Making entangled photons indistinguishable by a time lens Shivang Srivastava1Dmitri B. Horoshko1 2yand Mikhail I. Kolobov1z 1Univ. Lille CNRS UMR 8523 - PhLAM - Physique des Lasers Atomes et Mol ecules F-59000 Lille France

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Making entangled photons indistinguishable by a time lens

Shivang Srivastava,1, ∗Dmitri B. Horoshko,1, 2, †and Mikhail I. Kolobov1, ‡

1Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Mol´ecules, F-59000 Lille, France

2B. I. Stepanov Institute of Physics, NASB, Nezavisimosti Ave 68, Minsk 220072 Belarus

(Dated: March 8, 2023)

We propose an application of quantum temporal imaging to restoring the indistinguishability of

the signal and the idler photons produced in type-II spontaneous parametric down-conversion with

a pulsed broadband pump. It is known that in this case, the signal and the idler photons have

diﬀerent spectral and temporal properties. This eﬀect deteriorates their indistinguishability and

the visibility of the Hong-Ou-Mandel interference, respectively. We demonstrate that inserting a

time lens in one arm of the interferometer and choosing properly its magniﬁcation factor restores

perfect indistinguishability of the signal and the idler photons and provides 100% visibility of the

Hong-Ou-Mandel interference in the limit of high focal group delay dispersion of the time lens.

I. INTRODUCTION

Classical temporal imaging is a technique of manipu-

lation of ultrafast temporal optical waveforms similar to

the manipulation of spatial wavefronts in conventional

spatial imaging [1,2]. It is based upon the so-called

space-time duality or the mathematical equivalence of

the equations describing the propagation of the temporal

pulses in dispersive media and the diﬀraction of the spa-

tial wavefronts in free space. Temporal imaging was ﬁrst

discovered in purely electrical systems, then extended

to optics [3], and later converted into all-optical tech-

nologies using the development of non-linear optics and

ultrashort-pulse lasers [4–6]. In the past two decades,

classical temporal imaging has become a very popular

tool for manipulating ultrafast temporal waveforms with

numerous applications such as temporal stretching of ul-

trafast waveforms and compression of slow waveforms to

sub-picosecond time scales, temporal microscopes, time

reversal, and optical phase conjugation.

One of the key elements in classical temporal imag-

ing is a time lens which introduces a quadratic time

phase modulation into an input waveform, similar to the

quadratic phase factor in the transverse spatial dimen-

sion, introduced by a conventional lens. Nowadays, opti-

cal time lenses are based on electro-optic phase modula-

tion (EOPM) [4,7,8], cross-phase modulation [9,10],

sum-frequency generation (SFG) [11–15], or four-wave

mixing (FWM)[16–19]. A temporal magniﬁcation fac-

tor of the order of 100 times has been experimentally

realized.

Quantum temporal imaging is a recent topic of re-

search which brings the ideas from the spatial quantum

imaging [20–22] into the temporal domain. Quantum

temporal imaging searches for such manipulations of non-

classical temporal waveforms which preserve their non-

classical properties such as squeezing, entanglement, or

∗shivang.srivastava@cnrs.fr

†horoshko@ifanbel.bas-net.by

‡mikhail.kolobov@univ-lille.fr

nonclassical photon statistics. Some works have already

been published on this subject. Schemes for optical wave-

form conversion preserving the nonclassical properties

such as entanglement [23] and for aberration-corrected

quantum temporal imaging of a coherent state [24] have

been proposed. Spectral bandwidth compression of light

at a single-photon level has been experimentally demon-

strated by SFG [25] and EOPM [26–28]. Temporal imag-

ing in the single-photon regime has been demonstrated

with an atomic-cloud-based quantum memory [29,30].

Quantum temporal imaging has been demonstrated for

one photon of an entangled photon pair by SFG [31] and

for both photons by EOPM [32]. Quantum temporal

imaging of broadband squeezed light was studied for SFG

[33–35], and FWM-based [36] lenses. In Ref. [37], it was

demonstrated that a time lens can preserve nonclassical

eﬀects such as antibunching and sub-Poissonian statis-

tics of photons. In Ref. [38] a temporal magniﬁcation of

two weak coherent pulses with a picosecond-scale delay

by FWM was reported.

