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Improved generation of correlated photon pairs

from monolayer WS

based on bound states

in the continuum

TIECHENG WANG,ZHIXIN LI, AND XIANGDONG ZHANG*

Beijing Key Laboratory of Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology,

Beijing 100081, China

*Corresponding author: zhangxd@bit.edu.cn

Received 24 October 2018; revised 21 January 2019; accepted 22 January 2019; posted 23 January 2019 (Doc. ID 349001);

published 28 February 2019

Entangled photons are the fundamental resource in quantum information processing. How to produce them

efficiently has always been a matter of concern. Here we propose a new way to produce correlated photons effi-

ciently from monolayer WS

based on bound states in the continuum (BICs). The BICs of radiation modes in the

monolayer WS

are realized by designing the photonic crystal slab-WS

-slab structure. The generation efficiency

of correlated photon pairs from such a structure has been studied by using a rigorous quantum model of sponta-

neous parametric down-conversion with the plane wave expansion method. It is found that the generation effi-

ciency of correlated photon pairs is greatly improved if the signal and idler fields are located at the BICs

determined by the inverse scattering matrix of the structure. This is in contrast to the parametric down-conversion

process for the enhanced generation of nonlinear waves if the pump field is located at the BICs determined by the

scattering matrix of the structure. The generation rate of the correlated photon pairs can be improved by 7 orders

of magnitude in some designed structures. The generated quantum signals are sensitive to the wavelength and

exhibit narrowed relative line width, which is very beneficial for quantum information processing.

Chinese Laser Press

https://doi.org/10.1364/PRJ.7.000341

1. INTRODUCTION

In the past two decades, there has been a great deal of interest

in studying how to produce entangled photon pairs, because

they play a crucial role in quantum information processing

[1–3]. Many methods to produce such resource s have been de-

veloped [4–7]. A popular approach to generating entangled

photon pairs is based on the nonlinear process of parametric

down-conversion in naturally birefringent nonlinear crystals

such as β-barium borate (BBO) [8]. Other mechanisms, such

as using quantum dots, quasi-phase-matching in photonic

crystals, and periodically poled materials, have also been pro-

posed [9–18].

On the other hand, nonlinear optical properties of mono-

layer transition metal dichalcogenides (TMDC) have attracted

much attention in recent years because monolayer TMDC

as two-dimensional systems have ultra-high second-order non-

linear susceptibility [19–22]. For example, some investigations

have shown that the value of the effective second-order non-

linear susceptibility for the monolayer WS

is 3 orders of

magnitude larger than the values usually reported for other

nonlinear bulk crystals [23]. The question is whether the

ultra-high second-order nonlinear susceptibility in the mono-

layer TMDC can be used to produce entangled photons effi-

ciently. In fact, such an idea is constrained by weak interactions

between monolayer TMDC and light due to the single atom

thickness of the sample. The weak interactions block efficient

generation of nonlinear effects.

Fortunately, some methods to improve the interaction be-

tween monolayer materials and electromagne tic (EM) waves

have been proposed [24–27]. For example, bound states in the

continuum (BICs) can be utilized to improve this interaction

[28,29]. Analogous to the localized electrons with energy larger

than their potential barriers, light BICs have been realized

in recent years [30–34]. The BICs are known as embedded

trapped modes, which correspond to discrete eigenvalues co-

existing with extended modes of a continuous spectrum. They

have been shown to exist in the dielectric gratings, waveguide

structures, the surface of the object, photonic crystal slabs, and

some open subwavelength nanostructures [35–40]. Recently, it

has been demonstrated that nonlinear effects can be improved

greatly using these BICs [41].

Motivated by these investigations, in this work we explore

the possibility to improve the generation rate of correlated

Research Article

Vol. 7, No. 3 / March 2019 / Photonics Research 341

photon pairs based on the BICs. In order to calculate this gen-

eration rate in our model including a photonic crystal slab, we

extend the previous quantum theory of spontaneous parametric

down-conversion (SPDC) by using a plane wave expansion

method for the first time to our knowledge. We find that pho-

ton-pair generation is enhanced if the signal and idler fields are

located at the resonant state determined by the inverse scatter-

ing matrix of the structure and the generated fields possess a

narrowed relative line width and directivity. It is noted that

we do not perform the comparison between our monolayer

source of WS

with those SPDC sources, such as periodically

poled lithium niobate (PPLN) waveguides and periodically

poled KTiOPO

(PPKTP) crystal. This is because it is not suit-

able for comparing the source of the monolayer atom with

those of bulk SPDC sources. However, we compare the case

based on the BICs with those of bare monolayers.

2. THEORY AND METHOD

We consider a three-layer structure consisting of a photonic

crystal slab, a monolayer WS

, and a dielectric slab as shown

in Figs. 1(a) and 1(b). The monolayer WS

is put at the inter-

face between the photonic crystal slab and the dielectric slab.

