# Quantum entanglement and phase transition in a two-dimensional photon-photon
pair model

Jian-Jun Zhang<EMAIL_ADDRESS>Jian-Hui Yuan  Jun-Pei zhang  Ze Cheng
School of Physics, Huazhong University of Science and Technology, Wuhan
430074, China

###### Abstract

We propose a two-dimensional model consisting of photons and photon pairs. In
the model, the mixed gas of photons and photon pairs is formally equivalent to
a two-dimensional system of massive bosons with non-vanishing chemical
potential, which implies the existence of two possible condensate phases.
Using the variational method, we discuss the quantum phase transition of the
mixed gas and obtain the critical coupling line analytically. Moreover, we
also find that the phase transition of the photon gas can be interpreted as
second harmonic generation. We then discuss the entanglement between photons
and photon pairs. Additionally, we also illustrate how the entanglement
between photons and photon pairs can be associated with the phase transition
of the system.

###### keywords:

Bose-Einstein condensate, Quantum phase transition, Entanglement

##  1 Introduction

Bose-Einstein condensate (BEC) is the remarkable state of matter that
spontaneously emerges when a system of bosons becomes cold enough that a
significant fraction of them condenses into a single quantum state to minimize
the system’s free energy. Particles in that state then act collectively as a
coherent wave. The phase transition for an atomic gas was first predicted by
Einstein in 1924 and experimentally confirmed with the discovery of superfluid
helium-4 in 1938. Obviously, atoms aren’t the only option for a BEC. In recent
years, with the development of techniques, the phenomenon of BEC was observed
in several physical system [1-9], including exciton polaritions, solid-state
quasiparticles and so on. We know that photons are the simplest of bosons, so
that it would seem that they could in principle undergo this kind of
condensation. The difficulty is that in the usual blackbody configuration,
which consists of an empty three-dimensional (3D) cavity, the photon is
massless and its chemical potential is zero, so that the BEC of photons under
these circumstances would seem to be impossible. However, very recently, J.
Klaers, etc. have overcome both obstacles using a simple approach [10,11]: By
confining laser light within a two-dimensional (2D) cavity bounded by two
concave mirrors, they create the conditions required for light to thermally
equilibrate as a gas of conserved particles rather than as ordinary blackbody
radiation.

What is more, it is well known that there are many fascinated optical effects
in the nonlinear medium, for instance, reduced fluctuation in one quadrature
(squeezing) [12], sub-Poissonian statistics of the radiation field [13], or
the collapse-revivals phenomenon [14]. Especially, in the nonlinear medium, a
photon from the laser beam can couple with other photons to form a photon-pair
(PP) [15-18]. The essence of PP has been investigated by many authors [19-21].
However, inspired by the experimental discovery of BEC of photons, in this
letter we construct another interesting 2D model consisting of photons and
PPs. In this model, the mixed system of photons and PPs is formally equivalent
to a 2D gas of massive bosons with non-vanishing chemical potential, which
implies the existence of two possible condensate phases, the mixed photon-PP
condensate phase and the pure PP condensate phase. By means of a variational
method we investigate the quantum phase transition of the mixed photon gas.
Especially, we find that the quantum phase transition of the photon gas can be
interpreted as second harmonic generation. We then discuss the entanglement
between photons and PPs. By investigating the entanglement in the ground state
and the dynamics of entanglement, we also illustrate how the entanglement
between photons and PPs can be associated with the phase transition of the
system. The investigation of these questions is important both for its
connection with quantum optics and for its practical applications to harmonic
generation and quantum information.

The remainder of this paper is organized as follows: In Sec. II, we
theoretically investigate the phenomenon of BEC of photons and PPs in a 2D
optical microcavity. The entanglement between photons and PPs is investigated
in Sec. III. Finally, we make a simple conclusion.

##  2 Bose-Einstein condensation of photons and photon pairs

Figure 1: Scheme of the optical microcavity: The microcavity consists of two
curved-mirrors with high reflectivity. A filter filled with a dye solution is
inserted into it, in which photons are repeatedly absorbed and re-emitted by
the dye molecules. The other part of the cavity is filled with a Kerr-like
nonlinear medium.

###  2.1 Description of the photon pair

We start with the description of the PP. History speaking, the essence of the
PP is presently still under discussion [19-21], and there exist many different
ways to obtain it. However, here we will use the standard procedure [22] in
the construction of harmonic generation to derive the PP. We know that the
presence of an electromagnetic field in the nonlinear material causes a
polarization of the medium and the polarization can be expanded in powers of
the instantaneous electric field:

$\displaystyle{\bf{P}}({\bf{r}},t)=\chi^{(1)}{\bf{E}}({\bf{r}},t)+\chi^{(2)}{\bf{E}}^{2}({\bf{r}},t)+....$
(1)

Here, the first term defines the usual linear susceptibility, and the second
term defines the lowest order nonlinear susceptibility. Ignoring the high
order parts (i.e. only expend the polarization to second order in electric
field $E$), we find that the Hamiltonian describing the interaction of the
radiation field with the dielectric medium is decomposed into two terms:

$\displaystyle\begin{array}[]{l}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}H_{{\mathop{\rm
int}}}=H_{line}+H_{nonline}\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}H_{line}=-\int{\chi^{(1)}{\bf{E}}^{2}({\bf{r}},t)}d{\bf{r}}\\\
H_{nonline}=-\int{\chi^{(2)}{\bf{E}}^{3}({\bf{r}},t)}d{\bf{r}}\\\
.\end{array}$ (6)

where $H_{line}$ represents the energy of the linear interaction and
$H_{nonline}$ the nonlinear interaction.

