# Electromagnetically Induced Transparencies with Two Transverse Bose-Einstein
Condensates in a Four-Mirror Cavity

Kashif Ammar Yasir<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal
University, Jinhua 321004, China.  Zhaoxin Liang  Department of Physics,
Zhejiang Normal University, Jinhua 321004, China.  Gao Xianlong  Department of
Physics, Zhejiang Normal University, Jinhua 321004, China.  Wu-Ming Liu
<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100190, China.  Songshan Lake Materials Laboratory, Dongguan,
Guangdong 523808, China.

###### Abstract

We investigate electromagnetically induced transparencies with two transverse
Bose-Einstein condensates in four-mirror optical cavity, driven by a strong
pump laser and a weak probe laser. The cavity mode, after getting split from
beam splitter, interacts with two independent Bose-Einstein Condensates
transversely trapped in the arms of the cavity along $x$-axis and $y$-axis.
The interaction of intra-cavity optical mode excites momentum side modes in
Bose-Einstein Condensates, which then mimic as two atomic mirrors coupled
through cavity field. We show that the probe field photons transition through
the atomic mirrors yields to two coupled electromagnetically induced
transparency windows, which only exist when both atomic states are coupled
with the cavity. Further, the strength of these novel electromagnetically
induced transparencies gets increased with an increase in atom-cavity
coupling. Furthermore, we investigate the behavior of Fano resonances and
dynamics of fast and slow light. We illustrate that the Fano line shapes and
dynamics of slow light can be enhanced by strengthening the interaction
between atomic states and cavity mode. Our findings not only contribute to the
quantum nonlinear optics of complex systems but also provide a platform to
test multi-dimensional atomic states in a single system.

##  I Introduction

Quantum nonlinear optics – a quantum interpretation of nonlinear interactions
of light at single photon level Ref1 ; Ref2 ; Ref3  – offers a stunning
platform to practically demonstrate information process with quantum
computation Ref4 ; Ref5 ; Ref6 . To demonstrate such controlled nonlinear
optics, strong and highly nonlinear photon-photon interactions are essential
Ref1 . The electromagnetically induced transparency (EIT) – a manifestation of
quantum interference occurring between two different transitional pathways of
photons from a single state Ref7  – is proven to be the best example of such
nonlinear quantum interactions Ref8 ; Ref9 . For this, the coherently driven
cavity quantum electrodynamics (cQED) provides an effective foundation to
engineer and manipulate the phenomena of quantum nonlinear optics, because of
the strong-coupling regime of light and matter Ref10 . This coherent and
strongly coupled environment of the cQED has enabled the researchers to
demonstrate the phenomenon of EIT in versatile configurations, like EIT with
ionic coulomb crystals Ref11 , with a single atom Ref12 ; Ref13 , and with
Rydberg blockades Ref14 ; Ref15 ; Ref16 . The crucial implication of quantum
interference occurs when they stop or slow-down Ref18  the transmitting light
in the EIT interval providing a platform for the creation of optical storage
devices Ref4 ; Ref7 .

The mechanical characteristics of light in optomechanical system Ref18 ;
Ref291 ; Ref29 ; Ref48 ; Ref23 ; Ref49 ; Ref50 ; Ref51 ; Ref52  yield to
phonon induced transparencies Ref19 ; Ref20 , which can be referred as
optomechanically induced transparency (OMIT) Ref21 ; Ref22 . The coupling
produced by the mechanical effects of light between multiple oscillators,
notably mirrors and ultra-cold atomic states, further lead to the concept of
multiple EITs Ref24 ; Ref25 ; Ref26 ; Ref27 ; Ref28 . These transparencies
occur because of quantum interferences in multiple transitional pathways at
intermediate states of the system. However, the recent discussions on Fano
resonances – a phenomenon of quantum nonlinear interaction that consequently
occurs because of the off-resonant interference Ref280 ; Ref2800  – in a four
mirror optomechanical cavity, with two vibrating mirrors, have raised another
aspect of engineering hybrid systems Ref281 ; Ref282 . Although, the
phenomenon of quantum nonlinear optics has been discussed in complex systems
Ref283 ; Ref284 ; Ref285 ; Ref286 , but the demonstration to configure ultra-
cold atomic states in such multidimensional hybrid system is essential,
especially with respect to quantum nonlinear optics.

