#  REVIEW OF MODERN PHYSICS Theory for the beam splitter in quantum optics:
quantum entanglement of photons and their statistics, HOM effect.

D.N. Makarov∗
Northern (Arctic) Federal University, Arkhangelsk, 163002, Russia
E-mail<EMAIL_ADDRESS>
###### Abstract

The theory of the beam splitter (BS) in quantum optics is well developed and
based on fairly simple mathematical and physical foundations. This theory has
been developed for any type of BS and is based on the constancy of the
reflection coefficients $R$ (or the transmission coefficient, where $R+T=1$)
and the phase shift $\phi$. It has recently been shown that the constancy of
these coefficients cannot always be satisfied for a waveguide BS, where $R$
and $\phi$ depend in a special way on photon frequencies. Based on this, this
review systematizes the concept of BS in quantum optics into “Conventional”
and frequency-dependent BS, and also presents the theory of such BS. It is
shown that the quantum entanglement, photon statistics at the output ports,
and the Hong-Ou-Mandel (HOM) effect for such BS can be very different. Taking
into account the fact that the waveguide BS is currently acquiring an
important role in quantum technologies due to the possibility of its
miniaturization, this review will be useful not only for theoreticians, but
also for experimenters.

Keywords: Beam splitter, waveguide beam splitter, quantum entanglement,
photons, reflection coefficient, phase shift, photon statistics, Hong-Ou-
Mandel effect.

##  1 Introduction

The beam splitter (BS) is one of the main devices not only in classical
optics, but also in quantum optics. A beam splitter is an optical device that
splits a beam of light into a transmitted and a reflected beam. This is the
most important device of many optical and measuring systems. For example, such
systems can be interferometers: Michelson-Morley, Mach-Zehnder and Hong-Ou-
Mandel [1, 2, 3, 4]. Despite its simple purpose - to separate the incident
beam, the beam splitter in quantum optics has a much broader meaning [1, 2].
In quantum optics, two modes of the electromagnetic field are usually
considered (two input and output ports), because even if 1 input port remains
unused, it should be considered as an input for vacuum fluctuations [5]. The
main value in quantum optics is the quantum states of the electromagnetic
field at the output ports of the beam splitter. Depending on the reflection
coefficient $R$ (similar to the transmission coefficient $T$, where $R+T=1$)
and the input states of the electromagnetic field, the quantum states of
interest can be obtained at the output ports of the BS. This can be used in
many applications of quantum technologies. For example, BS is used in linear
optical quantum computing Indeed, Knill et al. showed in 2001 that it is
possible to create a universal quantum computer using only BS, phase shifts,
photodetectors and single photon sources (KLM protocol)[6]. Also, using BS,
you can create quantum entanglement between the input modes of electromagnetic
fields [4, 7, 8], simulate quantum transport [9] and determine the degree of
identity of photons [4, 10], etc. BS is an integral part of quantum metrology
[11] and quantum information [12], including two- and multi-photon
interference [4, 13].

In this review, we will consider two-port beam splitters, since they are the
most important and frequently used in quantum technologies. It is well known
that such beam splitters can be of various types and properties. By types, BS
are divided into ways of their manufacture, for example, a cubic BS or a
waveguide BS, see Fig. 1. The BS can be a glass plate with a silver or
dielectric coating, a glass cube with a coating in a diagonal plane, two glass
plates with parallel planes, between which a coating is located, or a coating
deposited on films. A waveguide BS is two waveguides brought close enough
together so that the electromagnetic fields overlap; in this case it is a
directional coupler (eg [14, 15]). Waveguide BS have an advantage over many
types of beam splitters because they are much smaller than them, and also have
many other advantages [16, 7, 9]. Subdivision by properties usually means the
use of polarized or non-polarized BS or other unique properties. In quantum
optics, the BS can generally be represented as (independent of the type of BS)
Fig.1(a), and the schematic designations depend on their types, e.g. Fig.1(b),
see [17].

Figure 1: In Fig.1(a) shows a BS scheme with two input ports and two output
ports, where annihilation operators 1 and 2 modes on the input ports represent
$\hat{a}_{1}$ and $\hat{a}_{2}$ respectively, and on the BS output ports
represent $\hat{b}_{1}$ and $\hat{b}_{2}$ respectively. The BS is represented
as a “black box” in which “mixing” of electromagnetic field input modes takes
place. Fig.1(b) shows the BS with free space optics, i.e., cubic BS (top) and
fibre optics, i.e., waveguide BS (bottom).

Without going into the technical details and properties of these beam
splitters (there are quite a few reviews on this topic, see eg. [7]), we can
say that in quantum optics there are two main parameters in BS, this is its
reflection coefficient $R$ (or transmission coefficient $T$) and phase shift
$\phi$. Usually, to study the properties of electromagnetic waves (hereinafter
referred to as photons) at the output ports of the BS or in technical devices
where the BS is a component, in quantum optics it is considered that $R$ and
$\phi$ are constant values, see for example, [18, 19, 20, 7, 8, 11, 21]. This
means that for finding quantum states of photons at the BS output ports these
quantities do not change, i.e. their definite values are always given. For
example, in the Hong-Oy-Mandel (HOM) effect these values $R=T=1/2$ and the
effect is independent of phase $\phi$. Of course, these values $R$ and $\phi$
depend in general on the wavelength (or frequency) of the incident field
(regardless of the BS type), i.e. $R=R(\lambda)$, $\phi=\phi(\lambda)$, then
this dependence should have no effect on photon states at the BS output ports
if we fix their values. This has always seemed an obvious fact and has not
been studied before. Anyway, in general case $R$ and $\phi$ are not constant
quantities and the dependence of $R$ and $\phi$ on frequencies is such that it
can affect the photon states at the BS output ports. If $R$ and $\phi$ depend
on frequencies in this way, then the quantum states at the BS output ports
differ with respect to the case of quantum states at constant $R$ and $\phi$.
It follows that quantum entanglement at such BS will be different from the
case with constants $R$ and $\phi$. As it turns out, such BS is a waveguide
beam splitter (the term “Fiber-optic splitter” is often used) which differs
from many other types of BS by this property.

Recently [17, 22, 23] the theory of a frequency dependent waveguide BS has
been presented. In these papers, it was shown that if the BS is represented as
a coupled waveguide, then the coefficients $R$ and $T$ depend on the
frequencies of photons entering both ports of the BS. Taking into account the
frequency dependence of the coefficients $R$ and $T$, many known theories can
be modified, for example, the HOM interference theory [24, 25] or quantum
entanglement of photons based on a beam splitter [22, 23]. It should be added
that such a frequency dependence of the coefficients $R$ and $T$ is inherent
only in a waveguide beam splitter.

Thus, there is a need for a review on this topic, where BSs in quantum optics
will be systematized into frequency-dependent and non-frequency-dependent
“conventional” and based on such systematization, the quantum entanglement of
photons at the output ports of the beam splitter is considered. In this
review, such a systematization is carried out and not only the quantum
entanglement of photons on such beam splitters is considered, but also the
statistical properties of photons at the output ports and the HOM effect.

##  2 Beam splitter in quantum optics

Since the BS produces separation of incoming beams, the quantum state of
photons at the BS output ports is
$|\Psi_{out}\rangle=e^{i{\hat{H}}t_{BS}}|\Psi_{in}\rangle$, where ${\hat{H}}$
is the Hamiltonian of the quantized electromagnetic field interacting with
matter, $t_{BS}$ is interaction time, $|\Psi_{in}\rangle$ is the initial state
of electromagnetic field. It should be added that ${\hat{H}}$ can be quite
complex depending on the type of BS. In general, $|\Psi_{in}\rangle$ can be
represented as [26, 13]

$\displaystyle|\Psi_{in}\rangle=\sum_{s_{1},s_{2}}\frac{C_{s_{1},s_{2}}}{\sqrt{s_{1}!s_{2}!}}{{\hat{a}_{1}}^{\dagger}}{}^{s_{1}}{{\hat{a}_{2}}^{\dagger}}{}^{s_{2}}|0\rangle_{1}|0\rangle_{2},$
(1)

where the 1st and 2nd mode creation operators respectively represent
${\hat{a}_{1}}^{\dagger}$ and ${\hat{a}_{2}}^{\dagger}$, $s_{1}$ and $s_{2}$
are the quantum numbers of the 1st and 2nd modes, respectively (or the number
of photons in the modes), $C_{s_{1},s_{2}}$ are the expansion coefficients
defining the initial state, $|0\rangle_{1}|0\rangle_{2}$ the vacuum states for
modes 1 and 2 respectively (for convenience we will write
$|0\rangle_{1}|0\rangle_{2}\to|0\rangle$). It should be added that if the
initial states are in the Fock state, then the coefficient
$C_{s_{1},s_{2}}=1$. In this case, it is easy to show (up to an insignificant
phase) that [26, 1, 13]

$\displaystyle|\Psi_{out}\rangle=\sum_{s_{1},s_{2}}\frac{C_{s_{1},s_{2}}}{\sqrt{s_{1}!s_{2}!}}{{\hat{b}_{1}}^{\dagger}}{}^{s_{1}}{{\hat{b}_{2}}^{\dagger}}{}^{s_{2}}|0\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\
{\hat{b}_{k}}^{\dagger}=e^{i{\hat{H}}t_{BS}}{\hat{a}_{k}}^{\dagger}e^{-i{\hat{H}}t_{BS}}\leavevmode\nobreak\
(k=1,2),$ (2)

where ${\hat{b}_{1}}^{\dagger},{\hat{b}_{2}}^{\dagger}$ are the creation
operators at the output ports of the BS for modes 1 and 2, respectively. It
turns out that these operators can be found in general terms without even
considering a particular type of BS, just on the basis of general physical
considerations.

It is well known that lossless two-mode BS (with two input and output ports)
in quantum optics is described by the unitary matrix $U_{BS}$, which has the
form [5, 18, 19, 27]

$\displaystyle\begin{pmatrix}\hat{b}_{1}\\\ \
\hat{b}_{2}\end{pmatrix}=U_{BS}\begin{pmatrix}\hat{a}_{1}\\\ \
\hat{a}_{2}\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ U_{BS}=\begin{pmatrix}\sqrt{T}&e^{i\phi}\sqrt{R}\\\ \
-e^{-i\phi}\sqrt{R}&\sqrt{T}\end{pmatrix};$ (3)

where the 1st and 2nd mode annihilation operators respectively represent
$\hat{a}_{1}$ and $\hat{a}_{2}$, and at the BS output ports $\hat{b}_{1}$ and
$\hat{b}_{2}$; $T$ and $R$ are the transmittance and reflectance, respectively
($R+T=1$), and $\phi$ is the phase shift. It should be added that in the
literature one often sees the use of the BS matrix $U_{BS}$ in various
representations. The most commonly used representation is when the phase shift
$\phi=\pm\pi/2$, the second representation often encountered is when $\phi=0$,
in these cases the $U_{BS}$ matrix is respectively

$\displaystyle U_{BS}=\begin{pmatrix}\sqrt{T}&\pm i\sqrt{R}\\\ \ \pm
i\sqrt{R}&\sqrt{T}\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ U_{BS}=\begin{pmatrix}\sqrt{T}&\sqrt{R}\\\ \
-\sqrt{R}&\sqrt{T}\end{pmatrix}.$ (4)

In fact, both representations can only be used when the result is independent
of the phase shift $\phi$. As will be shown below, many characteristics
studied in quantum optics, such as the quantum entanglement of photons at the
BS output ports, do not really depend on the phase. In spite of this we will
use the general Eq. (4).

