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Continuous-variable optical quantum-state tomography

A. I. Lvovsky*

Department of Physics and Astronomy, University of Calgary, Calgary, Alberta,

Canada T2N 1N4

M. G. Raymer

Department of Physics and Oregon Center for Optics, University of Oregon,Eugene, Oregon 97403, USA

Published 16 March 2009

This review covers the latest developments in continuous-variable quantum-state tomography ofoptical fields and photons, placing a special emphasis on its practical aspects and applications inquantum-information technology. Optical homodyne tomography is reviewed as a method ofreconstructing the state of light in a given optical mode. A range of relevant practical topics isdiscussed, such as state-reconstruction algorithms with emphasis on the maximum-likelihoodtechnique, the technology of time-domain homodyne detection, mode-matching issues, andengineering of complex quantum states of light. The paper also surveys quantum-state tomography forthe transverse spatial state spatial mode of the field in the special case of fields containing preciselyone photon.

DOI: 10.1103/RevModPhys.81.299 PACS numbers: 03.65.Wj, 42.50.Dv

CONTENTS

I. Introduction 299

A. The concept of quantum tomography 299

B. Quantum tomography of light 301

1. Optical homodyne tomography 301

2. Optical mode tomography 302

II. The Principles of Homodyne Tomography 302

A. Balanced homodyne detection 302

B. Wigner function 304

III. Reconstruction Algorithms 304

A. State reconstruction via inverse linear

transformation 305

1. Inverse Radon transformation 305

2. Pattern functions 305

B. Maximum-likelihood reconstruction 306

1. Why maximum likelihood? 306

2. Classical algorithm 307

3. The discrete quantum case 307

4. Iterative scheme for homodyne tomography 308

5. Error handling 309

C. Maximum-entropy reconstruction 309

IV. Technical Aspects 310

A. Time-domain homodyne detection 310

B. Matching the mode of the local oscillator 311

1. The advanced wave 311

2. Decorrelating photons 312

3. The continuous-wave case 313

4. Strong pumping mode 314

V. Applications in Quantum Technology 314

A. Two-mode squeezed vacuum 315

1. Description of the state 315

2. Homodyne characterization 316

B. Fock state tomography 317

C. The optical qubit 318

1. The dual-rail qubit 318

2. Nonlocality of the single photon 318

3. Remote state preparation using the

nonlocal single-photon state 319

4. Teleportation using the nonlocal

single-photon state 319

5. Quantum-optical catalysis 319

D. Schrdinger cats and kittens 3191. Preparation by photon subtraction 320

2. Generating Schrdinger cats from Fock

states 321

E. Photon-added states 322

1. Single-photon-added coherent states 322

2. Single-photon-added thermal states 323

VI. Spatial Quantum-State Tomography 323

A. Spatial mode of the one-photon field 323

B. Noninterferometric reconstruction 324

C. Interferometric reconstruction by wave-front

inversion 325

VII. Summary and Outlook 327

Acknowledgments 328References 328

I. INTRODUCTION

A. The concept of quantum tomography

A quantum state is what one knows about a physicalsystem. The known information is codified in a state vec-tor , or in a density operator , in a way that enablesthe observer to make the best possible statistical predic-tions about any future interactions including measure-

*[email protected]@uoregon.edu

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ments involving the system. Such a definition has acomfortable interpretation within information theory,and so it appears natural in the context of research inquantum information QI.

Imagine that an experimentalist, Alice, uses a well-characterized procedure to prepare an individual par-ticle in a particular physical state. Since Alice possessesthe information about the procedure, she can make defi-nite predictions about the particles behavior under vari-ous conditions, and is thus fully aware of the particlesstate.

Now suppose Alice sends the prepared particle to an-other party, Bob, who is not aware of the preparationprocedure, but wishes to determine the state of the par-ticle. By making observations on the particle, Bob canobtain information about the physical state prepared byAlice by observing how it interacts with other well-characterized systems, such as a measurementapparatus.1 The amount and nature of this informationdepend strongly on whether the particle is macroscopicor microscopic. In the macroscopic, classical case, Bobcan observe the individual particles trajectory without

disturbing it, and determine its state.In quantum mechanics, on the contrary, it is impos-

sible to learn the quantum state of any individual physi-cal system. Each observation, no matter how subtle, willdisturb its state just enough to prevent further observa-tions from yielding enough information for a state deter-mination DAriano and Yuen, 1996. This is the basis ofquantum key distribution for cryptography Bennett andBrassard, 1984.

If Alice provides Bob with an ensemble of identicallyprepared systems, then he can measure the same vari-able for each system, and build up a histogram of out-comes, from which a probability density can be esti-

mated. According to the Born rule of quantum theory,this measured probability density will equal the squaremodulus of the state-vector coefficients, represented inthe state-space basis corresponding to the measuring ap-paratus. This by itself will not yet yield the full stateinformation since the phase of the complex state-vectorcoefficients will be lost.

As an example, measure the position x of each of100 000 identically prepared electrons, which can moveonly in one dimension. This yields an estimate of theposition probability density, or the square modulusx2 of the Schrdinger wave function. If the wavefunction has the form xexpix, where x is a

spatially dependent phase, then we will need more infor-mation than simply x2 in order to know the wavefunction. If we are able to measure the momentum p ofa second group of identically prepared electrons, then

we can estimate the probability density p2, where

p = xexp ixp/dx 1is the Fourier transform of the spatial wave function. Ifwe know a priori that the ensemble can be described bya pure state, then we can determine, by numerical meth-ods, the complex wave function x, up to certain sym-metry transformations such as a complex conjugationjust from these two types of measurement. This is a clas-sic example of phase retrieval Gerchberg and Saxton,1972.

In the typical case, however, we do not know ahead oftime if the systems state is pure or mixed. Then we mustmake many sets of measurements on many suben-sembles, each time modifying the apparatus so that setsof projection statistics associated with a different basiscan be acquired. One can then combine these results toreconstruct the density matrix of the state Raymer,1997a. The data do not yield the state directly, butrather indirectly through data analysis i.e., a logical in-ference process. This is the basis of quantum-state to-mography QST. A set of observables whose measure-

ments provide tomographically complete informationabout a quantum system is called a quorum Fano,1957.

Niels Bohr 1958 seems to have had an intuitive ideaof QST when he said, A completeness of descriptionlike that aimed at in classical physics is provided by thepossibility of taking every conceivable arrangement intoaccount. A more rigorous concept was developed intheoretical proposals Newton and Young, 1968; Bandand Park, 1970, 1971, 1979; Bertrand and Bertrand,1987; Vogel and Risken, 1989, followed by the first ex-periments determining the quantum state of a light fieldSmithey, Beck, Cooper, Raymer, et al., 1993; Smithey,

Beck, Raymer, et al., 1993. Nowadays, quantum tomog-raphy has been applied to a variety of quantum systemsand has become a standard tool in QI research Parisand Reh~ek, 2004.

To continue the example of an electron moving in onedimension, a quorum of variables can be constructed bymeasuring different groups of electrons positions x af-ter a variable length of time has passed. For example, infree space this is x=x +pt/m, where m is the electronmass. The wave function at time t is

x,t = Gx,x;txdx , 2where

Gx,x; t = m2it

exp imx x22t

3is the quantum propagator appropriate to the waveequation for the particle. The experimentally estimatedprobability densities prx , t = x , t2, for all tpositiveand negative, provide sufficient information to invertEq. 2 and determine the complex state function xassuming the functions prx , t are measured with veryhigh signal-to-noise ratio. Note that Eq. 2 can be in-

1We can interpret the quantum state as a belief, or confidencelevel, that a person has in his or her knowledge and ability topredict future outcomes concerning the physical systemFuchs, 2002. No measurements can, generally speaking, pro-vide full information on Alices preparation procedure.

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terpreted as a generalization of Eq. 1. As such, it cor-responds to a change of basis.

If the state is not known beforehand to be pure thatis, the physical systems state is entangled with someother system, then it is described by a density matrix,x1 ,x2 ; t,

x1,x2;t = dx1dx2G*x1,x1; tGx2,x2 ;tx1,x2;0. 4

Through inversion of Eq. 4, the set of the measuredprobability functions prx ; t =x ,x , t determines thedensity matrix x1 ,x2 ; 0.

This procedure works in principle for a Schrdingerequation with an arbitrary, known potential-energy func-tion. Such a method was proposed Raymer et al., 1994;Janicke and Wilkens, 1995; Leonhardt and Raymer,1996; Raymer, 1997b and implemented Kurtsiefer etal., 1997 for the transverse spatial state of an ensembleof helium atoms and the classical light beam McAlisteret al., 1995.

An important, recently introduced extension to QSTis quantum-process tomography QPT. Consider aquantum black box, which subjects each incomingquantum state to some unknown transformation. Thegoal of QPT is to determine this transformation so theoutput of the black box can be predicted for an arbitraryinput.

A general quantum process is a trace-preserving, posi-tive linear map on the linear space LH of all densitymatrices over Hilbert space H. The process can thus becharacterized by feeding it with each of the dimH2 el-ements of a spanning set ofLH and performing QSTon the outputs.

