Design of mid-infrared entangled photon sources using lithium niobate Jin-Long Zhu1Wen-Xin Zhu1Xiao-Tao Shi1Chen-Tao Zhang1Xiangying Hao1yZi-Xiang Yang1zand Rui-Bo Jin1x_2

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Design of mid-infrared entangled photon sources using lithium niobate

Jin-Long Zhu,1, ∗Wen-Xin Zhu,1, ∗Xiao-Tao Shi,1, ∗Chen-Tao

Zhang,1Xiangying Hao,1, †Zi-Xiang Yang,1, ‡and Rui-Bo Jin1, §

1Hubei Key Laboratory of Optical Information and Pattern Recognition,

Wuhan Institute of Technology, Wuhan 430205, PR China

The mid-infrared (MIR) band entangled photon source is vital for the next generation of quantum

communication, quantum imaging, and quantum sensing. However, the current entangled states are

mainly prepared in visible or near-infrared bands. It is still lack of high-quality entangled photon

sources in the MIR band. In this work, we optimize the poling sequence of lithium niobate to prepare

two kinds of typical entangled states, the Hermit-Gaussian state and the comb-like entangled state

at 3.2 µm. We have also calculated the photon pair rates and estimated the eﬀect of fabrication

resolution in the schemes. Our approach will provide entangled photon sources with excellent

performance for the study of quantum information in the MIR band.

I. INTRODUCTION

The entangled photon source in the mid-infrared

(MIR) band (approximately 2-20 µm) is promising for

the next generation of quantum communication, quan-

tum imaging, and quantum sensing[1,2]. In quan-

tum communication, the entangled photon source in the

wavelength between 3 µm and 5 µm covers the atmo-

spheric transmission window, which has higher trans-

parency than that in the near-infrared band and is ben-

eﬁcial for free-space quantum communications [3]. In

quantum imaging, room temperature objects emit light

at MIR wavelengths, therefore the MIR band entangled

photon sources are compatible with novel applications in

infrared thermal imaging [4,5]. In quantum sensing, the

MIR band entangled photon source has strong absorp-

tion bands of a variety of gases, which leads to essential

applications in gas quantum sensing [6]. With the help

of entanglement, the sensing precision may be improved

from the shot noise limit to the Heisenberg limit [7].

Spontaneous parametric down-conversion (SPDC) is

one of the widely used methods to prepare an entan-

gled photon source. Recently, several works have investi-

gated the generation of entangled photons in MIR range

from an SPDC process in the nonlinear crystal. From

the theoretical side, in 2016, Lee et al reported a scheme

for the generation of polarization-entangled state from

periodically poled potassium niobate (PPKN) covering

3.2 to 4.8 µm [8]; In 2018, McCracken et al numerically

investigated six novel nonlinear crsystals in order to gen-

erate MIR single photons [9]; In 2020, Kundys et al nu-

merically studied the reconﬁgurable MIR single-photon

sources in PMN-0.38PT crystal at 5.6 µm [10]; In 2021,

Wei et al theoretically investigated the preparation of

MIR spectrally uncorrelated biphotons from an SPDC

process using doped lithium niobate (LN) crystals [11];

∗These authors contributed equally to this work.

†xyhao.321@163.com

‡yangzixiangyzx@foxmail.com

§jin@wit.edu.cn

These schemes for single photon can also be upgraded

to prepare entangled photons. From the experimental

side, in 2020, Prabhakar et al demonstrated an entan-

gled photons source at 2.1 µm generated from type-II

PPLN crystals [12].

PPLN is one of the most promising crystals for MIR

entangled photon source, not only for its large nonlin-

ear coeﬃcient and wide transparency range [13,14], but

also for its group velocity-matched (GVM) wavelengths,

which are in the MIR range [11]. However, from the view-

point of quantum state engineering, the previous entan-

gled source with PPLN is still not optimal [11,12]. For

a standard PPLN crystal, the phase matching function

(PMF) has a “sinc” distribution, which has side lobes

and will harm the spectral purity of heralded single pho-

ton. To overcome the problem of side lobes, one useful

method is to adopt the “customized poling” instead of

“periodical poling”[15–18].

Many previous works have been devoted to the opti-

mization of a poling period in a periodically poled potas-

sium titanyl phosphate (PPKTP) crystal at 1550 nm,

and the optimization can be divided into three categories:

(1) the optimization of poling order: in 2011, Branczyk

et al proposed and experimentally demonstrated the ﬁrst

optimization design of KTP by arranging the poling order

[19], and this approach was further improved by Kaneda

et al in 2021 [20]. (2) the optimization of duty cycle:

in 2013, Dixon et al proposed to design the duty cycle

of KTP [21], which was veriﬁed experimentally in 2017

[22]; In 2019, Cui et al adopted the Adam algorithm in

a machine learning framework to optimize the duty cy-

cle [23]. In 2022, Cai et al optimized the duty cycle of

LN using the particle swarm algorithm [24]; (3) the op-

timization of domain sequence: in 2016, Tambasco et al

optimized the domain sequence in the unit of dual do-

main blocks [25]; In the same year, Dosseva et al pro-

posed to optimize the sequence of single domain blocks

using simulated annealing algorithm [26]; In 2017, Graf-

ﬁtti et al theoretically optimized the domain blocks with

sub-coherence-length [15], and then veriﬁed experimen-

tally in 2018 [27,28]. Recently, frequency-bin entangle-

ment generated by domain-engineered down-conversion

arXiv:2210.12466v1 [quant-ph] 22 Oct 2022

has been demonstrated theoretically [16] and experimen-

tally [17,18]. These optimization works were mainly fo-

cused on PPKTP and at 1550 nm wavelength. However,

for the MIR band, PPKTP crystal is no longer applicable

because it no longer meets the GVM condition.