It is well known that the electric field operator in a microcavity can be
expanded in terms of normal modes [23] as

$\displaystyle{\bf{E}}(r,t)=i\sum\limits_{\bf{k}}{{\bf{e}}_{\bf{k}}}\left({\frac{{\hbar\omega_{\bf{k}}}}{{2V\varepsilon}}}\right)^{1/2}\left({\hat{a}_{\bf{k}}e^{-i\omega_{\bf{k}}t+i{\bf{k}}\cdot{\bf{r}}}+\hat{a}_{\bf{k}}^{+}e^{i\omega_{\bf{k}}t-i{\bf{k}}\cdot{\bf{r}}}}\right),$
(7)

where $\hat{a}_{\bf{k}}$ and $\hat{a}_{\bf{k}}^{+}$ are the annihilation and
creation operators of photons with frequency $\omega_{\bf{k}}$, and they all
obey the usual boson commutation rules. $V$ is the normalization volume,
$\varepsilon$ is the dielectric constant of the medium and ${\bf{e}}_{\bf{k}}$
is the unit polarization vector with the usual polarization indices omitted
for simplicity. Substituting (3) into (2), for the linear interaction part, we
find that it consists of two processes, dissipation and two-photon
absorption(or emission). Here, dissipation is essentially also a two-photon
process, in which one photon is absorbed by the medium, meanwhile another one
is emitted. The linear interaction can be ignored, if the incident photon
field frequency $\omega_{0}$ is well below the electronic transition
frequencies of the medium. In that case, we need only consider the nonlinear
interaction, which has the simple form

$\displaystyle
H_{nonline}=\frac{\hbar}{{\sqrt{V}}}\sum\limits_{{\bf{k,k^{\prime}}}}{\chi_{{\bf{k}},{\bf{k^{\prime}}}}\left({\hat{b}_{{\bf{k}}+{\bf{k^{\prime}}}}^{+}\hat{a}_{\bf{k}}\hat{a}_{{\bf{k^{\prime}}}}+H.c.}\right)},$
(8)

under the requirements of phase matching. Above, the operator $\hat{a}$
represents the normal photons, $\hat{b}$ represents the coupling PP, and where
$\chi_{{\bf{k}},{\bf{k^{\prime}}}}$ is the coupling matrix element. The
interaction energy in (4) consists of two terms. The first term
$b_{{\bf{k}}+{\bf{k^{\prime}}}}^{+}a_{\bf{k}}a_{{\bf{k^{\prime}}}}$ describes
the process in which two normal photon with wave-vector ${\bf{k}}$ and
${\bf{k^{\prime}}}$ couple into a PP with wave-vector
${\bf{K}}={\bf{k}}+{\bf{k^{\prime}}}$, and the second term describe the
opposite process. The energy is conserved in both the processes.

###  2.2 Free-photon dispersion relation inside the optical microcavity

In this letter, we restrict out investigation inside a 2D optical microcavity.
The microcavity, as shown in Fig. 1, consists of two curved dielectric mirrors
with high reflectivity (about 99.9), which ensure prefect reflection of the
longitudinal component of the electromagnetic field within the cavity. In
addition, the transverse size of the cavity is much larger than its
longitudinal one.

We know that for a free photon, its frequency as a function of transversal
($k_{r}$) and longitudinal ($k_{z}$) wave number is
$\omega=c\left[{k_{z}^{2}+k_{r}^{2}}\right]^{1/2}$. However, in the case of
photons confined inside the microcavity, the vanishing of the electric field
at the reflecting surfaces of the curved-mirrors imposes a quantization
condition on the longitudinal mode number $k_{z}$, $k_{z}=n\pi/D(r)$, where
$n$ is an integer and where $D(r)=D_{0}-2(R-\sqrt{R^{2}-r^{2}})$ is the
separation of two curved-mirrors at distance $r$ from the optical axis, with
$D_{0}$ the mirror separation at distance $r=0$ and $R$ the radius of
curvature.