It should be added that in reality photons are not monochromatic and the
frequency distribution [1, 5] must be taken into account. In this case the
initial wave function of photons will be

$\displaystyle|\Psi_{in}\rangle=\sum_{s_{1},s_{2}}\frac{C_{s_{1},s_{2}}}{\sqrt{s_{1}!s_{2}!}}\int\phi(\omega_{1},\omega_{2}){{\hat{a}_{1}}^{\dagger}}{}^{s_{1}}{{\hat{a}_{2}}^{\dagger}}{}^{s_{2}}|0\rangle
d\omega_{1}d\omega_{2},$ (5)

where $\phi(\omega_{1},\omega_{2})$ is the joint spectral amplitude (JSA) of
the two-mode wave function. Given the frequency distribution, the output is
($\int|\phi(\omega_{1},\omega_{2})|^{2}d\omega_{1}d\omega_{2}=1$)

$\displaystyle|\Psi_{out}\rangle=\sum_{s_{1},s_{2}}\frac{C_{s_{1},s_{2}}}{\sqrt{s_{1}!s_{2}!}}\int\phi(\omega_{1},\omega_{2}){{\hat{b}_{1}}^{\dagger}}{}^{s_{1}}{{\hat{b}_{2}}^{\dagger}}{}^{s_{2}}|0\rangle
d\omega_{1}d\omega_{2}.$ (6)

###  2.1 Basic expressions for beam splitters of any type

Consider the beam splitter in general terms without defining its type. Let us
set the problem to find the matrix $U_{BS}$ on the basis of general physical
considerations. This problem in quantum optics has been solved quite a long
time ago, see eg [18, 19, 27]. The result has essentially always been the same
- this BS matrix has been obtained in the Eq.(3). Let us present such a
conclusion here in the simplest version. For this purpose we will disregard
the effects of polarisation, misalignment and imperfect beam collimation, as
these can be incorporated into BS theory quite easily. The input
(${\hat{a}_{1}},{\hat{a}_{2}}$) and output (${\hat{b}_{1}},{\hat{b}_{2}}$)
annihilation operators, at some chosen frequency for linear BS can be
represented as

$\displaystyle\begin{pmatrix}\hat{b}_{1}\\\ \
\hat{b}_{2}\end{pmatrix}=\begin{pmatrix}U_{11}&U_{12}\\\ \
U_{21}&U_{22}\end{pmatrix}\begin{pmatrix}\hat{a}_{1}\\\ \
\hat{a}_{2}\end{pmatrix}.$ (7)

It should be added that the representation as Eq.(7) is a direct consequence
of the conservation of particles (photons) before and after their passage
through BS. If photons before hitting the BS were $s_{1}+s_{2}$ and after
passing through the BS became $n+m$, then for the number of photons to be
conserved $s_{1}+s_{2}=n+m$ requires that the transformation matrix be in the
form Eq. (7) [26, 19]. The conversion matrix $U_{BS}$ contains the elements
$U_{i,j}$ ($i,j=1,2$), which are generally complex and must be represented as
$U_{i,j}=|U_{i,j}|e^{i\phi_{i,j}}$. The well-known bosonic commutation
relations at the output of the BS must also hold, i.e.
$[\hat{b}_{i},{{\hat{b}_{j}}^{\dagger}}]=\delta_{i,j}$. This commutation
relation leads to the relations

$\displaystyle|U_{1,1}|^{2}+|U_{1,2}|^{2}=1,\leavevmode\nobreak\
\leavevmode\nobreak\ |U_{2,1}|^{2}+|U_{2,2}|^{2}=1,\leavevmode\nobreak\
\leavevmode\nobreak\ U_{1,1}U^{*}_{2,1}+U_{1,2}U^{*}_{2,2}=0.$ (8)

The last relation in Eq. (6) can be simplified by reducing to 2 equations, we
get

$\displaystyle|U_{1,1}||U_{2,1}|=|U_{1,2}||U_{2,2}|,\leavevmode\nobreak\
\leavevmode\nobreak\ \phi_{1,1}-\phi_{1,2}=\phi_{2,1}-\phi_{2,2}\pm\pi.$ (9)

As a result, from the first two ratios in Eq.(8) and from the first relation
in Eq.(9) it can be deduced that
$|U_{2,1}|^{2}=|U_{1,2}|^{2}=R=\sin^{2}\theta,\leavevmode\nobreak\
\leavevmode\nobreak\ |U_{1,1}|^{2}=|U_{2,2}|^{2}=T=\cos^{2}\theta$, where
$0\leq\theta\leq\pi/2$. In this case, one can see that the relation $R+T=1$.
From the point of view of physics, these coefficients $R$ and $T$ are the
coefficients of reflection and transmission, respectively. The four-phase
dependence can be reduced to a two-phase dependence by making phase
substitutions in the form:
$\phi_{\tau}=1/2(\phi_{11}-\phi_{22});\phi_{\rho}=1/2(\phi_{12}-\phi_{21}\mp\pi);\phi_{0}=1/2(\phi_{11}+\phi_{22})$.
In this case, the BS matrix Eq. (7) will be

$\displaystyle U_{BS}=e^{i\phi_{0}}\begin{pmatrix}\cos\theta
e^{i\phi_{\tau}}&\sin\theta e^{i\phi_{\rho}}\\\ \ -\sin\theta
e^{-i\phi_{\rho}}&\cos\theta e^{-i\phi_{\tau}}.\end{pmatrix}$ (10)

In Eq. (10) has only two phases left, since $\phi_{0}$ is the phase
multiplier, which is known to be negligible. We can see that by using the BS
matrix in the form of Eq. (10), the operators $\hat{b}_{1},\hat{b}_{2}$ can be
further simplified, we obtain:
$\hat{b}_{1}=e^{i\phi_{\tau}}(\cos\theta\hat{a}_{1}+\sin\theta
e^{i(\phi_{\rho}-\phi_{\tau})}\hat{a}_{2})$;
$\hat{b}_{2}=e^{-i\phi_{\tau}}(-\sin\theta
e^{-i(\phi_{\rho}-\phi_{\tau})}\hat{a}_{1}+\cos\theta\hat{a}_{2})$. Again we
have got irrelevant phase multipliers in front of the brackets, which can be
disregarded. Introducing the notion of phase difference
$\phi=\phi_{\rho}-\phi_{\tau}$, we obtain the BS matrix $U_{BS}$ in the form
Eq.(3).

Representation of the wave function at the BS output ports in the case of
monochromatic beam Eq. (3) and non-monochromatic Eq. (6) is based on the
number of photons in each mode of the electromagnetic field. Since the
electromagnetic field is two-mode, an alternative representation of Eqs. (3)
and (6) is possible, which is based not on the number of photons in each mode,
but on the angular momentum $L$, where also there are 2 quantum numbers, which
are the orbital momentum $l$ and the projection of orbital momentum on a
chosen axis $-l\leq m\leq l$ [19, 28]. For this we can introduce three new
operators ${\hat{L}}_{i}$ ($i=1,2,3$) which satisfy the commutation rules for
angular momentum
$[{\hat{L}}_{i},{\hat{L}}_{j}]=i\epsilon_{i,j,k}{\hat{L}}_{k}]$
($\epsilon_{i,j,k}$ is a Levi-Civita tensor), where

$\displaystyle{\hat{L}}_{1}=\frac{1}{2}\left({{\hat{a}_{1}}^{\dagger}}\hat{a}_{2}+{{\hat{a}_{2}}^{\dagger}}\hat{a}_{1}\right),$
$\displaystyle{\hat{L}}_{2}=\frac{1}{2i}\left({{\hat{a}_{1}}^{\dagger}}\hat{a}_{2}-{{\hat{a}_{2}}^{\dagger}}\hat{a}_{1}\right),$
$\displaystyle{\hat{L}}_{3}=\frac{1}{2}\left({{\hat{a}_{1}}^{\dagger}}\hat{a}_{1}-{{\hat{a}_{2}}^{\dagger}}\hat{a}_{2}\right).$
(11)

The square, and projection, of the angular momentum are related to the boson
number operators via

$\displaystyle{\hat{L}}^{2}=\sum^{3}_{i=1}{\hat{L}}^{2}_{i}={\hat{l}}({\hat{l}}+1),\leavevmode\nobreak\
\leavevmode\nobreak\ {\hat{L}}_{3}={\hat{m}},$
$\displaystyle{\hat{l}}=\frac{1}{2}\left({\hat{n}}_{1}+{\hat{n}}_{2}\right),\leavevmode\nobreak\
\leavevmode\nobreak\
{\hat{m}}=\frac{1}{2}\left({\hat{n}}_{1}-{\hat{n}}_{2}\right),$ (12)

where ${\hat{n}}_{1},{\hat{n}}_{2}$ are operators of the number of particles
(photons) in modes 1 and 2 respectively. The BS output ports will be
accordingly

$\displaystyle{\hat{L}}^{{}^{\prime}}_{1}=\frac{1}{2}\left({{\hat{b}_{1}}^{\dagger}}\hat{b}_{2}+{{\hat{b}_{2}}^{\dagger}}\hat{b}_{1}\right),$
$\displaystyle{\hat{L}}^{{}^{\prime}}_{2}=\frac{1}{2i}\left({{\hat{b}_{1}}^{\dagger}}\hat{b}_{2}-{{\hat{b}_{2}}^{\dagger}}\hat{b}_{1}\right),$
$\displaystyle{\hat{L}}^{{}^{\prime}}_{3}=\frac{1}{2}\left({{\hat{b}_{1}}^{\dagger}}\hat{b}_{1}-{{\hat{b}_{2}}^{\dagger}}\hat{b}_{2}\right).$
(13)

Thus, it is necessary to introduce a unitary operator ${\hat{U}}$ that is
defined by the expression
${\hat{L}}^{{}^{\prime}}_{i}={\hat{U}}{\hat{L}}_{i}{{\hat{U}}^{\dagger}}$.
Since this operator is responsible for the transformation of the angular
momentum, which is responsible for the symmetry during rotations, the general
form of such an operator should be of the form
${\hat{U}}=e^{-i\Phi{\hat{L}}_{3}}e^{-i\Theta{\hat{L}}_{2}}e^{-i\Psi{\hat{L}}_{3}}$,
where the parameters ($\Phi,\Theta,\Psi$) are quantum-mechanical counterparts
to the classical Euler angles. Each angle has its own physical meaning, which
we will try to define next. Using the Baker-Campbell-Hausdorff formula you can
get [19]

$\displaystyle\begin{pmatrix}\hat{b}_{1}\\\ \
\hat{b}_{2}\end{pmatrix}=U\begin{pmatrix}\hat{a}_{1}\\\ \
\hat{a}_{2}\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
U=\begin{pmatrix}\cos(\theta/2)e^{i(\Psi+\Phi)/2}&\sin(\theta/2)e^{i(\Psi-\Phi)/2}\\\
\ -\sin(\theta/2)e^{-i(\Psi-\Phi)/2}&\cos\theta
e^{-i(\Psi+\Phi)/2}.\end{pmatrix}$ (14)

Let’s compare the obtained Eq. (14) c Eq. (10), then we get (up to an
insignificant phase)

$\displaystyle\Theta=2\theta,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Phi+\Psi=2\phi_{\tau},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Psi-\Phi=2\phi_{\rho}.$ (15)

The result is

$\displaystyle{\hat{U}}=e^{-i(\phi_{\tau}-\phi_{\rho}){\hat{L}}_{1}}e^{-2iarccos\sqrt{1-R}\leavevmode\nobreak\
{\hat{L}}_{2}}e^{-i(\phi_{\tau}+\phi_{\rho}){\hat{L}}_{3}}.$ (16)

If we introduce step-up and step-down operators
${\hat{L}}_{\pm}={\hat{L}}_{1}\pm i{\hat{L}}_{2}$
(${\hat{L}}_{+}={{\hat{a}}^{\dagger}}_{1}{\hat{a}}_{2},\leavevmode\nobreak\
{\hat{L}}_{-}={{\hat{a}}^{\dagger}}_{2}{\hat{a}}_{1}$), then the operator
${\hat{U}}$ can be represented as

$\displaystyle{\hat{U}}={{\hat{D}}^{\dagger}}e^{-2i\phi_{\tau}{\hat{L}}_{3}},\leavevmode\nobreak\
\leavevmode\nobreak\ {\hat{D}}=e^{\zeta{\hat{L}}_{+}-\zeta^{*}{\hat{L}}_{-}},$
(17)

where $\zeta=arccos\sqrt{1-R}\leavevmode\nobreak\ e^{i\phi}$ (where
$\phi=\phi_{\rho}-\phi_{\tau}$), and the operators ${\hat{L}}_{\pm}$ have
known properties ${\hat{L}}_{\pm}|l,m\rangle=\\{l(l+1)-m(m\pm
1){\\}}^{1/2}|l,m\pm 1\rangle$. Eq.(17) can be further simplified by ignoring
the irrelevant phase factor which will give the operator
$e^{-2i\phi_{\tau}{\hat{L}}_{3}}$, since
$e^{-2i\phi_{\tau}{\hat{L}}_{3}}|l,m\rangle=e^{-2i\phi_{\tau}m}|l,m\rangle$,
then we get that ${\hat{U}}={{\hat{D}}^{\dagger}}$. As a result, using Eq.(17)
we can find the wave function on the BS output ports
$|\Psi_{out}\rangle={\hat{U}}^{\dagger}|\Psi_{in}\rangle$.