Such a direct approach to QPT was first proposed byPoyatos et al. 1997 as well as Chuang and Nielsen1997 and experimentally realized on one-qubit telepor-tation Nielsen et al., 1998, a system formed by the vi-brational levels of atoms in an optical lattice Myrskog etal., 2005 and on a two-qubit controlled-NOT gate Childset al., 2001; OBrien et al., 2004. In an alternative ap-proach, information about the process is recovered byperforming tomography on the output state of a singlemaximally entangled bipartite input with the processacting on one of the two input subspaces DAriano andLo Presti, 2001. Experimental results based on the lat-ter scheme were obtained for a general single-qubit gate

Altepeter et al., 2003; De Martini et al., 2003.Further study of QPT is beyond the scope of this pa-per. A comprehensive review on the subject has recentlybeen given by Mohseni et al. 2008.

B. Quantum tomography of light

The current interest in QST is motivated by recentdevelopments in QI processing, which requires inter aliaa technique for detailed characterization of quantumstates involved Paris and Reh~ek, 2004. Additionally,significant progress has been made in measurement

technologies, which now allow experimenters to mea-sure a set of observables sufficiently diverse to allowreliable state reconstruction from the data.

Among many physical systems in which QI processingcan be implemented, light is of particular significancebecause it is mobile and thus irreplaceable as an infor-mation carrier in quantum communication networks. Inthis paper, we review the methods for QST of opticalfields.

Even specialized to light, quantum tomography is toovast a field to be fully covered in a single review paper.Here we choose to concentrate on optical QST, whichinvolves measuring continuous degrees of freedom: fieldamplitude and/or spatial distribution. We study twoclosely related tomographic problems. The first dealswith the case in which the mode of the field is known orchosen a priori, and the state of this mode is to be de-termined. The second deals with the case in which thefull field is known to contain exactly one photon, but themanner in which this photon is distributed among spatialand spectral modes is to be determined.

1. Optical homodyne tomography

The Hamiltonian of an electromagnetic mode isequivalent to that of the harmonic oscillator. Quantumstates of light in this mode can thus be reconstructedsimilarly to motional states of massive particles dis-cussed in the previous section. This is done by measur-ing quantum noise statistics of the field amplitudes atdifferent optical phases Leonhardt, 1997. The proce-dure of this reconstruction is known as optical homo-dyne tomography OHT.

It is interesting, in the context of this paper, that ho-modyne tomography was the first experimental demon-stration of optical QST. Using balanced homodyne de-tection BHD, Smithey, Beck, Raymer, et al. 1993measured a set of probability densities for the quadra-ture amplitudes of a squeezed state of light. These his-tograms were inverted using the inverse Radon trans-form, familiar from medical tomographic imaging, toyield a reconstructed Wigner distribution and densitymatrix for a squeezed state of light. This 1993 paperintroduced the term tomography into quantum optics.

OHT is the subject of the next four sections of this

paper. In Secs. II and III, we discuss the concept of ho-modyne tomography and the methods of reconstructingthe states Wigner function and density matrix from a setof experimental data. Special attention is paid to thelikelihood-maximization technique, which is now mostcommonly used. Section IV is devoted to technical is-sues arising in experimental OHT. In Sec. V, we discussapplications of OHT in experiments of engineering andcharacterizing specific quantum states of light, such asphotons, qubits, and Schrdinger cat states.

In the context of applications, it is instructive to com-pare OHT with another technique: determining the

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quantum state of a system of dual-rail optical qubits2 bymeasuring relative photon number statistics in eachmode and in their various linear superpositions. Due toits relative simplicity, this approach has been applied inmany experiments see Altepeter et al. 2004 for a re-view.

A textbook example of the above is the work ofJames et al. 2001. In this experiment, the polarizationstate of a pair of entangled photons A and B generatedin type-II parametric down-conversion was analyzed bymeasuring photon coincidence count statistics in 16 po-larization projections. Tomographic analysis has re-vealed the photons to be almost perfectly in the state

=1

2HAVB + VAHB , 5

where Hand Vindicate horizontal and vertical polariza-tion. It is tempting to say that such a state has high en-tanglement.

This analysis does not reveal, however, that the pho-ton pair is generated not on demand, but with some

probability 2, which is usually low. A more completerepresentation of the state of the optical modes analyzedcould be

= 0AH0AV0BH0BV + 1AH0AV1BH0BV

+ 0AH1AV0BH1BV + O2 , 6

where, for example, 1AV indicates a one-photon statepresent in the vertical polarization mode of channel A.The bipartite entanglement is of the order 2 log,which is much less than 1.

Equations 5 and 6 reveal a significant limitation ofthe photon-counting approach. This method works well

if it is a priori known that the modes involved are in oneof the qubit basis states or their linear combination. Inpractice, however, this is not always the case: photonscan be lost, or multiple photons can be present where weexpect only one. Such events compromise the perfor-mance of quantum logical gates, but usually go unrecog-nized by the photon-counting approach; they are simplyeliminated from the analysis.3 As a result, one character-izes not the true quantum state of the carrier modes, butits projection onto the qubit subspace of the optical Hil-bert space. This may lead to a false estimation of gateperformance benchmarks van Enk et al., 2007.

OHT, on the contrary, permits complete characteriza-

tion of the field state in a particular spatiotemporalmode, taking into account the entire Hilbert space ofquantum optical states. It thus provides more reliable

information about the performance and scalability of anoptical QI processor. However, it is also more techni-cally involved as it requires matched local oscillators,sophisticated detection electronics, larger positiveoperator-valued measures POVMs, and measurementdata sets. It is thus less suitable for characterizing mul-timode states: to date, the largest qubit systems mea-sured using photon counting contained six qubits Lu etal., 2007 while with OHT, only single dual-rail qubits

were reconstructed Babichev, Appel, and Lvovsky,2004.

2. Optical mode tomography

An altogether different use of QST arises when a lightfield is known to contain a definite number of photons,but their distribution over spatial and/or spectral modesis unknown. If the set of modes is discrete e.g., in thecase of polarization qubits, characterization can bedone using the photon-counting method discussed aboveAltepeter et al., 2004. But if the distribution of lightparticles over electromagnetic modes is described by a

continuous degree of freedom, methods of continuous-variable QST become irreplaceable.

The problem of reconstructing the modal distributionof a field state is largely analogous to determining thespatial wave function r of a massive particle, as de-scribed by Eq. 2. It turns out that the procedure out-lined via Eqs. 14 also applies to QST for an en-semble of single photons, and is in fact quite similar tothe method of OHT. This is consistent with the adoptionof a definition for a photons spatial wave function Sipe,1995; Bialynicki-Birula, 1996; Smith and Raymer, 2007,in which the Schrdinger equation is replaced by theMaxwell equations since a photon cannot be strictly lo-

calized in space, some subtleties must be taken into ac-count. In Sec. VI, we analyze various techniques andrecent experimental progress in reconstructing spatialoptical modes of single photons as well as entangledpairs.

II. THE PRINCIPLES OF HOMODYNE TOMOGRAPHY

A. Balanced homodyne detection

The technique of balanced homodyne detectionBHD and homodyne tomography has been extensivelydescribed in the literature, for example, in the textbook

of Leonhardt 1997 and in recent reviews by Raymerand Beck 2004, Zhang 2004, and Zavatta, Viciani,and Bellini 2006. Here we present only a brief intro-duction with concentration on theoretical aspects ofmode matching between the local oscillator and the sig-nal field.

Figure 1 illustrates balanced homodyne detection,which is a means to measure the amplitude of any phasecomponent of a light mode. In BHD, the weak signal

field ESt which may be multimode and a strong co-

herent local oscillator LO field ELt are overlapped at

2In the dual-rail qubit, the logical value is assigned to asingle photon being in one of two orthogonal modes A or B:

0= 1A ,0B, 1 = 0A , 1B, where the right-hand side is writtenin the photon number Fock basis for each mode.

3An important exception is the work by Chou et al. 2005,where the photon-counting approach is used, but the vacuumcontribution is accounted for when evaluating the entangle-ment of a dual-rail qubit.


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a 50% reflecting beam splitter, and the two interferedfields are detected, temporally integrated, and sub-tracted.

The signal electric field operator is written as a sum ofpositive- and negative-frequency parts, which are conju-

gates of one another, ES = E

S+ + ES

. The positive-frequency part can be decomposed into plane waves ac-cording to Diracs quantization scheme,

E

S

+r,t = i

j j

20Vbjj expikj r ijt , 7

where j, kj, and j are, respectively, the mode fre-quency, wave vector, and the unit polarization vector;the creation and annihilation operators obey the com-

mutator bj, bj =jj and are defined in some large vol-

ume V which may be taken to infinity later. It is con-venient to consider the signal field in the paraxialapproximation with z the propagation axis. In this casej ckjz and the polarization j is along either x or y.