The photonic crystal slab consists of a square lattice of air holes

introduced into a high-index dielectric medium, and the cor-

responding lattice constant is denoted by l. The thickness, rel-

ative permittivity, and relative permeability of this high-

index medium are denoted by d

, ε

, and μ

, respectively.

The thickness, relative permittivity, and relative permeability

of the dielectric slab are represented by d

, ε

, and μ

, respec-

tively. The thickness, relative permittivity, relative permeability,

and second-order susceptibility of the monolayer WS

are de-

scribed by d

, ε

, μ

, and χ

2

. All these materials are taken to

be nonmagnetic. The photonic crystal slab and dielectric slab

are taken to be linear in order to investigate the nonlinear effect

of monolayer WS

. As one component of TMDC monolayers,

the relative permittivity of monolayers WS

can be calculated

by the permittivity of TMDC monolayer ε

TMDC

, which is

equal to a superposition of N Lorentzian functions [42,43],

TMDC

 1 

k1

−ω

−iωγ

, where ω is the angular fre-

quency of the EM wave; f

, ω

, and γ

represent the oscillator

strength, resonance frequency, and spectral width of the kth

oscillator, respectively; and the values of these model para-

meters are provided in Ref. [44].

A. Theory for the Spontaneous Parametric Down-

Conversion in the Photonic Crystal Slab-Monolayer

-Slab Structure

In contrast to the graphene, TMDC monolayers are noncen-

trosymmetric and there fore the second-order nonlinear effect is

allowed. Based on the symmetry properties of their space group

, it can be shown that the structure of their quadratical sus-

ceptibility tensor χ

2

yields only one independent and nonvan-

ishing component [4,5]:

2

 χ

2

yyy

 −χ

2

yxx

 −χ

2

xyx

 −χ

2

xxy

, (1)

where x represents the zigzag direction of the monolayer and y

is the orthogonal armchair direction. According to Ref. [23],

the effective thickness of the monolayer WS

is taken as

 0.618 nm. Figure 1(b) shows the process of photon-pair

generation in the three-layer structure. When a pump field with

the angular frequency ω

is incident on the structure, the signal

and idler fields with angular frequencies ω

and ω

, respectively,

are generated simultaneously due to the second-order nonlinear

effect of the monolayer WS

, the incident angle of the pump

wave is denoted by θ

, and the irradiated angles of the signal

and idler fields are taken as θ

and θ

. The nonlinear interact ion

in the three-layer structure is described by a Hamiltonian H

int

t:

int

tε

d~r

α, β, γ

χ

2

αβγ



p,α

~r, t

−

s,β

~r, t

−

i,γ

~r, tH:c:,

(2)

where ε

is the permittivity of air and χ

2

αβγ

represent the com-

ponents of susceptibility tensor χ

2

of the monolayer WS

Here α, β, and γ denote the direction (x or y), E



p,α

~r, t is

the positive frequency electric field for the classical strong pump

wave at the monolayer WS

, and



s,β

~r, t and



i,γ

~r, tdenote

the corresponding electric field operators for the generated pho-

tons with frequencies ω

and ω

. E

−

p,α

~r, t,

−

s,β

~r, t, and

−

i,γ

~r, t denote the corresponding negative frequency electric

fields, which are the conjugated terms of the positive frequency

fields. H.c. stands for a Hermitian conjugated term.

The plane-wave expansion method is utilized to study the

photon-pair generation [45]. Because of the periodicity of the

three-layer structure, the pump field can be expanded in plane

waves as



p,α

~r, t

E

pFmn,α

expiβ

pmn

zE

pBmn,α

exp−iβ

pmn

z

× expik

pmnx

x  k

pmny

y − iω

t





pmn,α

z, ω

expik

pmnx

x  k

pmny

y − iω

t,

(3)

pmn



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

− k

pmnx

− k

pmny

, k

 ω

nω

∕c, (4)

Fig. 1. (a) Diagram of the photonic crystal slab-monolayer

-slab. The air holes are arranged in a square lattice with lattice

constant l and the radius of the holes is r. The thicknesses of the pho-

tonic crystal slab and dielectric slab are denoted by d

and d

, and the

monolayer WS

is put at the interface between the photonic crystal

slab and the dielectric slab. (b) Schematic of the photon-pair gener-

ation process in the three-layer structure. The pump beam with fre-

quency ω

and angle θ

is incident on the three-layer structure, and

due to the second-order nonlinear effect of the monolayer WS

, the

signal field with frequency ω

and angle θ

and the idler field with

frequency ω

and angle θ

are generated.

342 Vol. 7, No. 3 / March 2019 / Photonics Research

Research Article

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