In the present work, we consider to fix the longitudinal mode number of
photons by inserting a circular filter into the cavity. The filter is filled
with a dye solution, in which photons are repeatedly absorbed and re-emitted
by the dye molecules. Thus, it also plays the role of photon reservoir. We
know that the longitudinal size of the cavity (i.e., the distance between the
mirrors) is very small. The small distance $D(r)$ between the mirrors causes a
large frequency spacing between adjacent longitudinal modes, comparable with
the spectral width of the dye. Modify spontaneous emission such that the
emission of photons with a given longitudinal mode number, $n=q$ in our case,
dominates over other emission processes. In this way, the longitudinal mode
number is frozen out. For fixed longitudinal mode number $q$ and in paraxial
approximation ($r\ll R$, $k_{r}\ll k_{z}$), we also find that the dispersion
relation of photons approximatively becomes $\omega\approx q\pi
c/D_{0}+ck_{r}^{2}D_{0}/2q\pi$. The above frequency-wavevector relation, upon
multiplication by $\hbar$, becomes the energy-momentum relation for the photon

$\displaystyle E\approx
m_{{\rm{ph}}}c^{2}+\frac{{(p_{r})^{2}}}{{2m_{{\rm{ph}}}}},$ (9)

where $m_{{\rm{ph}}}=\hbar q\pi/D_{0}c=\hbar\omega_{eff}/c^{2}$ is the
effective mass of the confined photons. At low temperatures, it is convenient
to redefine the zero of energy, so that only the effective kinetic energy,

$\displaystyle E\approx\frac{{(p_{r})^{2}}}{{2m_{{\rm{ph}}}}},$ (10)

remains. The above analysis shows that for the photon confined inside the 2D
microcavity, it is formally equivalent to a general boson having an effective
mass $m_{{\rm{ph}}}=\hbar\omega_{eff}/c^{2}$, that is moving in the transverse
resonator plane.

Furthermore, we here consider the case that the microcavity (except the filter
part) is filled with a Kerr nonlinear medium exhibiting significant third-
order optical nonlinearity. Due to the nonlinear effect, photons can couple
into PPs. If we connect the non-vanishing effective photon mass to the
previous analysis of the PPs, in this case we then can rewrite the nonlinear
interaction $H_{nonline}$ as

$\displaystyle
H_{nonline}=\frac{\hbar}{{\sqrt{S}}}\sum\limits_{{\bf{k}}_{r}{\bf{,k^{\prime}}}_{r}}{\chi_{{\bf{k}}_{r},{\bf{k^{\prime}}}_{r}}\left({b_{{\bf{k}}_{r}+{\bf{k^{\prime}}}_{r}}^{+}a_{{\bf{k}}_{r}}a_{{\bf{k^{\prime}}}_{r}}+H.c.}\right)}$
(11)

where $S$ is the surface area of the 2D cavity, and where $a_{{\bf{k}}_{r}}$
and $a_{{\bf{k^{\prime}}}_{r}}$ are the annihilation operators of massive
photons with transverse wavevectors ${\bf{k}}_{r}$ and
${\bf{k}}_{r^{\prime}}$, respectively, and
$b_{{\bf{k}}_{r}+{\bf{k^{\prime}}}_{r}}^{+}$ are the creation operator of the
massive PPs with transverse wave-vector
$K_{r}={\bf{k}}_{r}+{\bf{k^{\prime}}}_{r}$. Here, it should be remarked that
the existence of effective photon mass makes the thermodynamics of this 2D
mixed gas of photons and PPs different from the usual 3D photon gas. For the
2D system, thermalization is achieved in a photon-number-conserving way
($N=N_{a}+2N_{b}$) with nonvanishing chemical potential $\mu$, by multiple
scattering with the dye molecules, which acts as heat bath and equilibrates
the transverse modal degrees of freedom of the photon gas to the temperature
of dye molecules.

###  2.3 BEC of photons and photon pairs

In virtue of the above analysis, we consider the following basic Hamiltonian,
to give a simple model of PP formation (with $\hbar=1$ throughout this letter)

$\displaystyle H_{\mu}=H_{a}+H_{b}+H_{ab},$ (12)

with

$\displaystyle\begin{array}[]{l}H_{a}=\sum\limits_{\bf{k}}{\frac{{{\bf{k}}^{2}}}{{2m_{ph}}}a_{\bf{k}}^{+}a_{\bf{k}}}+\frac{{u_{aa}}}{S}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}},{\bf{k^{\prime\prime}}}}{a_{{\bf{k}}+{\bf{k^{\prime}}}-{\bf{k^{\prime\prime}}}}^{+}a_{{\bf{k^{\prime\prime}}}}^{+}}a_{{\bf{k^{\prime}}}}a_{\bf{k}}\\\
H_{b}=\sum\limits_{\bf{k}}{\left({\frac{{{\bf{k}}^{2}}}{{4m_{ph}}}-2\mu}\right)b_{\bf{k}}^{+}b_{\bf{k}}}+\frac{{u_{bb}}}{S}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}},{\bf{k^{\prime\prime}}}}{b_{{\bf{k}}+{\bf{k^{\prime}}}-{\bf{k^{\prime\prime}}}}^{+}b_{{\bf{k^{\prime\prime}}}}^{+}}b_{{\bf{k^{\prime}}}}b_{\bf{k}}\\\
H_{ab}=\frac{{u_{a,b}}}{S}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}}}{a_{\bf{k}}^{+}b_{{\bf{k^{\prime}}}}^{+}}b_{{\bf{k^{\prime}}}}a_{\bf{k}}-\frac{1}{{\sqrt{S}}}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}}}{\chi_{{\bf{k}},{\bf{k^{\prime}}}}(b_{{\bf{k}}+{\bf{k^{\prime}}}}^{+}}a_{{\bf{k^{\prime}}}}a_{\bf{k}}+H.c.)\\\
\end{array}$ (16)

