,underthis prescriptionandassumingfreeevolutiongeneratedbythe Hamiltonianinequation(2.114),eachmode-sector,

i,is dynamicallydecoupled fromany othersector, meaning thatthere isnomixing betweendifferent wave-packet modes, fi; however, there is mixing between the k-modes containedwithin agiven sector One can think of the different degrees of mixing as "external" versus

"internal" dynamics, where the external dynamics describes the interactions between thedifferent sectors (between wave-packets) and the internal dynamics describes the interac-tions occurring within a given sector (an individual wave-packet) It is often the case that

26

the external dynamics are physicallymore relevant (think of scattering betweenM initiallyindependent wave-packets), and one must simply take care that the internal dynamics arenot affecting the external dynamics inanimpactful manner.

Beforeweconcludethissection,I wouldliketointroducethequadratureoperators forthewave-packet f, and make some comments which will, hopefully, permit an easy transitionintothe formalismof thenextsection Definethe quadratureoperators,Xˆf and Pˆf,as,

ˆ

Xf def

=

r12

ˆ

af +ˆa†f (2.131)ˆ

Pf def

= −i

r12

ˆ

af −ˆa†f (2.132)Observethat,

ℏωf

2

ˆ

Pf2+Xˆf2=ℏωf(ˆa†fˆaf +1/2), (2.133)which is reminiscent ofthe single, quantum-harmonic oscillator See equation (2.41) Theseoperators satisfy,

[Xˆf,Pˆf]=i, (2.134)foralltimesandundergo timeevolutionviathe Hamiltonianofequation(2.114),leadingto,

ˆ

Xf(t)=

s1

2ωf

ˆ

af(t)+ˆa†f(t) (2.135)ˆ

af(t)−aˆ†f(t), (2.136)

with ˆaf(t) given by equation (2.128) By inspecting equations (2.108) and (2.109), we seethat thequadrature operators,Xˆ andPˆ,are quitesimilartothe canonical operators,ϕˆandˆ

π Duetotheir closecorrespondencewiththe canonicaloperators,IwillrefertoXˆ and Pˆasthe canonicaloperatorsthemselves In theforthcoming sections,I use thislanguage withoutdiscretion

2.2 Photon dynamicsOurdiscussiononphotondynamicswillberestrictedtoquadraticinteractionsandsymplectictransformations For generic remarks regarding systems and states associated with suchdynamics, see Serafini’s monograph [24] (which I highly recommend) A large part of theliteratureinquantumopticsfocusesonpreciselythissubsetofquadraticinteractions Hence,for anquantum-opticalperspective,seee.g [21, 22] From here on, unlessstated otherwise,

I set ℏ=1

27

Some notationOuranalyseshastodowithafinitesetofinteractingquantumoscillators(ormodes) Hence,

if we considern modes, then we are dealingwith a Fock space

[ˆxi,pˆj]=iδij (2.137)[ˆxi,xˆj]=[ˆpi,pˆj]=0, (2.138)where δij isthe Kroneckerdelta-function They are related tothe annihilation and creationoperators, ˆa and ˆa†, via

ˆ

x= √1

2 ˆa+ˆa

†
and pˆ=−√i

2 ˆa−ˆa

†
(2.139)

These canonicaloperators are actuallythe (dimensionless) quadrature operatorsof the fieldintroduced in the preceding section, however, I will simply refer to them as the canonicaloperators

It will beefficient to definea 2n×1 vector of canonical operators,

ˆrdef= (ˆx1,pˆ1, ,xˆn,pˆn)⊺, (2.140)which compactly describes all the field modes and their canonical momenta.5 From thecanonical commutationrelations above,one can express the commutationrelations betweenallmodes via,

[ˆri,ˆrj]=iΩij, (2.141)where Ωij are the matrix coefficients of the 2n×2n symplectic form,

inˆi The extra number, n/2,isdue, essentially,to thevacuum of eachmode contributing afactor of 1/2tothe energy of the field A similar relationwas found in equation (2.133)

c-5 We borrow this notation from Serafini [24].