The LO field is treated classically, and at each photo-diode face z= 0 is assumed to be a strong coherentpulse propagating along the z axis,

EL+r,t = i L20VLLvLx,ygLt

expikLz iLt , 8

where the coherent-state amplitude is L = Lei, and

vLx ,ygLt is the normalized spatiotemporal mode.The local oscillator and the signal fields meet at a

beam splitter, where they undergo the transformation

EL,E

SEL + ES2 , EL E

S

2 . 9

The difference of the numbers of photoelectrons re-corded in the two beam splitter outputs is then, assum-ing a perfect detection efficiency see Raymer et al.1995 and Raymer and Beck 2004 for details and moregeneral considerations,

N = det

t

0V

c2ELE

sdtdxdy = Lae

i+ aei,

10

where the integration is over the detector sensitive areaand the measurement time t. Assuming that the above

fully accommodate the local oscillator pulse, all integra-tion limits in Eq. 10 can be assumed infinite. The pho-ton creation operator a associated with the detectedspatiotemporal mode is given by

a = j

Cjbj, 11

where the Cjs equal the Fourier coefficients for the LO

pulse,

Cj= L* j vL* x,ygL* t expikjxx + ikjyy ickjz kLtdtdxdy . 12

When using a pulsed LO field ELt, the concept of alight mode needs to be generalized beyond the commonconception as a monochromatic wave. As first discussedby Titulaer and Glauber 1966, and extended by Smithand Raymer 2007, a polychromatic light wave packetcan be considered a mode with a well defined spatial-temporal shape, whose quantum state is described in the

usual way using photon creation and annihilation opera-tors. For example, a one-photon wave-packet state iscreated by 1a = a

vac more on this in Sec. VI.A. Themeaning of Eqs. 11 and 12 is that the BHD detectsthe state of the electromagnetic field in the spatial-temporal mode defined by the LO pulse Smithey, Beck,Raymer, et al., 1993; Raymer et al., 1995; Raymer andBeck, 2004. This allows temporal and spatial selectivity,or gating, of the signal field not the signal intensity.This gating technique linear-optical sampling has appli-cation in ultrafast signal characterization Dorrer et al.,2003; Raymer and Beck, 2004.

As usual, the modes annihilation operator can be

expressed as a sum of Hermitian operators a= eiQ+ iP /2, called quadrature amplitudes, with

4 Q, P

= i. For zero phase, Q, Pare denoted Q , P, respectively

so Q= Q cos + P sin , and are analogous to positionand momentum variables for a massive harmonic oscil-lator. For the LO phase equal to , BHD measures thequadrature amplitude

N /L2 = aei+ aei/2 = Q. 13

According to quantum mechanics, the probability den-sity for observing the quadrature equal to Q for the

field in the signal mode given by the density operator is

4This convention, in which =1, is consistent with the stan-dard quantum-mechanical commutator between the position

and momentum. Some use Q,P= i /2 or Q, P=2i. Theadvantage of the former convention is that the annihilation

operator becomes a= eiQ+ iP. The latter convention pro-

vides that the vacuum state noise 0Q20= 0P

20=1. Allquadrature-dependent plots used have been rescaled to com-

ply with the convention Q, P= i.

FIG. 1. Balanced homodyne detection.


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prQ, = Q,Q,, 14

where Q, is the quadrature eigenstate with eigen-value Q. These probability densities, also known asmarginal distributions, are histograms of the field ampli-tude noise samples measured with the homodyne detec-tor. The optical phase plays the role of time in Eqs.14. When it is varied over one complete cycle,quadrature amplitudes Q form a quorum for QST Vo-

gel and Risken, 1989.In a practical experiment, the photodiodes in the ho-

modyne detector are not 100% efficient, i.e., they do nottransform every incident photon into a photoelectron.This leads to a distortion of the quadrature noise behav-ior which needs to be compensated for in the recon-structed state. We present, without derivation, a gener-alization of the above expression for detectors with anonunitary quantum efficiency Raymer et al., 1995;Raymer and Beck, 2004,

prQ, =: exp Q / Q2/2222 : , 15where 22 =1 / and the angular brackets indicate aquantum expectation value. The double dots indicatenormal operator ordering annihilation operators to theright of creation operators.

B. Wigner function

Because the optical state reconstructed using tomog-raphy is generally nonpure, its canonical representationis in the form of a density matrix, either in the quadra-ture basis or in the photon-number Fock basis. In thecase of homodyne tomography, it is convenient to rep-

resent the reconstructed state in the form of the phase-space quasiprobability density, the Wigner functionWigner, 1932,

WQ,P =1

2

Q + 12 QQ 12 QeiPQdQ.

16

This object uniquely defines the state and, at the sametime, is directly related to the quadrature histograms14 and 15 measured experimentally Raymer et al.,1995; Raymer and Beck, 2004 via the integral

prQ, = +

+

Q Q cos

Psin WdetQ,PdQdP

=

WdetQcos Psin ,Qsin

+ Pcos dP. 17

In other words, the histogram prQ, is the integralprojection of the Wigner function onto a vertical planeoriented at angle to the Q axis Fig. 2. The detectedWigner function Wdet corresponds to the ideal Wigner

function 16 for a loss-free detector, and for a detectorwith quantum efficiency it is obtained from the lattervia a convolution Leonhardt and Paul, 1993; Kuhn etal., 1994; Raymer et al., 1995; Leonhardt, 1997,

WdetQ,P =1

1

+

+

WQ,P

exp Q Q

2 +P P

2

1 dQdP. 18

III. RECONSTRUCTION ALGORITHMS

A homodyne tomography experiment yields a set ofpairs Qm ,m, which can be binned up to form marginaldistributions prQ , for several local oscillator phases.Our next task is to develop mathematical methods thatcan be used to convert the experimental data into thestates density matrix and/or Wigner function. This is the

subject of the current section.Mathematical methods of OHT can be divided into

two categories. The so-called inverse linear transformtechniques Sec. III.A use the fact that the experimen-tally measured marginal distributions are integral pro-jections of the Wigner function. Because integration is alinear operation, one can reverse it and reconstruct theWigner function from the set of marginals in a proce-dure that somewhat resembles solving a system of linearequations of the form 17. We discuss this and relatedmethods in Sec. III.A; they have been reviewed in moredetail by DAriano 1997, Leonhardt 1997, Welsch etal. 1999, Paris and Reh~ek 2004, and Raymer and

Beck 2004.For reasons discussed later Sec. III.B, inverse linear

transform methods are rarely used in modern OHT.More frequently, we employ methods of statistical infer-ence, whose classical versions have been developed intraditional statistics and data analysis Paris and Re-h~ek, 2004, Chaps. 2, 3, 6, and 10. A popular method islikelihood maximization MaxLik, which looks for themost probable density matrix that will generate the ob-served data. It is discussed in detail in Sec. III.B. An-other statistical inference method, entropy maximiza-tion, is reviewed in Sec. III.C.

W ,P( )Q

QP

pr( )Q ,

Q

FIG. 2. Color online The Wigner function. The experimen-tally measured field quadrature probability density prQ, isthe integral projection of the Wigner function WQ ,P onto avertical plane defined by the phase of the local oscillator.


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present optimal form Leonhardt et al. 1996; Leonhardt,1997; DAriano et al., 2004. We combine Eqs. 19 and22 to write

TrA = 0

+

prQ,FAQ,dQd

= FAQ,Q,, 23

where averaging is meant in the statistical sense over allacquired values of Q,, and

FAQ, =1

+

KQ cos + Psin Q

WAQ,PdQdP 24

is the sampling function. Given a specific operator A, thefunction FA does not depend on the experimental histo-gram prQ,, but only on the operator itself. It thusneeds to be calculated only once, prior to the experi-ment, and substituted into Eq. 23 once the data be-come available.

Specializing to the Fock basis FmnQ ,= 1/eimnMmnQ, with MmnQ the so-called pat-tern functions DAriano et al., 1994; Paul et al., 1995;Leonhardt and Raymer, 1996,

MmnQ = P

+ mxnx

Q x2dx , 25

where

nx = nx = 11/4 Hnx2nn! exp

x2

2 26

are the Fock state wave functionsthat is, wave func-

tions of energy eigenstates of a harmonic oscillator. Hndenote the Hermite polynomials. Figure 3a shows anexample of calculating the density matrix using the pat-tern function method.

Efficient numerical algorithms for computing the pat-tern functions were given by Leonhardt et al. 1996 andLeonhardt 1997. In our experience, the most practicalalgorithm involves the irregular wave functions nx,which are alternative, non-normalizable solutions of thetime-independent Schrdinger equation for the har-monic oscillator. These functions obey a recursion

n+1x =1

2n + 2xnx nx 27

with

0x = 3/4 exp x2

2erfix 28

and are all readily expressed through the error functionerfix. Once the desired number of irregular wave func-tions have been calculated, the pattern functions are ob-tained using

Mmnx = mxnx/x for nmnxmx/x for nm.

29

The pattern function method can be extended to di-rect sampling, or quantum estimation Munroe et al.,1995; Paul et al., 1995, of the expectation value of anyoperator directly without first reconstructing the state.In many cases, this requires fewer probability functionsto be measured, since less complete information is beingasked for. Indeed, since Wigner functions are linear withrespect to their generating operators, we conclude from

Eq. 24 that for any operator A

=Amnmn,

FAQ, = m,n

FmnQ,Amn

=1

m,n

nAmMmnqexpim n . 30

The expectation value A=TrA can then be calcu-lated according to Eq. 23.

For example, if we desire the photon-number prob-

ability prj, we choose Aj= jj. Then FAjQ ,= 1 /MjjQ, which is independent of phase , and Eq.

23 becomes Munroe et al., 1995

prj = Trjj =

dQMjjQprQ,. 31

This is a convenient result, since only a single probabil-ity function needs to be measured, while sweeping orrandomizing the phase. Demonstrations of this tech-nique have been given by Munroe et al. 1995, Schilleret al. 1996, as well as Raymer and Beck 2004.

