Slide 1

Alexander LvovskyTHREE WAYS TO SKIN A CAT

CHARACTERIZE A QUANTUM OPTICAL BLACK BOX

1OutlineIntroduction: coherent-state quantum process tomographyMethod 1: approximating the P functionMethod 2: integration by partsMethod 3: maximum-likelihood reconstructionIn classical electronicsConstructing any complex circuit requires precise knowledge of each components operationWhy we need process tomography

This knowledge is acquired by means of network analyzersMeasure the components response to simple sinusoidal signalsCan calculate the components response to arbitrary signals

In quantum information processingIf we want to construct a complex quantum circuit, we need the same knowledge

Quantum process tomographySend certain probe quantum states into the quantum black box and measure the outputCan calculate what the black box will do to any other quantum stateWhy we need process tomography

Quantum processesGeneral propertiesPositive mappingTrace preserving or decreasingNot always linear in the quantum Hilbert space

Example: decoherence |1 |1 |2 |2but |1 + |2 |11| + |22|Always linear in density matrix space

Quantum process tomographyMethodologyThe approachA set of probe states {ri} must form a spanning set in the space of density matricesSubject each ri to the process, measure E(ri)Any arbitrary state r can be decomposed: Linearity Process output for an arbitrary state can be determinedChallengesNumbers to be determined = (Dimension of the Hilbert space)4Process on a single qubit 16 Process on two qubits 256 Need to prepare multiple, complex quantum states of light All work so far restricted to discrete Hilbert spaces of very low dimension

The main ideaDecomposition into coherent statesCoherent states form a basis in the space of optical density matricesGlauber-Sudarshan P-representation (Nobel Physics Prize 2005)

Application to process tomographySuppose we know the effect of the process E(|aa|) on each coherent stateThen we can predict the effect on any other state

The good newsCoherent states are readily available from a laser. No nonclassical light neededComplete tomography

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders and A. L., Science 322, 563 (2008)The process tensorFock basis representation of the processSince

it is enough to know for all relevant photon numbers m, n,because then

The process tensor

contains full information about the process

Expressing the process tensor using the P function

In practice: reconstructed up to some nmax

Method 1Approximating the P functionThe P-function[Glauber,1963; Sudarshan, 1963]What is it?Deconvolution of the states Wigner function with the Wigner function of the vacuum state

Example

Wigner function of a coherent stateP-function of a coherent state

=*

The P-function [Glauber,1963; Sudarshan, 1963]What about nonclassical states?Their Wigner functions typically have finer features than W0(a)The P-function exists only in the generalized senseThe solution [Klauder, 1966] Any state can be infinitely well approximated by a state with a nice P function by means of low pass filtering

Example: squeezed vacuumBounded Fourier transformof the P-functionRegularized P-functionWigner function from experimental dataWigner function from approximated P-function12Practical issuesNeed to choose the cut-off point L in the Fourier domainCant test the process for infinitely strong coherent states must choose some amaxThere is a continuum of as process cannot be tested for every coherent state must interpolateProcess not guaranteed to be physical (positive, trace preserving)Many processes are phase-invariant

it is sufficient to perform measurements only for as on the real axis

Example of application: Memory for light as a quantum process

M. Lobino, C. Kupchak, E. Figueroa and A. L., PRL 102, 203601 (2009)

Process reconstructionThe experimentInput: coherent states up to amax=10; 8 different amplitudesOutput quantum state reconstruction by maximum likelihoodProcess assumed phase invariantInterpolationHow memory affects the stateAbsorptionPhase shift (because of two-photon detuning)Amplitude noisePhase noise (laser phase lock?)Process reconstruction:the result for photon number statesEach color: diagonal elements of the output density matrix for a given input photon number stateWe can tell what happens to the Fock states without having to prepare themLet us now verify by storing nonclassical states

Zero 2-photon detuning540 kHz 2-photon detuningoutput photonsinput photonsoutput photonsinput photonsExperiments on storing nonclassical lightExisting workL. Hau, 1999: slow lightM. Fleischauer, M. Lukin, 2000: original theoretical idea for light storageM. Lukin, D. Wadsworth et al., 2001: storage and retrieval of a classical stateA. Kuzmich et al., M. Lukin et al., 2005: storage and retrieval of single photonsJ. Kimble et al., 2007: storage and retrieval of entanglementM. Kozuma et al., A. Lvovsky et al., 2008: memory for squeezed vacuum= Various states of light stored, retrieved, and measuredShortcomingsComplicatedDo not answer how an arbitrary state of light is preserved in a quantum storage apparatus.Coherent-state process tomography resolves both shortcomings!

