# The Formation of Photon-Molecules in Nanoscale Waveguides

Hashem Zoubi<EMAIL_ADDRESS>Department of Physics, Holon Institute of
Technology, Holon 5810201, Israel

(21 July, 2021)

###### Abstract

We investigate the formation of photon bound states in a system of interacting
photons inside nanoscale wires. The photons interact through the exchange of
vibrational modes induced along the waveguide mainly due to radiation
pressure. The problem of many-body photons is treated in using the formalism
of contour Green’s functions under the T-matrix approximation. The complex
pole of the T-matrix is a signature for the appearance of photon-molecules.
The analysis of such singularity provides the critical temperature at which
the T-matrix approximation breaks down and photon-molecules appear. For
strongly interacting slow photons the amplitude of the photon-molecule
wavefunction acquires a significant quantum nonlinear phase inside the
nanowire. Photon bound-states can be implemented for quantum information
processing as quantum logic gates, e.g. for $\pi$ phase shift the photon-
molecule is shown to serve as a Z-controlled gate.

##  I Introduction

The noninteracting nature of photons makes them efficient for long distance
communication but nonefficient for information processing. The interaction
among photons is the corner stone for their physical implementation in quantum
information processing and quantum logic gates [1]. Quantum nonlinear optics
involving single photons is a recent and hot topic with importance for
fundamental physics and applications [2], e.g., for photonic switches, optical
modulation and manipulation, the generation of single photons on demand,
nonlinear spectroscopy, memory devices and transistors, with impact on
physical and biological sciences [3, 4, 5, 6]. On the other hand, in classical
nonlinear optics propagating light in a medium can modify the optical
properties of the material, e.g., by producing intensity dependent refractive
index, which implies powerful lasers. Therefore, conventional nonlinear optics
found to be negligible at the level of individual photons [7, 8], and hence
the need for efficient new mechanisms is appealing.

Remarkable advances in the search for the realization of strong interactions
among single photons have been accumulated in the last decades. Cavity Quantum
Electrodynamics (QED) have been among the first experiments for achieving
effective photon-photon interactions by enhancing the light-matter coupling in
localizing atoms inside high finesse cavities [9, 10, 11, 6]. Cooling and
trapping single atoms within a cavity is a complex mission, hence cavity QED
experiments have been successfully extended into solid state systems including
quantum dots in semiconductors [12, 13, 14, 15] or nitrogen-vacancy centers in
diamond [16]. However, in a recent successful experiment a deterministic
photon-photon quantum gate has been realized for a single atom in an optical
resonator [17, 18]. Even though, the discreteness of the cavity spectrum gives
rise to output photons with narrow bands, besides dephasing and decay effects
of the excited electronic states, which put limitations on the quantum
nonlinear optics performance.

In order to overcome the limitations imposed by confining the photons in high
quality optical cavity, the search has been turned to strong atom-photon
coupling in cavity free environment [19, 20, 21, 22]. For example, in using
Rydberg atoms in a dense medium [23, 24, 25, 26, 27] by exploiting Rydberg
blockade phenomena [28] in the Electromagnetic Induced Transparency (EIT)
scheme [29, 30]. In such systems the significant enhancement of photon-photon
interactions is mainly due to the achievement of slow light using EIT
environment with extremely narrow transparency band [31]. Slowing and stopping
the light has been observed in cold and ultracold boson gases [3, 32, 33]. The
interaction between photons is exploited to demonstrate a photon–photon
quantum gate, extending the potential of Rydberg systems as a platform for
quantum communication and quantum networking [34].

In parallel, solid-state set-ups of optical fibers [35, 36, 37, 38] and
photonic crystals [39, 40, 41] have received significant interest, as they can
be easily integrated into all-optical on-chip platforms. In particular optical
fibers can realize tunable delays of optical signals with the possibility of
achieving fast and slow light in a comparatively wide bandwidth [42, 43, 44].
The most efficient nonlinear process inside optical fibers is Stimulated
Brillouin Scattering (SBS), that is the scattering of optical photons by long
lived acoustic phonons commonly induced by electrostriction [45]. Recent
progress in the fabrication of nanoscale waveguides, in which the wavelength
of the light becomes larger than the waveguide dimension, achieved a
breakthrough in SBS [46, 47, 48]. In this regime the coupling of photons and
phonons is significantly enhanced due to radiation pressure dominating over
electrostriction [49, 50, 51] with significant implications for the field of
continuum quantum optomechanics [52, 53, 54, 55].

We have explored the possibility of achieving a significant nonlinear phase
shift among photons propagating in nanoscale waveguides. The interaction among
photons is mediated by vibrational modes and induced through SBS, where an
effective photon-photon interaction Hamiltonian is derived [56]. Moreover, we
have introduced a configuration for slowing down photons by several orders of
magnitude via SBS involving sound waves and pump fields. We extracted the
conditions for maintaining vanishing amplitude gain or loss for slowly
propagating photons while keeping the influence of thermal phonons to the
minimum.

In the present paper we search for the possibility of the formation of exotic
photon molecules. Such issue has been addressed before in the context of an
atomic ensemble with Rydberg blockade where attractive photon-photon
interactions can be achieved [25, 57, 58]. Here we aim to examine the
possibility of the formation of two-photon bound states that is induced
through the exchange of phonons. To this end we exploit our previous results
where effective photon-photon coupling have been derived and that found to be
attractive or repulsive [51, 56]. We use the tool of contour Green’s functions
and we derive a hierarchy of equations that can be truncated by the appeal to
the T-matrix approximation [59, 60, 61, 62, 63]. The T-matrix approximation
allows us to treat the scattering of particles in many-body systems, that is
in a medium of interacting particles. The breakdown of the T-matrix
approximation, which appears as a singularity in the solution, indicates the
formation of bound states. We concentrate in the case of slow photons
propagating in one-dimensional nanoscale wires, and we look for the critical
temperature at which photon molecules form. The interactions among two
counter-propagating photons give rise to an accumulated quantum nonlinear
phase, and we show how such phase can be exploited in order to implement
photon molecules as quantum logic gates.

After presenting the Hamiltonian of interacting photons in section 2, we
introduce the contour Green’s functions in the complex plane, where the
functions allow considering quantum and ensemble averages on equal footing. In
section 3 we derive the equations of motion and solve them in applying the
T-matrix approximation. We treat the one-dimensional case where a complex pole
appears in the T-matrix that represents the appearance of photon molecules.
The photon bound states are discussed in section 4, and the equation of motion
of the state amplitude is solved to yield a quantum nonlinear phase that shown
to be useful for achieving quantum logic gates. The conclusion is given in
section 5. In the appendix we analyze the properties of two-point correlators
in Keldysh space.

##  II Interacting photons

We consider propagating optical photons in nanoscale waveguides (see Fig.1),
where the photons can propagate to the left or to the right with effective
group velocity $v_{e}$. In our previous work we introduced a configuration for
controlling the group velocity and to achieve relatively slow photons by
exploiting the coupling between photons and acoustic phonons that assisted by
additional pump fields [56]. The photons are shown to propagate without gain
or loss along the waveguide with a linear dispersion of
$\omega_{k}=\omega_{0}+v_{e}k$ in the appropriate region with wavenumber $k$,
and $\omega_{0}$ appears due to the transverse nanoscale confinement of
dimension in the range of hundred nanometers, (as seen in Fig.2). Slow photons
are necessary for achieving manifest phenomena of nonlinear quantum optics in
waveguides of several centimeter length [51]. The photons found to interact by
exchanging optical phonons (vibrational modes), that is the mechanism for
achieving photon bound states, and which is the main concern in the present
paper. The effective photon-photon coupling, $v$, is detunable and can be
positive or negative that indicates a repulsive or an attractive interaction
among the photons [56]. In typical nanoscale waveguides one can achieve
photon-photon coupling per meter of about $1$ MHz, and effective group
velocity of $10^{5}$ m/s, as have been shown by us in [56].