Above, $H_{a}$ and $H_{b}$ denote the pure photon and PP contributions, and
$H_{ab}$ refers to the interaction between them. In the dilute gas limit
$u_{a,a}$, $u_{b,b}$, $u_{a,b}$ are proportional to the two-body s-wave
photon-photon, photon-PP, and PP-PP scattering lengths [23], respectively, and
$\chi_{{\bf{k}},{\bf{k^{\prime}}}}$ characterizes the coupling strength,
encoding that PPs are composed of two massive photons. Note that in (9) we
ignore the chemical potential term $\mu N_{a}$ from the Hamiltonian [24].
Additionally, in (9), we also drop the subscript of the transverse wave-
vectors of photons and PPs, i.e. we take ${\bf{k}}_{r}={\bf{k}}$.

It is well known that for a general massive Boson-Boson pairs gas, at absolute
zero temperature, there exists a BEC consisting of two possible condensate
phases [25-27]: (i) Both the single boson and the pair of bosons are
condensed. (ii) The pair of bosons are condensed but the single boson is not.
Now we know that in the case of photons (or PPs) confined inside the
microcavity, the subsystem of photons (or PPs) is formally equivalent to a 2D
gas of massive bosons with non-vanishing chemical potential. Thus, the feature
should also survive for the 2D mixed gas of photons and PPs. In the condensate
phase, a macroscopic number of particles occupy the zero-momentum state, and
it is useful to separate out the condensate modes from the Hamiltonian. Follow
the process, we find that at the BEC state, the grand canonical Hamiltonian of
the mixed system has the form

$\displaystyle H_{\mu}=H_{0}+\delta H,$ (17)

where

$\displaystyle\begin{array}[]{l}H_{0}=-\mu_{b}b^{+}b+\frac{{u_{aa}}}{S}\left({a^{+}a}\right)^{2}+\frac{{u_{bb}}}{S}\left({b^{+}b}\right)^{2}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\frac{{u_{ab}}}{S}a^{+}ab^{+}b-\frac{\chi}{{\sqrt{S}}}\left({a^{+}a^{+}b+b^{+}aa}\right)\\\
\end{array}$ (20)

is the condensate part of the Hamiltonian with $\mu_{b}=2\mu+u_{bb}/S$ the
modified chemical potential of PPs, and where $\delta
H=H\left({a_{\bf{k}},b_{\bf{k}}}\right)$ is the perturbation part and it is a
complex function of the non-condensate modes. Here, we mention that at the
condensate state we only need consider the case of single mode coupling, thus
we can treat the coupling matrix element $\chi$ as an adjustable constant.
Note that in (11) we also drop the zero-momentum subscript of the creation and
annihilation operators of photons and PPs. In the present work, we mainly aim
to investigate the phenomenon of BEC of photons and PPs, thus, hereafter we
will ignore the perturbation part and approximately write the Hamiltonian as
the form $H_{\mu}\approx H_{0}$. The Hamiltonian commutes with the total
photon number $N=a^{+}a+2b^{+}b$ and $n=N/S$ is the particle density.

Up to now we have not made a careful distinction between the two possible
condensate phases. However, for the mixed system, working out the ground-state
phase diagram is very important. Here, we intend to employ the variational
principle method for finding the ground-state configurations of the present
system and examining their dependence from the microscopic parameters. In
other word, we aim to work out the ground-state phase diagram of the mixed
system starting from the study the semiclassical equation. Clearly, we know
that the chemical potential $\mu$ of the mixed system allows the total photon
number (whether free or bound into PPs) to fluctuate around some constant
average value $N$, then the total number of photons need only be conserved on
the average value. For convenience, hereafter we assume that $N$ is an even
number and thus $M=N/2$ denotes the maximum number of the PPs. Furthermore, in
this case we also introduce a new operation, namely the double photon creation
operation with the relation

$\displaystyle\begin{array}[]{l}\left({c^{+}}\right)^{m}\left|0\right\rangle\equiv\sqrt{\frac{{m!}}{{\left({2m}\right)!}}}\left({a^{+}}\right)^{2m}\left|0\right\rangle\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}c^{+}c\left|0\right\rangle\equiv 2a^{+}a\left|0\right\rangle\\\
\end{array},$ (23)

where $\left|0\right\rangle$ is the vacuum state. Using the new operator, we
construct the Gross-Pitaevskii (GP) states [28]