This result is quite interesting scientifically because without solving the
Schrodinger equation we can find the wave function $|\Psi_{out}\rangle$ at the
BS output ports knowing $R$ and phase shift $\phi$. Moreover, we are not
interested in the type of BS and this solution is suitable for all two-port
linear BS. Of course, with such a general consideration, the “nature” of the
appearance of coefficients $R$ and $\phi$ in quantum optics is not clear,
since they have been introduced as some constants having the properties of
these coefficients. In spite of this, the “nature” of the appearance of these
coefficients is well studied in classical optics for various types of BS. For
different types of BS, these coefficients can be calculated using classical
optics, where the dependence $R=R(\lambda)$ (or in the frequency
representation $R=R(\omega)$) appears. It should be added that the phase shift
$\phi$ does not affect the intensity at the BS output ports, so it may not be
considered. For example, for a cubic BS, which is often used in quantum optics
experiments, the dependence can be represented in Fig. 2.

Figure 2: A sample of an non-polarizing cubic beam splitter (50:50), with an
operating wavelength range of (400 - 700) nm, BS dimensions of 5 mm, is
presented. The figures on the right show the dependence of reflection
coefficient $R$ and transmittance $T$ as a function of wavelength $\lambda$.
BS fabrication accuracy: $T=47\pm 10\%$, $R=47\pm 10\%$, $R+T>90\%$ [29].

It should be added that to find the wave function $\Psi_{out}$, taking into
account the non-monochromaticity of photons, using Eq. (6) there is a rather
interesting case, which is not discussed in the textbook and review
literature, but which is fundamentally important for waveguide BS. Non-
monochromatic photons are known to have some frequency distribution
$\phi(\omega_{1},\omega_{2})$, i.e. the frequencies $\omega_{1},\omega_{2}$
are “blurred” and the amount of blurring is determined by the variance
$\sigma$. It should be added that the optical sources (including single photon
sources) that are commonly used are such that $\sigma\ll\omega$. In order to
find $\Psi_{out}$ it is necessary to integrate over the frequencies
$\omega_{1},\omega_{2}$, see Eq. (6). With this integration, the coefficients
$R$ and $T$ can be assumed to be constant, since
$\\{R(\omega\pm\sigma)-R(\omega)\\}\to 0$. This is well seen from Fig. (2),
where by changing $\lambda$ by a small value $\Delta\lambda\ll\lambda$, the
coefficients $R$ and $T$ will also change by a small value (i.e. $\Delta R\ll
R$), whose change can be neglected. So by choosing constant values of $R$ and
$T$ we do not make a mistake when $R$ and $T$ are smooth (no jumps) functions
with respect to frequencies. Of course, it is reasonable to use such smooth
functions and at first sight there can be no other options. As will be shown
in the next section, that such functions appear in the case of waveguide BS
and this will significantly affect the statistical properties of photons at
the BS output ports. Moreover, quantum entanglement of photons will strongly
depend on it, which certainly changes the relation to the waveguide BS as a
source of quantum entangled photons. Thus, in this review, BSs where $R$ and
$\phi$ are assumed to be constant will be considered, which will be called
“conventional” beam splitters, and where $R$ and $\phi$ are not constant,
called frequency-dependent BSs. So far, there is information in the literature
about only one frequency-dependent type of BS, which is the waveguide BS.

###  2.2 “Conventional” waveguide beam splitter

It should be added that all the expressions obtained above for “conventional”
BS, are suitable for waveguide BS. Nevertheless, we will show how in theory
two coupled waveguides begin to exhibit BS properties. Consider two waveguides
connected (coupled) to each other in a certain area of space, see Fig.3.

Figure 3: 3D representation of coupled waveguides, where $\Psi_{in}$ and
$\Psi_{out}$ are the input and output (recorded by $D_{1}$ and $D_{2}$
detectors) wave function of photons, respectively. (b): Schematic 2D
representation of such a BS, see [22].

The theory of coupled waveguides is now quite well developed [30]. This theory
is based on coupling coefficients between neighbouring waveguides. It is
generally believed that by setting these coupling coefficients one can
regulate the photon states at the output ports of the waveguides. In fact,
this theory is applicable for identical and monochromatic photons incident on
incoming ports of waveguides. Or as an alternative representation: coupling
coefficients depend on frequencies but how they do not matter since these
coefficients are given by constants. As it has been shown recently [24, 25,
21, 17, 22], on the basis of exact quantum-mechanical consideration, that
states of photons at output ports of waveguides have strong dependence on
their frequencies and to set coupling coefficients by constant values is not
always correct and it can carry large errors. Waveguide theory (more exactly
it is a BS theory) under exact quantum-mechanical consideration in the
particular case of monochromatic and identical photons turns into a waveguide
theory with constant coefficients. Let us first introduce here the theory of
waveguide BS based on constant coupling coefficients.

It is well known that for coupled waveguides one can compose a system of two
equations [30]

$\displaystyle\frac{d\hat{a}_{1}}{dz}=-i(\beta_{1}+K_{11})\hat{a}_{1}-iK_{12}\hat{a}_{2},$
$\displaystyle\frac{d\hat{a}_{2}}{dz}=-i(\beta_{2}+K_{22})\hat{a}_{2}-iK_{21}\hat{a}_{1},$
(18)

where $\beta_{1},\beta_{2}$ these are the propagation constants for modes 1
and 2 respectively; $K_{12},K_{21}$ and $K_{11},K_{22}$ are the mutual and the
self coupling coefficients, respectively; $z$ is the coordinate at which the
field characteristics are determined. Moreover, if the coupled waveguide
system is lossless, self-consistency requires that the Eq. (18) satisfy the
power conservation law, i.e. at the input and output ports the number of
photons does not change. Because Eq. (18) is independent of the initial
condition, the coupling coefficients obey the following relations
$K_{12}=K^{*}_{21}=\kappa$ and $K_{11},K_{22}$ have to be real. The solution
of Eq.(18) is conveniently found in the form
$\hat{a}_{i}\to\hat{a}_{i}e^{-i/2(\beta_{1}+\beta_{2}+K_{11}+K_{22})z}$. In
this form, the coordinate $z$ will have the meaning of the waveguides coupling
region length. Using this form we obtain

$\displaystyle\frac{d\hat{a}_{1}}{dz}=-i\delta\hat{a}_{1}-i\kappa\hat{a}_{2},$
$\displaystyle\frac{d\hat{a}_{2}}{dz}=+i\delta\hat{a}_{2}-i\kappa\hat{a}_{1},$
(19)

where $\delta=1/2(\beta_{1}-\beta_{2}+K_{11}-K_{22})$. In this form, solve Eq.
(19) is simple enough, eg [30]. If we adopt new notation in the form
$\hat{a}_{i}(0)=\hat{a}_{i}$ and $\hat{a}_{i}(z)=\hat{b}_{i}$, then we get

$\displaystyle\begin{pmatrix}\hat{b}_{1}\\\ \
\hat{b}_{2}\end{pmatrix}=\begin{pmatrix}u_{11}&u_{12}\\\ \
u_{21}&u_{22}\end{pmatrix}\begin{pmatrix}\hat{a}_{1}\\\ \
\hat{a}_{2}\end{pmatrix},$ (20)

where $u_{11}=u^{*}_{22}=\cos(Sz)-i\cos\eta\sin(Sz)$ and
$t_{12}=t_{21}=-i\sin\eta\sin(Sz)$, and the parameters
$S=\sqrt{\delta^{2}+\kappa^{2}},\leavevmode\nobreak\ \tan\eta=\kappa/\delta$.
On ports 1 and 2 output, the probabilities of detecting photons would be,
respectively
$P_{1}=\cos^{2}(Sz)+\cos^{2}\eta\sin^{2}(Sz),P_{2}=\sin^{2}\eta\sin^{2}(Sz)$.
The maximum probability value on the 2nd output port is
$(P_{2})_{max}=\sin^{2}\eta$. If photons propagate in waveguides without loss
and the number of photons is conserved, then $(P_{2})_{max}=1$, which means
$\eta=\pm\pi/2$. In this case we will get

$\displaystyle\begin{pmatrix}\hat{b}_{1}\\\ \
\hat{b}_{2}\end{pmatrix}=\begin{pmatrix}\cos(\kappa z)&\pm i\sin(\kappa z)\\\
\ \pm i\sin(\kappa z)&\cos(\kappa z)\end{pmatrix}\begin{pmatrix}\hat{a}_{1}\\\
\ \hat{a}_{2}\end{pmatrix}.$ (21)

If we put $\sqrt{T}=\cos(\kappa z)$, then the result is in Eq. (21) is the
same as the beam splitter, see Eq.(4). In this consideration, we have not only
shown that coupled waveguides have BS property, but also found in simple form
reflections $R=\sin^{2}(\kappa z)$ [14].

Obviously, the case considered here is based on coupling coefficients, but
does not reveal the “nature” of the appearance of these coefficients.
Moreover, it is quite obvious, even by analogy with “conventional” BS, that
there must be a dependence of reflection coefficients $R$ on frequencies of
the incident field, i.e. $R=R(\omega)$. In order to find this dependence it is
necessary to consider a quantum mechanical system in which two quantized modes
of electromagnetic field interact with atoms of matter in coupled waveguides.
Such a consideration was made recently in the papers [24, 25, 21, 17, 22] and
the dependence not only of $R=R(\omega)$, but also of the phase shift
$\phi=\phi(\omega)$ was found. Let us now turn to this theory.

###  2.3 Frequency-dependent waveguide beam splitter

Consider two coupled waveguides, along each of which photons propagate. These
photons interact with atomic electrons. Let us represent the electromagnetic
field of photons through the transverse vector potential ${\bf A}$ in the
Coulomb gauge $div{\bf A}=0$ [1, 5], then the Hamiltonian of such system will
be

$\displaystyle\left\\{{\hat{H}}_{1}+{\hat{H}}_{2}+\frac{1}{2}\sum_{a}\left({\hat{\bf
p}}_{a}+\frac{1}{c}\hat{\bf A}_{a}\right)^{2}+\sum_{a}U({\bf
r}_{a})\right\\}\Psi=i\frac{\partial\Psi}{\partial t},$ (22)

where ${\hat{H}}_{i}=\omega_{i}\hat{a_{i}}^{+}\hat{a_{i}}$ is the Hamiltonian
operator for the first ($i=1$) and second ($i=2$) modes ($\omega_{i}$ is the
frequency and $\hat{a_{i}}$ is the photon annihilation operator in mode number
$i$); $U({\bf r}_{a})$ is the atomic potential acting on the electron with
number $a$; ${\hat{\bf p}}_{a}$ is the momentum operator of the electron with
number $a$; $\hat{\bf A}_{a}=\hat{\bf A}_{1,a}+\hat{\bf A}_{2,a}$, where
$\hat{\bf A}_{i,a}=\sqrt{\frac{2\pi c^{2}}{\omega_{i}V}}{\bf
u}_{i}(\hat{a_{i}}^{+}+\hat{a_{i}})$ is the vector potential in dipole
approximation for mode $i$, acting on an electron with number $a$ ($c$ is the
speed of light, $V$ is the modal volume, i.e. it is the Fourier transform of
the vector potential ${\bf A}$ enclosed in a finite volume $V$, ${\bf u}_{i}$
is the polarization of the mode numbered $i$) [1, 5]; the sum $\sum_{a}$ in
Eq.(22) is satisfied over all electrons of the two coupled waveguides
interacting with photons. As a result, the Hamiltonian of equation (22) will
be

$\displaystyle{\hat{H}}=\sum^{2}_{i=1}\left\\{\omega_{i}\hat{a_{i}}^{+}\hat{a_{i}}+\overline{N}\frac{\beta^{2}_{i}}{2}q^{2}_{i}+\beta_{i}q_{i}{\bf
u}_{i}\sum_{a}{\hat{\bf p}_{a}}\right\\}+$
$\displaystyle\overline{N}\beta_{1}\beta_{2}q_{1}q_{2}{\bf u}_{1}{\bf
u}_{2}+\sum_{a}\frac{{\hat{\bf p}}^{2}_{a}}{2}+\sum_{a}U({\bf r}_{a}),$ (23)

where $\beta_{i}=\sqrt{\frac{4\pi}{\omega_{i}V}}$, and the quantity
$\overline{N}=\sum_{a}$ is the number of electrons involved in the interaction
with photons in two coupled waveguides. As can be seen, equation (23)
corresponds to the equation for coupled harmonic oscillators interacting with
electrons of a polyatomic system. A similar system was considered in [21], we
obtain

