Introduction To Quantum Optics [Theory]

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P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18Introduction to Quantum Optics [Theory]olrtnocmuquantveitcepsrepPeter ZollerUniversity of Innsbruck and IQOQI, Austrian Academy of SciencesLectures 3 4

OutlineuibkQuantum Optical Systems & ControlLectures 1 2:Isolated / Driven Hamiltonian quantum optical systems Basic systems & concepts of quantum optics - an overview Example / Application: Ion Trap Quantum ComputerLectures 3 4:Open quantum optical systems [a modern perspective] Continuous measurement theory, Quantum StochasticSchrödinger Equation, master equation & quantum trajectories Example 1: Chiral / Cascaded Quantum Optical Systems& Quantum Many-Body Systems [Example 2: Entanglement by Dissipation]

Theory of Quantum Noise:Quantum Optical SystemsThe Quantum Stochastic Schrödinger Equation (QSSE) quantum operations, Kraus operators QSSE, master equations etc.- formal quantum information theory- quantum Markov processes- quantum opticsdrivesystembathlaserd dti [H, ] Lmaster equationΩ e⟩Γphoton g⟩3

Quantum OperationssystemenvironmentUquantum operation e 0 ihe 0 operation elements,Kraus operator Ref.: Nielsen & Chuang, Quantum Information and Quantum Computations4

Quantum d)probability5

Quantum noise & quantum optical systems decoherencestate preparationenvironmentsystem“bath”read outquantum operationsquantum opticscountstimeUsystemoutinü master equationü effect of observation on system:“preparation by quantum jumps”

LiteratureuibkThe Quantum World of Ultra-Cold Atoms and Light:Book I: Foundations of Quantum OpticsBook II: The Physics of Quantum-Optical DevicesBook III: Ultra-cold Atomsby Crispin W Gardiner and Peter ZollerQuantum NoiseA Handbook of Markovian and Non-Markovian Quantum StochasticMethods with Applications to Quantum Opticsby Crispin W Gardiner and Peter Zoller

Standard Model’ of Quantum Optics: System armonic oscillatorssystem operator driven two-level system spontaneous emissionquantum jumpoperatorlaserphoton detector

Standard Model’ of Quantum Optics: System armonic oscillatorssystem operatorsystem frequencyreservoir bandwidth B

Examples driven two-level system undergoing spontaneous emissionlaserphoton detector damped cavity modecountstimeoutinHsys fl!0 a † aphoton detector

Time evolution of the system environmentUSchrödinger equation:system environment

Time evolution of the system environmentQuestions: We do not observe the environment: reduced density operatormaster equation:üdecoherenceU üpreparation of the system(e.g. laser cooling, optical pumping)We measure the environment: continuous measurementconditional wave function:Ucountstimeü counting statisticsü effect of observation on systemevolution (e.g. preparation of the(single quantum) system)

Integration of the Schrödinger Equation technical step: interaction picture with respect to bath:interaction pictureSchrödinger equation 1b(t ) : p2ºˆ!0 #!0 #b(!)e i (! !0 )t d !noise operatorshi†b(t ), b (s) s (t s)e i !0 (t s)“white noise on scale 1/B”system frequencyreservoir bandwidth Brotating frame

We integrate the Schrödinger equation in small time steps0time tQ.: size of time step? hierarchy of time scales (see below: "coarse graining")

1st time step0 time tFirst time step: we start with the first interval and expand U(Δt) tosecond order in Δtfirst order in Δt

1st time step0 time tFirst time step: to first order in Δtone photonno photontimetime

1st time step0 First time step: to first order in Δttime t

1st time steptime

1st time step:summary0 time tSummary of first time step: to first order in Δtoperation elementstimetimeU

1st time stepDiscussion: We do not read the detector: reduced density operatorU( t)no photonUone photon

1st time stepDiscussion: We read the detector:U( t)“click”quantum jumpoperatorU“k”

1st time stepDiscussion: We read the detector:U( t)“no click”decaying normU“k”

2nd time step etc.0 time tSecond and more time steps:stroboscopicintegrationü Note: remember commute in different time slots