Although the techniques of OHT have been general-ized to fields involving more than one optical mode spa-tial, polarization, or temporal Opatrn et al., 1996;

Raymer et al., 1996; DAriano et al., 2000, their practicalapplication is challenging. This is a particular case inwhich direct sampling is handy. We can apply it if ourtask is to determine the expectation values of certainobservables, but full reconstruction of the multimodestate is not necessary. An example is the acquisition ofcorrelated photon-number statistics of two-mode fieldsMcAlister and Raymer, 1997b; Vasilyev et al., 2000;Blansett et al., 2001, 2005; Voss et al., 2002.

B. Maximum-likelihood reconstruction

1. Why maximum likelihood?

Quantum-state reconstruction can never be perfect,due to statistical and systematic uncertainties in the es-timation of the measured statistical distributions. Inboth discrete- and continuous-variable domains, inverselinear transformation methods work well only whenthese uncertainties are negligible, i.e., in the limit of avery large number of data and very precise measure-ments. Otherwise, the errors in the right-hand side ofthe system of linear equations we are trying to solve canlead to inaccurate, even seemingly unphysical, featuresin the reconstructed state. For example, negative values




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may be found on the diagonal of the reconstructed den-sity matrix and its trace is not guaranteed to equal 1Fig. 3a.

In the case of continuous-variable tomography, thereis an additional complication: a harmonic oscillator is aquantum system of infinite dimension, and no finiteamount of measurement data will constitute a quorum.In order to achieve reconstruction, one needs to makecertain assumptions that limit the number of free param-

eters defining the state in question. For example, thefiltered back-projection imposes low pass filtering ontothe Fourier image of the Wigner function, i.e., assumesthe ensemble to possess a certain amount of classical-ity Vogel, 2000. Such smoothing reduces the accuracyof the reconstruction Herman, 1980; Leonhardt, 1997and introduces characteristic ripples Breitenbach et al.,1997 on the reconstructed phase-space density Fig.3a.

Although errors cannot be eliminated completely, wewould like a reconstruction method that guarantees aphysically plausible ensemble and minimizes artifacts.This requirement is satisfied by the maximum likelihood

MaxLik approach, which aims to find, among the vari-ety of all possible density matrices, the one that maxi-mizes the probability of obtaining the given experimen-tal data set and is physically plausible. Because thismethod is relatively new, but is rapidly gaining popular-ity, we present here its relatively detailed description.An even more comprehensive review on quantum Max-Lik limited to the discrete domain has been given byHradil et al. 2004.

2. Classical algorithm

We begin with a discussion of the classical

expectation-maximization method. Consider a certainsystem characterized by a set of parameters r such thatri0 and iri =1, which we need to determine. We areallowed to subject the system to a measurement with arandom outcome. The probability of each possible resultindexed by j is related to r linearly,

prrj = i

rihij, 32

where all hij are known positive numbers. The measure-ment is repeated Ntimes, of which each outcome occurs

fj times. The goal is to infer the parameter set r from the

set of measurement results f.The ideal inference is the one that satisfies the system

of linear equations

fj/N= i

rihij. 33

However, a solution to this system exists only if thenumber of parameters is larger than the number ofequations. Otherwise, we have to settle for less: find thedistribution r which would maximize the probabilitylikelihood

Lr = j

prrjfj 34

of the observed measurement result. This approach hasa very large variety of applications ranging from imagedeblurring to investment portfolio optimization.

The maximum-likelihood parameter set is determinedby the so-called expectation-maximization EM algo-rithm, which consists of sequential iterations Dempster

et al., 1977; Vardi and Lee, 1993

rin+1 = ri

nj

hijrjn

prrnj, 35

initialized with some positive vector r. Each single itera-tion step is known to increase the likelihood. Further-more, because the likelihood is a convex function, i.e.,for any two distributions r1 and r2 holds

L r1 + r22

Lr1 + Lr22

,

the iterations will approach the global likelihood maxi-

mum.

3. The discrete quantum case

A quantum tomographic procedure can be associatedwith a positive operator-valued measure POVM, witheach possible measurement result described by a posi-

tive operator j, which occurs with a probability

prj = Trj. 36

Here, again, we are dealing with a linear inversion prob-lem, because the probabilities are proportional to thedensity-matrix elements. However, the latter are notnecessarily positive not even real and their sum is notequal to 1, so the EM algorithm in its original form hasonly limited application to the quantum case.

In order to reconstruct a quantum state, we introducethe non-negative operator

R =1

N

j

fj

prjj. 37

As shown by Hradil 1997, the state that maximizes thelikelihood 34 obeys the extremal equation

R00 = 0R0 = 0 38

as well as

R00R0 = 0. 39

One can intuitively understand these equations as fol-lows: when is the maximum-likelihood state, we have

fj/N prj, so the operator R becomes jj, which is nor-mally unity.

The analogy to the classical scheme would suggest an

iterative procedure k+1= Rkk based on Eq. 38.However, unfortunately, such iteration does not pre-serve positivity of the density matrix unless it is guaran-




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teed to be diagonal in some constant basis, in which casethe iteration reduces to Eq. 35such as in Banaszek1998a, 1998b. A possible solution is to apply theexpectation-maximization iteration to the diagonalizeddensity matrix followed by rediagonalization Reh~eket al., 2001; Artiles et al., 2005.

A more common approach to constructing the itera-tive algorithm relies on Eq. 39 Hradil et al., 2004. We

choose some initial density matrix as, e.g., 0

=N

1

,and apply repetitive iterations

k+1 = NRkkRk , 40

where Ndenotes normalization to a unitary trace. Here-after, we refer to this scheme as the RR algorithm.

This iteration ensures positivity of the density matrixand has shown fast convergence in a variety of experi-ments. However, there is no guarantee of a monotonicincrease of the likelihood in every iteration; on the con-trary, there exists a counterexample Reh~ek et al.,2007. There remains a risk that the algorithm could failfor a particular experiment.

The remedy against this risk has been proposed byHradil et al. 2004 and further elaborated by Reh~ek etal. 2007. These authors present a diluted linear itera-tion

k+1 = N I+ R1 +

kI+ R

1 + , 41

which depends on a single parameter that determinesthe length of the step in the parameter space associ-ated with one iteration. For , the iteration becomesRR. On the other hand, in the limit of 0, there isproof that the likelihood will monotonically increase andthe iterations will converge to the maximum-likelihood

state. We thus obtain a reserve algorithm for the case inwhich the likelihood fails to increase in the RR itera-tion. In practice, however, this situation is not likely.

In some tomography schemes, one or more possiblemeasurement results may not be accessible and, conse-

quently, G jj is not equal to the unity operator.Then the extremal map 39 should be replaced by

G1R00R0G1 = 0 42

to avoid biased results Reh~ek et al., 2001; Hradil et al.2006; Mogilevtsev et al., 2007. This issue may becomesignificant in homodyne tomography reconstruction

which we discuss next.

4. Iterative scheme for homodyne tomography

The applications of MaxLik to homodyne tomographyhave been pioneered by Banaszek 1998a, 1998b, whoreconstructed the photon-number distribution the diag-onal density-matrix elements that correspond to aphase-randomized optical ensemble from a MonteCarlo simulated data set by means of the classical EMalgorithm. This idea was then extended to reconstruct-ing the Wigner function point by point Banaszek, 1999

by applying phase-dependent shifts to the experimentaldata. In a subsequent publication, Banaszek et al. 1999discussed direct MaxLik estimation of the density ma-trix, but presented no specific algorithm. More recently,the RR iterative algorithm was adapted to OHTLvovsky, 2004 and has since been frequently used inexperiments on homodyne reconstruction. We describethis adaptation below.

For a given local oscillator phase , the probability todetect a particular quadrature value Q is proportionalto

prQ, TrQ, , 43

where Q, is the projector onto this quadratureeigenstate, expressed in the Fock basis as

mQ,n = mQ,Q,n , 44

where the wave function m Q, = eimmQ is given

by Eq. 26.Because a homodyne measurement generates a num-

ber from a continuous range, one cannot apply the itera-tive scheme 40 directly to the experimental data. Oneway to deal with this difficulty is to discretize the data bybinning it up according to and Q and counting thenumber of events fQ,belonging to each bin. In this way,a number of histograms, which represent the marginaldistributions of the desired Wigner function, can be con-structed. They can then be used to implement the itera-tive reconstruction procedure.

However, discretization of continuous experimentaldata will inevitably lead to a loss of precision.6 To lowerthis loss, one needs to reduce the size of each bin andincrease the number of bins. In the limiting case of infi-

nitely small bins, fQ, takes on the values of either 0 or1, so the likelihood of a data set Qi ,i is given by

L = i

prQi,i, 45

and the iteration operator 37 becomes

R = i

Qi,i

prQi,i, 46

where i= 1 , . . . ,N enumerates individual measurements.The iterative scheme 40 can now be applied to find thedensity matrix which maximizes the likelihood 45.

In practice, the iteration algorithm is executed withthe density matrix in the photon-number representation.In order to limit the number of unknown parameters, wetruncate the Hilbert space by excluding Fock termsabove a certain threshold. This is equivalent to assumingthat the signal field intensity is limited. In many experi-

6In fact, recent research shows the precision loss due tobinning to be insignificant. On the other hand, binninggreatly reduces the number of data and thus expedites theiterative reconstruction algorithm Mogilevtsev 2007.




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mental situations, application of this assumption is bet-ter justified than the low-pass filtering used in the fil-tered back-projection algorithm.