28

Quadratic Hamiltonians and symplectic transformationsForinteractions betweenvariousmodes, wewill restrictourselves toquadratic Hamiltonians– implying, e.g., that the equations of motion for the fields are linear in the field variablesthough generally coupled In what follows, I will discuss dynamics in a highly simplisticmanner,makingnoreferencetothefieldvariablesthemselves,andwillrestrictthediscussiononlytothe transformationsof thequadrature variablesintroduced above I will notprovidesupport for this approach immediately,but will return tothis matterlater.

Dynamics and the Bloch-Messiah decompositionConsider a general, time-independent quadraticHamiltonian,

ˆrk(τ)=(SH)kiˆri, (2.145)where the 2n×2n symplectic matrix,

SHdef= exp (ΩHτ), (2.146)hasbeendefined Here, ΩisthesymplecticformandHistheHamiltonianmatrix Equation(2.145) shows, quite compactly, that the canonical operators at time τ are just a linearcombinations of the canonical operators atτ =0

Proof We prove the preceding equations Recall the quantum version of Hamilton’s tions [see, e.g., equation (2.55)] Then, considerfollowing setof equalities,

of the canonical form was used This is a first order differential equation, with a solutiongiven byequation (2.145)if SH =exp(ΩHτ)

6 Note that I have restricted to purely quadratic Hamiltonians This is a minor point, and I will come back to it.

29

Recall, from quantum theory, that Hamiltonian dynamics corresponds to evolving thesystem by aunitary operator, Uˆτ =exp(−iHˆτ),such that Uˆ†

τˆr ˆUτ =ˆr(τ) (Heisenberg tion) It thusfollows that, if the Hamiltonian is quadratic,then

evolu-ˆ

Uτ†ˆr ˆUτ =SHˆr (2.147)This correspondencebetweenunitary evolutionand symplectictransformationsallows ustodiscard discussion of the former infavor of the latter, which effectively reducesevolution tosimple matrix multiplication

With thiscorrespondenceinhand,itisalsouseful toknowwhatgenerallyhappensif,forexample, we concatenate successive symplectic transformations For example, first evolve,

in the Heisenberg picture via a unitary Sˆ1, then Sˆ2 then Sˆ3 etc., such that QN

n =1Sˆn givesthe full unitary process One can then show that this concatenation of unitary processescorresponds to the transformation,

− 1 etc.) However, atthe levelof symplecticmatrices, {Sn}, the evolutionfollows the actual sequence of events, from 1 to N, similar to the Schrödinger evolution ofquantum states

Proof We prove the preceding equationfor asequenceof two unitary operators,Sˆ1 and Sˆ2.The full result follows by induction Consider the unitary operator, Uˆ = Sˆ2Sˆ1, where theunitary operatorSˆl has acorresponding symplecticmatrix, Sl,and considerthe Heisenbergevolution of the canonical operator,ˆrk Then,the following setof equalities hold,

=(S2)k j(S1)j iˆri

=(S2S1rˆ)k.For the third and fourth equalities, we used the relationbetween Heisenberg evolution andsymplectic transformations [equation (2.147)] and then moved the matrix coefficients of S2

around freely The remaining equalities were found by the same sort of shuffling For the

30

final equality, weequated the sumover matrixelements tothe productof the matrices withproper ordering.