$\displaystyle{\hat{H}}=\sum^{2}_{i=1}\left\\{\frac{\sqrt{A_{i}}}{2}\left({\hat{P}}^{2}_{i}+y^{2}_{i}\right)+{\bf
D}_{i}y_{i}\sum_{a}{\hat{\bf p}_{a}}\right\\}+\sum_{a}\frac{{\hat{\bf
p}}^{2}_{a}}{2}+\sum_{a}U({\bf r}_{a}),$ (24)

where $y_{1}=A^{1/4}_{1}\left(q_{1}/\sqrt{\omega_{1}}\cos\alpha-
q_{2}/\sqrt{\omega_{2}}\sin\alpha\right)$ and
$y_{2}=A^{1/4}_{2}\left(q_{1}/\sqrt{\omega_{1}}\sin\alpha+q_{2}/\sqrt{\omega_{2}}\cos\alpha\right)$
is new variables; ${\hat{P}}_{i}=-i\partial/\partial y_{i}$; ${\bf
D}_{1}=\beta_{1}A^{-1/4}_{1}\sqrt{\omega_{1}}{\bf
u}_{1}\cos\alpha-\beta_{2}A^{-1/4}_{1}\sqrt{\omega_{2}}{\bf u}_{2}\sin\alpha$;
${\bf D}_{2}=\beta_{1}A^{-1/4}_{2}\sqrt{\omega_{1}}{\bf
u}_{1}\sin\alpha+\beta_{2}A^{-1/4}_{2}\sqrt{\omega_{2}}{\bf u}_{2}\cos\alpha$.
The study [31] showed that $\tan(2\alpha)=C/(B_{2}-B_{1})$, where
$C=2\overline{N}\beta_{1}\beta_{2}\sqrt{\omega_{1}\omega_{2}}{\bf u}_{1}{\bf
u}_{2}$; $B_{i}=(\omega_{i}+\overline{N}\beta^{2}_{i})\omega_{i}$ and
$A_{i}=B_{i}+(-1)^{i}C/2\tan\alpha$. Obviously, the value of $\beta_{i}$ is
very small in the case of single-photon interaction, see, eg [32], where
$\beta^{2}/\omega\ll 1$, even in the case of strong focusing. In this case,
the quantities ${\bf D}_{1}$ and ${\bf D}_{2}$ are negligible. This is an
obvious fact, since these quantities are responsible for various inelastic
transitions of electrons in an atom under the action of photons, which are
usually negligible in lossless BS. As a result, the dynamics of two photons in
BS will be described by the wave function

$\displaystyle|\Psi(t_{BS})\rangle=e^{-i{\hat{H}_{BS}}t_{BS}}|\Psi(0)\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{H}_{BS}=\sum^{2}_{i=1}\frac{\sqrt{A_{i}}}{2}\left\\{{\hat{P}}^{2}_{i}+y^{2}_{i}\right\\},$
(25)

where $t_{BS}$ is the photon interaction time in BS and $|\Psi(0)\rangle$ is
the initial state of the photons before entering the BS. It should be added
that the representation of the solution to the general Eq. (22) in the form
Eq. (25) allows us to determine in an analytical form the evolution of the
operators $\hat{a_{1}}$ and $\hat{a}_{2}$ in Eq.(3). Often, to calculate the
required quantities, one needs electric field operators ${\hat{E}}_{1}(t_{1})$
and ${\hat{E}}_{2}(t_{2})$ at time instants $t_{1}$ and $t_{2}$ on the first
and second detectors, respectively. To do this, we need to find the evolution
(in BS, as well as from BS to detectors) of the operators ${\hat{E}}_{01}(0)$
and ${\hat{E}}_{02}(0)$ for modes 1 and 2, respectively

$\displaystyle{\hat{E}}_{1}(t_{1})=e^{i{\hat{H}_{0}}t_{1}}e^{i{\hat{H}_{BS}}t_{BS}}{\hat{E}}_{01}(0)e^{-i{\hat{H}_{BS}}t_{BS}}e^{-i{\hat{H}_{0}}t_{1}},$
$\displaystyle{\hat{E}}_{2}(t_{2})=e^{i{\hat{H}_{0}}t_{2}}e^{i{\hat{H}_{BS}}t_{BS}}{\hat{E}}_{02}(0)e^{-i{\hat{H}_{BS}}t_{BS}}e^{-i{\hat{H}_{0}}t_{2}},$
(26)

where ${\hat{H}_{0}}=\sum^{2}_{i=1}\omega_{i}\hat{a_{i}}^{\dagger}\hat{a_{i}}$
is the Hamiltonian of photons outside of BS. Because
${\hat{E}}_{01}(0)\propto{\hat{a}}_{1}$, and
${\hat{E}}_{02}(0)\propto{\hat{a}}_{2}$ (see, eg [1, 5]), it is more
convenient to consider not the electric field operators, but the photon
creation and annihilation operators before entering BS (${\hat{a}}_{1}$ and
${\hat{a}}_{2}$) and on the detectors (${\hat{b}}_{1}$ and ${\hat{b}}_{2}$).
To this end, we replace
${\hat{E}}_{01}(0)\to{\hat{a}}_{1},{\hat{E}}_{02}(0)\to{\hat{a}}_{2}$ and
${\hat{E}}_{1}(t_{1})\to{\hat{b}}_{1}(t_{1}),{\hat{E}}_{2}(t_{2})\to{\hat{b}}_{2}(t_{2})$.
Including into account the time delay $\delta\tau$ for the spatial
displacement of BS from the equilibrium position and the time delay $\tau$
between 1 and 2 mods (see Fig. 4), we get (see supplementary materials in
[24])

$\displaystyle{\hat{b}}_{1}(t_{1})=\sqrt{T}e^{-i\omega_{1}(t_{1}-\tau)}{\hat{a}}_{1}+e^{i\phi}\sqrt{R}e^{-i\omega_{2}(t_{1}+\delta\tau/2)}{\hat{a}}_{2},$
$\displaystyle{\hat{b}}_{2}(t_{2})=\sqrt{T}e^{-i\omega_{2}t_{2}}{\hat{a}}_{2}-e^{-i\phi}\sqrt{R}e^{-i\omega_{1}(t_{2}-\delta\tau/2-\tau)}{\hat{a}}_{1},$
(27)

where

$\displaystyle R=\frac{\sin^{2}\left(\Omega
t_{BS}/2\sqrt{1+\varepsilon^{2}}\right)}{(1+\varepsilon^{2})};\leavevmode\nobreak\
\cos\phi=-\varepsilon\sqrt{\frac{R}{T}},$ $\displaystyle
T=1-R,\leavevmode\nobreak\ \leavevmode\nobreak\
\Omega=\frac{8\pi\overline{N}{\bf u}_{1}{\bf
u}_{2}}{(\omega_{1}+\omega_{2})V};\leavevmode\nobreak\
\varepsilon=\frac{\omega_{2}-\omega_{1}}{\Omega}.$ (28)

Figure 4: Schematic representation of a waveguide BS, where $D_{1},D_{2}$ are
first and second detectors respectively; $\tau$ is the time delay between
modes 1 and 2, and $\delta\tau$ is the time delay caused by the spatial shift
of the BS from its equilibrium position.

From (27) it can be seen that the matrix BS that is $U_{BS}$ completely
corresponds to the matrix Eq.(3) (needless to say, for $t_{1}=t_{2}=t$ and for
$\delta\tau=\tau=0$). In addition, $T$ and $R$ are symmetric, i.e. if we
change the first to the second mode $\omega_{1}\to\omega_{2},{\bf
u}_{1}\to{\bf u}_{2}$ and vice versa, then, as anticipated, the coefficients
will not alter. If we assume that photons are completely identical and
monochromatic then $\varepsilon=0$ (in reality this does not happen) and we
obtain the “conventional” waveguide BS described above (at
$\kappa=\Omega/(2v)$, $z=vt_{BS}$, where $v$ is the wave speed in the
waveguide). This means that this particular case will fully coincide with
Eq.(21). It can be seen that the theory presented here is general, where the
frequency dependencies of the reflection coefficient $R$ and phase shift
$\phi$ are found.

From Eq. (28) it can be seen that there is a new quantity characterizing BS
and it is $\Omega$. It can be qualitatively estimated if we take into account
that most of quantum effects are observed at
$\omega_{2}-\omega_{1}\ll\omega_{1},\omega_{2}$. We will then be interested in
the frequencies at $\omega_{2}\approx\omega_{1}$; then $\Omega=4\pi
n/\omega_{0}{\bf u}_{1}{\bf u}_{2}$, where
$\omega_{0}=(\omega_{2}+\omega_{1})/2$, $n=\overline{N}/V$. In this
representation $\omega_{0}$ can be thought of as a constant and $n$ as the
effective concentration. Moreover, even if the condition
$\omega_{2}-\omega_{1}\ll\omega_{1},\omega_{2}$ is not met, and
$\omega_{2}-\omega_{1}\sim\omega_{1},\omega_{2}$, then $\omega_{0}$ can be
considered approximately constant when integrated over frequencies in Eq. (6).
Note that $n$ is a characteristic of the waveguide BS, since it determines the
coupling of effectively interacting $\overline{N}$ electrons to photons in the
volume $V$. For example, if we consider the BS as a solid in which photons
interact in a volume $V$, then $n$ is the concentration of electrons in the
solid. Since it is known that the concentration of atoms in a solid does not
vary much, we get $\Omega\sim(10^{14}-10^{17})rad/s$ for the optical
frequencies ($\omega_{0}\approx 10^{15}rad/s$) in this case (if of course
${\bf u}_{1}{\bf u}_{2}\sim 1$). It should be added that for this type of BS,
the frequency $\Omega$ can be represented by the known value for plasma
frequency $\omega_{p}$, then $\Omega={\bf u}_{1}{\bf
u}_{2}\omega^{2}_{p}/\omega_{0}$. In a waveguide BS, the number of effectively
interacting electrons $\overline{N}$ in the volume $V$ is smaller than in a
solid, hence $\Omega$ will be smaller. In waveguide BS one can adjust $\Omega$
by changing the effective interaction between photons.

From Eq. (28) we can see that $R=R(\omega_{1},\omega_{2})$ and
$\phi=\phi(\omega_{1},\omega_{2})$, i.e. depends on frequencies of 1 and 2
modes. This is the main difference between waveguide BS and the other type of
waveguide, where the reflection coefficient $R$ depends on the frequency of
one mode. This result causes the reflection coefficient $R$ to become
“sensitive” to changes in any of the frequencies in the mode. Considering that
the $\Omega$ parameter is quite small, the dependence of the reflection
coefficient $R$ on the frequency can have a dramatic change. In other words,
such a dependence has a resonance character at
$(\omega_{2}-\omega_{1})\sim\Omega$. Indeed, by changing
$R(\omega_{1}\pm\sigma_{1},\omega_{2}\pm\sigma_{2})$ the reflection
coefficient value will change greatly at
$\sigma_{1}\sim\Omega,\sigma_{2}\sim\Omega$, where $\sigma$ is the variance
that was described above. Otherwise, if
$\sigma_{1}\ll\Omega,\sigma_{2}\ll\Omega$, then photons in the modes can be
considered monochromatic and the resonance dependence
$R(\omega_{1},\omega_{2})$ disappears, which means in this case the reflection
coefficient $R$ can be considered constant and such waveguide BS becomes
“conventional”[17].

##  3 Quantum entanglement of photons on a beam splitter

As is well known, quantum entanglement is a phenomenon where the quantum state
of each group of particles cannot be described independently of the state of
the others, including when the particles are separated by a large distance.
Quantum entanglement is now one of the most important research topics [33]. In
particular, quantum communication protocols such as quantum cryptography [34],
quantum dense coding [35], quantum computing algorithms [36] and quantum state
teleportation [37, 38] can be explained by entangled states. One of the
promising prospects for using quantum entanglement is to build a quantum
computer that will surpass modern classical computers by many orders of
magnitude. Moreover, the beam splitter could be used as a source of quantum
entangled photons and be the basic building block of a quantum computer [6].
Without going into details of this topic, it is worth saying that a rather
serious problem is the calculation of quantum entanglement for multiparticle
systems, where different measures of quantum entanglement [33], such as
Concurrence, Negativity, etc., are used. For calculations of two-component
systems, the quantum entanglement calculation is simpler and is based on well-
known measures of quantum entanglement - these are von Neumann entropy,
Schmidt parameter, etc. [33].