Figure 3 compares the inverse linear transform andMaxLik reconstruction methods in application to the ex-perimental data from Lvovsky and Mlynek 2002. Thedata set consists of 14 152 quadrature samples of an en-semble approximating a coherent superposition of thesingle-photon and vacuum states. We see that the Max-

Lik method eliminates unphysical features and artefactsthat are present in the inverse Radon reconstruction.

5. Error handling

A homodyne detector of nonunitary efficiency canbe modeled by a perfect detector preceded by an ab-sorber. In transmission through this absorber, photonscan be lost, and the optical state undergoes a so-calledgeneralized Bernoulli transformation Leonhardt, 1997.If is known, the Bernoulli transformation can be in-corporated into the matrices of the POVM elements

Q

, Banaszek et al., 1999; Lvovsky, 2004. These

operators can then be used to construct the matrix R, sothe iterative algorithm will automatically yield the den-sity matrix corrected for detector inefficiencies.

Theoretically, it is also possible to correct for the de-tector inefficiencies by applying the inverted Bernoullitransformation after an efficiency-uncorrected densitymatrix has been reconstructed Kiss et al., 1995. How-ever, this may give rise to unphysically large density-matrix elements associated with high photon numbers.Similar concerns about possible numerical instabilityarise when the detector inefficiency is being accountedfor in the pattern-function reconstruction Kiss et al.,

1995. With the inefficiency correction incorporated, asdescribed above, into the MaxLik reconstruction proce-dure, this issue does not arise Banaszek, 1998b.

Another source of error in OHT MaxLik estimationcan be the incomplete character of the homodyne mea-

surements: the sum of the projection operators G

= iQi ,i is not an identity operator even in thetruncated Fock space. Mogilevtsev et al. 2007 foundthat the deviation can be quite significant. This issue canbe resolved by employing the iteration based on the bi-ased extremal equation 42 instead of Eq. 39, such asin the experimental work by Fernholz et al. 2008.

Finally, we discuss statistical uncertainties of the re-

constructed density matrix. In generic MaxLik algo-rithms, they are typically estimated as an inverse of theFisher information matrix Rao et al., 1945; Cramr,1946. This method can be generalized to the quantumcase Usami et al., 2003; Hradil et al., 2004. In applica-tion to OHT, calculating the Fisher information appearsquite complicated due to a large number of independentmeasurements involved.

A sensible alternative is offered by a clumsy, yetsimple and robust bootstrap method. One simulates thequadrature data that would be associated with the esti-mated density matrix ML if it were the true state. One

then generates a large number of random sets of homo-dyne data according to Eq. 43, then applies theMaxLik reconstruction scheme to each set, and obtains aseries of density matrices k, each of which approxi-mates the original matrix ML. The average differenceMLkk evaluates the statistical uncertainty associ-ated with the reconstructed density matrix. Reh~eket al. 2008 argue that the bootstrap method of errorestimation is less precise than that based on the Fisherinformation.

C. Maximum-entropy reconstruction

The maximum-entropy MaxEnt method is applied inthe situation opposite to that of the MaxLik approach,namely, when the number of equations in system 33i.e., the number of available data is smaller than thenumber of unknown parameters. In this case, the solu-tion is not unique, and MaxEnt looks for the least biasedsolution, i.e., the one that maximizes the von Neumannentropy S =Tr log . Because in OHT one usuallycollects a large 104 106 number of data points, theMaxEnt method is not commonly used. Here we give itsbrief overview; more details can be found in Buek2004.

Consider an OHT experiment, where the acquiredquadrature-phase data Qi ,i are binned into a rect-angular array of dimension NQN. As discussed aboveSec. III.B.3, the number of occurrences Gmn in eachbin m , n, associated with the quadrature value Qn andphase m, is proportional to the expectation value of the

observable mn = QmnQm

n.We seek the state that satisfies the MaxEnt principle

while fulfilling the conditionsTr = 1, Trmn = Gmn . 47

This state is given by

ME = Nexp 0n n=1

NQ

m=1

N

mnmn , 48

where n is the photon-number operator and s are theLagrange multipliers that are introduced to fulfill theconstraints 47. Note that the state 48 is guaranteed tobe positive and have trace 1.

The Lagrange multipliers can be found by minimizing

the deviation functionQ = n TrMEn

2

+ n=1

NQ

m=1

N

Gmn TrMEmn2. 49

Here n is the mean photon number which can be de-termined, for example, from the phase-averaged quadra-ture variance. Similarly to the MaxLik case, all calcula-tions are performed in the Hilbert space truncated by acertain maximum photon number. Because there are norestrictions imposed on the Lagrange multipliers, the


http://-/?-http://-/?-


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minimum of Q can be found by means of a genericoptimization algorithm.

Buek et al. 1996, as well as Buek and Drobn2000, have elaborated application of this method tohomodyne tomography and performed reconstruction ofvarious simulated data sets. They found the results to besignificantly better than those obtained by inverse lineartransform, particularly in situations of incomplete to-mographic data marginal distributions available for asmall number of phases or measured on short intervals.

IV. TECHNICAL ASPECTS

A. Time-domain homodyne detection

When homodyne detection was first introduced toquantum optical measurements in the mid-1980s, it wasused for evaluating field quadrature noise rather thanfull state tomography. Such measurements are conve-nient to perform in the frequency domain, observing acertain spectral component usually around 110 MHz,where technical noise is minimized of the photocurrentdifference signal using an electronic spectral analyzer.Frequency-domain detection was used, for example, toobserve quadrature squeezing Slusher et al., 1985; Wu etal., 1986.

Quantum-information applications require measure-ment of optical modes that are localized in time. Homo-dyning has to be performed in the time domain: differ-ence photocurrent is observed in real time andintegrated over the desired temporal mode to obtain asingle value of a field quadrature. Repeated measure-ments produce a quantum probability distribution asso-ciated with this quadrature.

In this section, we discuss the design of time-domain

balanced detectors that operate with pulsed local oscil-lators. The first such detector was implemented bySmithey et al. 1992 and Smithey, Beck, Raymer, et al.1993 in their original quantum tomography experi-ments. Among subsequent schemes we note that ofHansen et al. 2001, which features a higher bandwidthand a signal-to-noise ratio, as well as that of Zavatta etal. 2002, exhibiting a further significant bandwidth in-crease at a cost of somewhat poorer noise characteris-tics.

Figure 4a shows the main elements of the circuit ofHansen et al. 2001, which are typical for todays pulsed,time-domain HDs. A pair of high-efficiency photodiodes

are wired in series to subtract their output currents, andthis difference signal is amplified by a charge-sensitivetransimpedance amplifier, followed by a pulse-formingnetwork. The optics in front of the photodiodes permitsvariable attenuation of the input to each photodiode al-ternatively, a setting with two polarizing beam splittersand a half-wave plate between them can be used forcombining the local oscillator and the signal. Thoroughbalancing of the photodiodes photocurrents is essentialfor the proper operation of the BHD.

With each LO pulse, the detector produces a burst ofamplified subtraction photocurrent Fig. 4b. Because

the response time of the detector is much slower thanthe width of the laser pulse, the generated signal is pro-portional to the time integral of the photocurrent overthe pulse duration. This is a single sample of the fieldquadrature noise in the spatiotemporal optical mode ofthe local oscillator pulse.

To prove that the pulsed noise generated by the ho-modyne detector with a vacuum signal input is indeedthe shot noise, one needs to verify that the output rmsnoise scales as the square root of the LO power7 Fig.4c. This is a signature distinguishing the shot noisefrom the classical noise proportional to the local oscil-lator intensity and the electronic noise which is con-stant Bachor and Ralph, 2004.

Design of time-domain BHD is more technically chal-lenging than its frequency-domain counterpart. First, theelectronics must ensure time separation of responses toindividual laser pulses. The shot-noise difference chargemust be low-noise amplified within a bandwidth exceed-

ing the local oscillator pulse repetition rate. Second, pre-cise subtraction of photocurrent is necessary in order toeliminate the classical noise of the local oscillator. Thereis a competition between this requirement, which iseasier satisfied at lower LO energies, and that of a suf-ficiently strong subtraction signal N, which increaseswith the LO power. The compromise is reached on thescale of N 10

3 106 photoelectons. Finally, the mea-

7This follows from Eq. 13, because L =NLO and Q varyon the scale of 1.

FIG. 4. Color online Balanced detector for time domain ho-modyne tomography. a Electro-optical scheme. b A super-position of multiple oscilloscope traces of the detector output.Each pulse produces a time-resolved quantum noise sample.c rms peak amplitude of the noise pulses as a function of the

LO power showing the expected square root power depen-dence up to the LO intensities of 3108 photons per localoscillator pulse. Filled squares show the measured noise vari-ances, open squares have the electronic noise background cor-responding to 730 electrons/pulse subtracted. From Hansen etal., 2001.




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sured quadrature values must not be influenced by low-frequency noises. The detector must thus provide ul-tralow noise, high subtraction, and flat amplificationprofile in the entire frequency range from almost dc toat least the LO pulse repetition rate.

A typical dilemma faced by a BHD designer is atradeoff between the signal-to-noise more precisely,shot-to-electronic noise ratio and the bandwidthRaymer and Beck, 2004. An amplifier with a higher

bandwidth usually exhibits poorer noise characteristicsNicholson, 1974; Radeka, 1988. An additional band-width limitation arises from the intrinsic capacitance ofphotodiodes, which may cause instability in the amplifi-cation circuit. Technologically, this capacitance is deter-mined by the thickness of the photodiode p-i-n junction;reducing this thickness compromises the quantum effi-ciency. A homodyne detector with a time resolution ca-pable of accommodating a typical repetition rate of amode-locked, pulsed Ti:sapphire laser around 80 MHzwas demonstrated by Zavatta et al. 2002 and Zavatta,Viciani, and Bellini 2006.

