Creation, Storage, and On-Demand Releaseof Optical Quantum States with a Negative Wigner Function

Jun-ichi Yoshikawa,1, ∗ Kenzo Makino,1 Shintaro Kurata,1 Peter van Loock,2 and Akira Furusawa1, †

1Department of Applied Physics, School of Engineering,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

2Institute of Physics, Staudingerweg 7, Johannes Gutenberg-Universitat Mainz, 55099 Mainz, Germany(Dated: May 11, 2018)

Highly nonclassical quantum states of light, characterized by Wigner functions with negative val-ues, have been created so far only in a heralded fashion. In this case, the desired output emergesrarely and randomly from a quantum-state generator. An important example is the heralded pro-duction of high-purity single-photon states, typically based on some nonlinear optical interaction. Incontrast, on-demand single-photon sources were also reported, exploiting the quantized level struc-ture of matter systems. These sources, however, lead to highly impure output states, composedmostly of vacuum. While such impure states may still exhibit certain single-photon-like featuressuch as anti-bunching, they are not enough nonclassical for advanced quantum information pro-cessing. On the other hand, the intrinsic randomness of pure, heralded states can be circumventedby first storing and then releasing them on demand. Here we propose such a controlled release,and we experimentally demonstrate it for heralded single photons. We employ two optical cavities,where the photons are both created and stored inside one cavity, and finally released through adynamical tuning of the other cavity. We demonstrate storage times of up to 300 ns, while keepingthe single-photon purity around 50% after storage. This is the first demonstration of a negativeWigner function at the output of an on-demand photon source or a quantum memory. In principle,our storage system is compatible with all kinds of nonclassical states, including those known to beessential for many advanced quantum information protocols.

Subject Areas: Quantum Information, Quantum Physics, Photonics

I. INTRODUCTION

Representing flying quantum information, photons areideally suited for communication between the stationsof a quantum network [1, 2]. Among various photonicquantum states, single-photon states are the most fun-damental. In principle, single-photon-based qubits wouldallow for scalable quantum computation based on linear-optical circuits, auxiliary quantum states of single andmany photons, and photon counters [3, 4]. However, non-classical states beyond single photons are also expectedto be useful for future advanced quantum informationprocessing [5–12].

Single-photon sources are often divided into two types[13], where one corresponds to transitions in electronicenergy levels accompanied by a photon emission, andthe other is implemented by probabilistically generatingphoton pairs, and detecting one photon and thus herald-ing another one. The first type of sources allow for anon-demand emission of photons, which has been demon-strated with various matter systems [13, 14], such astrapped atoms, defects in diamonds, and semiconductorquantum dots. However, these systems typically share acommon disadvantage, namely a very low efficiency forcollecting the emitted photons in a specific single spatialmode, and so it remains unclear whether a photon emis-

∗ yoshikawa@ap.t.u-tokyo.ac.jp† akiraf@ap.t.u-tokyo.ac.jp

sion has actually happened until the photon is detected[14]. So far, only the class of heralded sources offers highfidelity and flexibility, though at the expense of their in-trinsic randomness.

The photon pairs of the heralded sources are typicallyproduced via an optical nonlinearity, namely either para-metric down conversion [15] or four-wave mixing [16].Since this only requires a perturbative coupling betweenlight and matter, heralded photons can be emitted in awell-defined mode with good purity and optical coher-ence. As a consequence, a negative Wigner function hasbeen observed so far only with this type of source [17, 18].Moreover, measuring parts of a nonlinear optical resourcestate also allows for the heralded preparation of quan-tum states beyond single photons or qubits [6–10]. Fi-nally, in contrast to the highly restrictive wavelengthsof existing on-demand sources, basically determined bythe corresponding matter energy levels, the heralded ap-proach allows for a broader range of wavelengths, in par-ticular, including the telecom wavelength [19]. Owingto these advantages, the most advanced quantum infor-mation experiments have been demonstrated with suchheralding schemes [8–12]. On the negative side, however,since those events when a photon is heralded are totallyrandom and uncontrollable, the probability for creatingmany such heralded photons simultaneously, as requiredfor instance in linear-optical quantum computation [3, 4],drops exponentially with the number of photons — unlessefficient quantum memories are available.

Therefore, a possible solution to overcome the intrinsic

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FIG. 1. Schematic diagrams of storage and release. (a) Experimental setup and beam paths at the storage stage. (b)Conceptual diagram in the frequency domain at the storage stage. (c) Experimental setup and beam paths at the release stage.(d) Conceptual diagram in the frequency domain at the release stage. Ti:Sa, titanium-sapphire laser; CW, continuous wave;SHG, second harmonic generator; AOFS, acousto-optic frequency shifter; NOPO, non-degenerate optical parametric oscillator;PPKTP, periodically-poled KTiOPO4 crystal for parametric down conversion; EOM, electro-optic modulator; APD, siliconavalanche photodiode; LO, optical local oscillator; FPGA, field-programmable gate array to process logic signals; MC, memorycavity; SC, shutter cavity; FSR, free spectral range.

randomness of the heralded schemes is to store a heraldedstate until it is required [20]. Indeed, single photons werestored in matter systems in previous experiments [21–23].However, by introducing such buffer memories, some ofthe disadvantages of matter systems again emerge, suchas low photon purity and limited tunability of the opti-cal wavelength (though the memory efficiencies are ac-tually improving [24]). As a remedy, several all-opticalexperiments have been reported, however, none of themdemonstrated high purity [25–29]. Moreover, each hadtheir own additional disadvantages: when the opticaldelay lines were combined with optical switches, onlydiscrete delays were possible [26]; when the dynamicalcontrol of the Q factor of a photonic-crystal nanocavitywas successful, the memory time scale was still only sub-nanoseconds [27–29]. In any case, it appears generallyinefficient to first create flying photons and then couplethem with stationary memories. In fact, using atomic en-semble memories, Duan et al. proposed a scalable quan-tum repeater with linear optics by smartly unifying theheralding and storage (write-in) processes [30, 31]. Ourscheme relies upon a similar one-step mechanism, how-ever, in an all-optical fashion, where a heralded photonis created and automatically stored in an optical cavity.

Here, we propose the storage and controlled releaseof heralded quantum states by concatenating two cav-ities, and we experimentally demonstrate our scheme

for the case of heralded single photons. We succeededfor the first time in acquiring, effectively on demand,a quantum feature as strong as the negativity of theWigner function. This feature is essentially different fromthe more conventional single-photon character of anti-bunching [13], which is immune to linear optical lossesand hence valid even for sources with very low purity.The Wigner-function negativity rules out any descrip-tion in terms of a classical phase-space distribution, andthus is important in quantum computing [5]. Moreover,we stress that our system is much more general than asimple photon gun. It is rather an all-optical unificationof a (potentially continuous-variable, high-dimensional)entanglement generator and a quantum memory. In fact,our heralding mechanism can produce quantum statesof light more general than single photons [8–10], and allsuch states can be potentially stored in our system. Inprinciple, our system may function as a universal quan-tum memory, where optical, flying quantum informationof arbitrary dimension is written into the memory viaunconditional continuous-variable quantum teleportation[11, 12], and later recalled on demand.

