Quantum Storage of Single-Photon and Two-Photon
Fock States with an All-Optical Quantum Memory
M. Bouillard, G. Boucher, J. Ferrer Ortas, B. Pointard, Rosa Tualle-Brouri

To cite this version:
M. Bouillard, G. Boucher, J. Ferrer Ortas, B. Pointard, Rosa Tualle-Brouri. Quantum Storage ofSingle-Photon and Two-Photon Fock States with an All-Optical Quantum Memory. Physical Review
Letters, 2019, 122 (21), �10.1103/PhysRevLett.122.210501�. �hal-03345704�

HAL Id: hal-03345704
https://hal-iogs.archives-ouvertes.fr/hal-03345704
Submitted on 15 Mar 2024

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoires
publics ou privés.

Quantum storage of single-photon and two-photon Fock states with an all-optical quantum memory
M. Bouillard, G. Boucher, J. Ferrer Ortas, B. Pointard, and R. Tualle-Brouri
Laboratoire Charles Fabry, Institut d’Optique Graduate School,
Université Paris-Saclay, 2 Avenue Augustin Fresnel, 91127, Palaiseau, France(Dated: October 18, 2018)arXiv:1810.07419v1 [quant-ph] 17 Oct 2018

Quantum memories are a crucial element towards efficient quantum protocols. In the continuous variables
domain, such memories need to have near unity efficiencies. Moreover, one needs to store complex quantum
states exhibiting negative Wigner functions after storage. We report the implementation of an all optical quantum
memory: the storage of single and two-photon Fock states with high fidelities has been realized. The Wigner
functions of the reconstructed states shows negativity after ∼ 0.75 µs and ∼ 0.61 µs respectively for the singlephoton and two-photon Fock states. This is, to our knowledge, the first demonstration of the storage of nonGaussian states with more than one photon, representing a key step towards hybrid quantum protocols.

A main bottleneck towards large-scale quantum-optics protocols concerns the ability to provide simultaneously several
resources issued from non-deterministic protocols. In thiscontext, efficient quantum memories are an essential feature
for the implementation of sophisticated operations. Over thelast decade, numerous memories have been realized in orderto store quantum states using vapor cells [1], cold atoms [2–4], crystals [5, 6]... Those memories usually exhibit long storage time (in the ms scale), high fidelities but low efficiency,
making them good candidates for discrete variables protocols,
where quantum states are usually measured via correlationor anti-correlation between multiple detection events. Whenconsidering hybrid quantum protocols, with the use of continuous variables, the states are usually measured with homodyne or heterodyne detections. Unlike the discrete variables
case, the measurement of field quadratures always leads to a
result, regardless of the presence of photons in the observed
optical mode: there is no “lost modes” in that case, and theefficiency has to be taken as one. Failed extractions have to
be considered as vacuum, leading to a substantial drop of the
fidelity. For that reason, most quantum memories developed
so far are not suitable for continuous variables.

Continuous variables quantum protocols rely on nonGaussian states containing more than one photon [7–9], anda main asset of optical cavity-based quantum memories istheir potentiality to store such states with high fidelity [10].
A key parameter of the generation of non-Gaussian states isthe negativity of their Wigner functions (defined as their minimal value), which can be used as a quantitative witness of
non-classicality. Thus, the negativity of the stored states is an
important parameter for characterizing quantum memories. In
this paper, we experimentally demonstrate the quantum storage, in an optical cavity, of Fock states up to the two-photon
level exhibiting negative Wigner functions after dozens ofround trips. To our knowledge, this experiment represents thefirst storage of a non-Gaussian quantum state containing morethat one photon, which is a crucial step towards quantum information protocols.

I.