Figure 1: The nanoscale wire of radius $a$ of several hundreds of nanometers
and length of several centimeters. The fiber made of dielectric material with
refractive index $n$ larger than that of the surrounding air.

The total Hamiltonian is given by $\hat{H}=\hat{H}_{0}+\hat{H}_{I}$. The free
part Hamiltonian in real space is

$\hat{H}_{0}=\int d{\bf x}d{\bf x}^{\prime}\hat{\psi}^{\dagger}({\bf
x})\langle{\bf x}|\hat{h}|{\bf x}^{\prime}\rangle\hat{\psi}({\bf
x}^{\prime}),$ (1)

where $\langle{\bf x}|\hat{h}|{\bf x}^{\prime}\rangle=\delta({\bf x}-{\bf
x}^{\prime})h({\bf x},-i{\bf\nabla})$, and the interaction part Hamiltonian is

$\hat{H}_{I}=\frac{1}{2}\int d{\bf x}d{\bf x}^{\prime}~{}v({\bf x},{\bf
x}^{\prime})~{}\hat{\psi}^{\dagger}({\bf x})\hat{\psi}^{\dagger}({\bf
x}^{\prime})\hat{\psi}({\bf x}^{\prime})\hat{\psi}({\bf x}),$ (2)

where the interaction potential obeys $v({\bf x},{\bf x}^{\prime})=v({\bf
x}^{\prime},{\bf x})$. Here $\hat{\psi}({\bf x})$ and
$\hat{\psi}^{\dagger}({\bf x})$ are the field annihilation and creation
operators, respectively. For bosons the operators obey the commutation
relations

$\left[\hat{\psi}({\bf x}),\hat{\psi}({\bf
y})\right]=\left[\hat{\psi}^{\dagger}({\bf x}),\hat{\psi}^{\dagger}({\bf
y})\right]=0,\ \ \ \left[\hat{\psi}({\bf x}),\hat{\psi}^{\dagger}({\bf
y})\right]=\delta({\bf x}-{\bf y}).$ (3)

For linear dispersion we get $\langle{\bf x}|\hat{h}|{\bf
x}^{\prime}\rangle=\delta({\bf x}-{\bf
x}^{\prime})(\omega_{0}-iv_{e}{\bf\nabla})$, and the local interaction
potential is $v({\bf x},{\bf x}^{\prime})=v\delta({\bf x}-{\bf x}^{\prime})$.

Figure 2: The photon dispersion $\omega_{k}$ as a function of wavenumber $k$,
that includes two branches of photons propagating to the left and the right of
the nanoscale waveguide. Here $\omega_{0}$ is a minimum frequency appears due
to transverse confinement.

###  II.1 The Contour Green’s Functions

The many-particle system of interacting photons can be treated in using the
tool of Green’s functions [59, 60, 61, 62, 63]. We adopt a unified framework
of the contour formalism that allows us treating time-dependent problems and
statistical averages at finite temperature of thermal equilibrium. The contour
n-particle Green’s function is defined by [63]

$G_{n}(1,\cdots,n;1^{\prime},\cdots,n^{\prime})=\frac{1}{i^{n}}\frac{\text{Tr}\left[{\cal
T}\left\\{e^{-i\int_{\gamma}d\bar{z}\hat{H}(\bar{z})}\hat{\psi}_{H}(1)\cdots\hat{\psi}_{H}(n)\hat{\psi}_{H}^{\dagger}(n^{\prime})\cdots\hat{\psi}_{H}^{\dagger}(1^{\prime})\right\\}\right]}{\text{Tr}\left[{\cal
T}\left\\{e^{-i\int_{\gamma}d\bar{z}\hat{H}(\bar{z})}\right\\}\right]}.$ (4)

The integral is along the contour $\gamma$ that appears in figure (3) in the
complex plane, where a general point on the contour is denoted by $z$. The
horizontal part contains the forward branch $\gamma_{-}$ from time $t_{0}$ up
to time $t$, and the backward branch $\gamma_{+}$ from $t$ back to $t_{0}$,
along the real time axis. Then, a point on branch $\gamma_{-}$ at time
$t^{\prime}$ is denoted by $z^{\prime}=t^{\prime}_{-}$ and on branch
$\gamma_{+}$ by $z^{\prime}=t^{\prime}_{+}$. The horizontal part is extended
to infinity without affecting the result and is called the Schwinger-Keldysh
contour. The vertical part $\gamma_{M}$ of the contour starts at $z_{a}=t_{0}$
and end at $z_{b}=t_{0}-i\beta$ along the imaginary axis, with the constrain
$z_{b}-z_{a}=i\beta$, where at temperature $T$ we have
$\beta=\frac{1}{k_{B}T}$.

The field operators $\psi_{H}(j)$ and $\hat{\psi}_{H}^{\dagger}(j)$ are in the
contour Heisenberg picture, where we used the short notation $(j\equiv{\bf
x}_{j},z_{j})$. For an operator $\hat{O}(z)$ with explicit time dependent in
the contour Heisenberg picture we have
$\hat{O}_{H}(z)=\hat{U}(z_{\text{i}},z)\hat{O}(z)\hat{U}(z,z_{\text{i}})$,
where $z_{\text{i}}$ is the initial point of the contour, (and $z_{\text{f}}$
is the final point). The contour evolution operator, for $z_{2}$ later than
$z_{1}$, is defined by

$\hat{U}(z_{2},z_{1})={\cal
T}\left\\{e^{-i\int_{z_{1}}^{z_{2}}dz^{\prime}\hat{H}(z^{\prime})}\right\\},$
(5)

where the sign ${\cal T}$ stands for the contour time ordering. The result
holds also for the case of time independent operator $\hat{O}$, but we keep
the contour argument of the operator in order to specify its position on the
contour, especially under the contour time ordering ${\cal T}$.

The n-particle Green’s functions obey the boundary condition

$G_{n}(1,\cdots,{\bf
x}_{k},z_{\text{i}},\cdots,n;1^{\prime},\cdots,n^{\prime})=G_{n}(1,\cdots,{\bf
x}_{k},z_{\text{f}},\cdots,n;1^{\prime},\cdots,n^{\prime}),$ (6)

known as the Kubo-Martin-Schwinger relation.

Figure 3: The contour $\gamma\equiv\gamma_{-}\oplus\gamma_{+}\oplus\gamma^{M}$
in the complex plane is presented. The horizontal part along the real axis
contains two branches, the forward branch $\gamma_{-}$ from $t_{0}$ up to $t$,
and the backward branch $\gamma_{+}$ from $t$ back to $t_{0}$, where the
contour is extended to infinity. The vertical branch $\gamma_{M}$ is along the
imaginary axis from $t_{0}$ to $t_{0}-i\beta$.

On the horizontal part of the contour, if $z$ lies on the $\gamma_{-}$ and
$\gamma_{+}$ branches, in the n-particle Green’s function Eq.(4) we get the
time dependent quantum average. For a physical observable we have
$\hat{O}(z^{\prime}=t^{\prime}_{\pm})\equiv\hat{O}(t^{\prime})$, e.g. for the
Hamiltonian we have
$\hat{H}(z^{\prime}=t^{\prime}_{\pm})\equiv\hat{H}(t^{\prime})$. For the field
operators in the Schrodinger picture we have $\hat{\psi}({\bf
x},z^{\prime}=t_{\pm})\equiv\hat{\psi}({\bf x})$ and
$\hat{\psi}^{\dagger}({\bf
x},z^{\prime}=t_{\pm})\equiv\hat{\psi}^{\dagger}({\bf x})$, and in the
Heisenberg picture we have $\hat{\psi}_{H}({\bf
x},z)\rightarrow\hat{\psi}_{H}({\bf x},t)$ and $\hat{\psi}_{H}^{\dagger}({\bf
x},z)\rightarrow\hat{\psi}_{H}^{\dagger}({\bf x},t)$. Moreover, concerning our
previous Hamiltonian we have $h(z=t_{\pm})=h(t)$ and $v({\bf x},{\bf
x}^{\prime},z=t_{\pm})=v({\bf x},{\bf x}^{\prime},t)$.