2nd time step etc.0 time tSecond and more time steps:stroboscopicintegrationIto Quantum Stochastic Schrödinger Equation & Master Equation Ito Quantum Stochastic Schrödinger Equation †(I) dt (t) iHsys dt dB (t)c (t) ( (0) sys vac )with Ito rulesB(t) B † (t) vac t vac dB(t)dB † (t) dt Reduced system density operator: (t) : TrB (t) (t) Master Equation: Lindblad form (t) i [Hsys , (t)]1 2c (t)c†2c† c (t) (t)c† c

Wave function of the system environment: entangled state click:no click:

Tracing over the environment we obtain the master equationU( t)master equationü Lindblad formü coarse grained time derivativedProof:(t) TrB . . . dB(t) dt 0 . dB † (t) dB † (t) dB(t) 0dB(t)dB † (t) dt2

Some Simple Examples

Example 1: Two-level atom spontaneous emissionMaster Equation two-level systemlaser Hamiltonian and in the rotating framea quantum jump (detection of an emission) prepares the atom in theground stateprobability for click in time interval (t,t dt] master equation (Optical Bloch Equations)

Example 2: Two-level atomEvolution conditional to observationphoton count trajectory (single run)Ω e⟩Γphotodetector g⟩timeEvolution of the atom, given this counting trajectory?conditional time evolution / wave function

Example 2: Two-level atomEvolution conditional to observationFig.: typical quantum trajectory (upper state population)click:“quantum jump” effect ofdetecting a photon on system sys (t)i!p no click: sys (t)i eiHeff t sys (0)i tsys (t)iwith Wigner-WeisskopfHamiltonianµ 1Heff !eg i æee . . .2

Example 2: Two-level atomEvolution conditional to observationFig.: typical quantum trajectory (upper state population) t Monte Carlo wave function simulationstochastic wavefunction (t)sys(dim d)reduced density matrix (t) h DMRG wave function simulationsys (t)i h sys (t) istdensity matrixsys (t)(dim dd)

Example 3: Two-level atomEvolution conditional to observationinitial state: e⟩photodetector g⟩timeatom c (0)i cg gi ce eiOutcome of experiment:We observe NO photon up to time tQuestion: what is the state of the atomconditional to this observation after time t?Answer: c (t)i eiHeff t/ ! gi for t ! 1 c (0)icg gi ce e k. . .kk. . .kt/2 eiWe learn that the system is in the ground state

poor man’s way of creating entanglementPreparation of 2 atoms in a Bell state via measurementlaseratom Aatom Aatom Blow efficiencyphotodetectorsatom Blaser- Weak (short) laser pulse, so that the excitation probability is small.- If no detection, pump back and start again.- If detection, an entangled state is created.

Process:atom Aatom Batom Aatom B

Engineered Dissipation — ExamplesOpen Quantum Many-Body SystemsExample 1:Chiral Quantum OpticsExample 2:Entanglement by Dissipation [& Ion Experiment]uibk

P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18Example 1:Chiral Quantum OpticsTheory: Cascaded Quantum Systemschiralphotonicchannelnode unidirectional couplings appearnaturally in nanophotonic devices

P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18'Chiral' Quantum Optics & NanophotonicsQuantum communicationwith chiral edge stateWhat is Chiral Quantum Optics?qubit 2 photonic nanostructure atoms, spin, 2D topologyqubit 1Ref.: topological quantum optics, Perczel, Lukin et al., PRL 2017; Hafezi, Segev,

P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18'Chiral' Quantum Optics & Nanophotonicschiral coupling between light and quantum emittersmetry,mysybdetprotecynot topologNanophotonic devices: chirality appears naturally atoms & nanofibersatoms & CQEDquantum dots &photonic nanostructuresP. Lodahl, A. Rauschenbeutel, PZ et al., Nature Review 2017