Suppression of homodyne detector electronic noise is

important for quantum-state reconstruction. As shownby Appel et al. 2007, the presence of the noise leads toan equivalent optical loss of 1/ S, where S is the detec-tors shot-to-electronic noise ratio at the particular localoscillator power.

We note that time-domain homodyne detection findsits applications not only in quantum tomography, butalso in other fields of quantum and classical technology.One example is shot-noise-limited absorption measure-ments at subnanowatt power levels achievable thanks tothe very low technical noise Hood et al., 2000. Anotheris ultrafast, ultrasensetive linear optical sampling forcharacterizing fiber optical systems Dorrer et al., 2003.

Time-domain homodyning is also an essential element ofcontinuous-variable quantum cryptography Grosshansand Grander, 2002; Silberhorn et al., 2002; Funk, 2004;Lodewyck et al., 2007.

B. Matching the mode of the local oscillator

1. The advanced wave

In homodyne detection, the spatiotemporal opticalmode to be measured is determined by that of the localoscillator. In this way, OHT provides indirect informa-tion on the modal structure of the signal field. This is

useful for evaluating quantum optical information pro-cessing systems, which require that interacting opticalqubits be prepared in identical, pure optical modes. Onthe other hand, achieving the mode matching betweenthe local oscillator and the signal, or even preparing thesignal state in a well-defined, pure spatiotemporal mode,can be challenging. In this section, we discuss the mode-matching techniques, specializing to a particular case ofthe signal state being a heralded single photon.

In order to prepare a heralded photon, a parametricdown-conversion PDC setup is pumped relativelyweakly so it generates, on average, much less than a

single photon pair per laser pulse or the inverse PDCbandwidth. The two generated photons are separatedinto two emission channels according to their propaga-tion direction, wavelength, and/or polarization. Detec-tion of a photon in one of the emission channels labeledtrigger or idler causes the state of the photon pair tocollapse, projecting the quantum state in the remainingsignal channel into a single-photon state Fig. 5a.Proposed and tested experimentally by Hong and Man-del 1986 as well as Grangier et al. 1986, this techniquehas become a workhorse for many quantum optics ex-periments.

The biphoton is a complex entangled state with manyparameters spectrum, direction, polarization, etc. ofthe two photons highly correlated,

st = s,t,ks,kt1s,ks1t,ktdsdtdksdkt,50

where and k denote the frequencies and wave vectorsof the signal and trigger photons. If the trigger photon ismeasured with any uncertainty in one of these param-eters, the signal photon will be prepared in a nonpurestate,

s = TrtTt,ktsisi , 51

where Tt, kt is the transmission function of the filtersin the trigger channel defining the measurement uncer-tainty.

Formation of the heralded mode has been illustratedby the heuristic concept of advanced waves proposed byKlyshko 1988a, 1988b, 1988c and further advanced byAichele et al. 2002. According to this concept, the trig-ger photon detector is replaced with a fictitious lightsource, which, at the moment of detection, produces aclassical incoherent electromagnetic wave traveling

FIG. 5. Parametric down-conversion and the advanced wavemodel. a Preparation of single photons by conditional mea-surements on a biphoton state. b The Klyshko advancedwave model. The trigger detector is replaced with an incoher-ent light source, which generates an incoherent advanced wavepropagating backwards in space and time. Nonlinear interac-tion of this wave with the pump produces a difference-frequency pulse that mimics that of the conditionally preparedphoton. c In an experiment, a laser beam, aligned for maxi-mum transmission through all the filters, can model the ad-vanced wave. From Aichele et al., 2002.




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backwards in space and time see Fig. 5b. Whenpropagating through the trigger channel filters, the ad-vanced wave acquires some degree of spatiotemporalcoherence, quantified by the filters width. It then entersthe down-conversion crystal and undergoes nonlinearinteraction with the pump pulse, generating adifference-frequency pulse. This pulse turns out to becompletely identical, in its modal characteristics, to theoptical mode of the conditionally prepared single pho-ton Aichele et al., 2002, and is thus helpful for visual-izing many of its properties.

Suppose, for example, that the pump is pulsed fem-tosecond or picosecond. Because the timing jitter of thephoton counter event is typically on the order of a nano-second, the exact moment when the photon pair hasbeen emitted is uncertain, so the advanced wave can beassumed continuous in time. The nonlinear interactionbetween the advanced wave and the pump is, however,restricted by the spatiotemporal window determined bythe coherent pump pulse. If the latter is much narrowerthan the coherence time and coherence width of the ad-vanced wave, the difference-frequency pulse will be al-

most transform limited, in both the spatial and temporaldimensions.

We conclude that narrow spatial and spectral filteringof the trigger photon can be used to obtain the signalphoton in a pure spatiotemporal mode. To the best ofour knowledge, for the first time this fact was stated byZukowski et al. 1995, independently by Rarity 1995,and later confirmed in a more detailed study by Ou1997. Specifically in the context of OHT, theoreticaltreatment was given by Grosshans and Grangier 2001as well as by Aichele et al. 2002.

Mode purity of the signal photon does not by itselfguarantee its matching to the local oscillator. The ad-

vanced wave model suggests the following experimentalprocedure for achieving this matching. Although the ad-vanced wave propagates backwards in space and timeand is thus a purely imaginary object, it can be modeledby a forward-going alignment beam inserted into thetrigger channel so that it overlaps spatially and tempo-rally with the pump beam inside the crystal and passesthrough the optical filters Fig. 5c. Nonlinear interac-tion of such an alignment beam with the pump wave willproduce difference frequency generation into a spa-tiotemporal mode similar albeit no longer completelyidentical to that of the conditionally prepared singlephoton. If one observes and optimizes the interference

pattern between this wave and the local oscillator, onecan be sure that, after blocking the alignment beam, themode of the signal photon will be matched to that of thelocal oscillator Aichele et al., 2002.

Controlling the spatial mode of the signal photon issimplified if a single-mode optical fiber is used as anoptical filter instead of a pinhole arrangement. Such afilter automatically selects a pure spatial mode in thetrigger channel, which transforms to a spatially pure sig-nal photon. It is also advantageous in terms of the pairproduction rate Ourjoumtsev, Tualle-Brouri, Laurat, etal. 2006, supporting material. Unfortunately, there is

no similar arrangement possible for the spectral tempo-ral mode matching.

2. Decorrelating photons

Reducing the spectral linewidth of the trigger filterwill improve the mode purity of the heralded photons,but also reduce their production rate. This compromise

would be avoided if we could arrange the PDC setup insuch a way that the trigger and signal photons in theoutput of the down-converter are uncorrelated: thefunction in Eq. 50 could be written as

s,t = ss tt. 52

In this case, detection of any photon in the trigger chan-nel signifies that the signal photon has been emitted intoa pure spatiotemporal mode defined by the function s.

The first detailed theoretical inquiry into preparationof uncorrelated down-conversion spectra was made byGrice et al. 2001, based on general theoretical analysisof Keller and Rubin 1997. This theory was further

elaborated by URen et al. 2005, 2007. The configura-tion of the correlation function depends primarily onthe energy-conservation condition

s + t= p 53

and the phase-matching condition

ks + kt kp. 54

For any generated pair of photons with parameterss , ks ,t, kt, there must exist a pump photon p , kpfor which the above equations are satisfied.

Suppose the PDC occurs in an almost collinear con-figuration and the crystal is aligned so that Eqs. 53 and

54 hold simultaneously for the central pump frequencyp0 and some signal and idler frequencies s0 and t0,respectively. The frequency and wave vector are con-nected through dispersion relations d=vgrdk, wherevgr is the waves group velocity. Neglecting dispersionorders higher than 1 analysis beyond this approxima-tion has been made by URen et al. 2005, we cast Eq.54 into the form

s s0vgr,s

+t t0vgr,t

=p p0vgr,p

. 55

Equations 53 and 55, plotted in the s ,t plane with

p =p0, form straight lines crossing at s0 ,t0.Considered more accurately, the lines defined by theseequations are not infinitely narrow. This is because thepump is pulsed, so it contains photons not only at p0,but in a finite frequency range p p0/p deter-mined by the inverse pump pulse width p. The phase-matching condition also has a tolerance ks + kt kp/ Lc, where Lc is the crystal length that limits theregion of nonlinear optical interaction. The down-conversion spectrum is thus determined by the overlapof two band-shaped areas in the s ,t plane Figs. 6aand 6b.




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As can be seen, the spectrum does not automaticallyuphold Eq. 52. However, by choosing the crystal lengthand other parameters of PDC, one can engineer the tiltangle and width of the spectral region in which phasematching is satisfied, such that the overlap area can beexpressed in the product form 52, implying uncorre-lated photon spectra. The condition that has to be ful-filled takes the form Grice et al., 2001

1

2= 0.048L2 1

vgr,p

1

vgr,s 1

vgr,p

1

vgr,t , 56

where L is the crystal length and is the pump spectrumwidth.