If $z$ lies on $\gamma_{M}$, the vertical part of the contour, in the
n-particle Green’s function Eq.(4) we get the ensemble average at thermal
equilibrium. Now $\hat{O}(z)$ is the same at every point, and we use
$\hat{O}^{M}\equiv\hat{O}(z\in\gamma^{M})$, where we choose
$\hat{O}^{M}=\hat{O}(t_{0})$. For the Hamiltonian we have
$\hat{H}^{M}=\hat{H}(z\in\gamma^{M})$, where
$\hat{H}^{M}=\hat{H}(t_{0})-\mu\hat{N}$, with the number operator
$\hat{N}=\int d{\bf x}~{}\hat{\psi}^{\dagger}({\bf x})\hat{\psi}({\bf x})$,
and $\mu$ is the chemical potential. The field operators on the contour
$\gamma_{M}$ are constant, then we have $\hat{\psi}({\bf
x},z\in\gamma^{M})\equiv\hat{\psi}({\bf x})$ and $\hat{\psi}^{\dagger}({\bf
x},z\in\gamma^{M})\equiv\hat{\psi}^{\dagger}({\bf x})$. In our case of the
above Hamiltonian we have $h(z\in\gamma^{M})=h^{M}$, where $h^{M}=h-\mu$, and
$v({\bf x},{\bf x}^{\prime},z\in\gamma^{M})=v^{M}({\bf x},{\bf x}^{\prime})$.

The one-particle Green’s function, $G(1;1^{\prime})\equiv
G_{1}(1;1^{\prime})$, equation of motion reads

$\left[i\frac{d}{dz_{1}}-h(1)\right]G(1;1^{\prime})=\delta(1;1^{\prime})+i\int
d2~{}v(1;2)G_{2}(1,2;1^{\prime},2^{+}),$ (7)

and the two-particle Green’s function equation of motion is

$\displaystyle\left[i\frac{d}{dz_{1}}-h(1)\right]G_{2}(1,2;1^{\prime},2^{\prime})$
$\displaystyle=$
$\displaystyle\delta(1;1^{\prime})G(2;2^{\prime})\pm\delta(1;2^{\prime})G(2;1^{\prime})$
$\displaystyle+$ $\displaystyle i\int
d3~{}v(1;3)G_{3}(1,2,3;1^{\prime},2^{\prime},3^{+}),$

and so on for higher order Green’s functions. We get a system of coupled
differential equations that is known as the Martin-Schwinger hierarchy for the
Green’s functions. The delta function is defined by
$\delta(j;k)\equiv\delta(z_{j};z_{k})\delta({\bf x}_{j}-{\bf x}_{k})$, where
$\delta(z,z^{\prime})$ is zero everywhere except in $z=z^{\prime}$. On the
contour $\gamma$ the $\delta$-function is zero if $z$ and $z^{\prime}$ lie on
different branches, namely $\delta(t_{\pm},t_{\mp})=0$. Due to the orientation
of the contour we have $\delta(t_{-},t^{\prime}_{-})=\delta(t-t^{\prime})$ and
$\delta(t_{+},t^{\prime}_{+})=-\delta(t-t^{\prime})$ where
$\delta(t,t^{\prime})$ is the real axis $\delta$-function. On the vertical
branch we get
$\delta(t_{0}-i\tau,t_{0}-i\tau^{\prime})=i\delta(\tau-\tau^{\prime})$. We
have for the diagonal Hamiltonian $\langle{\bf x}_{j}|\hat{h}(z_{j})|{\bf
x}_{k}\rangle=\delta({\bf x}_{j}-{\bf x}_{k})h(j)$, with $h(j)=h({\bf
x}_{j},-i{\bf\nabla}_{j},z_{j})$. Note that $h(z)=h^{M}=h-\mu$ on the vertical
part for $z=t_{0}-i\tau$, and $h(z)=h$ on the horizontal branches for
$z=t_{\pm}$. We introduced $v(j;k)\equiv\delta(z_{j};z_{k})v({\bf x}_{j},{\bf
x}_{k},z_{j})$. Explicitly we get $v({\bf x},z;{\bf
x}^{\prime},z^{\prime})=\delta(z,z^{\prime})v({\bf x},{\bf x}^{\prime},t)$ on
the horizontal branches of $\gamma$ with $z=t_{\pm}$, and $v({\bf x},z;{\bf
x}^{\prime},z^{\prime})=\delta(z,z^{\prime})v^{M}({\bf x},{\bf x}^{\prime})$
on the vertical part of $\gamma$. Furthermore, in the notation $(j^{+}={\bf
x}_{j},z_{j}^{+})$ the $z_{j}^{+}$ is infinitesimally later than $z_{j}$, (and
in $(j^{-}={\bf x}_{j},z_{j}^{-})$ the $z_{j}^{-}$ is infinitesimally earlier
than $z_{j}$). The analytical properties of two point correlators that belong
to the Keldysh space is presented in the appendix.

##  III The T-Matrix approximation

We obtain an infinite hierarchy of equations in which the equation of motion
for $G_{n}$ is related to $G_{n+1}$ and $G_{n-1}$. The hierarchy can be
truncated by appealing to appropriate approximations. We apply the T-matrix
approximation that holds for short range interactions in the limit of low
density particles. The two-particle Green’s function can be written as

$G_{2}(1,2;1^{\prime},2^{\prime})=G(1;1^{\prime})G(2;2^{\prime})\pm
G(1;2^{\prime})G(2;1^{\prime})+\Gamma(1,2;1^{\prime},2^{\prime}),$ (9)

where the first two terms give the Hartree-Fock approximation, and the last
term includes the $\Gamma$ correlation function. To the lowest order in the
interaction, the solution for the correlation function yields

$\Gamma(1,2;1^{\prime},2^{\prime})\approx i\int
d3d4~{}G_{0}(1;3)G_{0}(2;4)v(3;4)G_{2}(3,4;1^{\prime},2^{\prime}),$ (10)

where the one-particle non-interacting Green’s function, $G_{0}$, obeys
$\left[i\frac{d}{dz_{1}}-h(1)\right]G_{0}(1;1^{\prime})=\delta(1;1^{\prime})$.
The $G_{2}$ can be written as

$G_{2}(1,2;1^{\prime},2^{\prime})=\int
d3d4~{}S(1,2;3,4)\left[G(3;1^{\prime})G(4;2^{\prime})\pm
G(3;2^{\prime})G(4;1^{\prime})\right],$ (11)

where we define the $S$ function by

$S(1,2;3,4)=\delta(1;3)\delta(2;4)+i\int
d5d6~{}G_{0}(1;5)G_{0}(2;6)v(5;6)S(5,6;3;4).$ (12)

The $T_{0}$ matrix is defined by $T_{0}(1,2;1^{\prime},2^{\prime})\equiv
v(1;2)S(1,2;1^{\prime},2^{\prime})$. Multiplying $G_{2}$ by $v$ we get

$v(1;2)G_{2}(1,2;1^{\prime},2^{\prime})=\int
d3d4~{}T_{0}(1,2;3,4)\left[G(3;1^{\prime})G(4;2^{\prime})\pm
G(3;2^{\prime})G(4;1^{\prime})\right].$ (13)

Using the above definition of $S$, we obtain

$T_{0}(1,2;1^{\prime},2^{\prime})=\delta(1;1^{\prime})\delta(2;2^{\prime})v(1^{\prime},2^{\prime})+i\int
d3d4~{}T_{0}(1,2;3,4)G_{0}(3;1^{\prime})G_{0}(4;2^{\prime})v(1^{\prime};2^{\prime}).$
(14)

The T-matrix is a tool for studying the scattering of particles in quantum
mechanics. Here we use it in order to study the scattering of particles in
many-body problems, namely quantum scattering among particles in a medium of
interacting particles [59].