In this work, we focus on the MIR band and propose

two categories of entangled states at 3.2 µm, the Hermit-

Gaussian (H-G) entangled state and the comb-like entan-

gled state. We will explain how to realize these states by

optimizing the poling period of the LN crystal using the

domain sequence arrangement algorithm. This work is

expected to provide quantum entangled photon sources

with excellent performance for the study of quantum in-

formation in the MIR band.

II. THEORY

During the SPDC process, a pump photon with higher

frequency impinges on a nonlinear optical crystal, and

produces a pair of photons with lower frequency, often

referred to as a signal and a idler photon, collectively

called biphoton. Without considering higher order non-

linear eﬀects, the joint spectral amplitude (JSA) of down-

converted photons is given by the product of PMF and

pump envelope function (PEF)[29,30]:

f(ωs, ωi) = φ(ωs, ωi)×α(ωs+ωi),(1)

where PEF αis determined by the spectral distribution of

pump photon and is usually taken as a Gaussian function,

and PMF φis determined by the properties of the crystal.

For periodically poled crystals, PMF can be expressed as

a function of ∆k:

φ(∆k) = 1

LZL

exp[i∆kz]dz = sinc( ∆kL

2) exp(i∆kL

2),

(2)

where Lis the crystal length and ∆kis the diﬀerence

of the wave vector kp(s,i)=ωp(s,i)np(s,i)

c, ∆k=ωpnp

c−

ωsns

c−ωini

c−2π

Λ. Using wavelength as the variable, ∆k

can also be written as:

∆k= 2π×(np(λp)

λp−ns(λs)

λs−ni(λi)

λi−1

Λ),(3)

where np(s,i)and λp(s,i)are respectively the refractive

index and the wavelength of the pump (signal, idler)

photon. Λ is the poling period, it can be customarily

designed according to the target PMF, which will be dis-

cussed in Section IV.

Due to the energy conservation, ωs+ωi=ωp, so the

ridge direction of α(ωs+ωi) is always distributed along

the anti-diagonal direction. To make the φ(ωs, ωi) dis-

tributed along the diagonal direction, it is necessary to

consider the GVM condition [30]:

2V−1

p=V−1

s+V−1

i,(4)

where V−1

p(s,i)is the inverse of the group velocity of

the pump (signal, idler). Under the GVM condition,

α(ωs+ωi) is perpendicular to φ(ωs, ωi), and their prod-

uct may achieve a single Gaussian mode. For type-II

phase-matched PPLN (o →o+e, the pump and the sig-

nal is o-ray, and the idler is e-ray), the GVM condition

is satisﬁed at the wavelength of 3207.6 nm [11] according

to the Sellmeier equation [31]. The joint temporal am-

plitude (JTA) can be obtained by performing an inverse

Fourier transform (IFT) on the JSA:

g(ts, ti) = Z∞

−∞ Z∞

−∞

f(ωs, ωi) exp (iωsts+iωiti)dωsdωi.

(5)

The properties of the JSA and JTA can be measured

using the Hong-Ou-Mandel (HOM) interference [32], in

which the coincidence probability pas a function of time

delay τcan be calculated as [33]:

p(τ) = 1

2−1

2Z∞

−∞ Z∞

−∞ |f(ωs, ωi)|2cos (ωs−ωi)τdωsdωi.

(6)

III. ENTANGLED PHOTON SOURCES FOR

MIR WAVELENGTHS

For future applications in quantum communication,

quantum imaging, and quantum sensing in the MIR

band, we propose two kinds of typical entangled states.

The ﬁrst category is the H-G state, while the second cat-

egory is the comb-like entangled state. Both categories

are intrinsically high-dimensional entangled states and

can carry more information for quantum communications

[28,34]. Further, the high-dimensional entangled state

has a much narrower interference pattern in HOM inter-

ference, which is very helpful to improve the precision of

a quantum measurement [35].

Figure 1 shows the JSA, absolute of the JTA, and

HOM interference patterns for H-G states and comb-like

states. Figure 1 (a1) shows a typical three-mode JSA,

which can be obtained by choosing a PMF of second-

order Hermitian function, and a PEF of zero-order Her-

mitian function. Figure 1 (a2) is the absolute of the

JTA calculated using Eq.(5). Figure 1 (a3) is the simu-

lated HOM interference pattern using Eq.(6). Figure 1

(b1-b3) shows cases of a four-mode H-G state. Figure 1

(c,d,e,f) shows cases of ten-mode, four-mode, ﬁve-mode,

and sixteen-mode comb-like state. It is interesting to

notice in Fig.1 (a1-a2) and (b1-b2) that the H-G mode

entangled state has the same mode numbers in both fre-

quency domain and time domain. In addition, as shown

in Fig.1(c3) and (f3), the HOM interference pattern have

a much narrower interference valleys and peaks .

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Designofmid-infraredentangledphotonsourcesusinglithiumniobateJin-LongZhu,1,Wen-XinZhu,1,Xiao-TaoShi,1,Chen-TaoZhang,1XiangyingHao,1,yZi-XiangYang,1,zandRui-BoJin1,x1HubeiKeyLaboratoryofOpticalInformationandPatternRecognition,WuhanInstituteofTechnology,Wuhan430205,PRChinaThemid-infrared(MIR)banden...

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Design of mid-infrared entangled photon sources using lithium niobate Jin-Long Zhu1Wen-Xin Zhu1Xiao-Tao Shi1Chen-Tao Zhang1Xiangying Hao1yZi-Xiang Yang1zand Rui-Bo Jin1x_2

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