It is known that by Schmidt’s theorem [39, 40] the wave function
$|\Psi(t)\rangle$ of interacting systems 1 and 2 can be decomposed as
$|\Psi(t)\rangle=\sum_{k}\sqrt{\lambda_{k}(t)}|u_{k}(x_{1},t)\rangle|v_{k}(x_{2},t)\rangle$,
where $|u_{k}(x_{1},t)\rangle$ is the pure state wave function of system 1 and
$|v_{k}(x_{2},t)\rangle$ is the pure state wave function of system 2. Where
$\lambda_{k}$ is the Schmidt mode, which is the eigenvalue of the reduced
density matrix, i.e.
$\rho_{1}(x_{1},x^{{}^{\prime}}_{1},t)=\sum_{k}\lambda_{k}(t)u_{k}(x_{1},t)u^{*}_{k}(x^{{}^{\prime}}_{1},t)$
or
$\rho_{2}(x_{2},x^{{}^{\prime}}_{2},t)=\sum_{k}\lambda_{k}(t)v_{k}(x_{2},t)v^{*}_{k}(x^{{}^{\prime}}_{2},t)$.
If we find the Schmidt mode $\lambda_{k}$, we can calculate the quantum
entanglement of the system. To do this, we can use various measures of quantum
entanglement, such as the Schmidt parameter [39, 40]
$K=\left(\sum_{k}\lambda^{2}_{k}\right)^{-1}$ or von Neumann entropy [41, 42]
$S_{N}=-\sum_{k}\lambda_{k}\ln\left(\lambda_{k}\right)$. The main difficulty
in calculating quantum entanglement is to find $\lambda_{k}$ of the system in
question.

###  3.1 Quantum entanglement on a “conventional” beam splitter

Quantum entanglement on “conventional” BS is quite well studied and presented
in various works, see eg [20, 13, 21, 43, 44, 45, 46]. Let us present here the
simplest derivation of the quantum entanglement of the two-mode
electromagnetic field on BS. The initial states of photons $|\Psi_{in}\rangle$
in BS ports 1 and 2 will be chosen as Fock states. Consideration of other
initial states can be found in many works, but the basis of all works are Fock
states. In this case, from Eq.(2) we obtain

$\displaystyle|\Psi_{in}\rangle=\frac{1}{\sqrt{s_{1}!s_{2}!}}{{\hat{a}_{1}}^{\dagger}}{}^{s_{1}}{{\hat{a}_{2}}^{\dagger}}{}^{s_{2}}|0\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\
|\Psi_{out}\rangle=\frac{1}{\sqrt{s_{1}!s_{2}!}}{{\hat{b}_{1}}^{\dagger}}{}^{s_{1}}{{\hat{b}_{2}}^{\dagger}}{}^{s_{2}}|0\rangle.$
(29)

To find $\Psi_{out}$ you can use Eq.(17), as was done in [20] using the [19]
approach. This approach is not the easiest to calculate and analyse quantum
entanglement. The simplest approach has been proposed with [21], using the
[47] approach. As a result, it was obtained
$\lambda_{k}(R)=\left|c_{k,s_{1}+s_{2}-k}\right|^{2}$, where

$\displaystyle
c_{k,p}=\sum^{s_{1}+s_{2}}_{n=0}A^{s_{1},s_{2}}_{n,s_{1}+s_{2}-n}A^{*{k,p}}_{n,s_{1}+s_{2}-n}e^{-2in\leavevmode\nobreak\
{\arccos\left(\sqrt{1-R}\sin\phi\right)}},$ $\displaystyle
A^{k,p}_{n,m}=\frac{\mu^{k+n}\sqrt{m!n!}}{(1+\mu^{2})^{\frac{n+m}{2}}\sqrt{k!p!}}P^{(-(1+m+n),m-k)}_{n}\left(-\frac{2+\mu^{2}}{\mu^{2}}\right),$
$\displaystyle\mu=\sqrt{1+\frac{1-R}{R}\cos^{2}\phi}-\cos\phi\sqrt{\frac{1-R}{R}},$
(30)

where $P^{\alpha,\beta}_{\gamma}(x)$ are Jacobi polynomials, $k$ and $p$ are
the number of photons at the output ports 1 and 2 of the BS respectively, with
$k+p=s_{1}+s_{2}$. In the Eq.(30), no matter what value $\phi\in(0,\pi/2)$ we
choose, the value $\lambda_{k}(R)$ will not depend on $\phi$. This amazing
property of the Eq. (30) can be used depending on the tasks under
consideration. For example, if we choose $\phi=0$, then when summing by $n$ in
(30), the exponent is replaced by $(-1)^{n}$. if we choose $\phi=\pi/2$, then
all the dependence on the reflection coefficient $R$ will be concentrated in
the exponent. In this way, the relationship of quantum entanglement with the
reflection coefficient $R$ was found. Figure 5 shows the dependence of quantum
entanglement for the Von-Neumann entropy $S_{N}$ and the Schmidt parameter
$K$, depending on the reflection coefficient $R$.

Figure 5: The dependence of the von Neumann entropy $S_{N}$ in Figures (a) and
(b), and the Schmidt parameter in Figures (c) and (d) as a function of $R$ are
presented. The figures show the dependencies ($s_{1},s_{2}$) for different
values of the number of photons in 1 and 2 modes, respectively. For example,
when $s_{1}=1$ and $s_{2}=6$, notations (1,6) are introduced.

For example, for $s_{1}=1$ and $s_{2}=1$, quantum entanglement will be in the
form

$\displaystyle S_{N}=-(1-2R)^{2}\ln(1-2R)^{2}-4R(1-R)\ln\left(2R(1-R)\right),$
$\displaystyle K=\frac{1}{1-8R(1-R)(1-3R(1-R))}.$ (31)

For Eq. (31) it is interesting to find the value $R$ at which there will be
the maximum quantum entanglement and this value $R=1/2(1\pm 1/\sqrt{3})$. At
this value $R$, the quantum entanglement will be $S_{N}=\ln 3$ and $K=3$. This
is quite an interesting result, since at first glance it seems that the
maximum entanglement should be at $R=1/2$. Fig. 5 shows that quantum
entanglement strongly depends on the reflection coefficient $R$, but is always
zero at $R=0,1$. The larger the quantum numbers $s_{1},s_{2}$, the greater the
quantum entanglement. You can see that there are two maxima quantum
entanglement for different pairs $s_{1},s_{2}$, except for the case when one
of the quantum numbers is zero. No matter what measure of quantum entanglement
we use, all dependencies are similar.

We also present quantum entanglement for various values of $s_{1}=s_{2}=s$ in
Fig. 6. The calculations are given for $R\in(0,1/2)$, since for symmetric
values of $R\in(1/2,1)$, the results will be the same.

Figure 6: The dependence of the von Neumann $S_{N}$ entropy is shown in Fig.
(a), as well as the Schmidt parameter in Fig. (b) for various values of $R$
and $s$.

Figure 6 shows that the quantum entanglement at $s_{1}=s_{2}=s$ increases
significantly with increasing $s$. For the Schmidt parameter $K$ , this
dependence is close to linear. Besides being able to obtain simple expressions
for quantum entanglement, other physical characteristics can be calculated
using equation (30). For example, if one of the quantum numbers is zero,
choose $s_{2}=0$, then in a very simple form one can find the average of the
quantum numbers $k$ and $p$ (see Eq. (30)). The result is
$\overline{k}=\sum_{k}k\lambda_{k}(R)=s_{1}(1-R)$ and
$\overline{p}=\sum_{p}p\lambda_{p}(R)=s_{1}R$.

One of the interesting applications of the results obtained is the possibility
of obtaining the wave functions $\Psi_{out}$ Eq. (29) that have practical
applications by specifying the coefficient $R$. For example, we need a wave
function $\Psi(t)$ that defines the states of Holland-Burnett (HB)[48]. It is
well known that this wave function is of great interest in various fields of
physics, for example, in quantum metrology [49, 11]. To do this, we need to
select $R=1/2$ and $s_{1}=s_{2}=s$ (for even values $s$). As a result, we get
the wave function $\Psi_{out}$ in the form

$\displaystyle\Psi=\sum^{s}_{n=0}e^{2in\phi}\frac{\sqrt{(2n)!(2s-2n)!}}{2^{s}n!(s-n)!}|2n,2s-2n\rangle.$
(32)

It should be added that using the Eq. (32) one can obtain an expression for
quantum entanglement using the Schmidt parameter $K$ in the form [21]

$\displaystyle
K=\frac{\pi(s!)^{2}}{\Gamma(s+1/2)^{2}{{}_{4}}F_{3}(1/2,1/2,-s,-s;1,1/2-s,1/2-s;1)},$
(33)

where $\Gamma(x)$ is the gamma function,
${{}_{4}}F_{3}(x_{1},x_{2},x_{3},x_{4};y_{1},y_{2},y_{3};1)$ is the
generalized hypergeometric function. It should be added that the Eq.(33) has a
fairly simple approximation $K=s^{0.897}$. This clearly shows that quantum
entanglement is unbounded from above, which was well known earlier (the more
$s$, the more the quantum entanglement).

If we consider the case when in the initial state $\Psi_{in}$ the number of
photons in 1 mode $s_{1}$ and in the second one $s_{2}=0$. In this case the
expression for quantum entanglement in the form of the Schmidt parameter will
be

$\displaystyle
K=\frac{1}{(1-R)^{2s_{1}}{{}_{2}F}_{1}\left(-s_{1},-s_{1};1;\left(\frac{R}{1-R}\right)^{2}\right)},$
(34)

where ${{}_{2}F}_{1}(a,b;c;x)$ is Gaussian hypergeometric function. The
general dependence of the Eq. (34) can be seen from Fig.5 (b, d). Also,
analyzing the Eq.(34), you can get that the maximum of this function at
$R=1/2$. With this value of $R=1/2$, one can obtain a simpler expression for
quantum entanglement

$\displaystyle K_{max}=2^{2s_{1}}\frac{(s_{1}!)^{2}}{(2s_{1})!}.$ (35)

You can also find from the Eq. (35) the parameter $K$ for large values of the
quantum number $s_{1}$, we get $K_{max}(s_{1}\gg 1)\to\sqrt{\pi s_{1}}$. It
can be seen in this case that the quantum entanglement is unbounded from above
(the more $s_{1}$, the more the quantum entanglement).

It should be added that the cases of quantum entanglement given here using BS
are only some cases, in reality there are many more.

###  3.2 Quantum entanglement on a waveguide beam splitter

Quantum entanglement on a frequency-dependent waveguide BS was recently
considered in [22, 23]. In these papers, it was shown that quantum
entanglement would be defined in the same way as in the “Conventional”
waveguide BS, with the only difference that the Schmidt mode $\lambda_{k}$
must be replaced by $\lambda_{k}\to\Lambda_{k}$, where

$\displaystyle\Lambda_{k}=\int|\phi(\omega_{1},\omega_{2})|^{2}\lambda_{k}(R)d\omega_{1}d\omega_{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \lambda_{k}(R)=\left|c_{k,s_{1}+s_{2}-k}\right|^{2}.$
(36)

We will assume that there is no quantum entanglement of photons at the BS
input ports. In other words, we will consider the incoming photon states as
Fock states, but photons are not monochromatic. In this case, as is well
known, the photon wave function is factorizable, i.e.
$\Psi_{in}=\int\phi_{1}(\omega_{1})|s_{1}\rangle
d\omega_{1}\int\phi_{2}(\omega_{2})|s_{2}\rangle d\omega_{2}$, where
$\phi(\omega_{1},\omega_{2})=\phi_{1}(\omega_{1})\phi_{2}(\omega_{2})$. To
analyse quantum entanglement we will use von Neumann entropy
$S_{N}=-\sum_{k}\Lambda_{k}\ln\left(\Lambda_{k}\right)$. Next, let’s choose
$\phi_{i}(\omega_{i})$ ($i=1,2$) in the most commonly used form, this is the
Gaussian distribution

$\displaystyle\phi_{i}(\omega_{i})=\frac{1}{(2\pi)^{1/4}\sqrt{\sigma_{i}}}e^{-\frac{(\omega_{i}-\omega_{0i})^{2}}{4\sigma^{2}_{i}}},$
(37)

where $\omega_{0i}$ is the mean frequency and $\sigma_{i}^{2}$ is the
dispersion. Then we will assume that $\omega_{0i}/\sigma_{i}\gg 1$, which is
applicable to most photon sources.