$\displaystyle\left|{\psi_{{\rm{GP}}}}\right\rangle=\frac{1}{{\sqrt{M!}}}\left[{\alpha
c^{+}+\beta b^{+}}\right]^{M}\left|0\right\rangle$ (24)

as the trial macroscopic state. Here,
$\alpha=\left|\alpha\right|e^{i\theta_{a}}$ and
$\beta=\left|\beta\right|e^{i\theta_{b}}$ are complex amplitudes with
$\left|\alpha\right|^{2}=N_{a}/N$ and $\left|\beta\right|^{2}=2N_{b}/N$ the
photon and PP densities, respectively. $\theta_{a}$ and $\theta_{b}$ (real
valued) denoted the phases of each species. Obviously, the parameters $\alpha$
and $\beta$ satisfy the normalized condition
$\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$. With the help of the GP
state, the semiclassical model Hamiltonian $\bar{H}(\alpha,\beta)$ is given by

$\displaystyle\begin{array}[]{l}\bar{H}=\mathop{\lim}\limits_{N\to\infty}\frac{{\left\langle{\psi_{GP}}\right|H_{\mu}\left|{\psi_{GP}}\right\rangle}}{{M{\chi}\sqrt{2n}}}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\left[{-\mu_{b}\left|\beta\right|^{2}+2u_{aa}n\left|\alpha\right|^{4}+u_{bb}n\left|\beta\right|^{4}}\right./2\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{{\left.{+u_{ab}n\left|\alpha\right|^{2}\left|\beta\right|^{2}}\right]}\mathord{\left/{\vphantom{{\left.{+u_{ab}n\left|\alpha\right|^{2}\left|\beta\right|^{2}}\right]}{{\chi}\sqrt{2n}}}}\right.\kern-1.2pt}{{\chi}\sqrt{2n}}}-2\left|\alpha\right|^{2}\sqrt{\left|\beta\right|^{2}}\cos\theta\\\
\end{array},$ (28)

where $\theta=\theta_{b}-2\theta_{a}$ is the phase difference. Considering the
conserved condition $\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$, we
next introduce a new variables
$s=\left|\alpha\right|^{2}-\left|\beta\right|^{2}$. Using the new notation, we
rewrite the model Hamiltonian as

$\displaystyle\bar{H}=-\lambda s^{2}-2\gamma
s+\xi-\sqrt{2\left({1-s}\right)}(1+s)\cos\theta,$ (29)

with

$\begin{array}[]{l}\lambda=\frac{{\sqrt{2n}}}{{\chi}}\left({\frac{{u_{ab}}}{4}-\frac{{u_{aa}}}{2}-\frac{{u_{bb}}}{8}}\right)\\\
\gamma=\frac{{\sqrt{2n}}}{{\chi}}\left({\frac{{u_{bb}}}{8}-\frac{{u_{aa}}}{2}-\frac{{\mu_{b}}}{{4n}}}\right)\\\
\xi=\frac{{\sqrt{2n}}}{{\chi}}\left({\frac{{u_{aa}}}{2}+\frac{{u_{ab}}}{4}+\frac{{u_{bb}}}{8}-\frac{{\mu_{b}}}{n}}\right)\\\
\end{array}$.

According to the variational principle, we minimize the energy
$\bar{H}(\alpha,\beta)$ with $s$ and $\theta$ as variational parameters. We
then obtain the optimum values [i.e.($\bar{s}$, $\bar{\theta}$)] of parameters
for the ground state as follows:

$\displaystyle\left({\bar{s},\bar{\theta}}\right)=\left\\{\begin{array}[]{l}(-1,{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}\theta),{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\gamma-\lambda+1<0\\\ (s,{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}0{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}or{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}\pi),{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}\gamma-\lambda+1>0\\\ \end{array}\right.,$
(32)

where $-1<s<1$ is the solution of the equation $\lambda
s+\gamma=\left({3s-1}\right)/2\sqrt{2(1-s)}$ (the explicit value can be obtain
by graphical solution method, it is generally too messy to be shown here). The
result, together with the fact that the parameters $\left|\alpha\right|^{2}$
and $\left|\beta\right|^{2}$ denote the photon and PP densities, indicates
that when $\gamma-\lambda+1<0$ the system converts from the mixed photon-PP
phase to the pure PP phase. We therefore can interpret this line
$\gamma-\lambda+1=0$ as the threshold coupling for the formation of a
predominantly PP state. Here, it is to be mentioned that for the pure PP
phase, $\bar{s}=-1$ thus the relative phase $\theta$ cannot be defined.