In this paper we consider the application of quan-

tum temporal imaging to the light produced in

frequency-degenerate type-II spontaneous parametric

down-conversion (SPDC) pumped by a pulsed broadband

source. This type of SPDC was considered in Refs. [39–

41]. In particular, it was demonstrated that the temporal

and spectral properties of the signal and the idler down-

converted photons can be signiﬁcantly diﬀerent in the

case of a pulsed pump. This diﬀerence aﬀects the indis-

tinguishability of these photons and, consequently, dete-

riorates the visibility of the Hong-Ou-Mandel interference

[42]. One could try to increase the indistinguishability by

inserting a spectral ﬁlter in one of the arms of the inter-

ferometer. However, this ﬁltering will result in additional

losses of the SPDC light. We suggest a non-destructive

way of increasing the indistinguishability by using a time

lens in one of the arms instead of a ﬁlter. We demon-

strate that by choosing properly the magniﬁcation factor

of the temporal imaging system, one can achieve unit

visibility in the Hong-Ou-Mandel interferometry, thus,

reestablishing the perfect indistinguishability of the pho-

tons. We consider the case of two spectrally entangled

arXiv:2210.14964v2 [quant-ph] 7 Mar 2023

photons only. Such photons become indistinguishable

from one another after the time lens, however, they re-

main entangled and therefore distinguishable from other

photons having the same spectral and temporal shapes.

Note, that applications of single photons in boson sam-

plers [43,44] and quantum networks [45–47] require that

the photons are indistinguishable and disentangled.

We employ a more realistic treatment of the time lens

with respect to a usual approach found in the literature.

Precisely, instead of considering the ﬁeld transformation

by the time lens using a local time in a reference frame

traveling with the group velocity, we use the absolute

time in the laboratory reference frame. This description

allows us to evaluate correctly diﬀerent time delays in

the interferometric scheme used for observation of the

Hong-Ou-Mandel eﬀect. We clarify the role of the syn-

chronization condition between the pulsed photon-pair

source and the time lens and investigate the sensitivity

of the visibility of the Hong-Ou-Mandel interference with

respect to the precision of this synchronization.

The paper is organized as follows. In Sec. II we de-

scribe the SPDC process with a pulsed broadband pump

and provide the corresponding Heisenberg equations. We

evaluate the spectra and the average intensities of the sig-

nal and the idler waves. We also describe the time lens

and diﬀerent time delays in the interferometric scheme.

In Sec. III we evaluate explicitly the coincidence count-

ing rate in the Hong-Ou-Mandel interferometer and in-

vestigate the visibility of the interference pattern as a

function of various physical parameters of the problem.

In Sec. IV we summarize our results and give a short

outlook for future work.

II. SECOND-ORDER INTERFERENCE OF

PHOTON PAIRS

The scheme for the generation of photon pairs and ob-

servation of their second-order interference is shown in

Fig. 1. It consists of several parts, which are described

separately below. A quantum description of the ﬁeld

transformation in each optical element is done in the

Heisenberg picture, instead of the Schr¨odinger picture

traditionally employed for such schemes [39,40,42,48–

50], because the time-lens formalism is developed in the

Heisenberg picture [34,35]. In addition, the Heisenberg

picture formalism provides a natural extension to the

high-gain regime of parametric downconversion (PDC),

where multiple photon pairs are created at once.

A. Parametric downconversion

The model for the description of single-pass pulsed

PDC in a χ(2) nonlinear crystal in the Heisenberg pic-

ture is developed in Refs. [51–56] and here we adopt its

single-spatial-dimension version. We consider a crystal

slab of length L, inﬁnite in the transverse directions, cut

for type-II collinear phase matching. We take the xaxis

as the pump-beam propagation direction and choose the

zaxis so that the optical axis (axes) of the crystal lies

(lie) in the xz plane.

The pump is a plain wave polarized in either the yor z

direction. In the time domain, it is a Gaussian transform-

limited pulse whose maximum passes the position x= 0

at time t=t0. Its central frequency is denoted by ωp.