AN OPTICAL QUANTUM MEMORY FOR HYBRIDPROTOCOLS

A scheme of the quantum memory is represented in figure 1: it consists in a cavity with a thin film polarizing beam
splitter (PBS), represented as a cube in the figure 1. A Pockels
cell placed inside the cavity allows to switch the polarizationof the light. Typically, a light pulse enters into the cavity witha horizontal polarization, the Pockels cell allows to rotate thispolarization to vertical so that the pulse is then reflected by the
PBS, and thus stored in the cavity. To extract the light, one justhas to turn the polarization back to horizontal. The light canthus be delayed by an arbitrary number of round trips, with asynchronization that is naturally warranted, provided that thelength of the cavity (3.9 m) matches the repetition rate of thelaser (76 MHz).
Recently such kind of protocols have been realized in orderto synchronize single photons from two independent sourcesin the discrete regime [11]. In this regime, an issue could appear when the polarization degree of freedom is used to store
the light, as such cavities cannot store information encoded inpolarization. In the continuous variables domain, the information can be encoded using the quadratures degree of freedom, and the polarization is no longer matter of interest for
information encoding. However, with continuous variables,
no measurement can be ruled out: the efficiency of a quantum
memory thus has to be taken as unity, and the different losses
will reduce the fidelity of the output states.
When using continuous variables, the non-Gaussian states
are natural candidates for quantum protocols [7–9, 12–14]. In
that direction, complex quantum states with coherent superpositions of Fock states with more than one photons are essential. For instance, Schrödinger cat states are very promising
candidates for quantum computation and quantum information protocols, making them highly popular [15–21]. Therefore, the ability to store states with more than one photon witha good fidelity is a crucial step.
Interestingly, this architecture for the quantum memory
could allow not only to store quantum state but also to create
quantum superposition such as NOON states or Schrödinger
cat states. The Pockels cell in the cavity indeed offers thepossibility to act as a quater-wave plate. In that case, thePBS in the cavity will act as a symmetric beam-splitter and,if two photons with orthogonal polarization are simultane-

2

Figure 1: Scheme of the setup. Pairs of single-photon and two-photon states are produced by SPDC in a BiB3 O6 crystal placedin the OPA cavity where the pump beam is resonant. The heralding part of the photon pairs is spatially and spectrally filtered,and separated towards two SPCMs in order to announce the creation of the quantum states. The prepared state go through an
optical delay line (∼ 200 ns) to account for the electronical delay of the Pockels Cells. Thanks to the Pockels cell, the quantum
state is then stored inside the quantum memory for a given number of round trip, and sent to the homodyne detection forcharacterization. F: spectral and spatial filters, DL: delay line, PBS: polarizing beam-splitter, HWP: half-wave plate, Pol:Polarizer.
ously present in the cavity, this will lead to the generation of a
two-photon NOON state via photon coalescence. It has been
demonstrated [19, 22] that, from such a NOON state and witha homodyne measurement on one mode of the state, one canproject the other mode in a Schrödinger cat state. This scheme
could then allow to manipulate the state stored in the cavity,
and therefore avoiding the use of multiple quantum memories
and reducing the losses due to consecutive storages and extractions. The memory also allows to wait for the additional
required resource, which is a single-photon in this example.
The probability of success of such experiment is limited by
the cavity losses as well as by the production rate of the single photons. If one fix N as the maximum number of roundtrips allowed for the single-photon to keep a reasonable fidelity threshold, and if p|1i is the probability of producing asingle-photon Fock state by a source, the probability P of synchronizing two single-photons with the cavity then reads:
P = 1 − (1 − p|1i )N
II.

(1)

EXPERIMENTAL SETUP

A scheme of the experimental setup is shown in figure 1. Pulses from a Ti:sapphire laser with a central wavelength of 850 nm (temporal length: 2.6 ps - repetition rate:
76 MHz) are frequency doubled using a ∼ 3.9 m-long bowtie second-harmonic-generation cavity with a type I BiB3 O6
crystal [23, 24]. The cavity, of finesse 60, allows to enhance the non-linear effect, leading to a conversion efficiencyof 70 %. The output pulses at 425 nm are sent to another∼ 3.9 m-long bow-tie cavity of finesse ' 80, namely the op-tical parametric amplification (OPA) cavity. The intra-cavity
peak power reaches 40 kW. A second type I BiB3 O6 crystal placed inside the OPA cavity allows to produce pairs ofphotons by non-colinear spontaneous parametric down conversion (SPDC) [24].
As the phase matching of SPDC is broadband, the heralding part of the photon pairs, in the signal beam, is first spectrally and spatially filtered by the use of a single mode fiberand a grating combined with a slit. The filtered photons
are probabilistically separated by a beam-splitter followed by
two single-photon counting modules (SPCM, Perkin Elmer
SPCM-AQR-13) in order to detect the generation of single
and two-photon Fock states.
To account for the delay of the electronics (∼ 200 ns), theheralded state in the idler beam is sent through an opticaldelay-line consisting in an open 4.8 m long cavity made ofthree mirrors [24]. It allows to tune the delay from 32 to
288 ns by steps of 32 ns with optical losses lower than 1 %.
The H-polarized Fock state is then transmitted by the PBS of
the quantum memory and enters into the cavity. A 14 ns highvoltage pulse is applied to the Pockels cell in order to turnthe polarization of the light pulse to vertical, and the Fockstate is then stored inside the cavity. After a given numberof round trips, the quantum state is extracted out of the cavity by rotating again its polarization. The extracted state issent to a homodyne detection in order to be characterized,
the overall efficiency η of the homodyne detection being ofηHD = ηPD ηC = 77 ± 3% (where ηPD = 94 ± 2% is thequantum efficiency of the photodiodes, and ηC =C2 =81±1%
is the mode-matching efficiency, with C the contrast measured
between the idler beam and the local oscillator [25, 26]).
To avoid the saturation of our homodyne detection (band-