##  3 Entanglement between photons and photon pairs

###  3.1 Entanglement of the ground state

At the BEC state, one may consider the mixed gas of photons and PPs as a
bipartite system of two modes. For the present system, the entanglement of two
modes is always closely associated with the phase transition of the system.
Moreover, the two modes, be they spatially separated, and differing in some
internal quantum number, are clearly distinguishable subsystems. Thus, the
state of each mode can be characterized by its occupation number. By using the
fact that the total number of photons $N$ is constant, a general state of the
system (in the Heisenberg picture) can be written for even $N$ in terms of the
Fock states by

$\displaystyle\left|\psi\right\rangle=\sum\limits_{m=0}^{M}{c_{m}}\left|{2m,M-m}\right\rangle,$
(33)

where $m$ is the half population of particles in photon mode $a$, and $c_{m}$
is the coefficients of the state. In the Fock representation, the GP state
also can be reexpressed as
$\left|{\psi_{GP}}\right\rangle=\sum\limits_{m=0}^{M}{g_{m}}\left|{2m,M-m}\right\rangle$,
with coefficients
$g_{m}=\sqrt{\frac{{M!}}{{m!\left({M-m}\right)!}}}\alpha^{m}\beta^{M-m}$. The
standard measure of entanglement of the bipartite system is the entropy of
entanglement $S(\rho)$

$\displaystyle
S(\rho)=-\sum\limits_{m=0}^{M}{\left|{c_{m}}\right|^{2}}\log_{2}\left({\left|{c_{m}}\right|^{2}}\right),$
(34)

which is the von Neumann entropy of the reduced density operator of either of
the subsystems[29]. In the present system, the maximal entanglement also can
be obtain by optimizing the expression (18) with respect to
$\left|{c_{m}}\right|^{2}$. By imposing the normalization condition
$\sum\limits_{m=0}^{M}{\left|{c_{m}}\right|^{2}}=1$, we finally get
$S_{\max}=\log_{2}\left({M+1}\right)$, which is related to the dimension $M+1$
of the Hilbert space of the individual modes.

Using expression (18) and the coefficients $c_{m}$ obtained through exact
diagonalization of the Hamiltonian (11) as done in the atom-molecule model
[30], we plot in Fig. 2 the entropy of entanglement of the ground state as a
function of the parameters $\lambda$ and $\gamma$. We note that in this letter
we restrict our attentions to the repulsive case, i.e., we restrict
$\lambda\leq 0$ throughout the letter. From Fig. 2, we observe that the
entanglement entropy exhibits a sudden decrease close $\gamma-\lambda+1=0$.
This is indicative of the fact that across the line $\gamma-\lambda+1=0$ a
quantum phase transition occurs.

Figure 2: Variation in the entropy of entanglement of the ground state with
respect to the parameters $\lambda$ and $\gamma$. Here, we set $N=200$ and
$u_{aa}=u_{bb}/4=0.25$.

To gain more information associated with the quantum phase transition of the
present system, we also depict in Fig. 3 the entropy of entanglement (solid
line) and the expectation value (dashed line) of the scaled PP number operator
of the ground state as a function $\gamma$ for fixed parameter value
$\lambda$. From Fig. 3, we see that the average value of the number of PPs
increases as $\gamma$ increases. Especially, when $\gamma-\lambda+1<0$, the
average number of PPs is maximal. The result confirms that there indeed exists
a phase transition for the present system in the ground state. Furthermore, we
also find that the ground-state entanglement entropy is not maximal at the
critical line, i.e. in the region $\gamma-\lambda+1>0$, the system is always
strongly entangled. Ref. (28) gives the property responsible for the long-
range correlation. Additionally, we also consider that the trait is associated
with the symmetry-broking of the coupling term of the system. Due to the
asymmetric form of the coupling term, the pure photon condensation will be
forbidden. As a result, in the mixed condensate phase the imbalance
$\left({2N_{b}-N_{a}}\right)/N$ between the two modes is always very small,
which is responsible for the strongly entanglement.

Figure 3: The average photon pair occupation number and the entanglement
entropy for the ground state as a function of the $\gamma$ for fixed parameter
value $\lambda=0$. Here, we set $N=200$ and $u_{aa}=u_{bb}/4=0.25$.

In addition, if we connect the non-vanishing photon mass $m_{ph}$ to the
longitudinal wave number $k_{z}$ by the relation $m_{{\rm{ph}}}=\hbar k_{z}/c$
with $k_{z}=q\pi/D_{0}$, then we find that the quantum phase transition of the
photon system can be interpreted as second harmonic generation. When
$\gamma-\lambda+1<0$, almost all photons with frequency $\omega=ck_{z}$ couple
into PPs with frequency $\omega=2ck_{z}$. In this case, the entanglement
between the photons and PPs is very small, and the entropy of entanglement is
close to zero.