wsehgnyeyeyhdsyclgoysmy.esy-htrapped in the evanescentfield surrounding67the nanofiber3 OCTOBER 2014 VOL 346 ISSUE 6205waist of a tapered optical fiber. The atoms are located aboutNANOPHOTONICS200 nm above the nanofiber surface in two diametric onedimensionalarrays of potentialwells, with at most oneChiralnanophotonicwaveguideatom per trapping site. By using a red-detuned standinginterfacebased onrunningspin-orbitwave and a blue-detunedwave, localization of theatoms in the threeinteractionof(radial,lightazimuthal, and axial) directionsis achieved with trap frequencies of (200, 140, 315) kHz.Jan Petersen, Jürgen Volz,* Arno Rauschenbeutel*In order to drive transitions between the hyperfine groundControlling the flow of light with nanophotonic waveguides has the potential ofstates ofintegratedthe ationatoms,processing.weBecausestrong transverseconfinementof the guidedphotons,their internalspinandtheir ollowing,inmomentum get coupled. Using this spin-orbit interaction of light, we break the mirrorsymmetryof theourscatteringwith so-calleda gold nanoparticleon thesurface of a betweenwe limitstudyof lightto theclocktransitionnanophotonic waveguide and realize a chiral waveguide coupler in which the handednessstateslightjei j6Sthe; F ¼ 4;directionmF ¼in 0iand jgiWe controlj6S1 2 ;ofthethe incidentdeterminesthe waveguide.1 2propagationthe directionality of the scattering process and can direct up to 94% of the incoupledF into¼ a3;givenmFdirection.¼ 0i.OurThisjgiallows! jeitransitionexhibits ofonly alightapproachfor thecontrol and manipulationSCIENCE sciencemag.org3 OClight in optical waveguides and new designs of optical sensors.Nanofiber-based two-color dipole trap(a)he development of integrated electronic cir-(b) physical quanlight are no longer independentcuits laid the foundations for the informatities but are coupled (4, 5). In particular, the spintion age, which fundamentally changeddepends on the position in the transverse planemodern society. During the past decades,and on the propagation direction of light in theR. 2015)to as spin-orbit intransition from electronic to photonic inwaveguide—aneffect referredformationtransfertook place,nowadays,teraction of10,light(SOI). Thiseffect holds greatI. Söllner,P. Lodahlet onic circuits and waveguides promisepromises for the investigation of a large range ofto partially replace their electronic counterpartsphysical phenomena such as the spin-Hall effectand to enable radically new functionalities (1–3).(6, 7) and extraordinary momentum states (8)Thestrongconfinementoflightprovidedbysuchhas been observed for freely Lukin-Vuletic-Park,propagating lightAtoms Nanophotonics exp/theory:andKimble-Cirac-Chang,waveguides leads to large intensity gradients onfields (9, 10) in the case of total internal reflectionthe wavelength scale. In this strongly nonparaxial(11, 12), in plasmonic systems (13–15), and forregime, spin and orbital angular momentum ofradio frequency waves in metamaterials (16). Re-ToutOrozco, solid statechirality natural / generic feature of photonic nanostructuresFIG. 1 (color online). (a) Sketch of the experimental setup

Chiral Quantum Opticsfiber Lleft-moving photon Rright-moving photon ‘chiral’ atom-light interface:broken left-right symmetry L 6 Ruibk

Chiral Quantum Opticsfiber Rright-moving photon ‘chiral’ atom-light interface:broken left-right symmetry L 0; R‘chirality' open quantum systemuibk

Chiral Photon-Mediated Interactionsuibkopenboundariesfiber ‘chiral’ interactionsbroken left-right symmetryatoms only talk to atoms on the right

‘Chiral' Interactions How to Model? uibkinteractions mediated by photons- quantum optics we knowleft - rightsymmetric dipole-dipole interaction H ª æ 1 æ2 æ1 æ2by integrating out photons- chiral quantum opticsbroken left - rightsymmetry unidirectional interaction H ª æ æ1 2?Theory: ‘Cascaded Master equation’ open quantum system

Theory - Master Equationuibkopenboundariesfiber We integrate the photons out as ‘quantum reservoir’in Born-Markov approximation Master equation for reduced dynamics:density operator of atoms§i Ω̇ Hsys , Ω L Ωfl

1. Bidirectional Master Equation uibk openboundaries Master equation: symmetricdriven atomsΩ̇ 1D dipole-dipole i [Hsys sin(k x 1 x 2 )(æ æ æ1 22 æ1 ), Ω]X1 2 cos(k x i x j )(æi Ωæ j {æi æ j , Ω}).2i , j 1,2collective spontaneous emission“Dicke" master equation for 1D: D E Chang et al 2012 New J. Phys. 14 063003

2. Cascaded Master Equationuibk openboundaries Master equation: unidirectional†Ω̇ L Ω i (Heff Ω ΩHeff) æΩæ†Lindblad form non-Hermitian effective Hamiltonian Heff H1 H2 i æ1 æ1 æ2 æ2 2æ2 æ12 quantum jump operator: collective æ æ æ12 C.W. Gardiner, PRL 1993;H. Carmichael, PRL 1993general case:positionsN atoms,of thechiralatoms does not matterH. Pichler et al., PRA 2015