The pump is treated as an undepleted deterministic wave

and is described by a c-number function of space and time

coordinates. The positive frequency part of the pump

ﬁeld (in photon ﬂux units) can be written as

E(+)

p(t, x) = ˆα(Ω)eikp(Ω)x−i(ωp+Ω)tdΩ

2π,(1)

where Ω represents the frequency detuning from the cen-

tral frequency and the integration limits can be extended

to inﬁnity. The pump spectral amplitude α(Ω) is nonzero

in a limited band only and does not depend on x, since

the pump is undepleted. All variations of the pump wave

in the longitudinal direction are determined by the wave-

vector kp(Ω) = np(ωp+ Ω)(ωp+ Ω)/c, where np(ω) is the

refractive index corresponding to the polarization of the

pump and cis the speed of light in vacuum.

We assume that the pump pulse is a transform-limited

Gaussian pulse of full width at half maximum (FWHM)

τp, that is

E(+)

p(t, 0) = E0e−(t−t0)2/4σ2

t−iωpt,(2)

where τp= 2√2 ln 2σtand E0is the peak amplitude.

From this equation and Eq. (1) we ﬁnd

α(Ω) = E0

√π

Ωp

e−Ω2/4Ω2

p+iΩt0,(3)

where Ωp= 1/2σt.

As a result of the nonlinear transformation of the pump

ﬁeld in the crystal, a subharmonic ﬁeld emerges. In

the case of frequency-degenerate type-II phase match-

ing, considered here, there are two subharmonic waves

with the central frequency ω0=ωp/2, polarized in the

y(ordinary wave) and z(extraordinary wave) directions.

These waves are treated in the framework of quantum

theory and are described by Heisenberg operators. The

positive-frequency part of the Heisenberg ﬁeld operator

(in photon ﬂux units) can be written in the form of a

Fourier integral

E(+)

µ(t, x) = ˆµ(Ω, x)eikµ(Ω)x−i(ω0+Ω)tdΩ

2π,(4)

where µ(Ω, x) is the annihilation operator of a photon

at position xwith the frequency ω0+ Ω and polarization

along the yaxis for the ordinary (µ=o) wave or along

the zaxis for the extraordinary (µ=e) one, and kµ(Ω) =

nµ(ω0+ Ω)(ω0+ Ω)/c is the wave vector, with nµ(ω) the

refractive index corresponding to the polarization µ. The

pump extraordinary

ordinary

PBS

λ/2

type II PDC

δτ

Din

GDD

time

lens

Dout

GDD

FIG. 1. Scheme for generation of photon pairs and observation of their second-order interference. A pump pulse impinges on

a nonlinear crystal cut for type-II PDC. Two subharmonic pulses appear as ordinary and extraordinary waves in the crystal.

The pump is removed, while two subharmonic pulses are separated by a polarized beam splitter (PBS). Polarization of the

ordinary pulse is rotated by a half waveplate. Subsequently, the ordinary pulse passes through a temporal imaging system,

composed of an input dispersive medium, a time lens, and an output dispersive medium. The delay of the extraordinary pulse

δτ is controlled by a delay line. The pulses interfere at a beam splitter (BS) and are detected by single-photon detectors. A

coincidence is registered, if both detectors ﬁre in the same excitation cycle.

evolution of this operator along the crystal is described

by the spatial Heisenberg equation [57]

dxµ(Ω, x) = i

~µ(Ω, x), G(x),(5)

where the spatial Hamiltonian G(x) is given by the mo-

mentum transferred through the plane x[58] and equals

G(x) = χ

+∞

−∞

E(−)

p(t, x)ˆ

E(+)

o(t, x)ˆ

E(+)

e(t, x)dt+H.c., (6)

where χis the nonlinear coupling constant and

E(−)

p(t, x) = E(+)∗

p(t, x) is the negative-frequency part

of the ﬁeld. Substituting Eqs. (1), (4), and (6) into

Eq. (5), performing the integration, and using the canon-

ical equal-space commutation relations [58,59]