3(a) 0 round trips(0 ns)(b) 10 round trips(130 ns)(c) 20 round trips(260 ns)(d) 30 round trips(390 ns)(e) 40 round trips(420 ns)

Figure 2: Wigner function of the single-photon (top) and two-photon state (bottom) after different storage round trips: from
0 (a) to 40 (e) round trips. The Wigner functions are corrected from a 77 % detection efficiency. Inset: diagonal elements of thedensity matrices without (yellow) and with (purple) correction of the detection efficiency.
width of a few MHz) we reduce the repetition rate of the local
oscillator. To do so, another Pockels cell (PC) combined witha polarizer is used and work synchronously with the quantum memory. Thus a pulse from the local oscillator is sent
to the homodyne detection only when a quantum state is extracted from the quantum memory. As the production rate of
the Fock states is low compared to the bandwidth of the detector, the analysis and production rates are almost identical[24].
The equation 1 shows that the probability to synchronize
two quantum states is defined by the number of round trips allowed in the cavity and by the production rate of these quantum states. The losses per round-trip of the cavity are estimated to be 0.6 %, leading to a theoretical lifetime of a singlephoton of τ = 2.2 µs (defined as F = F0 e−t/τ , where F0 denotes the initial fidelity of the state [27]). The initial fidelity of
the Fock states (without storage) is another key parameter ofthe setup: the quality of the single-photon (two-photon) states
reaches 91 ± 4 % (85 ± 6 %) after correction from the detection efficiency η, along with a production rate of 150-250 kHz(∼ 200 Hz)[24].

III.

RESULTS AND DISCUSSION

To characterize the quantum memory, homodyne tomographies of the single-photon and two-photon states have beenrealized for different numbers of round trips inside the cavity. For a given number of round trips, 50 000 and 10 000 data

points have been acquired for respectively the single-photon
and two-photon measurements. As the Fock states are phaseinvariant, the phase of the homodyne detection was not measured. Once acquired, the density matrix and the Wigner function of the stored states, represented on the figure 2, are reconstructed using a maximum likelihood technique [28] with correction of the detection efficiency η. The diagonal elementsof the density matrices with and without correction of the efficiency are also shown in the insets of the figure 2. As they
increase with the number of round trips, the effect of losses in
the quantum memory reduces the fidelity of the stored statesleading to a decrease of the negativity of the Wigner function.
Both states tend towards a bi-dimensional Gaussian shape representing the Wigner function of the vacuum.
The values of the diagonal elements of the density matrix(corrected from the detection efficiency), which are the fidelities F|ni = hn| ρ |ni of the stored state ρ with the Fock state|ni, are represented on the figure 3 for the single-photon Fockstates (figure 3a : red) and the two-photon Fock state (figure
3b : blue) as a function of the number of round trips of storage. The lifetimes are estimated to be of 1.44 µs and 0.51 µs
for the single-photon and the two-photon states. Such a difference is explained by the quadratic effect of the losses on the
two-photon states. As it reveals the non-classicality of quantum states, the negativity of the Wigner function is a more
relevant criteria than the lifetime for the characterization ofmemories in the continuous variable domain. The negativityof the reconstructed Wigner functions is directly plotted on
the figure 4. Starting at -0.25±0.02 for the single-photon and

4

(a)

(b)