Fig. 7 let us represent the dependence of the von Neumann entropy $S_{N}$
depending on the dimensionless parameter $\Omega t_{BS}$ for identical
photons, i.e. for $\sigma_{1}=\sigma_{2}=\sigma$ and
$\omega_{01}=\omega_{02}=\omega_{0}$.

Figure 7: The dependence of the von Neumann entropy $S_{N}$ on the parameter
$\Omega t_{BS}$ is presented. In figures (a, b, c, d) the value of $S_{N}$ is
presented for $|1,1\rangle,|2,3\rangle,|4,2\rangle,|3,3\rangle$, respectively
(where $|s_{1},s_{2}\rangle$ are input states to 1 and 2 ports of the BS,
respectively). All figures show the results for $\sigma/\Omega=10$ (brown),
$\sigma/\Omega=5$ (blue),$\sigma/\Omega=3$ (red), $\sigma/\Omega=1$ (green),
$\sigma/\Omega=1/3$ (orange) from bottom to top, respectively. The thin curve
is made at $\sigma/\Omega=0$ (black).

We add that the results obtained for monochromatic photons, i.e. for
$\sigma\to 0$ (more precisely, for $\sigma/\Omega\ll 1$ such that $\sigma
t_{BS}\ll 1$) are in agreement with the results of quantum entanglement
“conventional” BS calculations, see for example [20, 21]. Indeed, if we use
the results for “conventional” BS with reflection coefficient
$R=\sin^{2}(\kappa z)$ and, for example, [20] to calculate quantum
entanglement (or the above method), we can get results in Fig. 7 for
$\sigma/\Omega=0$ (thin lines). Thus our result is a more general one
applicable to non-monochromatic photons. One can see that there is a big
difference between quantum entanglement of monochromatic and non-monochromatic
photons. Moreover, in the case of non-monochromatic photons, when
$\sigma/\Omega\sim 1$, the quantum entanglement is larger. One can also see
that in the case of non-monochromatic photons, when $\sigma/\Omega\gtrsim 1$
and for relatively large $\Omega t_{BS}$, the quantum entanglement becomes a
constant value. In the case of monochromatic photons quantum entanglement is a
periodic function.

As an example, let us consider in more detail the case of input photons in
states $|1,1\rangle$. This case is interesting because it realizes the case of
the Hong-Ou-Mandel (HOM)[4, 24] effect. To analyze this case, it is convenient
to present a contour plot for von Neumann entropy as a function of two
parameters $\sigma/\Omega$ and $\Omega t_{BS}$, see Fig. 8(a). Since at large
$\Omega t_{BS}$ (of course, the condition $\sigma t_{BS}\gg 1$ must also be
satisfied) the quantum entanglement tends to a constant value, in Fig. 8(b)
presents this constant value depending on the parameter $\sigma/\Omega$. It is
also possible to find an analytic dependence for quantum entanglement for the
case $|1,1\rangle$ as

$\displaystyle S_{N}=\ln\frac{2(1-{\rm J})^{{\rm J}-1}}{(2{\rm J})^{{\rm
J}}},\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
J}=1+\frac{3}{8}\left(\frac{\Omega}{\sigma}\right)^{2}-\frac{\sqrt{\pi}}{16}\left(\frac{\Omega}{\sigma}\right)^{3}\left\\{3+10\left(\frac{\sigma}{\Omega}\right)^{2}\right\\}{\rm
erf}\left(\frac{\Omega}{2\sigma}\right){e}^{\left(\frac{\Omega}{2\sigma}\right)^{2}},$
(38)

where erf is an error function. Should be added that, to get the Eq. (8) sine
and cosine terms (rapidly oscillating terms at $\Omega t_{BS}\to\infty$) must
be ignored when integrating over frequencies.

Figure 8: (a): A contour plot of the von Neumann entropy $S_{N}$ versus two
parameters $\sigma/\Omega$ and $\Omega t_{BS}$ for the input state
$|1,1\rangle$ is presented. (b): von Neumann enetropy $S_{N}$ is presented as
a function of $\sigma/\Omega$ in the limiting case $\Omega t_{BS}\to\infty$
for the input state $|1,1\rangle$, see [22]

It can be seen from the graphs obtained that the quantum entanglement has a
maximum, which will be at $\sigma/\Omega=0.44467$. It is also seen that the
quantum entanglement is large at $\sigma/\Omega\sim 1$, and as $\sigma/\Omega$
increases, it tends to zero. As an example, let us give another calculation of
quantum entanglement from the initial states $|0,2\rangle$ at the same
remaining parameters, see Fig. 9.

Figure 9: (a): The dependence of Von Neumann entropy $S_{N}$ on the parameter
$\Omega t_{BS}$ is presented for $\sigma/\Omega=10$ (brown); $\sigma/\Omega=5$
(blue); $\sigma/\Omega=3$ (red); $\sigma/\Omega=1$ (orange);
$\sigma/\Omega=1/3$ (green); $\sigma/\Omega=0$ (black). (b): A contour plot of
Von Neumann entropy $S_{N}$ from two system parameters $\Omega t_{BS}$ and
$\sigma/\Omega$ is presented. The inset is presented for $S_{N}$ at $\Omega
t_{BS}\to\infty$ depending on the parameter $\sigma/\Omega$. Input photons are
in the state $|0\rangle,|2\rangle$, see [23].

Fig. 9 shows that the Von Neumann entropy at $\sigma/\Omega\gtrsim 1$ is quite
different from the case of monochromatic photons $\sigma/\Omega=0$. Fig. 9 (b)
also shows that the large value of entropy is at $\sigma/\Omega\sim 1$. The
largest entropy value at $\Omega t_{BS}\to\infty$ would be $S_{N}=1.092$ at
$\sigma/\Omega=0.24$.

In summary, it can be concluded that the waveguide BS can be a good source of
quantum entangled photons. And the entanglement can be easily adjusted by
changing the $\Omega$ parameter, e.g. by separating or bringing the waveguides
closer together. It should be noted that a waveguide BS can be a source of
large quantum entangled photons. Indeed, such a source generates almost
maximum possible quantum entanglement at $\sigma/\Omega\sim 1$ and $\Omega
t_{BS}>1$. Recall that the maximum quantum entanglement for von Neumann
entropy in our case $S_{N}=\ln(1+s_{1}+s_{2})$ [20, 50]. If we consider
constants $R$ and $T$ (monochromatic photons) the quantum entanglement is a
periodic function of $\Omega t_{BS}$, and for large $\Omega t_{BS}\gg 1$ a
rapidly oscillating relation, which is bad for use in quantum technologies.

##  4 Photon statistics on the beam splitter

The statistical properties at the BS output ports are determined by the
probability $P_{k}$ to detect $k$ photons on port 1 and the probability
$P_{p}$ to detect $p$ photons on port 2. The statistical properties at the BS
output ports have been studied in many works [19, 21, 22, 20, 13]. Initial
states may depend on which photons are fed to the BS input ports. These states
may differ not only in the distribution of photons in each mode, but also in
the quantum entanglement of the electromagnetic modes of the incoming photons.
We will limit ourselves here to considering the most important example - these
are the incoming Fock states, since other cases can be represented based on
them.

###  4.1 Photon statistics on a “conventional” beam splitter

In the case of “conventional” BS, the statistical properties at the BS output
ports have been well studied in [19]. The simplest expressions for calculating
statistical properties have been derived in [21], which are defined by Eq.
(30), where the probability $P_{k}=\lambda_{k}$ and the probability
$P_{p}=\lambda_{s_{1}+s_{2}-k}$, since the number of photons is conserved,
i.e. $s_{1}+s_{2}=k+p$. As an example, let’s present the calculations for the
case $|1,1\rangle$ in Fig. 10 (a) and (b) and for the case $|0,2\rangle$ in
Fig. 10 (c) and (d) for the two reflection coefficient values
$R=1/2,1/2(1+1/\sqrt{3})$. The choice of these values $R$ is determined by the
fact that for $|1,1\rangle$ and $R=1/2$ the HOM effect occurs, and at
$R=1/2(1+1/\sqrt{3})$ the quantum entanglement value is maximum (see Eq.(31)).

Figure 10: The calculation of probability $P_{k}$ for the case $|1,1\rangle$
in Figures (a) at $R=1/2$ and (b) at $R=1/2(1+1/\sqrt{3})$ and for the case
$|0,2\rangle$ in Figures (c) at $R=1/2$ and (d) at $R=1/2(1+1/\sqrt{3})$ are
presented.

It can be seen that the HOM effect does not appear at the maximum entanglement
of photons, but at the maximum entanglement the distribution is equally
probable. It should also be added that calculating the probabilities for any
initial states is not difficult using Eq. 30).

###  4.2 Photon statistics on a waveguide beam splitter

Let us consider the photon statistics at the BS output ports. Expressions to
calculate the statistical properties were obtained in [22], which are defined
by Eqs. (30) and (36), where probability $P_{k}=\Lambda_{k}$ and probability
$P_{p}=\Lambda_{s_{1}+s_{2}-k}$. It should be added that when considering
monochromatic photons the coefficients $R$ and $\phi$ would be constant, then
following Eq. (36) $\Lambda_{k}=\lambda_{k}$. Thus, our consideration is
general and in the particular case coincides with “conventional” BS. Let us
compare the results in the case of monochromatic and non-monochromatic
photons. To do this let us choose BS length $z=vt_{BS}$ and frequency
$\Omega=2\kappa v$ (see below, after Eq. (28)) so that the reflection
coefficient for monochromatic and identical photons is $R=\sin^{2}(\Omega
t_{BS}/2)=1/2$. In this case we will choose $\Omega t_{BS}=5\pi/2$. Next we
will consider the same ($\sigma_{1}=\sigma_{2}=\sigma$ and
$\omega_{01}=\omega_{02}=\omega_{0}$) but not monochromatic photons, see Eq.
(37). We present for comparison the calculation results in the case of
identical monochromatic photons for $R=1/2$ (at $\Omega t_{BS}=5\pi/2$) and in
the case of identical but not monochromatic photons (at $\Omega
t_{BS}=5\pi/2$), see Fig. 11. From Fig. 11(a) shows that for states
$|1,1\rangle$ when $\sigma/\Omega=0$ the HOM effect [4] is realized, see Fig.
10 (a). This means that only photon pairs are detected at the first or second
detector (in the figure it is for $k=0$ and $k=2$) with probability $1/2$.
With increasing $\sigma/\Omega$ the HOM effect disappears and the photon
statistics change dramatically. It is quite interesting to look at photon
statistics at maximum quantum entanglement, see Fig. 11(b). These statistics
are different from the statistics of the HOM effect of Fig. 11(a) and from the
statistics in the case of monochromatic photons at maximum quantum
entanglement, see Fig. 10(b). At large $\sigma/\Omega\gg 1$, the probability
$\Lambda_{k}$ will tend to one for $k=1$. This means that photons will never
arrive at detectors in pairs. A similar analysis can quite easily be done for
any input states $|s_{1},s_{2}\rangle$. The commonality for all cases will be
a coincidence with the statistics for monochromatic photons at
$\sigma/\Omega=0$. Also, the general probability behaviour for any states
$|s_{1},s_{2}\rangle$ will be at $\sigma/\Omega\gg 1$. This is easily
explained because at $\Omega\to 0$ (same as $\sigma/\Omega\gg 1$), the
coupling between the waveguides in the BS weakens, which means that photons
propagate along their waveguides.

Figure 11: A histogram of the dependence of the probability $P_{k}$ of
detecting $k$ and $p=s_{1}+s_{2}-k$ (we consider the case $|1,1\rangle$, where
$s_{1}=s_{2}=1$) photons at the output of the first and second ports,
respectively, for different values of $\sigma/\Omega$.