2

II. WORKING PRINCIPLE

A schematic of our experimental setup with a diagramin the optical frequency domain is shown in Fig. 1. Insidea nondegenerate optical parametric oscillator (NOPO)operating far below threshold, signal and idler photonsare probabilistically created and simultaneously appear-ing in different longitudinal modes. We placed a shuttercavity (SC) at the exit of the NOPO. Then, a photoninside the NOPO passes through the SC only when it isresonant with the SC, while otherwise it remains insidethe NOPO. Therefore, the NOPO concurrently works asa memory cavity (MC). The switch for releasing a pho-ton is quickly realized by shifting the resonance frequencyof the SC using an electro-optic modulator (EOM). Atfirst, we set the SC resonant to an idler photon. Anyidler photon immediately escapes from the concatenatedcavities and is then sent to a photon detector to her-ald the presence of its partner signal photon inside theMC [Fig. 1(a,b)]. After the photon creation succeeded,the signal photon can be released by switching the SCwhenever wanted [Fig. 1(c,d)]. Note that an extensionof this technique to multi-photon, phase-sensitive statesappears to be remarkably straightforward [8–10]. Multi-ple clicks of the photon detector would project the intra-cavity signal state onto a multi-photon state, and, unlikeprevious demonstrations without memory [8–10], theseclicks would not have to occur simultaneously. Furthernote that the switching EOM, which is often lossy, isplaced outside the MC. This is another reason for thehigh purity here compared to a previous scheme basedon an optical delay loop that contained the switchingEOM inside the loop [26].

On a more fundamental level, that intuitive explana-tion for the high output purity relies on the physics ofgeneral coupled systems. More specifically, we may definethe damping ratio of the coupled cavities to determinewhether they are over-damped or under-damped, whichis eventually reflected by the pulse shape of the releasedsingle photons. Although this observation is certainly ofgreat interest, it is not directly related to the main exper-imental achievement to be reported here, which is ratherthe controlled timing for releasing high-purity single pho-tons. Therefore, we choose to leave the above issue forfuture works.

The experimental parameters used in our demonstra-tion are described in the following section and in theSupplemental Material [32].

III. EXPERIMENTAL RESULTS ANDMETHODS

The output quantum states were characterized by op-tical homodyne detection. Homodyne detection employsinterference with a strong local-oscillator light beam, andhence it indicates the photons’ coherence and their tun-ability to some specific wavelength. This is in strong

contrast to single photons verified by anti-bunching, forwhich two single-photon creations are necessary andquantum coherence is confirmed by the Hong-Ou-Mandeleffect [33].

The experimental results are shown in Fig. 2, arrangedin an array of rows and columns. The storage time wasset to 0 ns, 100 ns, 200 ns, and 300 ns in addition to theintrinsic delay of about 150 ns after occurrence of theheralding signal, corresponding to columns from left toright.

First, we focus on the leftmost column, correspondingto the intrinsic delay. Figure 2(a) shows the temporalshape of the wavepacket envelope ψ(t) of the releasedflying photons, which was estimated from the raw dataof homodyne detection [34] (details can be found in theSupplemental Material). The horizontal axis is a relativetime, where 0 ns corresponds to the events of the idlerphoton detection. After opening the shutter at about150 ns, the wavepacket amplitude suddenly increases andrapidly goes to zero, which represents the emission of thephoton. The photon pulse width is about 50 ns. In fact,it can be inferred from the experimental parameters thatthe photon wavepacket exhibits damped oscillations (cor-responding to under-damping). This is experimentallyindicated by the overshoot of the temporal shape downto negative values. Before the photon emission, thereare already nonzero values from around −50 ns, showingpre-leakage. In Fig. 2(b), the quadrature samples andtheir histogram corresponding to the envelope functionψ(t) are presented. There is a dip at the center of thehistogram, which is a characteristic of a single-photonstate [17, 18]. By performing maximum likelihood esti-mation on the quadrature distribution [35], we obtainedthe Wigner function [Fig. 2(c)] and the photon-numberdistribution [Fig. 2(d)], where a dip with a negative valueof −0.054 for the Wigner function occurs at the phase-space origin as a result of a 58.2% fraction for the single-photon component.

Next, we look at the other panels to see whetherthe photon wavepacket alters its shape and position forlonger storages. As shown in Fig. 2(a), the emissiontimes are correctly shifted. However, the shape of thewavepacket is independent of the storage time, exceptfor the pre-leakage. These facts together indicate thesuccess of our demonstration. The single-photon com-ponents were estimated as 54.6%, 53.1%, and 49.7%, for100 ns, 200 ns, and 300 ns, respectively. Although longerstorage renders the Wigner function positive, the singlephotons may still be useful depending on applicationsowing to small multi-photon components [36].

All our results are combined in Fig. 3 for an easy com-parison between the envelope function ψ(t) [Fig. 3(a)], itsabsolute square |ψ(t)|2 [Fig. 3(b)], and the single-photoncomponent [Fig. 3(c)]. The probability density for thepresence of a photon at time t is proportional to the ab-solute square |ψ(t)|2. From Fig. 3(b), we can infer thatthe contribution of the pre-leakage is very small. Thelifetime of the photon storage is of the order of 1 µs, as

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can be seen from the fitting curve in Fig. 3(c), which iscomparable with the lifetimes of some ensemble memories[24]. In principle, this lifetime can be further extendedby decreasing the intracavity losses. In the following, wewill now summarize our experimental methods.

A. Procedure

Our setup is shown in Fig. 1. The source laser is acontinuous-wave Ti:sapphire laser with a wavelength of860 nm. A part of the laser output is frequency-doubledto 430 nm, which is used as a pump beam for the NOPO.Two longitudinal modes of the NOPO at 860 nm, sep-arated by a free spectrum range (FSR), are used as thesignal and idler. A single longitudinal mode (idler) is se-lected by filter cavities, and a photon detection projects

the twin mode (signal) onto a single-photon state. Aftersome predetermined waiting time, a high voltage of 721 Vis applied to the EOM to release the signal photon. Itwas experimentally essential to avoid above-threshold os-cillation at the unused longitudinal modes of the NOPO.The photon generation rate was about 300 per secondwith a NOPO pumping power of 3 mW. The power ofthe LO was 18 mW, which produced an optical shot noiseabout 20 dB above the detector’s electronic noise.

B. Cavity Configuration

The bow-tie shaped MC contains a periodically-poledKTiOPO4 (PPKTP) crystal as a nonlinear medium tooperate as a NOPO; it has a round-trip length of 1.4 m(corresponding to the FSR of 2.2× 102 MHz), where the

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FIG. 3. Experimental dependencies on the storage time. (a)Estimated wavepacket envelopes of the released photons. Thestorage times are 0, 100, 200, and 300 ns, in addition to theintrinsic 150 ns. (b) Absolute square of the envelopes in a,proportional to the photon probability density with respect totime. (c) Decay of the single-photon component with respectto the storage time, including the intrinsic delay of 150 ns.Red, experimental values of 58.2%, 54.6%, 53.1%, and 49.7%,with error bars of ±0.5%. The error was roughly estimatedby the bootstrap method [39]. Blue, exponential fitting curveP (t) = P (0) exp(−t/τ), where P (0) = 62.6% and τ = 1.98µs.

round-trip optical loss was 0.2–0.3%. These values deter-mine the photon lifetime. The SC contains a RbTiOPO4

(RTP) EOM for the resonant-frequency shift; it has around-trip length of 0.7 m, where the round-trip opticalloss was 3%. The coupler reflectivity between the MCand the SC is 97%, while that between the SC and theexternal field is 83%.