hµ(Ω, x), †

ν(Ω0, x)i= 2πδµν δ(Ω −Ω0),(7)

we obtain the spatial evolution equations

do(Ω, x)

dx =κˆα(Ω + Ω0)†

e(Ω0, x)ei∆(Ω,Ω0)xdΩ0,

de(Ω, x)

dx =κˆα(Ω + Ω0)†

o(Ω0, x)ei∆(Ω0,Ω)xdΩ0,

(8)

where κ=iχ/2π~is the new coupling constant and

∆(Ω,Ω0) = kp(Ω + Ω0)−ko(Ω) −ke(Ω0) is the phase

mismatch for the three intracting waves.

In the low-gain regime, where the pump amplitude is

suﬃciently small, we can solve Eq. (8) perturbatively, by

substituting †

µ(Ω0, x)→†

µ(Ω0,0) under the integral. In

this way, we obtain for the ﬁelds at the crystal output

o(Ω, L) = o(Ω,0)

+κL ˆα(Ω + Ω0)†

e(Ω0,0)Φ(Ω,Ω0)dΩ0,

e(Ω, L) = e(Ω,0)

+κL ˆα(Ω + Ω0)†

o(Ω0,0)Φ(Ω0,Ω)dΩ0,

(9)

where Lis the crystal length and

Φ(Ω,Ω0) = ei∆(Ω,Ω0)L/2sinc[∆(Ω,Ω0)L/2] (10)

is the phase-matching function.

Substituting these solutions into Eq. (4), we obtain the

ﬁeld transformation from the crystal input face to its out-

put one. To write this transformation in a compact form,

we deﬁne the envelopes of the ordinary wave at the crys-

tal input and output faces as A0(t) = ˆ

E(+)

o(t, 0)eiω0tand

A1(t) = ˆ

E(+)

o(t, L)ei(ω0t−k0

oL)respectively, and those of

the extraordinary wave at the same positions as B0(t) =

E(+)

e(t, 0)eiω0tand B1(t) = ˆ

E(+)

e(t, L)ei(ω0t−k0

eL)respec-

tively. Here k0

µ=kµ(0). Using this notation, the ﬁeld

transformation in the crystal has the form of an integral

Bogoliubov transformation

A1(t) = ˆUA(t, t0)A0(t0)dt0+ˆVA(t, t0)B†

0(t0)dt0,(11)

B1(t) = ˆUB(t, t0)B0(t0)dt0+ˆVB(t, t0)A†

0(t0)dt0,(12)

where the Bogoliubov kernels are

UA(t, t0) = ˆei[ko(Ω)−k0

o]L+iΩ(t0−t)dΩ

2π,(13)

VA(t, t0) = ˆei[ko(Ω)−k0

o]L−i(Ω0t0+Ωt)J(Ω,Ω0)dΩdΩ0

2π,(14)

UB(t, t0) = ˆei[ke(Ω)−k0

e]L+iΩ(t0−t)dΩ

2π,(15)

VB(t, t0) = ˆei[ke(Ω0)−k0

e]L−i(Ω0t+Ωt0)J(Ω,Ω0)dΩdΩ0

2π.(16)

and

J(Ω,Ω0) = κLα(Ω + Ω0)Φ(Ω,Ω0) (17)

is the joint spectral amplitude (JSA) of the two generated

photons, or the biphoton [60].

B. Spectral and temporal shapes of the photons

The spectra of the ordinary and extraordinary waves

are Sµ(Ω) = h†

µ(Ω, L)µ(Ω, L)i, where µ=o, e. Substi-

tuting the solutions (9), applying the commutation rela-

tion (7), and setting to zero all normally ordered averages

at x= 0, we obtain

So(Ω) = 2πˆJ(Ω,Ω0)

2dΩ0,(18)

and a similar expression for Se(Ω) with J(Ω,Ω0) replaced

by J(Ω0,Ω).