There exist several theoretical proposals on shapingthe phase-matching region. Grice et al. 2001 found that

Eq. 56 is satisfied in a BBO crystal for degenerate col-linear type-II down-conversion if the pump wavelengthis set to 800 nm Figs. 6c and 6d. They calculated anumber of alternative down-conversion configurationswith decorrelated spectra. URen et al. 2003 proposedto implement PDC in a slightly noncollinear configura-tion and imposed additional restrictions onto the signal-idler spectrum by collecting only the photons emitted atcertain angles. In theoretical works by Walton et al.2003, 2004, down-conversion occurs in a nonlinearwaveguide, pumped almost orthogonally to the guideddirection. This also allows restricting the transverse

components of the signal and idler photon momenta, butwithout compromising the pair production rate and withthe possibility to choose the central wavelength of eachphoton. Torres et al. 2005 proposed to engineer thedown-conversion spectrum by employing a chirped,tilted pump and utilizing the Poynting vector walk-offeffect. Raymer et al. 2005 put forward the idea of plac-ing the down-conversion crystal into a microcavity,whose linewidth is much narrower than that allowed by

the energy conservation and phase-matching conditions.This leads to a spectrally uncorrelated biphoton in a spa-tial mode defined by the cavity. URen et al. 2006, 2007theoretically showed that the group delays can be con-trolled by means of a periodic assembly superlattice ofnonlinear crystals and birefringent spacers, and per-formed a proof-of-principle experiment to this effect, al-beit without actually achieving an uncorrelated spec-trum.

The only experimental demonstration of a virtuallyuncorrelated down-conversion spectrum to date hasbeen offered by Mosley et al. 2008. They used a rela-tively long potassium-dihydrogen-phosphate KDP

crystal and a pump wavelength of 415 nm. Under theseconditions, the pump will propagate with the samegroup velocity as the idler photon. Then the region al-lowed by the phase-matching condition Fig. 6b be-comes vertical and narrow, so the overlap region exhib-its almost no correlation. This was confirmed byobserving high-visibility Hong-Ou-Mandel interferenceof heralded signal photons from separate crystals.

3. The continuous-wave case

A completely different approach to producing her-alded photons must be taken if the pump is monochro-matic continuous and down-conversion occurs in anoptical cavity, such as in Neergaard-Nielsen et al. 2007.Spatial mode matching is simplified in this configurationbecause both photons are prepared in the spatial modeof the cavity. The biphoton spectrum is determined bythe cavity transmission spectrum: it consists of narrow afew MHz equidistant modes separated by the cavityfree spectral range FSR.8 If the down-converter ispumped at a frequency 20 where 0 coincides with oneof the cavity resonances, down-converted photons willbe generated at frequencies 0 nFSR Fig. 7.

Because the cavity itself acts as a spectral filter, addi-tional narrow filtering of the trigger channel is not nec-essary to generate a spectrally pure heralded photon. It

is sufficient to apply a spectral filter that would transmitone of the cavity modes e.g., t=0 FSR. In order todetermine the temporal mode of the heralded photon,we again resort to the advanced wave model. In contrastto the pulsed laser case, the timing uncertainty of thetrigger photon detection event is much smaller than theinverse spectral width of the biphoton determined bythe inverse cavity line width. The advanced wave can

8The free spectral range of a cavity equals the inverseroundtrip time of a photon inside the cavity.

FIG. 6. Shape of the biphoton correlation spectrum deter-mined by a the energy conservation condition 53, b ge-neric phase matching 55 in the case of collinear type II PDCin a BBO crystal pumped at 400 nm, c engineered phasematching collinear type II PDC in a BBO crystal pumped at800 nm, and d cumulative effect of conditions a and c.The inset in d shows the Schmidt decomposition of the bi-photon spectrum s ,i=mmumsvmi; with a propercombination of geometrical parameters of the experiment, theSchmidt decomposition contains only one term, i.e., the spec-trum d is uncorrelated. From Grice et al., 2001.




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thus be considered to be an infinitely short pulse. Thispulse enters the cavity and interacts with the pump, pro-ducing a difference-frequency pulse at s =0 +FSRwhich, in turn, gets filtered by the cavity. Because thecavity spectrum is approximately Lorentzian, the tempo-ral shape of the conditionally prepared mode is given bythe Fourier transform of the cavity line:

ft,tc = ettc, 57

where tc is the moment of the trigger event and is thehalf-width at half-maximum HWHM of the cavity line-width Fig. 8a.9

Homodyne detection of the field in this mode requiresa local oscillator pulse with the temporal shape gLtin Eq. 8 identical to ft, tc. Such a pulse can betailored from a continuous laser field by means ofacousto- or electro-optical amplitude modulation. An al-ternative, more practical, procedure consists of using acontinuous local oscillator gLt =1 and continuouslyacquiring the difference photocurrent It as a function

of time. The acquired photocurrent is then post-processed according to NItft, tcdt . As evidencedby Eq. 10, the integrated difference charge is the sameas that obtained with a pulsed LO. This idea was firstutilized by Neersgaard-Nielsen et al. 2006 and subse-quently by Neersgaard-Nielsen et al. 2007 and Wakui etal. 2007.

4. Strong pumping mode

We now discuss the regime of strong-pump PDC, suchthat the number of pairs generated within the time pe-riod corresponding to the inverse down-conversionbandwidth is not negligible. This case is complicated and

largely uninvestigated for the following reason: if thereexists any correlation in the joint time or spectral distri-butions of the signal and trigger photons, narrow filter-ing of the trigger channel does not guarantee purity ofthe signal state. For example, in the monochromatic-pump case, a trigger event at time tc heralds the pres-ence of a photon in the mode ft, tc but does not ensurethat no incoherent contributions are present from pho-

tons in nearby modes ft, tc Fig. 8b. In order toprepare a high-purity state, one needs to have a high-efficiency detector, which will not only trigger a pair pro-duction event, but will also ensure there are no moreevents nearby. If the state to be prepared is more com-plex than a Fock state, such as a Schrdinger kitten

state, the situation is even more complicated.The strong pumping case in the context of preparingsingle and multiple heralded photons was investigatedby Mlmer 2006 as well as Nielsen and Mlmer 2007a,2007b and Rohde et al. 2007. Sasaki and Suzuki 2006reported a comprehensive theoretical study employingmode expansion in the basis of prolate spherical func-tions, and they obtained analytical expressions for a fewlimiting cases.

Experimental handling of the mode mismatch in thestrong pumping case is, on the other hand, relativelystraightforward Ourjoumtsev, Tualle-Brouri, andGrangier, 2006; Ourjoumtsev, Tualle-Brouri, Laurat, et

al., 2006; Ourjoumtsev et al., 2007. One introduces anempiric probability that the state heralded by a click inthe trigger detector belongs to the mode analyzed by thehomodyne detector. With probability 1 , the heraldedstate belongs to an orthogonal mode, which is equivalentto a dark count event. The actual value ofcan be foundfrom the experimental data statistics.

V. APPLICATIONS IN QUANTUM TECHNOLOGY

Implementation of light for the purposes of QI tech-nology relies on our ability to synthesize, manipulate,and characterize various quantum states of the electro-

magnetic field. OHT is used in optical quantum informa-tion as a way to solve the last of the above tasks. In thissection, we discuss applications of OHT to discrete-variable quantum-optical information and quantum-optical technology in general. We review the new statesof light that have been created in the past few years,methods of their preparation, and their tomographic re-construction.

It is convenient to restrict this review by the temporalboundaries of the present century. The only nonclassicalstate of light investigated by OHT prior to 2001 was thesqueezed state Smithey, Beck, Cooper, et al., 1993;

9Equation 57 has to be modified if the weak pumping limitis not applicable Mlmer, 2006; Sasaki and Suzuki, 2006.

1

2

a)

b)

1

2

1

2

1

2

t

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

t

FIG. 8. Temporal modes of the trigger 1 and signal 2 pho-tons. Assuming that the time resolution of the trigger detectoris very high, the signal mode is defined by Eq. 57. Cases aand b correspond to the weak and strong pumping regimes,respectively. From Mlmer, 2006.

t s

triggerphoton

signalphoton

spectralfilter

FSR

FIG. 7. Color online Spectrum of the Fabry-Perot parametriccavity. If one of the spectral modes t is selected in the trig-ger channel, a click of the trigger detector heralds productionof the signal photon in a pure cavity mode at s =20 t.




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Breitenbach et al., 1997. The past few years, on the con-trary, have shown a technology boom, resulting in aplethora of new quantum optical states Table I, someof which are significant not only to QI technology, but tothe basic concepts of quantum physics.

A. Two-mode squeezed vacuum

1. Description of the state

Consider an idealized nondegenerate parametricamplifier in which photons are emitted into two well-

defined optical modes a and b Barnett and Knight,1985; Gerry and Knight, 2005. Parametric down-

conversion in this setting is governed by the Hamil-tonian H = iab ab, where is proportional to thesecond-order nonlinear susceptibility and the pump

field. Assuming the initial state of modes a and b isdouble vacuum 0, 0, the evolution for time t will resultin the two-mode squeezed vacuum twin-beam state

=1

cosh rn=0

tanh rnn,n, 58

where r=t. In the absence of absorption, the numbersof photons in the two modes are perfectly correlated.