C. Verification

While the homodyne detection is a phase-sensitivemeasurement, the single-photon state is a phase-insensitive state. This phase-insensitivity was confirmedby the invariance of the quadrature distribution with thescanned and unscanned phase of the local oscillator. Thephoton envelope ψ(t) is determined by the principal com-ponent analysis (PCA) [34]. Each quadrature sample isobtained by a weighted integral of the continuous ho-modyne data x(t) as

∫x(t)ψ(t)dt. When we replaced

the idler beam by other, uncorrelated light before thephoton detection, the observed signal state was almosta zero-photon state, which ensures that the original sig-nal field was indeed conditionally obtained via the idlerdetection.

Further information on the experimental methods canbe found in the Supplemental Material [32].

IV. CONCLUSION

In conclusion, we proposed and experimentally demon-strated the insertion of a quickly tunable cavity at theexit of a NOPO, effectively serving as a kind of quan-tum memory. After a heralded creation, storage, andon-demand release of single photons with this system,the output states exhibited strong nonclassicality, as be-came manifest through their negative Wigner functions.Our device works at normal temperature and pressure.Moreover, our scheme is compatible with hybrid quan-tum optical experiments, where photon and homodynedetections are employed at the same time, like quan-tum teleportation [11, 12] and entanglement distillation[37, 38]. Many advanced quantum demonstrations arepossible with our system in the future, e.g., the stor-age and release of multi-photon and/or phase-sensitivestates of light [8–10], the controlled interference of manyphotons simultaneously released from independent mem-ories [3, 4, 33], and the deterministic transfer of ar-bitrary quantum optical states into the memories viaunconditional continuous-variable quantum teleportation[11, 12].

ACKNOWLEDGMENTS

This work was partly supported by PDIS, GIA, G-COE, and APSA commissioned by the MEXT of Japan,FIRST initiated by the CSTP of Japan, and REFOSTof Japan. P.v.L. acknowledges support from QuOReP ofthe BMBF in Germany.

[1] N. Gisin and R. Thew, Quantum Communication, Nat.Photon. 1, 165–171 (2007).

[2] H.J. Kimble, The Quantum Internet, Nature (London)453, 1023–1030 (2008).

5

[3] E. Knill, R. Laflamme, and G.J. Milburn, A Scheme forEfficient Quantum Computation with Linear Optics, Na-ture (London) 409, 46–52 (2001).

[4] P. Kok, W.J. Munro, K. Nemoto, T.C. Ralph, J.P. Dowl-ing, and G.J. Milburn, Linear Optical Quantum Comput-ing with Photonic Qubits, Rev. Mod. Phys. 79, 135–174(2007).

[5] A. Mari and J. Eisert, Positive Wigner Functions RenderClassical Simulation of Quantum Computation Efficient,Phys. Rev. Lett. 109, 230503 (2012).

[6] D. Gottesman, A. Kitaev, and J. Preskill, Encoding aQubit in an Oscillator, Phys. Rev. A 64, 012310 (2001).

[7] P. Marek, R. Filip, and A. Furusawa, DeterministicImplementation of Weak Quantum Cubic Nonlinearity,Phys. Rev. A 84, 053802 (2011).

[8] E. Bimbard, N. Jain, A. MacRae, and A.I. Lvovsky,Quantum-Optical State Engineering up to the Two-Photon Level, Nat. Photon. 4, 243–247 (2010).

[9] M. Yukawa, K. Miyata, T. Mizuta, H. Yonezawa, P.Marek, R. Filip, and A. Furusawa, Generating Super-position of up-to Three Photons for Continuous VariableQuantum Information Processing, Opt. Exp. 21, 5529–5535 (2013).

[10] M. Yukawa, K. Miyata, H. Yonezawa, P. Marek, R.Filip, and A. Furusawa, Generation and Characteriza-tion of Resource State for Nonlinear Cubic Phase Gate,arXiv:1305.4336 [quant-ph] (2013).

[11] N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb,E. Huntington, and A. Furusawa, Teleportation of Non-classical Wave Packets of Light, Science 332, 330–333(2011).

[12] S. Takeda, T. Mizuta, M. Fuwa, P. van Loock, and A.Furusawa, Deterministic Quantum Teleportation of Pho-tonic Quantum Bits by a Hybrid Technique, Nature(London) 500, 315–318 (2013).

[13] M.D. Eisaman, J. Fan, A. Migdall, and S.V. Polyakov,Invited Review Article: Single-Photon Sources and De-tectors, Rev. Sci. Instrum. 82, 071101 (2011).

[14] G.S. Buller and R.J. Collins, Single-Photon Generationand Detection, Meas. Sci. Technol. 21, 012002 (2010).

[15] S.A. Castelletto and R.E. Scholten, Heralded Single Pho-ton Sources: a Route towards Quantum CommunicationTechnology and Photon Standards, Eur. Phys. J. Appl.Phys. 41, 181–194 (2008).

[16] J. Fulconis, O. Alibart, J.L. O’Brien, W.J. Wadsworth,and J.G. Rarity, Nonclassical Interference and Entangle-ment Generation Using a Photonic Crystal Fiber PairPhoton Source, Phys. Rev. Lett. 99, 120501 (2007).

[17] A.I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J.Mlynek, and S. Schiller, Quantum State Reconstructionof the Single-Photon Fock State, Phys. Rev. Lett. 87,050402 (2001).

[18] J.S. Neergaard-Nielsen, B. Melholt Nielsen, H. Taka-hashi, A.I. Vistnes, and E.S. Polzik, High Purity BrightSingle Photon Source, Opt. Exp. 15, 7940–7949 (2007).

[19] S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos,N. Gisin, and H. Zbinden, High-Quality AsynchronousHeralded Single-Photon Source at Telecom Wavelength,New J. Phys. 6, 163 (2004).

[20] C. Simon et al.,Quantum Memories – A Review Basedon the European Integrated Project “Qubit Applications(QAP)”, Eur. Phys. J. D 58, 1–22 (2010).

[21] M.D. Eisaman, A. Andre, F. Massou, M. Fleischhauer,A.S. Zibrov, and M.D. Lukin, Electromagnetically In-

duced Transparency with Tunable Single-Photon Pulses,Nature (London) 438, 837–841 (2005).

[22] T. Chaneliere, D.N. Matsukevich, S.D. Jenkins, S.-Y.Lan, T.A.B. Kennedy, and A. Kuzmich, Storage and Re-trieval of Single Photons Transmitted between RemoteQuantum Memories, Nature (London) 438, 833–836(2005).

[23] H.P. Specht, C. Nolleke, A. Reiserer, M. Uphoff, E.Figueroa, S. Ritter, and G. Rempe, A Single-Atom Quan-tum Memory, Nature (London) 473, 190–193 (2011).

[24] M.P. Hedges, J.J. Longdell, Y. Li, and M.J. Sellars, Ef-ficient Quantum Memory for Light, Nature (London)465, 1052–1056 (2010).

[25] A.I. Lvovsky, B.C. Sanders, and W. Tittel, Optical Quan-tum Memory, Nat. Photon. 3, 706–714 (2009).

[26] T.B. Pittman, B.C. Jacobs, and J.D. Franson, Sin-gle Photons on Pseudodemand from Stored ParametricDown-Conversion, Phys. Rev. A 66, 042303 (2002).

[27] Q. Xu, P. Dong, and M. Lipson, Breaking the delay-bandwidth limit in a photonic structure, Nat. Phys. 3,406–410 (2007).

[28] Y. Tanaka, J. Upham, T. Nagashima, T. Sugiya, T.Asano, and S. Noda, Dynamic Control of the Q Factor ina Photonic Crystal Nonocavity, Nat. Mater. 6, 862–865(2007).