Also consider the limiting case described above for quantum entanglement, this
is the case for large $\Omega t_{BS}$ (more precisely, the condition $\sigma
t_{BS}\gg 1$ must also be satisfied) for the input state $|1,1\rangle$. In
this case, it is easy to obtain the probability of detecting photons on both
detectors $P_{1,1}={\rm J}$, where ${\rm J}$ is represented in Eq. (38). The
probability of detecting pairs of photons on 1 or 2 detectors will then be
$P_{2,0}=P_{0,2}=1/2(1-P_{1,1})$. The results are presented in Fig.12. It is
interesting enough to note that the minimum of the function $min\\{P_{1,1}\\}$
or the maximum of $max\\{P_{2,0}=P_{0,2}\\}$ for $\sigma/\Omega=0.44029$
practically coincides with the maximum for quantum entanglement at
$\sigma/\Omega=0.44467$, see Fig.8(b). This means that the maximum of quantum
entanglement is realized when photons at the first and second detectors can be
registered with a minimum probability. Or, which is the same, when pairs of
photons can be recorded on detectors with maximum probability. Also in the
insets Fig.12(a),(b) the probabilities $P_{1,1}$ and $P_{2,0}=P_{0,2}$ are
presented, respectively, depending on two parameters of the studied system
$\sigma/\Omega$ and $\Omega t_{BS}$. It can be seen that taking into account
the non-monochromaticity of photons significantly changes the probabilities,
in comparison with monochromatic ones.

Figure 12: Figure (a) shows the probability $P_{1,1}$, Figure (b) shows the
probability $P_{2,0}=P_{0,2}$ for $\Omega t_{BS}\to\infty$ depending on the
$\sigma/\Omega$ parameter. Also in the insets Figure (a),(b) are contour plots
for the probability $P_{1,1}$ and $P_{2,0}=P_{0,2}$ are presented,
respectively, depending on two parameters of the studied system
$\sigma/\Omega$ and $\Omega t_{BS}$.

It should be added that such an analysis, for quantum entanglement and photon
statistics, is fairly easy to carry out for any photons input states
$|s_{1},s_{2}\rangle$, as well as various BS parameters $\Omega,t_{BS}$ and
the values of non-monochromaticity of photons $\sigma_{1},\sigma_{2}$.

##  5 Hong-Ou-Mandel effect

The HOM effect was first experimentally demonstrated by Hong et al. in 1987
[4]. The effect is thought to occur when two identical single-photon waves hit
the BS 1:1 (with reflection coefficients $R$ and transmittance $T$ close to
1/2), one at each input port. When the photons are identical, they will cancel
each other out. HOM interference appears in many cases, both in fundamental
studies of quantum mechanics and in practical implementations of quantum
technologies [51, 8]. The theoretical explanation of the HOM effect, based on
constant coefficients $R$ and $T$ and bosonic photon statistics, is quite
simple [1, 5]. In this interpretation we are not interested in what happens to
the incident photons in the BS. For this, we consider a lossless BS with
constant coefficients $R$ and $T$ (i.e. “conventional” BS) and the BS is the
source of two other photons obeying bosonic statistics. In this case, the wave
function at the output ports
$\Psi_{out}={{\hat{b}_{1}}^{\dagger}}{{\hat{b}_{2}}^{\dagger}}|0\rangle$,
where ${\hat{b}_{1}},{\hat{b}_{2}}$ is defined through the BS matrix as Eq.
(3). It’s easy to see that
${{\hat{b}_{1}}^{\dagger}}{{\hat{b}_{2}}^{\dagger}}=(T-R){{\hat{a}_{1}}^{\dagger}}{{\hat{a}_{2}}^{\dagger}}+\sqrt{RT}\left(e^{-i\phi}\left({\hat{a}_{2}}^{\dagger}\right)^{2}-e^{i\phi}\left({\hat{a}_{1}}^{\dagger}\right)^{2}\right)$.
Choosing the coefficients $R=T=1/2$ we get
${{\hat{b}_{1}}^{\dagger}}{{\hat{b}_{2}}^{\dagger}}=1/2\left(e^{-i\phi}\left({\hat{a}_{2}}^{\dagger}\right)^{2}-e^{i\phi}\left({\hat{a}_{1}}^{\dagger}\right)^{2}\right)$.
It means that photons get to detectors only in pairs, i.e. the probability
will be $P=1/2$ for each of detectors, that contradicts to the classical
representation of separation of two beams of light with coefficients
$R=T=1/2$. In the classical notion at $R=T=1/2$ there can be 4 variants:
1\. first and second photons fall on detectors 1 and 2 accordingly
2\. first and second photons fall on detectors 2 and 1 accordingly
3\. first and second photons fall on detector 1
4\. first and second photons fall on detector 2.
A total of 4 equally probable events, which obviously gives a probability of
$P=1/4$ for each of the 4 choices. Such phenomenon is called the HOM effect
(or HOM interference) and it is a good method to check quantum properties not
only of photons, but also of other particles. In other words, in the HOM
effect, the probability of photons hitting the first and second detector
$P_{1,2}=(R-T)^{2}$, with equal $R$ and $T$ will be zero. If we consider non-
monochromatic photons but identical (frequency dependent BS), it is easy to
see that
$P_{1,2}=\int|\phi(\omega_{1},\omega_{2})|^{2}\left(R(\omega_{1},\omega_{2})-T(\omega_{1},\omega_{2})\right)^{2}d\omega_{1}d\omega_{2}=\overline{(R-T)^{2}}$.
Choosing $\bar{R}=\bar{T}=1/2$ we can get
$P_{1,2}=4(\overline{T^{2}}-\overline{T}^{2})=4(\overline{R^{2}}-\overline{R}^{2})$,
i.e. there is a fluctuation of the reflectance $R$ and transmittance $T$,
which was not considered before. These issues will be discussed in detail
here.

###  5.1 Hong-Ou-Mandel effect on a “conventional” beam splitter

A schematic of the HOM interference experiment can be shown in Fig.4, with the
difference that we consider “conventional” BS. The detector coincidence rate
will drop to zero when identical input photons completely overlap in time.
This is called the Hong-Ou-Mandel dip, or HOM dip. Let us consider the general
case not restricted to identical and monochromatic photons and to simultaneous
registration by photon detectors. In this case we have to consider the output
state as (see Eq.(6))
$|\Psi_{out}\rangle=\int\phi(\omega_{1},\omega_{2}){{\hat{b}_{1}}^{\dagger}}{{\hat{b}_{2}}^{\dagger}}|0\rangle
d\omega_{1}d\omega_{2}$, where ${\hat{b}_{1}},{\hat{b}_{2}}$ is defined by
Eq.(27). It should be added that getting Eq. (27) to find
${\hat{b}_{1}}(t_{1}),{\hat{b}_{2}}(t_{2})$ quite easily if we consider
${\hat{b}_{k}}(t_{k})=e^{i{\hat{H}_{0}}t_{k}}{\hat{b}_{k}}(0)e^{-i{\hat{H}_{0}}t_{k}}$,
where ${\hat{b}_{k}}(0)={\hat{b}_{k}}$ from Eq. (3), where $k=(1,2)$. We will
assume the coefficients $R$ and $\phi$ to be constant, i.e. consider
“conventional” BS. Consider the probability $P_{1,2}$ of co-detection of
photons at 1 and 2 detectors (correlation between the two detectors). If our
coincidence gate window takes counts for time $T_{D}$, then the frequency of
coincidence between detectors 1 and 2 is proportional (see, e.g., [4, 52,
53]).

$\displaystyle
P_{1,2}\propto\int^{T_{D}/2}_{-T_{D}/2}\int^{T_{D}/2}_{-T_{D}/2}\langle{\hat{b}}^{\dagger}_{1}(t_{1}){\hat{b}}^{\dagger}_{2}(t_{2}){\hat{b}}_{1}(t_{1}){\hat{b}}_{2}(t_{2})\rangle
dt_{1}dt_{2}.$ (39)

Consider the case where the reaction time $\tau_{D}$ (time resolution) of
detectors $D_{1}$ and $D_{2}$ in the experiment is many times slower than
other time scales of the problem (reaction time is large), then in this case
$T_{D}\to\infty$. It should be added that the theory presented below is not
difficult to generalize to the case when the detector reaction times are
short, which is currently realized experimentally (eg [54, 55]). The
calculation of the correlation function (39) can be found in various papers
depending on the JSA function $\phi(\omega_{1},\omega_{2})$, eg [4, 52, 53,
56, 54, 57]. As a result, the correlation function in general form will be at
$R=T=1/2$

$\displaystyle P_{1,2}=\frac{1}{2}\left(1-{\rm
Re}\biggl{\\{}\int\phi(\omega_{1},\omega_{2})\phi^{*}(\omega_{2},\omega_{1})e^{-i(\omega_{2}-\omega_{1})(\delta\tau+\tau)}\biggl{\\}}\right)d\omega_{1}d\omega_{2},$
(40)

where $P_{1,2}$ is normalised so that at $R=0$ the probability is $P_{1,2}=1$
(without BS the probability of detection is $100\%$), which corresponds to the
standard normalisation in HOM theory. In Eq. (40) is usually replaced by
$\Delta\tau=\delta\tau+\tau$ corresponding to the time between detection of 1
and 2 photons. It can be seen that at $\Delta\tau=0$ the correlation function
$P_{1,2}=0$, only when the two photons are identical, i.e. at
$\phi(\omega_{1},\omega_{2})=\phi(\omega_{2},\omega_{1})$. Thus, the HOM
effect will be observed at photon identity and at $\Delta\tau=0$.

Here are examples of the calculated correlation functions $P_{1,2}$. Here we
give the result obtained in the very first paper where this effect was
demonstrated [4]

$\displaystyle
P_{1,2}=\frac{1}{2}\left(1-e^{-(\delta\omega\Delta\tau)^{2}}\right),$ (41)

where $\delta\omega$ is the bandwidth at Gaussian
$\phi(\omega_{0}/2+\omega,\omega_{0}/2+\omega)$. It should be added that
sources of single photons are often used in experiments, but their initial
state cannot always be determined by the Fock states. In other words, photons
incident on the BS can be quantum entangled. One of the most commonly used
sources of such photons is spontaneous parametric down-conversion (SPDC). In
the case of such photons, the spectral function $\phi(\omega_{1},\omega_{2})$
can have a complicated form. Quite a lot of such cases are considered in [56].

Here we have considered photons to demonstrate the HOM effect. This effect has
a deeper meaning than demonstrated here. First of all, this is due to the fact
that the HOM effect can be observed not only on photons, but also on other
particles of both bosonic and fermionic nature [58, 59, 60].

###  5.2 Hong-Ou-Mandel effect on a waveguide beam splitter

While the HOM effect theory for “conventional” BS is well known and described
in various literature, the HOM effect theory for frequency-dependent waveguide
BS has only recently appeared [24, 25]. This theory will be presented here in
more detail. Now the scheme of the HOM experiment will look like Fig. 4. The
calculation of the correlation function is carried out in the same way as it
is done in the case of “conventional” BS, with the only difference that we
consider the coefficients $R,T$ and $\phi$ to be frequency dependent and do
not fix their values. As a result, we get

$\displaystyle
P_{1,2}=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\left|\xi_{1}(t_{1},t_{2},\tau)-\xi_{2}(t_{1},t_{2},\tau,\delta\tau)\right|^{2}dt_{1}dt_{2}\leavevmode\nobreak\
,$
$\displaystyle\xi_{1}(t_{1},t_{2},\tau)=\frac{1}{2\pi}\int\phi(\omega_{1},\omega_{2})T(\omega_{1},\omega_{2})e^{-i\omega_{1}(t_{1}-\tau)}e^{-i\omega_{2}t_{2}}d\omega_{1}d\omega_{2}\leavevmode\nobreak\
,$
$\displaystyle\xi_{2}(t_{1},t_{2},\tau,\delta\tau)=\frac{1}{2\pi}\int\phi(\omega_{1},\omega_{2})R(\omega_{1},\omega_{2})e^{-i\omega_{2}(t_{1}+\delta\tau)}e^{-i\omega_{1}(t_{2}-\delta\tau-\tau)}d\omega_{1}d\omega_{2}\leavevmode\nobreak\
,$ (42)

where $P_{1,2}$ is normalized so that with $t_{BS}=0$ the probability is
$P_{1,2}=1$. We then obtain

$\displaystyle
P_{1,2}=\int\Biggr{(}|\phi(\omega_{1},\omega_{2})|^{2}\left(T^{2}(\omega_{1},\omega_{2})+R^{2}(\omega_{1},\omega_{2})\right)-$
$\displaystyle 2{\rm
Re}\biggr{\\{}\phi(\omega_{1},\omega_{2})\phi^{*}(\omega_{2},\omega_{1})T(\omega_{1},\omega_{2})R(\omega_{2},\omega_{1})e^{-i(\omega_{2}-\omega_{1})\Delta\tau}\biggl{\\}}\Biggl{)}d\omega_{1}d\omega_{2}.$
(43)

It should be added that if $T$ and $R$ are assumed to be independent of
frequencies and $T=R=1/2$, then Eq. (43) corresponds to the well-known
equation, eg [61, 62, 63], see above.