###  3.2 Dynamics of entanglement

In the above analysis, we have investigated the entanglement of the ground
state. We found that in the ground state, across the phase transition line the
entanglement entropy exhibits a sudden change. To gain a better understanding
of the influence of ground-state phase transition to entanglement, in this
subsection we investigate the dynamics of entanglement. In studying the
dynamics of the system, we first need express a general state in the form of
temporal evolution (i.e., need change the expression of a general state from
Heisenberg picture to Schrodinger picture). Following the standard procedure,
we can obtain

$\displaystyle\begin{array}[]{l}\left|{\psi\left(t\right)}\right\rangle=U\left(t\right)\left|{\psi\left(0\right)}\right\rangle\\\
=\sum\limits_{m=0}^{M}{c_{m}}\left(t\right)\left|{2m,M-m}\right\rangle\\\
\end{array},$ (37)

where,
$U\left(t\right)=\sum\limits_{n=0}^{M}{\left|{\psi_{n}}\right\rangle\left\langle{\psi_{n}}\right|}\exp\left({-iE_{n}t}\right)$
is the temporal operator with $\left|{\psi_{n}}\right\rangle$ the eigenstates
of the system having energy $E_{n}$, and
$\left|{\psi\left(0\right)}\right\rangle$ is the initial state. Here, the time
dependence of coefficients $c_{m}\left(t\right)$ are given by
$c_{m}\left(t\right)=\left\langle{2m,M-m}\right|U\left(t\right)\left|{\psi\left(0\right)}\right\rangle$.
Subsequently, the entanglement entropy given in (18) can be rewritten as
$S(\rho)=-\sum\limits_{m=0}^{M}{\left|{c_{m}\left(t\right)}\right|^{2}}\log_{2}\left({\left|{c_{m}\left(t\right)}\right|^{2}}\right)$.
In this case, the entanglement entropy depends on both the choice of initial
states and the value of microscopic parameters
$\left\\{{\lambda,\gamma}\right\\}$. At the present work, we consider that the
mixed system is in the BEC state, thus here choosing GP state as the initial
state is suitable. By adjusting the GP coefficients
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}$ and the
microscopic parameters $\left\\{{\lambda,\gamma}\right\\}$, in this subsection
we also want to know if the ground-state phase transition also characterizes
different dynamics.

The time evolution of the entanglement entropy for different initial state and
interaction parameters is shown in Fig. 4. From Fig.4 we observe the features
of quantum dynamics, such as the collapse and revival of oscillations and non-
periodic oscillations. Additionally, we also find that the amplitude of the
entanglement entropy is smaller in the region $\gamma-\lambda+1<0$ contrast
with in the region $\gamma-\lambda+1>0$. Especially, we note that the greater
the imbalance $\left|\beta\right|^{2}-\left|\alpha\right|^{2}$ between the two
modes in the initial state, the clearer the difference can be observed.

Figure 4: Time evolution of the entanglement entropy for different initial
states
$\left|{\psi\left({\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right)}\right\rangle$
and microscopic parameters $\left\\{{\lambda,\gamma}\right\\}$. Form top to
bottom the GP coefficients used are
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}=\left\\{{0.5,0.5}\right\\}$,
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}=\left\\{{0.25,0.75}\right\\}$
and
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}=\left\\{{0,1}\right\\}$.
From left to right the microscopic parameters used are
$\left\\{{\gamma,\lambda}\right\\}=\left\\{{-2,0}\right\\}$ and
$\left\\{{\gamma,\lambda}\right\\}=\left\\{{-0.5,0}\right\\}$, respectively.
Here, we set $N=20$ and $u_{aa}=u_{bb}/4=0.25$.

To understand the physical reason for the above phenomenon, we also need
rewrite the general state given in (19) in terms of the eigenstates of the
system
$\left|{\psi\left(t\right)}\right\rangle=\sum\limits_{n=0}^{M}{c(n,t)}\left|{\psi_{n}}\right\rangle$,
where
$\left|{c(n,t)}\right|^{2}=\left|{c(n,0)}\right|^{2}=\left|{\left\langle{\psi_{n}}\right|\exp(-iE_{n}t)\left|{\psi\left(0\right)}\right\rangle}\right|^{2}$
can be explained as the transition probability of the system from the initial
state $\left|{\psi\left(0\right)}\right\rangle$ to the corresponding energy
eigenstates $\left|{\psi_{n}}\right\rangle$ at any time $t$. We have already
known that for the ground state across the phase transition line
$\gamma-\lambda+1=0$ the entanglement entropy exhibits a sudden decrease. This
is why in the region $\gamma-\lambda+1<0$ the amplitude of the entanglement
entropy becomes smaller. In addition, in this phase transition region we also
investigative the dependence relation between the ground-state transition
probability $\left|{c\left({0,t}\right)}\right|^{2}$ and the imbalance
$\left|\beta\right|^{2}-\left|\alpha\right|^{2}$ of the initial state. The
result is shown in Fig. 5. From Fig. 5, it is obvious that with the increasing
of the initial-state imbalance the ground-state transition probability becomes
greater. Thus the greater imbalance between the two modes in the initial state
can lead the clearer difference of the amplitude for different region.

Figure 5: Variation of the ground-state transition probability
$\left|{c\left({0,t}\right)}\right|^{2}$ with respect to the initial-state
imbalance $\left|\beta\right|^{2}-\left|\alpha\right|^{2}$. Here, we set
$N=20$ and $u_{aa}=u_{bb}/4=0.25$.