The T-matrix obeys the relation
$T(1,2;1^{\prime},2^{\prime})=\delta(z_{1},z_{2})\delta(z^{\prime}_{1},z^{\prime}_{2})T({\bf
x}_{1},{\bf x}_{2},z_{1};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2},z^{\prime}_{1})$, where we dropped the $0$ from the T-matrix,
and in using the interaction property $v(j;k)=\delta(z_{j},z_{k})v({\bf
x}_{j},{\bf x}_{k})$, we get

$\displaystyle T({\bf x}_{1},{\bf x}_{2},z_{1};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2},z^{\prime}_{1})=\delta(z_{1},z^{\prime}_{1})\delta({\bf
x}_{1}-{\bf x}^{\prime}_{1})\delta({\bf x}_{2}-{\bf x}^{\prime}_{2})v({\bf
x}_{1},{\bf x}_{2})$ $\displaystyle+$ $\displaystyle\int d{\bf x}_{3}d{\bf
x}_{4}\int dz_{3}~{}T({\bf x}_{1},{\bf x}_{2},z_{1};{\bf x}_{3},{\bf
x}_{4},z_{3}){\cal G}_{2}({\bf x}_{3},{\bf x}_{4},z_{3};{\bf
x}^{\prime}_{1},{\bf x}^{\prime}_{2},z^{\prime}_{1})v({\bf
x}^{\prime}_{1};{\bf x}^{\prime}_{2}),$

where ${\cal G}_{2}({\bf x}_{3},{\bf x}_{4},z_{3};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2},z^{\prime}_{1})=iG({\bf x}_{3},z_{3};{\bf
x}^{\prime}_{1},z^{\prime}_{1})G({\bf x}_{4},z_{3};{\bf
x}^{\prime}_{2},z^{\prime}_{1})$. Note that the one-particle Green’s function
is the non-interacting one $G_{0}$. At thermal equilibrium the functions
depend only on the time difference, and the Fourier transform yields

$\displaystyle T({\bf x}_{1},{\bf x}_{2};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega)=\delta({\bf x}_{1}-{\bf x}^{\prime}_{1})\delta({\bf
x}_{2}-{\bf x}^{\prime}_{2})v({\bf x}_{1},{\bf x}_{2})$ $\displaystyle+$
$\displaystyle\int d{\bf x}_{3}d{\bf x}_{4}~{}T({\bf x}_{1},{\bf x}_{2};{\bf
x}_{3},{\bf x}_{4};\omega){\cal G}_{2}({\bf x}_{3},{\bf x}_{4};{\bf
x}^{\prime}_{1},{\bf x}^{\prime}_{2};\omega)v({\bf x}^{\prime}_{1};{\bf
x}^{\prime}_{2}).$

The T-matrix belong to the Keldysh space and contains a singular part (as
presented in the appendix). We treat now the retarded and advanced Keldysh
components of the T-matrix. We have

$T^{R/A}({\bf x}_{1},{\bf x}_{2};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega)=\delta({\bf x}_{1}-{\bf x}^{\prime}_{1})\delta({\bf
x}_{2}-{\bf x}^{\prime}_{2})v({\bf x}_{1},{\bf
x}_{2})+\int\frac{d\omega^{\prime}}{2\pi}\frac{{\cal B}({\bf x}_{1},{\bf
x}_{2};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega^{\prime})}{\omega-\omega^{\prime}\pm i\eta},$ (17)

where ${\cal B}({\bf x}_{1},{\bf x}_{2};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega)=i\left[T^{>}({\bf x}_{1},{\bf x}_{2};{\bf
x}^{\prime}_{1},{\bf x}^{\prime}_{2};\omega)-T^{<}({\bf x}_{1},{\bf
x}_{2};{\bf x}^{\prime}_{1},{\bf x}^{\prime}_{2};\omega)\right]$. Moreover, we
get the fluctuation-dissipation theorem $T^{>}({\bf x}_{1},{\bf x}_{2};{\bf
x}^{\prime}_{1},{\bf x}^{\prime}_{2};\omega)=e^{\beta(\omega-2\mu)}T^{<}({\bf
x}_{1},{\bf x}_{2};{\bf x}^{\prime}_{1},{\bf x}^{\prime}_{2};\omega)$. We
obtain $T^{<}({\bf x}_{1},{\bf x}_{2};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega)=-if(\omega-2\mu){\cal B}({\bf x}_{1},{\bf x}_{2};{\bf
x}^{\prime}_{1},{\bf x}^{\prime}_{2};\omega)$ and $T^{>}({\bf x}_{1},{\bf
x}_{2};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega)=-i\bar{f}(\omega-2\mu){\cal B}({\bf x}_{1},{\bf
x}_{2};{\bf x}^{\prime}_{1},{\bf x}^{\prime}_{2};\omega)$ with
$f(\epsilon)=\frac{1}{e^{\beta\epsilon}-1}$ and
$,\bar{f}(\epsilon)=1+f(\epsilon)$. In similar way, for ${\cal G}_{2}$ we get

${\cal G}_{2}^{R/A}({\bf x}_{3},{\bf x}_{4};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega)=i\int\frac{d\omega^{\prime}}{2\pi}\frac{{\cal
G}_{2}^{>}({\bf x}_{3},{\bf x}_{4};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega^{\prime})-{\cal G}_{2}^{<}({\bf x}_{3},{\bf x}_{4};{\bf
x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\omega^{\prime})}{\omega-\omega^{\prime}\pm i\eta}.$ (18)

The retarded ${\cal G}_{2}$ function is defined by

${\cal G}_{2}^{R}({\bf x}_{3},{\bf x}_{4};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};t_{3}-t^{\prime}_{1})=\theta(t_{3}-t^{\prime}_{1})\left\\{{\cal
G}_{2}^{>}({\bf x}_{3},{\bf x}_{4};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};t_{3}-t^{\prime}_{1})-{\cal G}_{2}^{<}({\bf x}_{3},{\bf
x}_{4};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};t_{3}-t^{\prime}_{1})\right\\},$ (19)

where as ${\cal G}_{2}^{\lessgtr}({\bf x}_{3},{\bf x}_{4};{\bf
x}^{\prime}_{1},{\bf x}^{\prime}_{2};t_{3}-t^{\prime}_{1})=iG^{\lessgtr}({\bf
x}_{3};{\bf x}^{\prime}_{1};t_{3}-t^{\prime}_{1})G^{\lessgtr}({\bf x}_{4};{\bf
x}^{\prime}_{2};t_{3}-t^{\prime}_{1})$, the Fourier transform gives

${\cal G}_{2}^{R}({\bf x}_{3},{\bf x}_{4};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\zeta)=i^{2}\int\frac{d\omega^{\prime}}{2\pi}\frac{d\omega^{\prime\prime}}{2\pi}\frac{\left\\{G^{>}({\bf
x}_{3};{\bf x}^{\prime}_{1};\omega^{\prime})G^{>}({\bf x}_{4};{\bf
x}^{\prime}_{2};\omega^{\prime\prime})-G^{<}({\bf x}_{3};{\bf
x}^{\prime}_{1};\omega^{\prime})G^{<}({\bf x}_{4};{\bf
x}^{\prime}_{2};\omega^{\prime\prime})\right\\}}{\zeta-\omega^{\prime}-\omega^{\prime\prime}},$
(20)

where $\zeta=\omega+i\eta$. The retarded T-matrix obeys

$\displaystyle T^{R}({\bf x}_{1},{\bf x}_{2};{\bf x}^{\prime}_{1},{\bf
x}^{\prime}_{2};\zeta)=\delta({\bf x}_{1}-{\bf x}^{\prime}_{1})\delta({\bf
x}_{2}-{\bf x}^{\prime}_{2})v({\bf x}_{1},{\bf x}_{2})$ $\displaystyle+$
$\displaystyle\int d{\bf x}_{3}d{\bf x}_{4}~{}T^{R}({\bf x}_{1},{\bf
x}_{2};{\bf x}_{3},{\bf x}_{4};\zeta){\cal G}_{2}^{R}({\bf x}_{3},{\bf
x}_{4};{\bf x}^{\prime}_{1},{\bf x}^{\prime}_{2};\zeta)v({\bf
x}^{\prime}_{1};{\bf x}^{\prime}_{2}).$

The retarded T-matrix is of big interest as it directly connected to the
scattering amplitude.