To ﬁnd these spectra analytically, we make two approx-

imations. The ﬁrst one is the approximation of linear dis-

persion in the crystal. This means that we consider only

linear terms in the dispersion law in the crystal, which

is justiﬁed for a not-too-long crystal [39,61]. Thus we

write kµ(Ω) ≈k0

µ+k0

µΩ, where k0

µ= (dkµ/dΩ)Ω=0 is

the inverse group velocity of the wave µ=p, o, e in the

nonlinear crystal. In this way we obtain ∆(Ω,Ω0)L/2≈

τoΩ+τeΩ0, where τo= (k0

p−k0

o)L/2 and τe= (k0

p−k0

e)L/2

are relative group delays of the ordinary and extraodi-

nary photons with respect to the pump at half crystal

length and we have also assumed a perfect phase match-

ing at degeneracy, k0

p−k0

o−k0

e= 0.

The second approximation is the replacement of the

sinc(x) function by a Gaussian function having the same

width at half maximum e−x2/2σ2

s, where σs= 1.61 [56,

61]. Applying both approximations, we write

Φ(Ω,Ω0)≈exp "−(τoΩ + τeΩ0)2

2σ2

+i(τoΩ + τeΩ0)#.

(19)

Substituting Eqs. (3), (19) and (17) into Eq. (18) and

a similar expression for Se(Ω) and performing integra-

tions (see Appendix A), we ﬁnd

Sµ(Ω) = √2πPb

σµ

e−Ω2/2σ2

µ,(20)

where the spectral standard deviation of the ordinary

photon is

σo=qσ2

s+ 2τ2

eΩ2

√2|τe−τo|,(21)

that of the extraordinary photon, σe, is obtained by a

replacement τo↔τe, and

Pb=ˆJ(Ω,Ω0)

2dΩdΩ0=√2π2(κLE0)2σs

Ωp|τe−τo|(22)

is the probability of biphoton generation per pump pulse.

In Fig. 2we show the modulus of the JSA for a βbar-

ium borate (BBO) crystal of length L= 2 cm, pumped at

405 nm by Gaussian pump pulses with a FWHM band-

width ∆λ= 0.2 nm, similar to the experimental setup

of Ref. [41], but with a longer crystal and longer pump

pulses. The pump bandwidth corresponds to Ωp= 0.98

rad/ps or τp= 1.21 ps. Using the Sellmeier equations for

the ordinary and extraordinary refractive indices of BBO

[62], we ﬁnd the angle between the pump and the opti-

cal axis θp= 41.42◦for a frequency-degenerate collinear

phase matching, τo= 0.76 ps and τe= 2.68 ps. For

these conditions, we ﬁnd σo= 1.48 rad/ps and σe= 0.71

rad/ps, which give the ratio σo/σe= 2.1, showing the

asymmetry in the spectral bandwidth of two generated

photons.

FIG. 2. Normalized contour plot of the JSA for two pho-

tons generated in a 2-cm-long BBO crystal pumped at 405

nm. The tilted shape of the JSA indicates the frequency-time

entanglement between the photons. The bandwidth of the

ordinary photon is more than two times higher than that of

the extraordinary one.

The temporal shapes of the photons are given by

their average intensities (in photon ﬂux units) Iµ(t) =

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摘要：

MakingentangledphotonsindistinguishablebyatimelensShivangSrivastava,1,DmitriB.Horoshko,1,2,yandMikhailI.Kolobov1,z1Univ.Lille,CNRS,UMR8523-PhLAM-PhysiquedesLasersAtomesetMolecules,F-59000Lille,France2B.I.StepanovInstituteofPhysics,NASB,NezavisimostiAve68,Minsk220072Belarus(Dated:March8,2023)Weprop...

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Making entangled photons indistinguishable by a time lens Shivang Srivastava1Dmitri B. Horoshko1 2yand Mikhail I. Kolobov1z 1Univ. Lille CNRS UMR 8523 - PhLAM - Physique des Lasers Atomes et Mol ecules F-59000 Lille France.pdf

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Making entangled photons indistinguishable by a time lens Shivang Srivastava1Dmitri B. Horoshko1 2yand Mikhail I. Kolobov1z 1Univ. Lille CNRS UMR 8523 - PhLAM - Physique des Lasers Atomes et Mol ecules F-59000 Lille France

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