Amplitudes of n-pair states obey thermal statistics.The entangled state 58 has the following wave func-

tion:

Q1,Q2 =1

exp 1

4e2rQ1 Q2

2

1

4e2rQ1 + Q2

2 59in the position basis and

P1,P2 =1

exp 14

e2rP1 + P22

1

4e2rP1 P2

2 60in the momentum basis. As evidenced by these equa-tions, although the individual variances of the quadra-ture observables in each mode are above that of thevacuum state, the positions associated with the twomodes are correlated, and the momenta are anticorre-lated in a highly nonclassical manner. In the limit r,state approaches that described by Einstein et al.1935. This state has many applications in quantum-

TABLE I. Quantum states recently characterized by OHT.

Reference State

Vasilyev et al. 2000 Two-mode squeezed vacuum

Wenger et al. 2005

DAuria et al. 2009

Lvovsky et al. 2001 Single-photon Fock state

Zavatta et al. 2004b 1Neergaard-Nielsen et al. 2007

Lvovsky and Babichev 2002 Displaced single photon

Lvovsky and Mlynek 2002 Single-rail qubit

Babichev et al. 2003 0+1

Babichev, Brezger, and Lvovsky 2004

Zavatta, DAngelo, Parigi, et al. 2006

Babichev, Appel, and Lvovsky 2004 Dual-rail qubit

DAngelo et al. 2006 0 ,1+1, 0

Zavatta et al. 2004a Photon-added coherent state

Zavatta et al. 2005 a

Zavatta et al. 2007 Photon-added thermal state

Parigi et al. 2007

Ourjoumtsev, Tualle-Brouri, and Grangier 2006 Two-photon Fock state 2

Wenger et al. 2004b Photon-subtracted squeezed

Neergaard-Nielsen et al. 2006 state Schrdinger kitten

Ourjoumtsev, Tualle-Brouri, Laurat, et al. 2006 1+3, where 1

Wakui et al. 2007

Ourjoumtsev et al. 2007 Squeezed Schrdinger cat




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information technology, for example, as a primary re-source for complete quantum teleportation Furusawa etal., 1998, quantum communication van Loock, 2002,imaging Lugiato et al., 2002, and process tomographyDAriano and Lo Presti, 2001.

Optical losses and spurious nonlinearities degrade thetwin-beam state, making the bipartite photon numbercorrelation imperfect and removing the minimum-uncertainty character of quadrature variances. A natu-

rally arising question in this context is whether, and towhich extent, this degraded state can still be used as anentangled resource in quantum-information technology.This question can be answered analytically if the two-mode state remains Gaussian.10 Bipartite Gaussianstates have been investigated extensively and their nec-essary and sufficient entanglement criteria have beenelaborated on Reid, 1989; Duan et al., 2000; Simon,2000.

Importantly, a Gaussian state is fully described by thefirst and second moments of its quadrature operators. Inmost practical cases, the first moments mean coherentamplitudes vanish. If a bipartite state is known to be

Gaussian, determining the 44 covariance matrixXiXj, where Xi= Q1 , P1 , Q2 , P2, is sufficient for itscomplete characterization and evaluation of its en-tanglement.

2. Homodyne characterization

As any bipartite state, the two-mode squeezedvacuum can be fully characterized by subjecting bothmodes to homodyne detection and measuring two-dimensional quadrature histograms for all phases ofboth local oscillators Fig. 9a. To the best of our

knowledge, such a complete experiment has not yetbeen performed, but has been approached in differentways.

The first homodyne measurement of the twin-beamstate was performed in a landmark work by Ou et al.1992. The experiment used a continuous-wave para-metric amplifier as the source and proceeded along thescheme of Fig. 9a. Quadrature correlations measuredin the two modes satisfied the Einstein-Podolsky-Rosennonlocality criterion introduced by Reid 1989.

This experiment, as well as many subsequent workse.g., Zhang et al., 2000; Schori et al., 2002; Bowen et al.,2004; Laurat et al., 2005, did not aim to fully character-ize the state, but only to demonstrate and quantify itsentanglement. For this purpose, it was sufficient to mea-sure quadrature noise correlations in the frequency do-main.

The first measurement on the twin-beam state thatcan be classified as OHT was performed by Vasilyev etal. 2000. The experiment, again, used the scheme ofFig. 9a, but employed a pulsed laser and featured time-

domain homodyne measurements, allowing direct sam-pling of the quadrature noise. The local oscillator phaseswere allowed to vary randomly, so no phase-sensitiveinformation was recovered, but photon number correla-tions between the two modes were determined Fig.

9c. Due to losses and complex modal structure of thedown-conversion output, this correlation was not perfectas predicted by Eq. 58. However, the observed statis-tics were clearly nonclassical.

More recently, two experiments exploited a simplifiedsetup with only one homodyne detector Fig. 9b tocharacterize the twin-beam state, along the lines of pro-posals by McAlister and Raymer 1997a, Richter 1997,Raymer and Funk 1999, and DAuria et al. 2005. InWenger et al. 2005, the mode mixer consisted of aphase shift applied to one of the twin beams and a sym-metric beam splitter. Homodyne detection was then ap-

10A Gaussian state is a state whose Wigner function is Gauss-ian. Linear losses, second-order nonlinearities and phase-spacedisplacements preserve the Gaussian character of a state.

NOPA

BHD

a)

NOPA

BHD b)

c)

)

Q

P

Q

P

modemixer

BHD

0123456

0

3

6

-0.05

0.00

0.05

0.10

0.150.20

0.25

0.30

-6 -4 -2 0 2 4 6

0.0

0.1

0.2

0.3

0.4

Corre

lation

d(n

-m)

Difference photon number (n-m)

FIG. 9. Color online Homodyne characterization of thetwin-beam state. a Scheme of the experimental setup for di-rect OHT. NOPA denotes nondegenerate optical parametricamplifier. b A simplified setup with one homodyne detec-tor; mode mixer is a linear-optical device see text. c Left:

joint photon-number probability distribution. Right: differencephoton-number distributions corresponding to the graph onthe left filled circles, experimental data; solid lines, theoretical

predictions; dashed lines, difference photon-number distribu-tions for two independent coherent states with the same totalmean number of photons. From Vasilyev et al., 2000. d Re-

constructed Wigner functions for modes c= a+ b /2 and d

= a b /2 show squeezing in position and momentum, re-spectively. From DAuria et al., 2009.




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plied to the beam splitter outputs, whose mode operator

can be written as a+ eib /2 where the phase is de-termined by the phase shifter. Additionally, single-mode quadrature distributions were measured by block-ing one of the PDC output modes. By measuringquadrature distributions at different values ofand dif-ferent local oscillator phases, the symmetric and Gauss-ian character of the bipartite state has been verified,

nonvanishing elements of the covariance matrix deter-mined, and entanglement quantified.In an experiment by DAuria et al. 2009, PDC took

place in a periodically poled KTiOPO4 crystal, with theproduced photons being identical in terms of both fre-quency and spatial mode, but having orthogonal polar-ization. In this way, the modes could be mixed by meansof a half- and/or quarter-wave plate, followed by a po-larizer. Quadrature noise statistics were measured for

modes a, b, c= a+ b /2, d= a b /2, e= ia+ b /2, f

= ia b /2, which was sufficient to assess the Gaussiannature of the state and to fully determine the covariancematrix Fig. 9d. From the latter, photon-number sta-

tistics similarly to the work ofVasilyev et al. 2000 dis-cussed earlier were evaluated.

B. Fock state tomography

The first non-Gaussian state to be studied by OHTLvovsky et al., 2001 is the single photon. This is notsurprising, given the role this state plays in basic andapplied quantum optics. Another important motivationfor this experiment was to demonstrate reconstructionof an optical state whose Wigner function takes on nega-tive values.

The schematic and results of the experiment are

shown in Fig. 10. The experiment employed a picosec-ond Ti:sapphire laser at a 790-nm wavelength. Pulsedsingle photons were prepared by conditional measure-ments on a biphoton state generated via parametricdown-conversion in the weak pumping regime. Narrowspatiotemporal filtering of the trigger photon was usedas outlined in Sec. IV.B. The field state in the signalchannel was characterized by means of optical homo-dyne tomography.

Remarkably, all imperfections of the experimentlosses in transmission of the signal photon, quantumefficiency of the HD, trigger dark counts, mode match-ing of the signal photon and the local oscillator, and spa-

tiotemporal coherence of the signal photon had a simi-lar effect on the reconstructed state: admixture of thevacuum 0 to the ideal Fock state 1,

meas = 11 + 1 00 . 61

The greater the efficiency , the deeper the well in theWigner function; classically impossible negative valuesare obtained when 0.5. The original 2001 experimentshowed =0.550.01; later, this value was improved to0.62.

An interesting feature of the optical single-photonstate reconstruction is that the technique of homodyne

tomography can be fully understood in the framework

of classical physics. This measurement could have beenconducted and interpreted by someone who does notbelieve in quantum mechanics. Yet the result of negativequasiprobabilities would appear absurd, incompatiblewith classical physicsthus providing strong evidence ofquantumness of our world.

A version of the single-photon Fock state tomographyexperiment, featuring a fast homodyne detector, allow-ing measurements at a full repetition rate of the pulsedlaser 82 MHz, was reported by Zavatta et al. 2004b.

Tomography of the two-photon Fock state 2 was re-ported by Ourjoumtsev, Tualle-Briori, and Grangier2006. The experimental arrangement is similar to that

of Fig. 10a, but the trigger channel is split and directedinto two single-photon detectors, whose simultaneousclick triggers a homodyne measurement. In order to ob-tain a sufficient rate of such events, the parametric gainhas to be non-negligible, which significantly complicatesthe analy

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