###  III.1 Momentum-Space Representation

As a first step, due to transnational symmetry, we change variables and use
the center of mass and relative positions, ${\bf X}=\frac{{\bf x}_{1}+{\bf
x}_{2}}{2}$ and ${\bf x}={\bf x}_{1}-{\bf x}_{2}$, to get

$\displaystyle T^{R}({\bf x},{\bf X};{\bf x}^{\prime},{\bf
X}^{\prime};\zeta)=\delta({\bf X}-{\bf X}^{\prime})\delta({\bf x}-{\bf
x}^{\prime})v({\bf x}^{\prime})$ $\displaystyle+$ $\displaystyle\int d\bar{\bf
x}d\bar{\bf X}~{}T^{R}({\bf x},{\bf X};\bar{\bf x},\bar{\bf X};\zeta){\cal
G}_{2}^{R}(\bar{\bf x},\bar{\bf X};{\bf x}^{\prime},{\bf
X}^{\prime};\zeta)v({\bf x}^{\prime}).$

Now we define the center of mass and relative momentum by ${\bf P}={\bf
p}_{1}+{\bf p}_{2}$ and ${\bf p}=\frac{{\bf p}_{1}-{\bf p}_{2}}{2}$,
respectively. The Fourier transform and its inverse are defined by

${\cal O}({\bf x})=\int\frac{d{\bf p}}{(2\pi)^{3}}e^{i{\bf p}\cdot{\bf
x}}{\cal O}({\bf p}),\ \ \ {\cal O}({\bf p})=\int d{\bf x}e^{-i{\bf
p}\cdot{\bf x}}{\cal O}({\bf x}),$ (23)

and we use the identity $\delta({\bf x})=\int\frac{d{\bf
p}}{(2\pi)^{3}}e^{i{\bf p}\cdot{\bf x}}$ and its inverse $\delta({\bf p})=\int
d{\bf x}e^{-i{\bf p}\cdot{\bf x}}$. The Fourier transform of the T-matrix
equation gives

$T^{R}({\bf p};{\bf p}^{\prime};{\bf P},\zeta)=v({\bf p}-{\bf
p}^{\prime})+\int\frac{d\bar{\bf p}}{(2\pi)^{3}}\frac{d\bar{\bf
p}^{\prime}}{(2\pi)^{3}}~{}T^{R}({\bf p};\bar{\bf p};{\bf P},\zeta){\cal
G}_{2}^{R}(\bar{\bf p};\bar{\bf p}^{\prime};{\bf P},\zeta)v(\bar{\bf
p}^{\prime}-{\bf p}^{\prime}).$ (24)

Here ${\bf p}$ is the initial momentum of one particle in the center of mass
system, and ${\bf p}^{\prime}$ is the final momentum of the scattered
particle. The center of mass momentum is conserved, where the initial center
of mass momentum ${\bf P}$ equals the final one ${\bf P^{\prime}}$, (that is
${\bf P}={\bf P}^{\prime}$). The momentum space coupling potential is

$v({\bf p}-{\bf p}^{\prime})=\int d{\bf r}^{\prime}e^{-i({\bf p}-{\bf
p}^{\prime})\cdot{\bf x}^{\prime}}v({\bf x}^{\prime}),$ (25)

and the ${\cal G}_{2}^{R}$ is ${\cal G}_{2}^{R}(\bar{\bf p},\bar{\bf
p}^{\prime};{\bf P},\zeta)=\delta(\bar{\bf p}-\bar{\bf
p}^{\prime})\Upsilon(\bar{\bf p};{\bf P}\zeta)$, where

$\displaystyle\Upsilon(\bar{\bf p};{\bf P},\zeta)$ $\displaystyle=$
$\displaystyle
i^{2}\int\frac{d\omega^{\prime}}{2\pi}\frac{d\omega^{\prime\prime}}{2\pi}\left\\{\frac{G^{>}(\frac{\bf
P}{2}+\bar{\bf p};\omega^{\prime})G^{>}(\frac{\bf P}{2}-\bar{\bf
p};\omega^{\prime\prime})}{\zeta-\omega^{\prime}-\omega^{\prime\prime}}\right.$
$\displaystyle-$ $\displaystyle\left.\frac{G^{<}(\frac{\bf P}{2}+\bar{\bf
p};\omega^{\prime})G^{<}(\frac{\bf P}{2}-\bar{\bf
p};\omega^{\prime\prime})}{\zeta-\omega^{\prime}-\omega^{\prime\prime}}\right\\}.$

We have

$T^{R}({\bf p};{\bf p}^{\prime};{\bf P},\zeta)=v({\bf p}-{\bf
p}^{\prime})+\int\frac{d\bar{\bf p}}{(2\pi)^{3}}~{}T^{R}({\bf p};\bar{\bf
p};{\bf P},\zeta)\Upsilon(\bar{\bf p};{\bf P},\zeta)v(\bar{\bf p}-{\bf
p}^{\prime}).$ (27)

We assume local interaction, in which $v({\bf x})=v\delta({\bf x})$, and the
Fourier transform gives $v({\bf p}-{\bf p}^{\prime})=v$. Now, in the center of
mass, the scattering is elastic where only the particle direction changes with
$|{\bf p}|=|{\bf p}^{\prime}|$. The T-matrix is depends only on the center of
mass momentum, where $T^{R}({\bf p};{\bf p}^{\prime};{\bf P},\zeta)\equiv
T^{R}({\bf P},\zeta)$. We achieve

$T^{R}({\bf P},\zeta)=\frac{v}{1-v\int\frac{d\bar{\bf
p}}{(2\pi)^{3}}~{}\Upsilon(\bar{\bf p};{\bf P},\zeta)}.$ (28)

Let us assume a small center of mass momentum. We can consider ${\bf P}\approx
0$, where we have counter-propagating particles such that the total momentum
is small, hence we have $T^{R}(\zeta)=\frac{v}{1-v\int\frac{d\bar{\bf
p}}{(2\pi)^{3}}~{}\Upsilon(\bar{\bf p};\zeta)}$. The greater and lesser one-
particle Green’s functions are $G^{<}(\bar{\bf p};\omega)=-if(\omega){\cal
A}(\bar{\bf p};\omega)$ and $G^{>}(\bar{\bf p};\omega)=-i\bar{f}(\omega){\cal
A}(\bar{\bf p};\omega)$, respectively, where
$f(\omega)=\frac{1}{e^{\beta(\omega-\mu)}-1}$ and
$\bar{f}(\omega)=1+f(\omega)=e^{\beta(\omega-\mu)}f(\omega)$. For non-
interacting spectral function we have ${\cal A}({\bf
p};\omega)=2\pi\delta(\omega-\epsilon({\bf p}))$, where $\epsilon({\bf p})$ is
the free particle dispersion. Using the dispersion symmetry $\epsilon({\bf
p})=\epsilon(-{\bf p})$, we get $\Upsilon(\bar{\bf
p};\zeta)=\frac{1+2f(\epsilon(\bar{\bf p}))}{\zeta-2\epsilon(\bar{\bf p})}$,
and we obtain $1+2f(\epsilon(\bar{\bf p}))=\coth\left(\beta\frac{\epsilon({\bf
p})-\mu}{2}\right)$. Finally we reach