[29] A.W. Elshaari, A. Aboketaf, and S.F. Preble, ControlledStorage of Light in Silicon Cavities, Opt. Exp. 18, 3014–3022 (2010).

[30] L.-M. Duan, M.D. Lukin, J.I. Cirac, and P. Zoller, Long-Distance Quantum Communication with Atomic Ensem-ble and Linear Optics, Nature (London) 414, 413–418(2001).

[31] N. Sangouard, C. Simon, H. de Riedmatten, and N.Gisin, Quantum Repeaters Based on Atomic Ensemblesand Linear Optics, Rev. Mod. Phys. 83, 33–80 (2011).

[32] See Supplemental Material.[33] K. Sanaka, A. Pawlis, T.D. Ladd, K. Lischka, and Y.

Yamamoto, Indistinguishable Photons from IndependentSemiconductor Nanostructures, Phys. Rev. Lett. 103,053601 (2009).

[34] H. Adbi and L.J. Williams, Principal Component Analy-sis, WIREs Comp Stat 2, 433–459 (2010).

[35] K. Banaszek, Maximum-Likelihood Estimation ofPhoton-Number Distribution from Homodyne Statistics,Phys. Rev. A 57, 5013–5015 (1998).

[36] R. Filip and L. Mista, Jr., Detecting Quantum States witha Positive Wigner Function beyond Mixtures of GaussianStates, Phys. Rev. Lett. 106, 200401 (2011).

[37] H. Takahashi, J.S. Neergaard-Nielsen, M. Takeuchi, M.Takeoka, K. Hayasaka, A. Furusawa, and S. Masahide,Entanglement Distillation from Gaussian Input States,Nat. Photon. 4, 178–181 (2010).

[38] J. Fiurasek, Distillation and Purification of SymmetricEntangled Gaussian States, Phys. Rev. A 82, 042331(2010).

[39] B. Efron and R.J. Tibshirani, An Introduction to theBootstrap, (Champman & Hall/CRC, 1994).

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Supplemental Material forCreation, Storage, and On-Demand Release

of Optical Quantum Stateswith a Negative Wigner Function

V. PRE-LEAKAGE DUE TO IMPERFECTIONOF THE SHUTTER DECOUPLING

In the analysis based on the principal component anal-ysis (PCA) in the main text, the pre-leakage of the sig-nal photon also contributes to the purity. The obtainedwavepackets are informative in the sense that they showthe actual shape of the photon wavepacket, however, theinclusion of the pre-leakage to the analysis reveals anoverestimation of the memory lifetime. An important fig-ure here is the probability of a single photon in a modetime-shifted in accordance with the storage time, becauseit contributes to a successful multi-photon interferencefrom independent memories.

In Figs. 4 and 5, we show the results for time-shiftedwavepackets. The wavepacket estimated from the re-sults of the intrinsic delay is the original one. Here,the time bin from −150 ns to 450 ns was clipped, andthe wavepacket then renormalized. Subsequently, it wasshifted by 100 ns, 200 ns, and 300 ns to analyze the cor-respondingly delayed results. No correction of losses wasperformed in the data processing.

We anticipate a reduction of the pre-leakage by increas-ing the number of the shutters serially. Another interest-ing option would be to monitor the pre-leakage by a pho-ton detector. If we detect a photon before opening theshutter, we can simply discard the stored light and starta new creation of quantum states. In this way, the puritywill be increased compared to the case of not monitoringthe pre-leakage.

VI. SUPPLEMENTAL METHODS

In this section, we will describe the experimental meth-ods and parameters, which are not included in the maintext, except for the quantum state reconstruction fromthe homodyne outcomes, which will be discussed sepa-rately in the next section.

The schematic setup is shown in Fig. 1 of the maintext. The contents of this section can be listed as follows:first, chopping of bright beams, second, filtering of theidler path, third, detuning of optical frequencies, fourth,data storage by the oscilloscope, fifth, other experimentalparameters.

A. Feedback Control of the Optical System andBeam Chopping

In order to keep the cavities resonant to the laserbeams, we utilized interference of auxiliary bright beams,

from which we obtained error signals for classical feed-back control. However, such bright beams, except for thelocal oscillator (LO) beam and the pump beam, can in-duce fake clicks or even break down the avalanche photo-diode (APD). Therefore, in the experiment, we switchedthe optical system cyclically between two phases: oneis the phase for actively controlling the optical systemwith classical feedback by using bright beams (classicalphase), and the other is the phase for manipulating thequantumness at the single-photon level by blocking thebright beams (quantum phase).

The blocking and unblocking of the bright beams wasachieved by switching on and off the diffraction via twoacousto-optic modulators (AOMs). One AOM upshiftedthe frequency, while the other AOM downshifted it tocancel out the frequency shift. The cavity lengths, whichdetermine the cavity resonances, were controlled by us-ing piezoelectric transducers (PZTs), except for the fastswitching of the shutter cavity (SC), which was achievedby an electro-optic modulator (EOM). During the quan-tum phase, the resonance of the cavities was held by keep-ing the voltages applied to the PZTs. We also placeda switching AOM just before the avalanche photodiode(APD), so that only the beam at the quantum phase canenter the APD. Note that these AOMs and PZTs, as wellas several EOMs that modulated the bright beams forthe Pound-Drever-Holl locking of the cavities, are omit-ted from Fig. 1 of the main text.

We repeated the quantum-classical phase changes at arate of 5 kHz and with a time ratio of 2 : 3. Thus, the300 counts per second for the single-photon generation inthe experiment effectively correspond to 750 counts persecond if we could maintain the quantum phase. Therepetition was controlled by the same FPGA that con-trolled the storage time after the heralding signals. Theheralding signals were gated by the FPGA, so that onlythose at the quantum phase drive the switching EOM ofthe SC.

B. The Idler Beam Path and Filter Cavities

We utilized three filter cavities. One was a bow-tieshaped ring cavity with a round-trip length of 250 mm,and the other two were Fabry-Perot cavities with around-trip lengths of several mm. The first ring cavityalso worked as a spatial separator of the signal and idlerbeams. These three cavities had different free spectralranges (FSRs) to prevent beams other than the targetidler from entering the APD. Between the two Fabry-Perot cavities, we inserted an optical isolator in order todecouple the two cavities. After the final filter cavity, asnoted above, we placed an AOM for switching the beampath. A spatial single mode was selected by a single-mode fiber before the APD to reduce fake clicks by straylight.

The transmission efficiencies of the filter cavities were90%, 76%, and 65%. That of the optical isolator was

7

0 1 2 3 4 5 6

00.0

0.0

0.1

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0.000.050.100.15

-0.05

0.00.20.40.60.81.0

0.2

0.4

0

2

3

4

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-1 1-2-3 2 3

0 200 400 600

0-1 1-2-3 2 3

10 4

0 1 2 3 4 5 6

0-1 1-2-3 2 3

0 200 400 600

0-1 1-2-3 2 3

0 1 2 3 4 5 6

0-1 1-2-3 2 3

0 200 400 600

0-1 1-2-3 2 3

0 1 2 3 4 5 6

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0 200 400 600

0-1 1-2-3 2 3

Photon Number

Quadrature

Quadrature

Time (ns)P

roba

bilit

yN

orm

aliz

ed F

requ

ency

Num

ber o

f Sam

ples

(1/

ns)(a)

(b)

(c)

(d)

ψ(t)

Photon Number Photon Number Photon Number

Quadrature Quadrature Quadrature

Quadrature Quadrature Quadrature

Time (ns) Time (ns) Time (ns)

FIG. 4. Experimental results for various storage times with time-shifted wavepackets. (a) Wavepacket envelopes. The idler-photon detection corresponds to 0 ns. (b) Samples of quadrature amplitudes (upper boxes), their normalized histograms (bluein lower boxes), and their fitting curves (red in lower boxes). (c) Sectional side view of the Wigner functions cutting throughthe phase-space origin. (d) Photon-number distributions. (c) and (d) correspond to the fitting curves in (b). The storage timesare 0, 100, 200, and 300 ns, in addition to an intrinsic delay of 150 ns, corresponding to the columns from left to right. Thevalues of the Wigner functions at the origin are −0.054, −0.020, −0.002, and 0.033 from left to right, respectively.