Method 2Integration by partsFinding the process tensorFock operators |nm|Process output: P function: Use integration by parts:How to process experimental dataMeasure density matrix of for a set of as using homodyne tomographyFit every element of with a polynomialElements of the process tensor are just coefficients of this polynomial!Advantages of this methodElimination of integration and the ugly P functionElimination of a potential source of error (lowpass filtering)Dramatic simplification of calculations

S. Rahimi-Keshari et al., New Journal of Physics 13, 013006 (2011)

experimental dataPractical issueWith experimental uncertainties, polynomial fitting is difficult.Fitting error increases with degree

Example: Creation and annihilation operatorsTwo fundamental operators of quantum optics

Non-unitary, non-trace preservingCan be approximated in experiment

Photon creation and annihilation.Experimental setupAnnihilation

A click indicates that a photon has been removed from |It must be included in the reconstruction formulaCreation

A click indicates that a down-conversion event has occurred and a photon added to |

Accounting for non-unitary traceTrace of the process output is given by the click probability

Photon creation operatoracting on a coherent state [see also A. Zavatta et al., Science 306, 660 (2004)]Initial coherent state

Photon-added coherent state

Behavior a 0: Fock state (highly nonclassical) a : coherent state (highly classical)

a increases Photon creation and annihilation.Process reconstructionAnnihilation

Creation

Method 3Maximum-likelihood iterationsFully statistical reconstriction[Most ideas from: Z. Hradil et al, in Quantum State Estimation (Springer, 2004)]Previous methodsExtremely tedious (P. K. Lam)Physicality of processtrace preservation, positivitynot guaranteedwould be nice to develop a fully statistical (MaxLik) reconstruction methodJamiolkowski isomorphismReplace the superoperator process by a state in extended Hilbert space

original Hilbert space (H)extension of Hilbert space (K)Then, for any probe coherent state input

Fully statistical reconstriction(continued)Homodyne measurement on output stateProjective measurement with operator

Probabilty to obtain a specific quadrature value X is

unknown stateprojective measurement

treat this as a new projector

Can apply iterative MaxLik state reconstruction procedure!quadraturephaseA. Anis and AL, New Journal of Physics 14, 105021 (2012)

Lagrange multiplier matrix to preserve trace

Handling non-trace-preserving processesE.g. photon creation and annihilationHeralded process. Success probability ga depends on the input stateIdea: introduce a fictitious state |No heralding event = projection onto |Modify L and R matrices accordingly

Photon creationProcess reconstruction video

Photon creation and annihilation.Process reconstructionAnnihilation

Creation

R. Kumar, E. Barrios, C. Kupchak, ALPRL (in press)All probe coherent states amplitudes 1!

Issue: nmax vs. amaxE.g. our experiment: nmax = 7. Which amax to choose?Too low: insufficient information about high photon number terms errors in high number terms of process tensor

Too high: input coherent states do not fit within the reconstruction space trace 1 unpredictable errors in process tensor Apparent solutionFirst reconstruct with higher nmax. Then eliminate high number terms Works with simulated data, not so well in real experiment

Photon creationnmax = 8, amax = 0.6 Photon creationnmax = 3, amax = 0.6 A. Anis and AL, New Journal of Physics 14, 105021 (2012) Coherent-state QPTSummaryBy studying what a quantum black box does to laser light, we can figure what it will do to any other stateComplete tomographyElimination of postselectionEasy to implement and process (3 different ways)Tested in several experiments

The three methodsSummaryMethod 1: approximating the P functionStraightforwardTediousRequires high amaxPhysicality of reconstructed process not guaranteedMethod 2: integration by partsEliminates integration and the ugly P functionEliminates a potential source of error (lowpass filtering)Dramatic simplification of calculationsPolynomial fitting can be finickyMethod 3: maximum-likelihood reconstructionGuarantees physicalityRequires low amaxComputationally intensiveUnresolved issues with reconstruction algorithm

Thanks!

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