The generalization toa concatenation of N unitary operatorsfollows by induction For,let Sˆ12 def

= Sˆ2Sˆ1 with corresponding symplectic matrix S12

def

= S2S1 and introduce anotherunitary evolution Sˆ3, with correspondingsymplectic matrix S3, suchthat Uˆ =Sˆ3Sˆ12 Fromabove, it follows that the unitary evolution of the canonical operator, ˆrk, equates to thematrix product (S3S12ˆr)k = (S3S2S1ˆr)k A similar relation thus holds for N such transfor-mations, by extension

The matrices S are calledsymplectic since theygenerally obey the relation,

S⊺ΩS =Ω, (2.149)which, e.g., musthold inorder for Hamiltoniandynamics topreservethe canonical commu-tation relations (2.141) Matrices which obey this relationformthe symplectic group I willnot delve into the mathematical particulars concerning this group, but I will point out onevery important observation That is, any symplectic transformation, on a finite number ofmodes, may be decomposed as a product of single-mode squeezers, phase shifters, and two-mode beamsplitters [25] This is known as the Bloch-Messiah decomposition To provide ashort description: Single-mode squeezers are single-mode transformationswhich reduce thevarianceinone quadrature ofthe mode andincreasethe varianceinthe otherquadrature ofthemode, allwhilesaturatingtheHeisenberguncertaintyrelation(forvacuuminputs) Thisprocess also corresponds to particle creation in a single mode On the other hand, beam-splitters correspond to “passive” scattering events between two modes, while phase shiftscorrespond tofree evolution At the level of symplectic transformations,beamsplitters andphase shifters induce orthogonal transformations (e.g rotations) and form a subgroup ofthe symplectic group Suchtransformations are passive, inthe sense that they preserveex-citation(photon)number Thiscanbe quicklyseen byexaminingthe invarianceofequation(2.143)whenthecanonicaloperatorsare subjecttoasymplectictransformation,O, obeyingthe orthogonality condition, O⊺O =I

Sketch of proof The decomposition of any (real) symplectic matrix into passive mations and single-mode squeezers can be made more precise First, observe that any realmatrix, S, admitsa spectral decompositionof the form,

transfor-S =O1DO2, (2.150)where D is apositive semi-definite, diagonal matrix, with the singularvalues of S along itsdiagonal, and {Ok} are orthogonal matrices (Ok⊺Ok = I) For S a symplectic matrix: (i){Ok} are orthogonal and symplectic7, and (ii) detS = 1, implying that the singular valuematrix, D, ispositive definite Furthermore,one can show that,

31

with dk >0 fromthe singularvaluedecomposition For, consider that,

SΩS⊺=(O1DO2) Ω (O2⊺DO⊺1)

=(O1D) Ω (DO1⊺),but SΩS⊺ = Ω, since S is symplectic Thus, using the fact that Ok are symplectic andorthogonal, itmust be sothat

DΩD=Ω.WritingD as adirect sum of two dimensionalblocks,

and substituting intothe previous relation, we find

which implies d′

k = 1/dk, as claimed Observe that D is just a direct sum of single-modesqueezingtransformationswith,e.g., dk =er k correspondingtosqueezing(anti-squeezing)inone quadrature of the mode k and 1/dk =e− r k corresponding toanti-squeezing (squeezing)

inthe other quadrature, whererk isthe squeezingstrength

Here, the orthogonaltransformations,O1 and O2,are elementsof thesymplecticonalgroupwhichisisomorphictotheunitarygroup,U(n) Hence,onecandirectlyinterpret

orthog-O1 and O2 as correspondingtosome finite-dimensionalunitary matrix Furthermore,it can

be shown that any finite-dimensional unitary matrix may be decomposed into a set of twomodebeamsplittersandphaseshiftersviatheso-calledReck-Zeilingerdecomposition[26] Itthusfollows that any symplectictransformationcan be decomposed asaset ofindependentsingle-mode squeezers sandwiched between two linear-optical circuits

Now to return to the physics of the matter One must question how well (or if at all)

do the simplistic dynamics thus posed correspond to physical scenarios of interest? Theanswer is quite well if, for instance, we restrict ourselves towell-posed scattering problems,which is what we will limit ourselves to in this thesis In this context, we imagine well-defined modes, and thus well-defined Fock spaces, “far away" from some interaction region(quotations because this applies to spatially and temporally localized interactions) Theseasymptotic regions, which I will call the in and out regions, are presumably described bysimple Hamiltonians, which can be easily handled and which define the underlying modestructure as well as well-defined in/out Fock spaces See Figure 2.1 for an illustration We