Figure 3: Evolution of the diagonal elements of the density matrix of the stored state for the single-photon (a) and two-photon
Fock state (b), as a function of the number of round trips (bottom axis) and of the storage time (top axis). The coloured bands
around the fits account for the uncertainty on the detection efficiency.
-0.10±0.02 for the two-photon state, the negativity is maintained over 50 round trips for both states. In both figures 3
and 4, error bars are deduced from the uncertainty on the detection efficiency η, which is the major error source in thisexperiment.
In order to evaluate the losses of the cavity, the evolutionof the density matrix were fitted using the equation developedin [29]:Nn

hn| ρ̂N |ni = (1 − p)


∞ 
X
n+i
i=0

n

(1 − (1 − p)N )ihn + i| ρ̂0 |n + ii (2)Where p is the losses per turn of the cavity and ρN is thedensity matrix of the states after N round trips of storage. The
fits of the data in figures 3 show a good agreement with experimental data points. The losses per round trip can be extracted from the fits, and are estimated to be of 1.0 % for the
single-photon and 1.3 % for the two-photon state experiment,
exhibiting a slight decrease for the two-photon states. Sucha difference could be explained by a longer acquisition time
for the two-photon states, making the results more sensitive
to experimental fluctuations. The losses per round trip do not
only include the optical losses of the cavity, but also the evolution for the spatial mode of the quantum state inside the cavity
as well as temporal mismatch between the cavity length andthe repetition rate of the laser, leading to temporal and spatial mismatches between the local oscillator and the quantum
state. The optical losses per round trip are estimated to beof 0.6 %, which is close to the experimental values, showing a good control of the spatio-temporal mode of the stored
quantum states. The fits allows the determination of the negativities for each number of round trips of storage. For the
single-photons, this negativity starts at −0.26 ± 0.03, and the
Wigner function exhibit negativity after 57±4 round trips ofstorage, corresponding to a 0.75 ± 0.05 µs storage time withpreservation of this non-classical feature. For the two-photon

Figure 4: Evolution of the negativity as a function of thenumber of round trips (bottom axis) and of the storage time
(top axis). The coloured bands around the fits account for theuncertainty on the detection efficiency.
states the Wigner functions remain negative until 46±6 roundtrips, leading to a storage time of 0.60 ± 0.08 µs.
With the lifetimes of the cavity, we can now estimate theprobability of synchronizing two single-photon Fock states(equation 1). This value is of 14 %, leading to a synchronization rate of two single-photon states of approximately 28 kHz,
corresponding to a rate enhancement of the synchronization
by a factor of about 53, which is to our knowledge the highest value reported for such kind of memory. Compared toour previous experiment with two independent single-photon
sources [19], this would correspond to an increase by morethan 4 order of magnitude. As it can be seen from equation 1,
the production rate is a key parameter for synchronizing two
quantum states. In order to increase the heralding rate, one
can increase the detection efficiency of the heralding part ofthe experiment (6 %), by using high-efficiency single-photon
detectors [30, 31] or by reducing the losses of the spectral fil-

5
tering (currently of ∼ 15 %), increasing significantly the success rate of the synchronization. As an example, an increaseof the detection efficiency by a factor 3 would give a successrate of ∼ 200 kHz for a success probability of ∼ 36%.
In conclusion, we have studied the storage of single andtwo-photon Fock states in an all-optical cavity, reporting what
is, to our knowledge, the first storage of a non-Gaussian statewith more than one photon. The stored quantum states show
negative Wigner functions after 57±4 and 46±6 round trips
for single and two-photon states, respectively, which corresponds to a storage time of respectively 0.75 ± 0.05 µs and0.60 ± 0.08 µs. Such cavities pave the way towards the synchronization of optical quantum states, which is a cornerstone
in the direction of quantum information protocols.