$T^{R}(\zeta)=\frac{v}{1-v\int\frac{d{\bf
p}}{(2\pi)^{3}}~{}\frac{\coth\left(\beta\frac{\epsilon({\bf
p})-\mu}{2}\right)}{\zeta-2\epsilon({\bf p})}}.$ (29)

###  III.2 One-Dimensional Attractive Photons

We consider now interacting photons propagating in one-dimensional nanoscale
waveguide. The coupling parameter is negative for attractive interaction,
where we make the change $v\rightarrow-v$, then in terms of the wavenumber

$T^{R}(\zeta)=\frac{-v}{1+v\int\frac{dk}{2\pi}~{}\frac{\coth\left(\beta\frac{\epsilon(k)-\mu}{2}\right)}{\zeta-2\epsilon(k)}}.$
(30)

The linear dispersion is $\epsilon(k)=\epsilon_{0}+v_{e}k$, where
$\epsilon_{0}$ is the minimum energy that appears due to transverse
confinement, and $v_{e}$ is the photon effective group velocity. The
interaction is limited to an energy band around the chemical potential where
$|\epsilon(k)-\mu|<\Delta$. The band width is taken to be of the order of the
phonon frequency $\Omega$. We change into the variable
$\epsilon=\epsilon(k)-\mu$, where $d\epsilon=v_{e}dk$, and we get

$T^{R}(\zeta)=\frac{-v}{1+\frac{v}{2\pi
v_{e}}\int_{-\Delta}^{+\Delta}d\epsilon~{}\frac{\coth\left(\beta\epsilon/2\right)}{\zeta-2\mu-2\epsilon}}.$
(31)

We evaluate the integral at the imaginary value of $\zeta-2\mu=i\eta$, where
in the limit $\eta\rightarrow 0$ we obtain

$T^{R}=\frac{-v}{1-g\int_{0}^{\Lambda}dx~{}\frac{\coth x}{x}},$ (32)

with $\Lambda=\beta\Delta/2$ and $g=\frac{v}{2\pi v_{e}}$, after making the
change of variable of $x=\beta\epsilon/2$.

The T-matrix measures the probability amplitude for adding a pair of particles
into the system and afterward removing a pair. A complex pole in the upper
half plane indicates that if a pair of particles with opposite momenta are
added at a certain time, the probability amplitude for removing such a pair
increases exponentially in time, and the T-matrix approximation breakdown. The
appearance of complex poles signals the formation of bound states (photon
molecules). In the limit of high temperature $T\rightarrow\infty$, where
$\beta\rightarrow 0$ and $x\rightarrow 0$, we have
$1<g\int_{0}^{\Lambda}dx\frac{\coth x}{x}$, as $\coth x\rightarrow\infty$.
Hence the T matrix contains no complex poles. On the other hand, in the limit
of low temperature $T\rightarrow 0$, where $\beta\rightarrow\infty$ and
$x\rightarrow\infty$, we have $1\geq g\int_{0}^{\Lambda}dx\frac{\coth x}{x}$,
as $\coth x\rightarrow 1$. Hence at low temperature complex poles can appear
in the T-matrix.

The critical temperature for the formation of bound states can be estimated
from the equality $1=g\int_{0}^{\Lambda}dx\frac{\coth x}{x}$, when the
denominator of the T-matrix vanishes. We are in the limit of $\Delta\ll\beta$,
and the integral is taken in a region where $\coth x\approx 1$. As we are in
the limit of low temperature one can integrate in the region of $x$ between
$1$ and $\Lambda$, where $1\approx g\int_{1}^{\Lambda}dx\frac{1}{x}$. Then we
get $1=g\ln\Lambda$, where we can write $\frac{2\pi
v_{e}}{v}=\ln\frac{\Delta}{2k_{B}T_{c}}$, that lead to the result of

$k_{B}T_{c}=\frac{\Delta}{2}e^{-\frac{2\pi v_{e}}{v}}.$ (33)

We choose the numbers such that $v/v_{e}=4\pi$, and the band width is fixed by
the vibrational mode frequency of about $\Omega=40$ GHz, with
$\Delta=\hbar\Omega$ [56]. Then we have
$k_{B}T_{c}=\frac{\hbar}{\sqrt{e}}~{}20\text{GHz}$, where the critical
temperature is $T_{c}\approx 0.1$ K.

##  IV Photon Molecules

After demonstrating the possibility of the formation of photon molecules in
the previous section, we represent here the photon bound states and show how
to implement them for quantum logic gates. We start with the three dimensional
case and concentrate later in the one dimensional waveguide case. We consider
two counter-propagating photons, $(a)$ and $(b)$, that form a molecule of
wavevector ${\bf K}$ and is characterized by a normalized wavefunction
$\chi({\bf x}_{a}-{\bf x}_{b})$. The photon molecule wavefunction is defined
by

$\phi_{\bf K}({\bf x}_{a},{\bf x}_{b})=\frac{1}{\sqrt{V}}~{}e^{i{\bf
K}\cdot({\bf x}_{a}+{\bf x}_{b})/2}\chi({\bf x}_{a}-{\bf x}_{b}),$ (34)

where $V=L^{3}$ is the system volume. Using transnational symmetry, then the
Fourier transform reads $\tilde{\chi}_{\bf k}=\frac{1}{\sqrt{V}}\int
d^{3}x~{}e^{-i{\bf k}\cdot{\bf x}}\chi({\bf x})$ with the inverse transform
$\chi({\bf x})=\frac{1}{\sqrt{V}}\sum_{\bf k}\tilde{\chi}_{\bf k}~{}e^{i{\bf
k}\cdot{\bf x}}$, to get

$\phi_{\bf K}({\bf x}_{a},{\bf x}_{b})=\frac{1}{V}\sum_{\bf
k}\tilde{\chi}_{\bf k}~{}e^{i\left(\frac{\bf K}{2}+{\bf k}\right)\cdot{\bf
x}_{a}}e^{i\left(\frac{\bf K}{2}-{\bf k}\right)\cdot{\bf x}_{b}}.$ (35)

We use the periodic boundary condition, then the wavenumber ${\bf
k}=(k_{x},k_{y},k_{z})$ takes the values $k_{i}=2\pi n_{i}/L$ with
$(i=x,y,z)$, and $n_{i}$ are integers. The normalization condition is given by
$\int d^{3}x\left|\chi({\bf x})\right|^{2}=\sum_{\bf k}\left|\tilde{\chi}_{\bf
k}\right|^{2}=1$.