We next consider the case of identical photons at $\Delta\tau=0$, in this case
$\phi(\omega_{1},\omega_{2})=\phi(\omega_{2},\omega_{1})$ and
$R(\omega_{1},\omega_{2})=R(\omega_{2},\omega_{1})$, and the quantity

$\displaystyle
P_{1,2}(\Delta\tau=0)=\overline{(T-R)^{2}}=\int|\phi(\omega_{1},\omega_{2})|^{2}\left(T(\omega_{1},\omega_{2})-R(\omega_{1},\omega_{2})\right)^{2}d\omega_{1}d\omega_{2}.$
(44)

If in (44) we choose $\overline{T}=\overline{R}=1/2$, then we get
$P_{1,2}=4(\overline{T^{2}}-\overline{T}^{2})=4(\overline{R^{2}}-\overline{R}^{2})$.
In other words, there is a mean-square fluctuation of the coefficients of
transmission $T$ and reflection $R$, which leads to a nonzero value of
$P_{1,2}$ in the case of identical photons. This conclusion is fundamental in
the theory of HOM interference and was not previously obtained. Also, from the
previously obtained Eqs. (43) and (44) it follows that the
$P_{1,2}(\Delta\tau\gg\tau_{c})=2\overline{T^{2}}=2\overline{R^{2}}$
($\tau_{c}$ is the coherence time).

Let us present the results of calculating the value of $P_{1,2}$ for the case

$\displaystyle\phi(\omega_{1},\omega_{2})=Ce^{-\frac{(\omega_{1}+\omega_{2}-\Omega_{p})^{2}}{2\sigma^{2}_{p}}}e^{-\frac{(\omega_{1}-\omega_{01})^{2}}{2\sigma^{2}_{1}}}e^{-\frac{(\omega_{2}-\omega_{02})^{2}}{2\sigma^{2}_{2}}}.$
(45)

We will be interested in the case applicable for most sources of photons
$\omega_{02}-\omega_{01}\ll\omega_{01},\omega_{02}$;
$\omega_{01}/\sigma_{1}\gg 1$; $\omega_{02}/\sigma_{2}\gg 1$, in this case the
normalization constant
$C=\frac{(\sigma^{2}_{1}+\sigma^{2}_{2}+\sigma^{2}_{p})^{1/4}}{\sqrt{\pi\sigma_{1}\sigma_{2}\sigma_{p}}}$.
The function (45) allows us to analyze the value of $P_{1,2}$ for two cases
that are of practical interest. The first case is spontaneous parametric down-
conversion (SPDC), for example, for
$\Omega_{p}=2\omega_{0};\omega_{0}=\omega_{01}=\omega_{02};\sigma_{1}=\sigma_{2}=\sigma$
is SPDC of type I, where $\sigma_{p}$ is the bandwidth of the pump beam,
$\omega_{0}$ and $\sigma$ are the central frequency and the bandwidth,
respectively, for both the signal and the idle beams [64]. The second case/on
the other hand if we consider $\sigma_{p}\rightarrow\infty$ in (45), then this
will be the case of Fock states (eg, [64, 56]). Indeed, in this case, in Eq.
(45), the $\phi(\omega_{1},\omega_{2})$ function will be factorized, which
corresponds to Fock states. Substituting (45) into (43) we obtain [24, 25, 17]

$\displaystyle
P_{1,2}=\int^{\infty}_{-\infty}\biggr{\\{}e^{-(y-\frac{\Delta\omega}{\Omega_{g}})^{2}}\left(T^{2}(y)+R^{2}(y)\right)-2Be^{-(\frac{\Delta\omega}{\Omega_{g}})^{2}}T(By)R(By)e^{-y^{2}}\cos\left(B\Delta\tau\Omega_{g}y\right)\biggr{\\}}\frac{dy}{\sqrt{\pi}},$
(46)

where
$B=A\sqrt{\frac{1+\frac{\sigma^{2}_{p}}{\sigma^{2}_{1}+\sigma^{2}_{2}}}{A^{2}+\frac{\sigma^{2}_{p}}{\sigma^{2}_{1}+\sigma^{2}_{2}}}}$
and $B\in(0,1)$;
$A=\frac{2\sigma_{1}\sigma_{2}}{\sigma^{2}_{1}+\sigma^{2}_{2}};\leavevmode\nobreak\
\Omega_{g}=\sqrt{\frac{4\sigma^{2}_{1}\sigma^{2}_{2}+(\sigma^{2}_{1}+\sigma^{2}_{2})\sigma^{2}_{p}}{\sigma^{2}_{1}+\sigma^{2}_{2}+\sigma^{2}_{p}}}$,

$\displaystyle\Delta\omega=\omega_{02}\frac{\sigma^{2}_{p}+2\sigma^{2}_{1}}{\sigma^{2}_{1}+\sigma^{2}_{2}+\sigma^{2}_{p}}-\omega_{01}\frac{\sigma^{2}_{p}+2\sigma^{2}_{2}}{\sigma^{2}_{1}+\sigma^{2}_{2}+\sigma^{2}_{p}}+\Omega_{p}\frac{\sigma^{2}_{2}-\sigma^{2}_{1}}{\sigma^{2}_{1}+\sigma^{2}_{2}+\sigma^{2}_{p}},$

where $T(y)$ and $R(y)$ are determined by the Eq.(28), with the only
difference being that

$\displaystyle\Omega=\frac{4\pi n}{\omega_{0}}{\bf u}_{1}{\bf
u}_{2},\leavevmode\nobreak\
\epsilon=\frac{\Omega_{g}}{\Omega}y,\leavevmode\nobreak\
\omega_{0}=\frac{2\sigma_{p}\sigma^{2}_{1}\sigma^{2}_{2}+\sigma^{2}_{p}(\sigma^{2}_{2}\omega_{01}+\sigma^{2}_{1}\omega_{02})}{4\sigma^{2}_{1}\sigma^{2}_{2}+(\sigma^{2}_{1}+\sigma^{2}_{2})\sigma^{2}_{p}}.$
(47)

If we assume that $\Omega_{g}/\Omega\ll 1$, then $T$ and $R$ become constant
quantities and such a BS becomes “conventional”, i.e. one can always choose
$T=R=1/2$. From estimates of $\Omega$ it can be seen that $\Omega_{g}$ can be
of the order of $\Omega$, so it is necessary to take into account fluctuations
of the coefficients R and T. Equation for $P_{1,2}$, in our case Eq. (46) at
constants $T=R=1/2$ is easily integrated and coincides with the known
$P_{1,2}=1/2(1-Be^{-(\Delta\omega/\Omega_{g})^{2}}e^{-1/4(B\Omega_{g}\Delta\tau)^{2}})$,
eg [56]. Next, consider what the value of $P_{1,2}$ as a function of
$\Delta\tau\Omega_{g}$ in the case $\sigma_{1}=\sigma_{2}=\sigma$ for
different values of $\Omega_{g}/\Omega$ and $\Delta\omega/\Omega_{g}$, but for
$\Omega t_{BS}$ such that $\overline{T}=\overline{R}=1/2$, see. Fig. 13: as
$\Omega_{g}/\Omega$ increases, the value of $P_{1,2}$ tends to one.

Figure 13: Dependence of $P_{1,2}$ on $\Delta\tau\Omega_{g}$ (HOM dip). Case
(a) corresponds to fully identical photons and cases (b), (c), (d) correspond
to non-identical photons. Visibility ${\rm
V}=\frac{P_{1,2}(\Delta\tau\gg\tau_{c})-P_{1,2}(\Delta\tau=0)}{P_{1,2}(\Delta\tau\gg\tau_{c})}={\rm
V}(\Omega_{g}/\Omega)$ depends on the parameter $\Omega_{g}/\Omega$ (line
colour corresponds to: red at $\Omega_{g}/\Omega=1$, green at
$\Omega_{g}/\Omega=0.5$, brown at $\Omega_{g}/\Omega=0.25$, blue at
$\Omega_{g}/\Omega=0$). The case $\Omega_{g}/\Omega=0$ and visibility ${\rm
V}(0)$ corresponds to the previously known HOM interference theory with
constant coefficients $T=R=1/2$.

In Fig. 13 also shows that when accounting for $T$ and $R$ from frequency,
$P_{1,2}$ can differ significantly from the previously known HOM interference
theory. This means that even if the photons are identical and the beam
splitter is perfectly balanced, then $P_{1,2}$ can differ significantly from
zero, which cannot be the case with “conventional” BS. This clarification is
very important, because using a frequency-dependent BS, even in the case of
identical photons and balanced BS, it is impossible to determine the degree of
photon identity using the HOM interferometer.

Let us give in more detail the case of the same photons and balanced BS, i.e.
$\sigma_{1}=\sigma_{2}=\sigma,\Delta\omega=0$ and $\Delta\tau=0$ the results
obtained here can essentially differ from the standard HOM theory, where, as
is well known, the visibility of $V=1$, see Fig. 14.

Figure 14: In Fig. (a) shows the dependence of the average reflection
coefficient $\overline{R}$ (see Eq. (44)) as a function of $\Omega_{g}/\Omega$
for large (asymptotic) values of $\Omega t_{BS}\gg 1$. The internal tab
contains $\overline{R}$ depending on $\Omega t_{BS}$ for
$\Omega_{g}/\Omega=1;2;5;10$ (top-down in figure). In Fig. (b) shows the
visibility of $\mathcal{V}$ at $\overline{R}=\overline{T}=1/2$ depending on
$\Omega_{g}/\Omega$ for five values of the lengths of the coupled waveguide,
which corresponds to $\Omega t_{BS}\approx 2;8;14;20;30$ (top-down in figure).

As can be seen from the figures, the standard HOM theory is applicable only
for $\Omega_{g}/\Omega\ll 1$. Differences from standard HOM theory can be very
large. This is especially noticeable in Fig. 14 (b), where the visibility of
$V$ changes very much. You can also see Fig. 14(a) (tab), that only for
certain $\Omega t_{BS}$ for given $\Omega_{g}/\Omega$ you can choose
$\overline{T}=\overline{R}=1/2$. Moreover, the number of choices $\Omega
t_{BS}$ is limited and depends on $\Omega_{g}/\Omega$, and therefore depends
on the length of the coupled waveguide (length limited from above). There are
no such limitations in the standard HOM theory [14]. For example, with
$\Omega_{g}/\Omega>2$, the HOM effect cannot be realized i.e.
$\overline{R}<1/2$.

##  6 Conclusion

Thus, in this review paper, we considered the BS in quantum optics, as well as
quantum entanglement and photon statistics at the BS output ports and the HOM
effect. The special purpose of this work was to present the theory of the beam
splitter in quantum optics and to systematize such BS into 2 types - these are
“conventional” and frequency dependent BS (waveguide BS). Based on this
systematization, the quantum entanglement of photons and their statistics, as
well as the HOM effect, are presented. It is shown that these two types of
beam splitters, despite their common matrix BS (see Eq. (3) ), can have
significantly different quantum entanglement, statistics, and HOM effect. Also
presented here is the theory of a frequency-dependent beam splitter based on a
waveguide BS, where the reflection coefficients $R$ and the phase shift $\phi$
are found. It is shown that the special dependence of $R$ and $\phi$ (see
Eq.(28) ) on the frequencies of incoming photons determines such unique
properties of the waveguide BS, which differ from “conventional” BS. Taking
into account the fact that the waveguide BS is currently acquiring an
important role in quantum technologies due to the possibility of its
miniaturization, this review will be useful not only for theoreticians, but
also for experimenters.

## Acknowledgements

_The study was supported by the Russian Science Foundation No. 20-72-10151;
Grant of the President of the Russian Federation No. MD-4260.2021.1.2; state
assignment of the Russian Federation No. 0793-2020-0005 and No.
FSRU-2021-0008_

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