88%. The diffraction efficiency of the AOM was 66%.The coupling efficiency to the single-mode fiber was 83%.The APD was an SPCM-AQRH-16 (Excelitas Technolo-gies), whose dark counts were 5 counts per second (takinginto account the time ratio, 13 counts per second). Notethat the losses on the idler line do not deteriorate thepurity of the heralded signal-photon states, but only de-crease the photon generation rate, including a relativeincrease of the dark-count contribution.

C. Optical Frequencies

The bright light source was a continuous wave (CW)Ti:sapphire laser operating at 860 nm, whose frequencyis denoted by ωsource. The optical frequency of the pump,signal, and idler beams for the nondegenerate optical

parametric oscillator (NOPO) are denoted by ωpump,ωsignal, and ωidler, respectively. From the energy con-servation condition of the parametric down conversion,we have ωpump ≈ ωsignal + ωidler. Here, since we musttake into account the finite bandwidth of each beam, weuse the approximately equal sign ‘≈’ instead of the equalsign ‘=’. For the characterization of the emitted lightwavepacket, we employed the output of the Ti:sapphirelaser without frequency shift as the LO for the homodynedetection, thus ωsignal ≈ ωLO = ωsource. The signal andidler frequencies were separated by nearly the FSR of theNOPO ∆ωNOPO, thus ωidler ≈ ωsource ±∆ωNOPO, whilewe did not distinguish plus or minus in the experiment.Therefore, the pump beam was the second harmonic ofthe light source with a frequency shift about the FSR,ωpump ≈ 2ωsource ±∆ωNOPO. The frequency of the sec-ond harmonic was shifted by an acousto-optic frequency

8

(a)

(b)

(c)

Storage Time (ns)

Time (ns)

0 100 200 300 400 500 600

-200 0 200 400 600 800

Time (ns)-200 0 200 400 600 800

0.0

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0.4

0.6

0.8

1.0

Pur

ity(1

/ns)

ψ(t)

| ψ(t)

|2(1

/ ns

)

FIG. 5. Experimental dependencies on the storage time withtime-shifted wavepackets. (a) Wavepacket envelopes. Thestorage times are 0, 100, 200, and 300 ns, in addition to theintrinsic 150 ns. (b) Absolute square of the envelopes in (a),proportional to the photon probability density with respect totime. (c) Decay of the single-photon component with respectto the storage time, including the intrinsic delay of 150 ns.Red, experimental values of 58.2%, 52.9%, 49.9%, and 44.8%,with error bars of ±0.5%. Blue, exponential fitting curveP (t) = P (0) exp(−t/τ), where P (0) = 65.9% and τ = 1.19µs.

shifter (AOFS) to produce the pump beam, as depictedin Fig. 1 of the main text.

However, as will be discussed below, we made use ofdetuning for various experimental reasons. The pumpfrequency shift ωpump−2ωsource was not exactly ∆ωNOPO,but detuned by 5 MHz. In addition, the signal frequencyωsignal was not exactly at the LO frequency ωsource, butdetuned by 300–500 kHz.

The former detuning was applied on the pump beamby the AOFS. The latter detuning corresponds to a shiftof the resonance of the NOPO. However, this detuningwas also realized via an AOFS, which is omitted fromFig. 1 of the main text. It was achieved by shifting thefrequency of the auxiliary bright beam utilized for thelocking of the NOPO.

Next, we explain the experimental reasons for the twodetunings.

1. Suppressing above-threshold oscillation

As mentioned in Sec. III of the main text, our NOPOstarts above-threshold oscillation at unused longitudinalmodes unless detuning is utilized. Since the longitudinalmodes are almost equally spaced by the FSR ∆ωNOPO

in the frequency domain, whenever a pair of ωsignal andωidler satisfies the energy conservation condition, thereare other pairs of longitudinal modes ωsignal +n∆ωNOPO

and ωidler−n∆ωNOPO with an integer n that also satisfythe energy conservation condition. Every longitudinalmode has a different finesse due to the frequency depen-dence of the SC. The higher the finesse is, the strongeris the intracavity vacuum field of the mode. As a result,every pair that satisfies the energy conservation has a dif-ferent gain of the down conversion, proportional to theproduct of the corresponding intra-cavity field strengths.In the experiment, we used the pair of a high-finesse mode(signal) and a low-finesse mode (idler) at the memoryexcitation (write-in) stage. However, almost inevitably,pairs exist that are composed of two high-finesse modes.Such pairs have much larger gain of the down conversionthan the selected signal and idler pair, thus the NOPOeasily goes beyond threshold for these pairs.

On the other hand, the high-finesse modes have muchnarrower bandwidth than the low-finesse modes. Thus,in principle, the down conversion gain becomes very smallwith a small detuning of the pump frequency for a pairof two high-finesse modes, while it remains almost un-changed for a pair of high- and low-finesse modes. By thistechnique, the above-threshold oscillation can be avoided.

Experimentally, a relatively large detuning of 5 MHzwas introduced to remove any above-threshold oscilla-tion. Another option would be to insert wavelength-selective components inside the MC or the SC. However,since such components are typically lossy, we did not ex-plore this option in our demonstration.

2. Reducing storage of scattered light

In the experiment, the LO beam for the homodyne de-tection was not switched. Therefore, the LO was beingscattered during the quantum phase, mainly at the pho-todiodes of the homodyne detector. We observed thatthe scattered LO, corresponding to a weak coherent stateat the single-photon level, was stored inside the MC andinduced phase-space displacements of the stored signalstates. This can be avoided though, with only a small de-terioration of the measured quality of the single-photonstates, by slightly detuning the signal-mode frequency ofthe MC from that of the LO. We detuned the signal fre-quency by 300 kHz–500 kHz, which is much smaller thanthe released photon bandwidth of about 20 MHz and thusdid not significantly degrade the photon quality observedby the homodyne detection. The detuning was achievedby a small shift of the MC round-trip length from the LOresonance, controlled by a PZT.

9

We did not take much care of the exact value of thedetuning, because the target quantum state was a phase-insensitive single-photon state. If we did want to write ina phase-sensitive state, since the stored state during itsstorage is expected to rotate in phase space relative to theLO frame at the detuning frequency, we would have toeither compensate this rotation or remove the displace-ments from the LO scattering according to the detuningcondition. However, we stress that the scattered light inour experiment was solely a technical problem of the ver-ification part of the setup, and not at all a fundamentaldeficiency of the memory system itself.

D. Data Storage

The heralding signals, which were processed by theFPGA, were not only to trigger the EOM driver, but alsoto trigger the oscilloscope storage of the homodyne signal.The oscilloscope captured the homodyne signal for 1 µsaround each trigger signal. The resolution of the analog-to-digital conversion was 8 bits and the sampling ratewas 1 GS/s. We repeatedly acquired the frame 4.3× 104

times for each condition.