The ket eigenstate is given by $|\phi_{\bf K}(a,b)\rangle=\sum_{\bf
k}\tilde{\chi}_{\bf k}\left|\left.a,\left(\frac{\bf K}{2}+{\bf
k}\right);b,\left(\frac{\bf K}{2}-{\bf k}\right)\right.\right\rangle$. The ket
can be created from the vacuum using the second quantization operators of
Fock’s space, by $|\phi_{\bf K}\rangle=\sum_{\bf k}\tilde{\chi}_{\bf
k}~{}\hat{a}_{\frac{\bf K}{2}+{\bf k}}^{\dagger}\hat{a}_{\frac{\bf K}{2}-{\bf
k}}^{\dagger}|\text{vac}\rangle$. We define the pair creation operator of a
molecule having a total wavenumber ${\bf K}$ by $\hat{A}_{\bf
K}^{\dagger}=\sum_{\bf k}\tilde{\chi}_{\bf k}~{}\hat{a}_{\frac{\bf K}{2}+{\bf
k}}^{\dagger}\hat{a}_{\frac{\bf K}{2}-{\bf k}}^{\dagger}$. The creation
operator is related to the field operator by $\hat{a}^{\dagger}_{\bf
k}=\frac{1}{\sqrt{V}}\int d^{2}x~{}e^{i{\bf k}\cdot{\bf
x}}\hat{\psi}^{\dagger}({\bf x})$, which lead to

$\hat{A}_{{\bf K}}^{\dagger}=\frac{1}{\sqrt{V}}\int d^{3}X~{}e^{i{\bf
K}\cdot{\bf X}}\int d^{3}x~{}{\chi}({\bf x})~{}\hat{\psi}^{\dagger}({\bf
X}+{\bf x}/2)\hat{\psi}^{\dagger}({\bf X}-{\bf x}/2).$ (36)

The pair field operator can be defined as $\hat{\Psi}^{\dagger}({\bf
X})=\frac{1}{\sqrt{V}}\sum_{\bf K}e^{-i{\bf K}\cdot{\bf X}}\hat{A}_{{\bf
K}}^{\dagger}$. We obtain, using the result $\sum_{\bf K}e^{i{\bf
K}\cdot\left({\bf X^{\prime}}-{\bf X}\right)}=V\delta({\bf X}-{\bf
X^{\prime}})$, the pair field creation operator

$\hat{\Psi}^{\dagger}({\bf X})=\int d^{3}x~{}{\chi}({\bf
x})~{}\hat{\psi}^{\dagger}({\bf X}+{\bf x}/2)\hat{\psi}^{\dagger}({\bf X}-{\bf
x}/2).$ (37)

###  IV.1 Quantum Nonlinear Phase

The wavefunction of a photon molecule build of two counter-propagating pair of
photons can accumulate a quantum nonlinear phase of the order of $\pi$ in the
appropriate condition. The phase is shown to be useful for the implementation
of the photon molecule as a quantum logic gate. We consider two counter
propagating photons, ($a$) and ($b$), of minimum frequency $\omega_{0}$, and
with effective group velocity $v_{e}$, inside one-dimensional nanoscale wires.
The free photon real space Hamiltonian is

$\hat{H}_{0}=\int dx\left\\{\omega_{0}\
\hat{\psi}_{a}^{\dagger}(x)\hat{\psi}_{a}(x)+iv_{e}\frac{\partial\hat{\psi}_{a}^{\dagger}(x)}{\partial
x}\hat{\psi}_{a}(x)+\omega_{0}\
\hat{\psi}_{b}^{\dagger}(x)\hat{\psi}_{b}(x)-iv_{e}\frac{\partial\hat{\psi}_{b}^{\dagger}(x)}{\partial
x}\hat{\psi}_{b}(x)\right\\},$ (38)

and the photon-photon interaction Hamiltonian of coupling parameter $v$ is
given by

$\hat{H}_{I}=\frac{v}{2}\int dxdx^{\prime}\ \delta(x-x^{\prime})\
\hat{\psi}_{a}^{\dagger}(x^{\prime})\hat{\psi}_{b}^{\dagger}(x)\hat{\psi}_{a}(x^{\prime})\hat{\psi}_{b}(x).$
(39)

The photon pair bound-state is given by

$|\Psi(t)\rangle=\int d(x_{1}-x_{2})\ \phi(x_{1}-x_{2},t)\
\hat{\psi}_{a}^{\dagger}(x_{1})\hat{\psi}_{b}^{\dagger}(x_{2})|\text{vac}\rangle.$
(40)

The state obeys the Schrodinger equation $i\frac{\partial}{\partial
t}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle$, and we get

$i\left\\{\frac{\partial}{\partial t}+v_{e}\left(\frac{\partial}{\partial
x_{1}}-\frac{\partial}{\partial
x_{2}}\right)\right\\}\phi(x_{1}-x_{2},t)=\left\\{2\omega_{0}+\frac{v}{2}\delta(x_{1}-x_{2})\right\\}\phi(x_{1}-x_{2},t).$
(41)

Moving into rotating frame by using
$\phi(x_{1}-x_{2},t)=\tilde{\phi}(x_{1}-x_{2},t)e^{-2i\omega_{0}t}$, gives

$i\left\\{\frac{\partial}{\partial t}+v_{e}\left(\frac{\partial}{\partial
x_{1}}-\frac{\partial}{\partial
x_{2}}\right)\right\\}\tilde{\phi}(x_{1}-x_{2},t)=\frac{v}{2}\delta(x_{1}-x_{2})\tilde{\phi}(x_{1}-x_{2},t).$
(42)

In term of the center of mass and relative positions,
$X=\frac{x_{1}+x_{2}}{2}$ and $x=x_{1}-x_{2}$, where

$\frac{\partial}{\partial X}=\frac{\partial}{\partial
x_{1}}+\frac{\partial}{\partial x_{2}},\ \ \ 2\frac{\partial}{\partial
x}=\frac{\partial}{\partial x_{1}}-\frac{\partial}{\partial x_{2}},$ (43)

we get

$i\left\\{\frac{\partial}{\partial t}+2v_{e}\frac{\partial}{\partial
x}\right\\}\tilde{\phi}(x,t)=\frac{v}{2}\delta(x)\tilde{\phi}(x,t).$ (44)

We apply the change of variables $\eta=x-2v_{e}t$ and $\xi=x$, where

$\frac{\partial}{\partial
X}=\frac{\partial}{\partial\eta}+\frac{\partial}{\partial\xi},\ \ \
\frac{\partial}{\partial t}=-2v_{e}\frac{\partial}{\partial\eta},$ (45)

and we obtain $\frac{\partial}{\partial t}+2v_{e}\frac{\partial}{\partial
x}=2v_{e}\frac{\partial}{\partial\xi}$, then the equation of motion

$\frac{\partial}{\partial\xi}\tilde{\phi}(\eta,\xi)=-i\frac{v}{4v_{e}}\delta(\xi)\tilde{\phi}(\eta,\xi),$
(46)

and the solution reads

$\tilde{\phi}(\eta,\xi)=\tilde{\phi}_{\text{in}}e^{-i\frac{v}{4v_{e}}\int
d\xi\delta(\xi)}=\tilde{\phi}_{\text{in}}e^{-i\frac{v}{4v_{e}}}.$ (47)

Table 1: Z-Controlled Gate Input State |  Output State
---|---
$|0,0\rangle$ | $\ |0,0\rangle$
$|1,0\rangle$ | $\ |1,0\rangle$
$|0,1\rangle$ | $\ |0,1\rangle$
$|1,1\rangle$ | -$|1,1\rangle$

Photon molecules can serve as a tool for the implementation of quantum logic
gates. We show how to achieve Z-controlled gate for the case of
$\vartheta=\pi$ where the nonlinear quantum phase is
$\vartheta=\frac{v}{4v_{e}}$, which is obtained for the previous value of
$v/v_{e}=4\pi$. If in the input we have zero photons at the two channels, that
is $|0,0\rangle$, the output is the same state of $|0,0\rangle$. If in the
input one channel includes one photon and the second is empty, that is
$|0,1\rangle$ or $|1,0\rangle$, then the output is again the same state,
$|0,1\rangle$ or $|1,0\rangle$. While for the case of two counter-propagating
photons, then the input state is $|1,1\rangle$, and now the output state
acquires a phase where $-|1,1\rangle$, as presented in table (1). The gate is
universal and in combination with photon Hadamard gates one can achieve all
requested quantum logic gates.