##  4 Conclusion

In this work, we have proposed a 2D model consisting of photons and PPs. In
the model, the mixed gas of photons and PPs is formally equivalent to a 2D
system of massive bosons with non-vanishing chemical potential, which implies
the existence of two possible condensate phase. Based on the GP state and
using the variational method, we have also discussed the quantum phase
transition of the mixed gas and have obtained the critical coupling line
analytically. Especially, we have found that the phase transition of the
photon gas can be interpreted as second harmonic generation. Moreover, by
investigating the entanglement entropy in the ground state and general state,
we have illustrated how the entanglement between photons and PPs can be
associated with the phase transition of the system.

## References

  * [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269 (1995) 198.
  * [2] K. B. Davis, M-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75 (1995) 3969.
  * [3] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78 (1997) 985.
  * [4] S. Jochim, _et al_ , Science 302 (2003) 2101.
  * [5] H. Deng, G. Weihs, J. Bloch, and Y. Yamamoto, Science 298 (2002) 199.
  * [6] J. Kasprzak, _et al_ , Nature 443 (2006) 409.
  * [7] R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, Science 316 (2007) 1007.
  * [8] S.O. Demokritov, _et al_ , Nature 443 (2006) 430.
  * [9] G. Ortiz and J. Dukelsky, Phys. Rev. A 72 (2005) 043611.
  * [10] J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, Nature 468 (2010) 545.
  * [11] J. Klaers, F. Vewinger, and M. Weitz, Nature Phys. 6 (2010) 512.
  * [12] D. F. Walls, and P. Zoller Phys. Rev. Lett. 47 (1981) 709.
  * [13] G. Rempe, F. Schmidt-Kaler, and H. Walther, Phys. Rev. Lett 64 (1990) 2783.
  * [14] O. V. Kibis, G. Ya. Slepyan, S. A. Maksimenko, and A. Hoffmann, Phys. Rev. Lett 102 (2009) 023601.
  * [15] D. F. Walls, and R. Barakat, Phys. Rev. A 1 (1970) 446.
  * [16] Z. Cheng, Phys. Rev. Lett. 67 (1991) 2788.
  * [17] T. Yamamoto, M. Koashi, S. K. Özdemir, and N. Imoto, Nature 421 (2003) 343.
  * [18] Z. D. Walton, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 70 (2004) 052317.
  * [19] W. Denk, J. H. Strickler, and W. W. Webb, Science 6 (1990) 73.
  * [20] T. Binoth, J.Ph. Guillet, E. Pilon, and M. Werlen, Eur. Phys. J. C 16 (2000) 311.
  * [21] Z. Cheng, J. Opt. Soc. Am. B 19 (2002) 1962.
  * [22] N. Bloembergen, Non-Linear Optics (W. A. Benjamin, Inc., New York, 1965)
  * [23] J. Klaers, J. Schmitt, T. Damm, F. Vewinger, and M. Weitz Appl. Phys. B 105 (2011) 17.
  * [24] It may be asked why this term can be ignored. The answer is connected with the fact that in this letter we only consider the infinite particle number case, i.e., we consider the total photon number $N$ as a constant. In the case, the chemical potential term $\mu N_{a}$ cannot give new restriction to the Hamiltonian. Especially, in the following analysis we will point out that at the BEC state the mixed gas of photons and PPs can convert from the mixed photon-PP condensate phase to the pure PP condensate phase. In the pure PP condensate phase, the term $\mu N_{a}$ is meaningless. We therefore ignore the restriction term $\mu N_{a}$ in the model Hamiltonian.
  * [25] L. Radzihovsky, J. Park, and P. B. Weichman, Phys. Rev. Lett. 92 (2004) 160402.
  * [26] M. W. J. Romans, R. A. Duine, S. Sachdev, and H. T. C. Stoof, Phys. Rev. Lett. 93 (2004) 020405;
  * [27] G. Santos, A. Tonel, A. Foerster, and J. Links, Phys. Rev. A 73 (2006) 023609; M. Duncan, A. Foerster, J. Links, E. Mattei, N. Oelkers, and A. P. Tonel, Nucl. Phys. B 767 (2007) 227; G. Santos, A. Tonel, A. Foerster, and J. Links, Phys. Rev. A 73 (2006) 023609.
  * [28] S. C. Li, J. Liu, and L. B. Fu, Phys. Rev. A 83, (2011) 042107; S. C. Li, and L. B. Fu, Phys. Rev. A 84, (2011) 023605.
  * [29] A. P. Hines, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 67, (2003) 013609.
  * [30] A. P. Tonel, J. Links, and A. Foerster, J. Phys. A:Math. Gen. 38, (2005) 1235; J. Links, H. Q. Zhou, R. H. McKenzie, and M. D. Gould, J. Phys. A 36 (2003) R63.