E. Other Parameters

In addition to the parameters given above and inSec. III of the main text, there are the following exper-imental parameters: the propagation efficiency from theexit of the SC to the homodyne detector was 97%, thequantum efficiency of the homodyne detector was 99%,and the interference visibility between the LO and theoutput beam of the SC was 99%.

The size of the second-order nonlinear optical PPKTPcrystals for the NOPO and the SHG was 10 mm in lengthand 1 mm by 1 mm in cross section. The radius ofcurvature of the two concave mirrors of the NOPO was150 mm, while the focal length of the lens inside the SCwas 500 mm.

The FPGA that controled the bright beam choppingand the delay of the trigger signal to shift the resonanceof the SC was Virtex-4 (Xilinx). The FPGA clock periodwas 10 ns, and thus, in our system, the photon emissiontimes were discretized into 10 ns intervals, whereas theoptical system itself would, of course, allow emissionscontinuous in time.

VII. VERIFICATION BY HOMODYNEDETECTION

As described above, we recorded a 1 µs frame of homo-dyne signal around each heralding signal. Every framecontained a single measurement result corresponding to asingle-photon state. However, every frame also contained

vacuum noise of other orthogonal modes. We must there-fore extract the information of the target single photonfrom such background noise, which is a big differencefrom click-by-click photon detection. In this section, wewill compare the homodyne and photon detections, anddiscuss how the information of the single-photon statecan be extracted from the homodyne signal.

The contents of this section are as follows. We willfirst review the mathematical description of the outputwavepacket, introducing the mode function. The modefunction is equivalent to the wavepacket envelope used inthe main text, however, here we choose a different termi-nology. Next, we will describe both the photon and thehomodyne detection, with a focus on optical coherence.Then, we explain how to estimate the mode function viathe PCA. Finally, we briefly describe how to representquantum states by Wigner functions.

A. Mathematical Description of a FlyingSingle-Photon State

Since the transverse modes and the polarization aredetermined by the LO, here we focus on the longitudinalmodes of the wavepackets. We use the time t as a con-tinuous degree of freedom in the longitudinal direction,which could also be, alternatively, the spatial coordinatez, with z = ct and c the speed of light. We define annihi-lation and creation operators at time t as a(t) and a†(t),respectively. Their commutation relation is,

[a(t), a†(t′)] =δ(t− t′), [a(t), a(t′)] =0, (1)

where δ(t) is the Dirac delta function. The photon den-sity operator at time t is defined as,

n(t) ≡ a†(t)a(t). (2)

Similarly, we define annihilation and creation operatorsfor a wavepacket mode specified by a mode function ψ(t)as follows,

a†ψ =

∫dtψ(t)a†(t), aψ =

∫dtψ∗(t)a(t), (3)

where the superscript ∗ denotes the complex conjugate.In the following, for simplicity, we often drop the argu-ment (t) of the mode function. The usual single-modebosonic commutation relations are equivalent to the nor-malization of the mode function (ψ,ψ) = 1, with theinner product of two mode functions,

(ψ, φ) ≡∫dtψ∗(t)φ(t). (4)

The corresponding photon-number operator is defined as,

nψ ≡ a†ψaψ. (5)

Note that∫dt|ψ(t)|2n(t) is very different from nψ. Re-

lated to this fact is the impossibility to determine the

10

photon number in a given wavepacket ψ in a simple wayusing photon detection, as will be discussed later.

An infinite set of orthonormal mode functions {ψk}k∈Nspecifies orthogonal modes such that

[aψk , a†ψ`

] =δk`, iff (ψk, ψ`) =δk`, (6)

where δk` is the Kronecker delta. We choose an infinitenumber of such orthonormal functions {ψk}k∈N to obtaina basis for the function space. Then, the creation oper-ator of an arbitrary mode ψ is expressed in this basisas,

a†ψ =∑k∈N

(ψk, ψ)a†ψk . (7)

Since the number operators {nψk}k∈N of orthonormalmodes are mutually commutative and independent, theyhave simultaneous eigenstates with independent eigenval-ues nk,

|n0 ψ0 , n1 ψ1 , n2 ψ2 , . . .〉 =a†n0

ψ0√n0!

a†n1

ψ1√n1!

a†n2

ψ2√n2!

. . . |∅〉, (8)

where |∅〉 is the zero-photon (vacuum) state, a(t)|∅〉 = 0for all t. An arbitrary pure state is a superposition ofsuch number states. This also means that the overallHilbert space H of the multi-mode quantum states can bedecomposed into a tensor product of single-mode Hilbertspaces,

H = Hψ0 ⊗Hψ1 ⊗Hψ2 ⊗ . . . . (9)

Operators acting on the total Hilbert space H may also

be decomposed into such tensor products, e.g., a†ψ0=

a†⊗ 1⊗ 1⊗ . . . where 1 is the identity operator. However,this does not hold in general. For example, a densityoperator % which cannot be expressed as a convex sumof tensor products describes an entangled (inseparable)state. The single-mode density operator ρψ0 is obtainedby tracing out all the orthogonal modes,

ρψ0 = Tr⊗k 6=0H

ψk %, (10)

where TrHX is the partial trace over a Hilbert space HX .In the following, we will use % for an overall density op-erator and ρ for a density operator reduced to a singlemode.

An ideal single-photon state, for which exactly onephoton is present in some wavepacket ψ, is expressed as

|1ψ〉 = a†ψ|∅〉 =

∫dtψ(t)a†(t)|∅〉. (11)

Realistic single-photon states, however, are effectivelymixtures of the above pure single-photon state and thevacuum state,

%ψ,p = p|1ψ〉〈1ψ|+ (1− p)|∅〉〈∅|. (12)

From the decomposition in Eq. (7), the reduced densityoperator of the mode ψ0 is,

ρψ0 = Tr⊗k 6=0H

ψk [p|1ψ〉〈1ψ|+ (1− p)|∅〉〈∅|]

= p|(ψ0, ψ)|2|1〉〈1|+ [1− p|(ψ0, ψ)|2]|0〉〈0|, (13)

where |n〉 is a single-mode n-photon state. As expected,the purity is maximized by setting the observed modefunction ψ0 equal to the ideal mode function of the singlephoton ψ. However, whenever these two functions aredifferent, the purity decreases depending on the modemismatch from p to p|(ψ0, ψ)|2.

For a single-photon state, the photon density with re-spect to time t is,

P (t) = Tr(n(t)|1ψ〉〈1ψ|) = |ψ(t)|2. (14)

Therefore, as mentioned in the main text, the probabilitydensity for the presence of a photon at time t is propor-tional to the absolute square of the mode function.

B. Homodyne Detection versus Photon Detection

A photon detector, such as an APD in the Geigermode, detects photons click by click in a particle-likemanner, whereas a homodyne detector observes thewave-like interference of the signal with a LO. Therefore,unlike photon detection, a single homodyne outcome cannever show the presence of a photon. However, from thestatistics of the outcomes, we can estimate the state –a technique known as quantum tomography. Homodynecharacterization is advantageous in several respects, suchas high quantum efficiencies and the full characterizationof a single optical mode. On the other hand, it is alsodemanding, because it requires a light source coherentwith the target field.

What we would like to stress here is that the sole ca-pability of homodyne verification is already an indicationfor the coherence of the obtained single photons. This isin strong contrast to photon-detection-based verificationrelying upon the observation of anti-bunching. The lat-ter negates the simultaneous presence of photons, but itcontains no information about the coherence.