P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18Theory Appendix QSSE for Cascaded / Chiral Systems Cascaded Master Equation

Cascaded Quantum SystemsCascaded quantum systems: first system drives in a unidirectionalcoupling a second quantum systemin 1 system 1:"source"out 1 in 2unidirectional couplingQuantum Stochastic Schrödinger EquationMaster Equationsystem 2:"drivensystem"out 2erheN 2 Cascaded Theory: C.W. Gardiner, PRL 1993; H. Carmichael, PRL 1993

Cascaded Quantum SystemsExample:countsin 1out 2out 1 photon counting timein 2unidirectional couplingsystem 1:"source"system 2:"driven system"

The Modelin 1system 1:"source"out 1 in 2unidirectional couplingsystem 2:"drivensystem"out 2HamiltonianH Hsys (1) Hsys (2) HB HintHB ˆ!0 #d ! ! b † (!)b(!)!0 #Interaction partHint only right running modeshi†† i !/c x 1 i !/cx 1i d ! 1 (!) b (!)ec 1 c 1 b(!)ehi †† i !/c x 2 i !/cx 2 i d ! 2 (!) b (!)ec 2 c 2 b(!)e unidirectional coupling(x 2 x 1 )

The Modelin 1system 1:"source"out 1 system 2:"drivensystem"in 2unidirectional couplingout 2time delayInteraction picturei p hip h ††††Hint (t ) i 1 b (t )c 1 b(t )c 1 i 2 b (t ø)c 2 b(t ø)c 2(ø ! 0 )where time ordering / delays reflects causality, and1b(t ) p2ºˆ # #d !b (!)e ı(! !0 )twhite noise operator

The Modelin 1system 1:"source"out 1 in 2unidirectional couplingsystem 2:"drivensystem"out 2Stratonovich Quantum Stochastic Schrödinger Equation with time delays(S)d (t )idt i Hsys (1) Hsys (2)ip h †† 1 b (t )c 1 b(t )c 1iop h †† 2 b (t ø)c 2 b(t ø)c 2 (t )i(ø ! 0 )time delaywhere time ordering / delays reflects causalityˆp1Scaling: i c i ! c i b(t) 2 d b ( )e ı(

Integrating the Schrödinger Eq.in 1system 1:"source"t 1/out 1 in 2system 2:"drivensystem"out 2timeFirst time step: (for time delay ø ! 0 )( ( t )i 1̂ i Hsys (1) t c 1 i Hsys (2) t c 2 ( i )2ˆ0 td t1ˆ0t2ˆ tb (t ) d t0ˆ0† t c 1† tˆb † (t ) d t c 2†0ˆb(t ) d t tb(t ) d tfirst systemsecond system0)h id t 2 b(t 1 )c 1† b(t 1 )c 2† b † (t 2 )c 1 b † (t 2 )c 2 . . . (0)iµ 1 †1 c 1 c 1 0 c 2† c 1 c 2† c 2 vaci t22first system emits, second absorbscausality & interaction53

Integrating the Schrödinger Equationt 1/in 1timeFirst time step: (for time delay ø ! 0 )nout 1 in 2system 1:"source"system 2:"drivensystem"out 2quantum c c c12jumpop ( t )i 1̂ i Heff t c B † (0) (0)i effective (non-Hermitian) system Hamiltoniancoherentinteraction:1 †1 †asymmetricHeff Hsys (1) Hsys (2) i c 1 c 1 i c 2 c 2 i c 2† c 122Ωquantum æ1 †1jump Hsys (1) Hsys (2) i c 1 c 2 c 2† c 1 i (c 1† c 2† )(c 1 c 2 )22 and more steps (as before in Lecture 2)54

Cascaded SystemsMaster Equation for Cascaded Quantum SystemsVersion 1:o2 nd1X†††Ω i [Hsys , Ω] 2c i Ωc i Ωc i c i c i c i Ωdt2 i 1no†† [c 2 , c 1 Ω] [Ωc 1 , c 2 ] asymmetric in 1 and 2countsin 1out 2out 1 photon countingtimein 2unidirectional couplingsystem 1:"source"system 2:"driven system"55