[1] K. F. Reim, P. Michelberger, K. C. Lee, J. Nunn, N. K. Langford, and I. A. Walmsley, Physical Review Letters 107 (2011),
10.1103/physrevlett.107.053603.
[2] E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi,
J. Stanojevic, A. Ourjoumtsev, and P. Grangier, Physical Review Letters 112 (2014), 10.1103/physrevlett.112.033601.
[3] V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino,
and J. Laurat, Nature Communications 6, 7706 (2015).
[4] P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Laurat, Nature Communications 9 (2018), 10.1038/s41467-01702775-8.
[5] P. Jobez, C. Laplane, N. Timoney, N. Gisin, A. Ferrier, P. Goldner, and M. Afzelius, Physical Review Letters 114 (2015),
10.1103/physrevlett.114.230502.
[6] E. Saglamyurek, J. Jin, V. B. Verma, M. D. Shaw, F. Marsili,
S. W. Nam, D. Oblak, and W. Tittel, Nature Photonics 9, 83(2015).
[7] T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro,and S. Glancy, Phys. Rev. A 68 (2003), 10.1103/physreva.68.042319.
[8] A. P. Lund, T. C. Ralph, and H. L. Haselgrove, Phys. Rev. Lett.
100 (2008), 10.1103/physrevlett.100.030503.
[9] S.-W. Lee and H. Jeong, Physical Review A 87 (2013),
10.1103/physreva.87.022326.
[10] K. Makino, Y. Hashimoto, J. i. Yoshikawa, H. Ohdan,
T. Toyama, P. van Loock, and A. Furusawa, Science Advances
2, e1501772 (2016).
[11] F. Kaneda, F. Xu, J. Chapman, and P. G. Kwiat, Optica 4, 1034
(2017).
[12] H. Jeong and M. S. Kim, Physical Review A 65 (2002),
10.1103/physreva.65.042305.
[13] S. Lloyd and S. L. Braunstein, Physical Review Letters 82, 1784
(1999).
[14] C. Silberhorn, T. C. Ralph, N. Lütkenhaus, and G. Leuchs,
Physical Review Letters 89 (2002), 10.1103/physrevlett.89.167901.
[15] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier,
Science 312, 83 (2006).
[16] A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier,
Nature 448, 784 (2007).
[17] S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M.
Raimond, and S. Haroche, Nature 455, 510 (2008).
[18] T. Gerrits, S. Glancy, T. S. Clement, B. Calkins, A. E. Lita, A. J.
Miller, A. L. Migdall, S. W. Nam, R. P. Mirin, and E. Knill,
Phys. Rev. A 82 (2010), 10.1103/physreva.82.031802.
[19] J. Etesse, M. Bouillard, B. Kanseri, and R. Tualle-Brouri, Phys.
Rev. Lett. 114, 193602 (2015).
[20] K. Huang, H. L. Jeannic, J. Ruaudel, V. Verma, M. Shaw,
F. Marsili, S. Nam, E. Wu, H. Zeng, Y.-C. Jeong, R. Filip,
O. Morin, and J. Laurat, Physical Review Letters 115 (2015),
10.1103/physrevlett.115.023602.
[21] D. V. Sychev, A. E. Ulanov, A. A. Pushkina, M. W. Richards,

I. A. Fedorov, and A. I. Lvovsky, Nature Photonics 11, 379(2017).
[22] J. Etesse, B. Kanseri, and R. Tualle-Brouri, Optics Express 22,30357 (2014).
[23] B. Kanseri, M. Bouillard, and R. Tualle-Brouri, Optics Communications 380, 148 (2016).
[24] M. Bouillard, G. Boucher, J. F. Ortas, B. Kanseri, and
R. Tualle-Brouri, (2018), 1807.05701v1.
[25] U. Leonhardt, Measuring the quantum state of light, Vol. 22(Cambridge university press, 1997).
[26] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek,
and S. Schiller, Phys. Rev. Lett. 87 (2001), 10.1103/physrevlett.87.050402.
[27] F = tr (ρ |ψi hψ|) where |ψi is the reference state and ρ theexperimental density matrix [32].
[28] A. I. Lvovsky, Journal of Optics B: Quantum and Semiclassical
Optics 6, S556 (2004).
[29] T. Kiss, U. Herzog, and U. Leonhardt, Physical Review A 52,
2433 (1995).
[30] H. L. Jeannic, V. B. Verma, A. Cavaillès, F. Marsili, M. D.
Shaw, K. Huang, O. Morin, S. W. Nam, and J. Laurat, Optics
Letters 41,, 5341 (2016), 1607.07459.
[31] D. Fukuda, G. Fujii, T. Numata, K. Amemiya, A. Yoshizawa,
H. Tsuchida, H. Fujino, H. Ishii, T. Itatani, S. Inoue, andT. Zama, Optics Express 19, 870 (2011).
[32] R. Jozsa, Journal of modern optics 41, 2315 (1994).