Another important point is that the anti-bunching isimmune to linear optical losses. On the other hand,the purity, which is given by the probability of the cor-responding wavepacket to contain exactly one photon,also directly represents the success probability for creat-ing single photons in the specific wavepackets coherently.The purity is degraded by optical losses, and in this sense,it is generally much more difficult to obtain good puritythan to obtain good anti-bunching.

1. Photon detection

Here we suppose that the speed of the photon detectoris high enough compared to the time scale of the photon

11

wavepacket. Then, the observable of the photon detec-tion at time t is well approximated by the photon den-sity operator n(t) ≡ a†(t)a(t). Given a density operator% containing at most one photon, the probability densityof the photon detection is P (t) = ηTr(n(t)%), where η isthe quantum efficiency of the detection system. Insertingthe density operator %ψ,p from Eq. (12), we obtain theprobability density to detect a photon at time t,

P (t) = pη|ψ(t)|2. (15)

Therefore, by taking statistics of the detection time t,we gain information about the mode function withoutits phase component, |ψ(t)|. The photon probability pis estimated from

∫dtP (t) if the quantum efficiency η

is known. However, as already mentioned, detecting aphoton with a probability density η|ψ(t)|2 can never be aproof for the presence of a photon in a wavepacket ψ. Forexample, we are not able to distinguish functions differentin phase, ψ(t) and ψ(t)eiθ(t). Note that an optical fre-quency shift by ω corresponds to setting θ(t) = ωt. Fur-thermore, we cannot even distinguish a pure state from adecohered mixed state. In order to show this, we considersmooth functions φτ (t) ≡ φ(t − τ) in order to deconvo-lute the photon density as |ψ(t)|2 =

∫dτp(τ)|φ(t− τ)|2,

where (φ, φ) = 1 and∫dτp(τ) = 1. Then, a mixed state,∫

dτp(τ)|1φτ 〉〈1φτ |, (16)

gives the same probability density P (t) = η|ψ(t)|2 as apure single-photon state |1ψ〉〈1ψ|. However, the single-photon purity with respect to the wavepacket ψ0 = ψ isdecreased from 1 to,

〈1|Tr⊗k 6=0Hψk

[∫dτp(τ)|1φτ 〉〈1φτ |

]|1〉

=

∫dτp(τ)|(φτ , ψ)|2. (17)

The discrepancy between a photon click and the actualpurity associated with some mode function ψ0 may be ex-plained as follows. The quantum state ρψ0 must be esti-mated from the observables on the mode ψ0, constructed

from aψ0 and a†ψ0. However, any observable constructed

from n(t) can never provide this due to the lack of inter-ference terms a†(t)a(t′) for t 6= t′. As will be discussedlater, in contrast, homodyne detection can provide ob-

servables that are constructed from aψ0and a†ψ0

, which

allows us to estimate ρψ0 .Since the coherence, as well as the stability of the fre-

quency, etc., is crucial for interferometric usage, theseproperties should be ensured experimentally. For thispurpose, in photon detection experiments, we typicallycreate two photons and check the beam splitter interfer-ence, by which we observe the photon bunching effect de-pending on their indistinguishability, i.e., whether theyhave identical frequencies, polarizations, and so forth;

but not whether some decoherence like in Eq. (16) oc-curs.

So far, we have discussed photon detection on fieldswhich contain at most one photon. In the experiment,also the effective vanishing of the multi-photon probabil-ity must be checked to verify the single-photon charac-ter. This can be achieved by observing the anti-bunchingeffect, which appears in the second-order photon correla-tion,

g(2)(τ) =〈: n(t)n(t+ τ) :〉〈n(t)〉〈n(t+ τ)〉

=〈a†(t)a†(t+ τ)a(t+ τ)a(t)〉〈a†(t)a(t)〉〈a†(t+ τ)a(t+ τ)〉

, (18)

where 〈X〉 is the expectation value of an observable X,

and : X : is the normally ordered form of an opera-tor X. It is obtained by using two photon detectors(the Hanbury-Brown and Twiss setup). If the multi-photon probability is zero at any instant of time, we haveg(2)(0) = 0. In contrast, when the photon probabilitiesare totally uncorrelated, g(2)(τ) = 1, which is typicallythe case for τ → ∞. Therefore, the anti-bunching istypically observed as a gap at τ = 0 in the correlationfunction g(2)(τ). A coherent state shows a Poissonianstatistics corresponding to g(2)(0) = 1, irrespective of itsaverage photon number, and we have g(2)(0) ≥ 1 for anyincoherent mixture of coherent states. Therefore, a mea-sure of the quality of a single-photon stream is, beyondthe classical limit of one, how much g(2)(0) gets closerto zero. Note that g(2)(τ) itself says nothing about thecoherence of the single photons, similar to the case of asingle-photon detector discussed above.

Another point is that g(2)(0) is the multi-photon prob-ability normalized by the single-photon probabilities,such that it quantifies the deviation from a random pho-ton emission. Therefore, g(2)(τ) remains unchanged un-der linear optical losses (i.e., the effect of a(t)→ √ηa(t) iscanceled between the numerator and denominator). Thisis both good and bad. On the one hand, it can ver-ify the nonclassicality of a very faint light beam, wherethe photon detection events only scarcely occur. Onthe other hand, it even validates very inefficient single-photon sources, even though these may be actually creat-ing wavepackets very close to vacuum states. Such inef-ficient, rarely successful single-photon generators are notuseful for many applications.

On the other hand, the single-photon purity measureshow much a given wavepacket ψ0 contains the idealsingle-photon state in a coherent manner. Especially,when its value exceeds one half, a stronger form of non-classicality, that is the negativity of the Wigner function,emerges. The density matrix and the Wigner function areusually obtained by homodyne tomography, as we discussnext.

12

2. Homodyne detection

The annihilation and creation operators a and a† inthe particle picture correspond to complex and complexconjugate amplitudes of the light field in the wave pic-ture. They are not observables, because they are not self-adjoint. However, the quadrature amplitudes are self-adjoint,

x ≡ a+ a†√2, p ≡ a− a

†

i√

2. (19)

Analogously to the quantum mechanical position andmomentum operators, x and p cannot be simultaneouslymeasured with perfect precision, since they do not com-mute with each other, [x, p] = i. The instantaneousoutcome of the homodyne detection is, for an ideal de-tector with infinite bandwidth, a quadrature at time t,x(t) = [a(t) + a†(t)]/

√2, as shown below. Note that re-

alistically, due to the finite bandwidth of the detector,we can never measure a truly instantaneous quadrature,which would have an infinitely large variance. However,since we are interested in a situation where the band-width of single photons is much narrower than that ofthe homodyne detector, we may faithfully equate an in-stantaneous homodyne outcome with an instantaneousquadrature outcome.

Homodyne detection measures the interference be-tween a LO beam and a signal light field. The LO has alarge coherent amplitude α = |α|eiθ. We consider a ro-tating frame, so that the complex amplitude of the LO isconstant. By combining at a beamsplitter the LO and aninstantaneous field represented by the annihilation op-erator a(t), we obtain the two output field amplitudes

aout-A(t) = (α+ a(t))/√

2 and aout-B(t) = (α− a(t))/√

2.We then detect their photocurrents and take the differ-ence of the two signals, which is proportional to a certainquadrature,

nout-A(t)− nout-B(t) ≈α∗a(t) + αa†(t)

=2|α|(x(t) cos θ + p(t) sin θ)

≡2|α|xθ(t). (20)

Therefore, by adjusting the phase θ of the LO, we canmeasure any quadrature at an arbitrary phase.