Cascaded SystemsMaster Equation for Cascaded Quantum SystemsVersion 2: Lindblad form 1 d††††Ω i Heff Ω ΩHeff 2cΩc c cΩ Ωc cdt2coherentinteractionwith jump operator c c 1 c 2 and 11 ††Heff Hsys i c 1 c 2 c 2 c 1 i c † c22countsin 1out 2out 1 photon countingtimein 2unidirectional couplingsystem 1:"source"system 2:"driven system"56

P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18End of Theory Appendix

Applications of Chiral Atom-Light Interfacesuibk1. Quantum Information: Chiral Quantum Networksopenboundariesfiberquantum state transfer protocol Æ g i1 Ø ei1 0ip g i1 ! g i1 Æ 0ip Ø 1ip g i1 ! g i1 0ip Æ g i2 Ø ei2qubit 1wavepacket in waveguidequbit 2 with chiral coupling in principle perfect state transfertheory: Cirac et al. 1997; Rabl et al. PRX 2017, Vermersch et al., PRL 2017, Gorshkov et al., PRA 2017exp: Ritter, Rempe et al, Nature 2012; Schoelkopf et al, 2017; Wallraff et al. 2018

Applications of Chiral Atom-Light Interfacesuibk2. Driven-Dissipative Many-Body Quantum Systems openboundaries Unique, pure steady state:Entanglement by dissipation /non-equilibrium quantum phasesproduct of pure quantum spin-dimers/EPRT. Ramos, H. Pichler, A.J. Daley, B Vermersch, P. Hauke, P.O. Guimond, K. Stannigel, P. Rabl, PZ,PRL 2014, PRA 2015, PRA 2017, PRL 2017 - see also: A Gorkov, D Chang,

Engineering Chiral Coupling (1): nano-photonicsspin-orbit coupling in nano-photonicsopenboundariesfiber Lleft-moving photon Rright-moving photon

Engineering Chiral Coupling (2): synthetic gauge field'meta-atom'master atomdipole-dipoleΦquantum reservoirfluxopenboundariesfiber Rtwo emitters: constructive / destructive interference could be implemented ‘as is’ withsuperconducting qubits / microwaveCan we do this in ‘free-space’ / no waveguide? 1D chiral

Simulating open system dynamics with spin wavesuibktwo system spins with smooth absorbing boundary' /6chiral casedimermagnons magnonsreservoirspins:real spacesystem:spinsmomentum

Engineered Dissipation — ExamplesOpen Quantum Many-Body SystemsExample 1:Chiral Quantum OpticsExample 2:Entanglement by Dissipation [& Ion Experiment]uibk

Entanglement by [Engineered] Dissipationtheory:Reviews M. Müller, S. Diehl, G. Pupillo, and P. Zoller,Engineered Open Systems and Quantum Simulations with Atomsand Ions, Advances in Atomic, Molecular, and Optical Physics (2012) C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. Imamoglu, P. Zoller, S.Diehl,Topology by dissipation, arXiv:1302.5135experiments:J. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller & R. BlattNature 470, 486 (2011)P. Schindler, M. Müller, D. Nigg, J. T. Barreiro, E. A. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, and R. Blatt,Nat. Phys. 9, 1 (2013).Krauter et al., Polzik & Cirac, PRL 2011 -- atomic ensembles64

Open System Dynamics [& Decoherence ] open system dynamicssystemnotobservedenvironmentcompletely positive maps:ΩE (Ω) XkE k ΩE k†Kraus operatorquantum control theory: open-loop [vs. closed loop measurement feedback]65

Entanglement from (Engineered) Dissipation open system dynamicssystem environmentnotobservedengineering Kraus operators:ΩE (Ω) XkE k ΩE k†! desired (pure)quantum state“cooling” into a pure state- non-unitary- deterministic66

Entanglement from (Engineered) Dissipation Markovianopen system dynamicssystem environmentnotobservedengineering Kraus operators:ΩE (Ω) XkE k ΩE k†master equation: ! i[H, ]X c c† desired (pure)quantum state“cooling” into a pure state- non-unitary- determinis

The Quantum World of Ultra-Cold Atoms and Light: Book I: Foundations of Quantum Optics Book II: The Physics of Quantum-Optical Devices Book III: Ultra-cold Atoms by Crispin W Gardiner and Peter Zoller Quantum Noise A Handbook of Markovian and Non-Markovian Quantum Stoch

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