##  V Conclusions

Quantum nonlinear optics is a hot topic and found big interest in the recent
years for its importance to both fundamental physics and applications in
quantum information processing. Several proposals have been suggested recently
for the realization of quantum nonlinear optics and mainly for achieving
strongly interacting photons, while for each proposal exists the advantages
and disadvantages. Nanoscale structures are solid components and then can be
easily integrated into all-optical platforms, the fact that makes them very
attractive. Many-body physics of photons can be treated in using different
techniques that rest on approximations. In the present paper we adopted the
method of contour Green’s functions that permits one to extract the system
physical properties. We reached a hierarchy of equations of motion for the
Green’s functions, where the equation for one-particle Green’s function
depends on the equation for two-particle Green’s function and so on for higher
orders. The system is unsolvable exactly and we applied the T-matrix
approximation that allows the truncation of the equations, and that can be
solved for the scattering problem in a medium of interacting particles.

In the paper we considered interacting slow photons in nanoscale wires, where
we used our previously derived Hamiltonian for the effective photon-photon
interaction that is mediated by phonons. We calculated the T-matrix and found
the complex pole at which the approximation breaks down. The singularity in
the T-matrix is the signature for the formation of photon bound states and
provides the critical temperature at which such phenomena can appear. The
photon bound state is represented by a ket state that is defined through two-
particle creation operators. The equation of motion for the state amplitude is
solved and results in nonlinear quantum phase shift that can be of the order
of $\pi$ in the appropriate region of physical numbers. The photon molecules
are shown to act as quantum logic gates by exploiting the nonlinear phase
shift.

The present work can be extended into more interesting effects of many-body
physics of photons in nanostructures. The contour Green’s function is a strong
tool that allows us to study further interesting phenomena, e.g. Bose-Einstein
condensation and superfluidity of photons. Moreover, interacting photons in
nanoscale structures is an ideal environment for studying non-equilibrium
behavior, and the contour Green’s function is the optimal tool for achieving
this aim. The implementation of interacting photons for quantum information
processing is an important step toward introducing further nanophotonic
quantum information components. Our setup allows us to achieve quantum
information processing and communication in using hybridized components
involving interacting photons. The issue we presented here can be extended
into a wide range of quantum computing devices of photons.

## Appendix

We introduce here some analytical properties of the the general two point
correlator that belongs to the Keldysh space. The two-point correlator is
defined by ${\cal C}(z,z^{\prime})=\text{Tr}\left[\hat{\rho}{\cal
T}\left\\{\hat{O}_{1}(z)\hat{O}_{2}(z^{\prime})\right\\}\right]$ for the two
operators $\hat{O}_{1}(z)$ and $\hat{O}_{2}(z^{\prime})$, where the matrix
operator is given by $\hat{\rho}=\frac{e^{-\beta\hat{H}^{M}}}{Z}$, with the
partition function $Z=\text{Tr}\left\\{e^{-\beta\hat{H}^{M}}\right\\}$. In
Keldysh space we can write the general form

${\cal C}(z,z^{\prime})={\cal
C}^{\delta}\delta(z,z^{\prime})+\theta(z,z^{\prime}){\cal
C}^{>}(z,z^{\prime})+\theta(z^{\prime},z){\cal C}^{<}(z,z^{\prime}),$ (48)

where ${\cal C}^{\delta}$ is the singular part, and ${\cal
C}^{>}(z,z^{\prime})=\text{Tr}\left[\hat{\rho}\hat{O}_{1}(z)\hat{O}_{2}(z^{\prime})\right]$,
with ${\cal
C}^{<}(z,z^{\prime})=\text{Tr}\left[\hat{\rho}\hat{O}_{2}(z^{\prime})\hat{O}_{1}(z)\right]$.
On the backward and forward branches, as
$\hat{O}_{i}(t_{+})=\hat{O}_{i}(t_{-})$ for $(i=1,2)$, we have ${\cal
C}^{\lessgtr}(t_{+},z^{\prime})={\cal C}^{\lessgtr}(t_{-},z^{\prime})$, and
${\cal C}^{\lessgtr}(z,t^{\prime}_{+})={\cal C}^{\lessgtr}(z,t^{\prime}_{-})$.
Also we have ${\cal C}^{\delta}(t_{+})={\cal C}^{\delta}(t_{-})\equiv{\cal
C}^{\delta}(t)$. In general, we have $\theta(z,z^{\prime})=1$ for $z$ later
than $z^{\prime}$ on the contour, and zero otherwise. Moreover, we have
$\delta(z,z^{\prime})=\frac{d}{dz}\theta(z,z^{\prime})=-\frac{d}{dz^{\prime}}\theta(z,z^{\prime})$.
We now define different Keldysh components that are functions of real
variables. When both arguments on the horizontal branches, the greater and
lesser Keldysh components are ${\cal C}^{>}(t,t^{\prime})\equiv{\cal
C}(t_{+},t^{\prime}_{-})$, and ${\cal C}^{<}(t,t^{\prime})\equiv{\cal
C}(t_{-},t^{\prime}_{+})$. Namely, the value of the contour function ${\cal
C}^{\lessgtr}(z,z^{\prime})$ is the real-time function ${\cal
C}^{\lessgtr}(t,t^{\prime})$. Furthermore, we define the retarded and advanced
Keldysh components of real time arguments

$\displaystyle{\cal C}^{R}(t,t^{\prime})$ $\displaystyle\equiv$
$\displaystyle{\cal
C}^{\delta}\delta(t-t^{\prime})+\theta(t-t^{\prime})\left[{\cal
C}^{>}(t,t^{\prime})-{\cal C}^{<}(t,t^{\prime})\right],$ (49)
$\displaystyle{\cal C}^{A}(t,t^{\prime})$ $\displaystyle\equiv$
$\displaystyle{\cal
C}^{\delta}\delta(t-t^{\prime})-\theta(t^{\prime}-t)\left[{\cal
C}^{>}(t,t^{\prime})-{\cal C}^{<}(t,t^{\prime})\right].$

The retarded component vanishes for $t<t^{\prime}$, while the advanced
component vanishes for $t>t^{\prime}$. Dropping the singular part, the Fourier
transform of the correlator, using ${\cal
O}(t_{1}-t_{2})=\int\frac{d\omega}{2\pi}e^{-i\omega(t_{1}-t_{2})}{\cal
O}(\omega)$ and the Heaviside identity
$\theta(t_{1}-t_{2})=i\int\frac{d\omega}{2\pi}\frac{e^{-i\omega(t_{1}-t_{2})}}{\omega+i\eta}$,
gives

${\cal C}^{R/A}(\omega)=\int\frac{d\omega^{\prime}}{2\pi}\frac{\hat{\cal
A}(\omega^{\prime})}{\omega-\omega^{\prime}\pm i\eta},$ (50)

where we defined the spectral function $\hat{\cal A}(\omega)=i\left[{\cal
C}^{>}(\omega)-{\cal C}^{<}(\omega)\right]$ or $\hat{\cal
A}(\omega)=i\left[{\cal C}^{R}(\omega)-{\cal C}^{A}(\omega)\right]$. We get
the fluctuation-dissipation theorem ${\cal
C}^{<}(\omega)=-if(\omega-\mu)\hat{\cal A}(\omega)$ and ${\cal
C}^{>}(\omega)=-i\bar{f}(\omega-\mu)\hat{\cal A}(\omega)$, where for bosons
$f(\omega)=\frac{1}{e^{\beta\omega}+1}$ and $\bar{f}(\omega)=1\pm
f(\omega)=e^{\beta\omega}f(\omega)$, with the relation ${\cal
C}^{>}(\omega)=e^{\beta(\omega-\mu)}{\cal C}^{<}(\omega)$.

## Acknowledgment

The work was supported by the Council for Higher Education in Israel via the
Maa’of Fellowship.

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