The quadrature values xψ of mode ψ can be expressedas an integral over the instantaneous quadratures,

xψ ≡aψ + a†ψ√

2

=

∫dtψ∗(t)a(t) + ψ(t)a†(t)√

2

=

∫dt{Re[ψ(t)]x(t) + Im[ψ(t)]p(t)}. (21)

Since we cannot simultaneously measure x(t) and p(t)at any time t, in general, we do not obtain a measure-ment outcome for xψ from the instantaneous outcomes.

However, if the mode function ψ is a real function in therotating frame of the LO, we obtain xψ by integratingover x(t) with a weight function of ψ(t). This condi-tion is the same as matching the frequencies of the signalmode and the LO.

The quadrature distribution of an arbitrary single-mode state ρψ0 =

∑m,n cm,n|m〉〈n| is sensitive to the

phases θ. Such a state can be reconstructed from thedistributions of the quadratures xθψ0

for various phasesθ. However, all photon-number states, including the vac-uum and single-photon states as well as their incoher-ent mixtures, have phase-insensitive quadrature distribu-tions, which exactly applies to our experiment. There-fore, we could reconstruct a diagonal density opera-tor ρψ0 =

∑n cn|n〉〈n|, containing the corresponding

photon-number distribution, from the quadrature distri-bution for some single phase. As described in Sec. IIIof the main text, the phase-insensitivity of our quantumstates was experimentally confirmed by checking that twodistributions coincide: one for which the LO phase θ wasslowly scanned, and another one for which it was notscanned. This phase scan was slow enough so that thephase remained constant within each photon pulse. Inthe following, we choose the measured observable to bex without loss of generality.

As can be found in standard textbooks, the quadraturedistributions of the vacuum and single-photon states are,

P (x = x; |0〉〈0|) =1√π

exp(−x2) ≡ P0(x),

P (x = x; |1〉〈1|) =2x2√π

exp(−x2) ≡ P1(x). (22)

Similarly, all multi-photon states have their own distri-butions. The distribution for an incoherent mixture ofphoton-number states can be decomposed into distribu-tions of pure photon-number states,

P (x = x;∑n

cn|nψ〉〈nψ|)

=∑n

cnP (x = x, |nψ〉〈nψ|). (23)

The photon-number distribution corresponds to the co-efficients of this decomposition. In the decomposition ofthe experimental distribution, we maximized the likeli-hood by using the downhill simplex method.

C. Mode Function Estimation by PrincipalComponent Analysis

As discussed above, if we know a priori the real modefunction ψ of the single photons, we can obtain the den-sity operator ρψ0 that maximizes the purity 〈1|ρψ0 |1〉 bysetting ψ0 = ψ. In the experiment, we estimated themode function ψ from the homodyne raw data. Thiswas possible, because only a single mode ψ was excited,

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and the other orthogonal modes were basically in vac-uum states. Therefore, we could easily observe the targetmode ψ0, since its photon number is maximized. Notethat this maximization of the photon number is equiva-lent to the maximization of the phase-insensitive quadra-ture variance,

2〈a†ψ0aψ0〉 = 〈x2ψ0

〉+ 〈p2ψ0〉 − 1. (24)

From the homodyne raw data, we can calculate theauto-covariance,

Vt,t′ ≡ 〈x(t)x(t′)〉. (25)

Then, optimization of the mode function ψ0 correspondsto finding the eigenfunction of the auto-covariance, whichthen gives the maximal eigenvalue. Therefore, the prob-lem is solved via the spectral decomposition of the auto-covariance. In the experiment, owing to the finite sam-pling rate and the finite frame length, the auto-covariancegives a finite-dimensional square matrix, whose diagonal-ized form serves our purposes. This method of decom-posing a mixed signal into independent signals is referredto as the principal component analysis (PCA).

Note that, in order to validate this method, the de-tector response must be sufficiently flat in the frequencyregion of interest. In addition, the PCA never discrimi-nates two components with the same variance. Therefore,in case there is some noise due to experimental imperfec-tions, with a variance close to that of a single-photonquadrature, the resulting function becomes in general acombination of that noise and the true function.

The quadrature samples are random, obeying somestatistics. Therefore, an estimation via the PCA maycapture the accidental deviation from the true distri-bution, which increases the eigenvalue, leading to someoverestimation of the average photon number. This querythen affects the precision of the estimation of the modefunction ψ0. We checked that the mode match |(ψ0, ψ

′0)|2

between two mode functions obtained from two differ-ent sets of quadrature samples was typically more than99%. Even when we used different sets of samples forthe estimation of mode ψ0 and for the estimation of thephoton-number distribution ρψ0 , the change in the ob-tained purity was negligible.

D. Wigner Function

A Wigner function W (x, p) of a single-mode quan-tum state is a two-dimensional quasi-probability distribu-tion in phase space, characterized by an operator algebra[x, p] = i,

W (x, p) =1

2π

∫ ∞−∞

exp(ipη)〈x− η

2|ρ|x+

η

2〉dη. (26)

It resembles a classical distribution function Wc(x, p) inthe phase space of a general position x and momentum p,where each point of the function Wc(x, p) has a meaning

of a joint probability density. Just like a classical dis-tribution function, the Wigner function gives the correctquadrature distributions via its marginal distributions,∫ ∞

−∞W (x, p)dp =P (x = x), (27)∫ ∞

−∞W (x, p)dx =P (p = p). (28)

The distribution of a quadrature xθ at an arbitrary phaseθ is obtained similarly.

The biggest difference between Wigner and classicaldistribution functions is that Wigner functions allow fornegative values. Due to the uncertainty principle, we cannever determine the conjugate variables x and p simulta-neously. Therefore, a single point of the Wigner functionW (x, p) alone cannot be interpreted as a joint probabil-ity; only when integrated, the meaning of a probabilitydensity arises, like in Eq. (28). For this reason, Wignerfunctions may take on negative values – a strong quantumfeature, because it is a consequence of the noncommuta-tive operator algebra. Note that even though W (x, p)may take on negative values, the integrated probabilitydensities never become negative.

In fact, a single-photon state is a prominent example ofsuch nonclassical states with a negative Wigner function.The Wigner functions for a single-photon and a vacuumstate are given by,

W|0〉〈0|(x, p) =1

πexp(−x2 − p2), (29)

W|1〉〈1|(x, p) =1

π(2x2 − 2p2 − 1) exp(−x2 − p2). (30)

As expected, they are rotationally symmetric in the x-pplane. The function is always positive for a vacuum state(being a Gaussian function), whereas it has a negative re-gion for a single-photon state. In particular, at the phase-space origin, the function value isW|0〉〈0|(0, 0) = 1/π for avacuum state, while it is W|1〉〈1|(0, 0) = −1/π for a single-photon state. Since the map between a Wigner functionand a density operator is linear, the Wigner function foran incoherent mixture of a vacuum state and a single-photon state is obtained by summing their Wigner func-tions with the corresponding weights,

Wp|1〉〈1|+(1−p)|0〉〈0|(x, p)

= pW|1〉〈1|(x, p) + (1− p)W|0〉〈0|(x, p). (31)

Thus, if the single-photon component exceeds one half,the total Wigner function has negative values around theorigin. More precisely, for even-photon-number states,we have W|2k〉〈2k|(0, 0) = 1/π, while for odd-photon-number states, W|2k+1〉〈2k+1|(0, 0) = −1/π. Therefore,if the contribution of multi-photon components, e.g.,a three-photon component, is no longer negligible, theWigner function can take on negative values even if thesingle-photon purity is below one half.

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