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Figure 2.1: Scattering of the “in-modes" to the “out-modes" Asymptotically, far in thepast/future and spatially far away, there exists a well-defined set of in/out modes Thein-modesscatter tothe out-modesbyimpingingon aninteraction region,which fills afinitespace-time volume of size V∆t Only quadratic interactions occur within this region, how-ever thespace-time dependenceof theseinteractions isgenerically complex,thus permittingmodes of different wavenumbers and/or frequencies tomix ina non-trivial way.

then formally view the dynamics – the scattering processes – as a mapping from the modes to the out-modes That is, we assume there exists a bijective map, Φ, such that

in-Φ : Fin → Fout, where Fin/out are the in/out Fock spaces just mentioned At the level

of quantum states, this corresponds to some unitary, Uˆin→out, that takes |ψ⟩in → |ψ⟩out,where |ψ⟩in ∈ Fin and |ψ⟩out ∈ Fout What, then, could this unitary be? At the level ofin/out canonical operators, which are well-defined by assumption, this unitary correspondsprecisely to a symplectic transformation, which gives the output canonical operators as alinear combination of input canonical operators and vice versa (bijection) This mustbe so

if the interactions are only quadratic; the equations of motion simply do not allow for anyotherpossibility Furthermore,since the scatteringprocess mustcorrespond toasymplectictransformation, it permits a decomposition into an array of linear-optical networks andsqueezers Therefore, to describe all physical processes of interest, at least the quadraticscattering-like interactions that we are restricting ourselves to, we need only provide anequivalentopticalnetwork,describedbyasetofappropriatelytunedparameters–squeezingstrengths, beamsplitter transmissivities, etc This is the approach I take in this thesis todescribe physicalphenomena

Before movingon, let me elaboratefurther onwhat I meanby “simpleHamiltonians"inthein/outregions Simplyput,ImeanthattheHamiltoniansare(locally)time-independentand also(butnot necessarily) not-too-complicatedintheir spatialdependence,i.e itiseasy

todiagonalizesuchHamiltoniansandconstructFockspacesfromwell-definedvacuumstates

in these regions This amounts to finding the so-called normal modes of the Hamiltonian,

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which provides a natural decomposition of the Hamiltonian into a set of independent monic oscillators inthe in/outregions; i.e.,

Xk2+Pˆk2, (2.151)

with{ωk}being thesetofnaturalfrequenciesforthemodes The correspondingFockspacescan then be built with respect to this set of modes, which again, provide an unambiguousnotionofgroundstate(orvacuum)asymptotically Thenotionthatsuchadecompositionex-istsisinteresting, anditholdsquitegenerically fortime-independentquadraticinteractions,

as the following sketch shows

Proof We sketch the existence of the above decomposition for time-independent quadraticHamiltonians Itrelies onthe so-called normal mode decomposition(or “Williamson’s theo-rem”)foranypositive-definite,realmatrix,whichweapplyherewithoutjustificationorproof

of the theorem Consider aquadratic, stable (i.e there existsa groundstate) Hamiltonian,which can be written as

ˆ

H =Hijˆriˆrj/2,perequation(2.144),where ˆrisanarbitrarybutwell-definedvectorofcanonical(quadrature)operators Now,byassumptionofstability,theHamiltonianmatrix,H,isapositive-definite,real matrix, and thus admits a normal-mode decomposition byWilliamson’s theorem, suchthat,

Xk2+Pˆk2,

which was to be shown Hence, any time-independent quadratic Hamiltonian can be composed into a set of independent quantum harmonic oscillators (normal modes), whichvibrate atthe naturalfrequencies {ωk}

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Photons

Weare almostinthepositiontodiscussthequantumofelectromagneticfield–i.e.,... producinga quantum theory fromthe classicaltheory, which Ireproducehere for convenience:

• Given the classical, Hamiltonian description of the system, ‘go over’ to the quantumtheory bycanonical