In this paper, we investigate EITs with two Bose-Einstein condensates (BECs)
transversely coupled to a four mirror cavity, driven by a pump and a probe
laser. The high-$Q$ (quality) factor of the cavity generates strong cavity
mode which couples both atomic states after getting split from the beam
splitter (BS) Ref281 ; Ref282 ; Ref283 ; Ref284 ; Ref285 ; Ref286 . The
optomechanical effects of strong cavity mode, after interacting with atomic
degrees of freedom, cause the excitation of momentum side-modes in BECs,
acting like two atomic oscillators coupled with each other through cavity
field Ref292 . We show that the quantum interference, occurring from both
atomic states during probe-cavity transitions, generate two novel EIT windows
in cavity transmission and which only exist when both atomic states are
coupled with cavity. The coupling strengths of atomic degrees of freedom with
system also provide tunability for the Fano line shapes occurring in off-
resonant domain. Further, we show the dynamics of fast and slow light induced
by transversely coupled BECs and illustrate that the increase in coupling
strengths will robustly slow down the transmitting probe light.

The manuscript is arranged as follows: In Section II, the detailed system
description as well as the mathematical formulation of the considered model
are discussed. The Section III provides the results and discussion about the
emergence of EIT in the four mirror cavity with BECs. The Section IV
illustrates and explains the behavior of Fano resonance. The section V
contains the results and discussion about dynamics of fast and slow light.
Finally, Section VI contains the conclusion of the work.

##  II System Description and Mathematical Modeling

Figure 1: The schematic diagram of a four mirror cavity with two transversely
coupled Bose-Einstein condensates, oriented along $x$-axis and $y$-axis. Two
lasers, an external pump and a probe, drive the cavity and generate coupling
between atomic states after splitting from beam splitter (BS). $\eta$
($\omega_{E}$) and $\eta_{p}$ ($\omega_{p}$) correspond to the intensities
(frequencies) of the external pump and probe lasers, respectively.

We consider a four mirror cavity containing transversely located two BECs
along $x$-axis and $y$-axis, as illustrated in Fig. 1, unlike the previous
investigations where two moving-end mirrors were considered in four mirror
optomechanical systems without BECs Ref281 ; Ref282 . External pump laser,
with intensity $\eta$ and frequency $\omega_{E}$, and probe laser, with
intensity $\eta_{p}$ and frequency $\omega_{p}$, longitudinally ($x$-axis)
drive the cavity with detuning $\Delta_{p}=\omega_{E}-\omega_{p}$. Due to high
quality ($Q$-) factor of the cavity mirrors, the intra-cavity photons perform
multiple round trips inside cavity building a strong cavity mode with detuning
$\Delta_{c}=\omega_{E}-\omega_{c}$, where $\omega_{c}$ is the frequency of
cavity mode. The cavity mode, after getting equally split from the ($1/2$) BS,
interacts with the BECs transversely located in the two arms of the cavity and
excites momentum side-modes. These momentum side-modes of BECs can be
analogically considered as two atomic mirrors coupled with each other through
cavity field. $\kappa$ accommodates the effective cavity mode decay rate,
including the photon leakage from the BS to the bottom mirror (oriented along
$-x$-axis) of the cavity.

We consider two one-dimensional BECs having quantized motion along $x$-axis
and $y$-axis. By considering large atom-field detuning $\Delta_{a}$, we
eliminate the internal excited state behaviors of BECs, which ultimately leads
to the suppression of spontaneous emission. Further by assuming that the BECs
are dilute enough, the atom-atom interactions can be ignored. Considering it,
we formulated the total Hamiltonian as Ref22 ; Ref281 ,

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\sum_{\sigma=x,y}\int
d{\sigma}\hat{\boldsymbol{\psi}}_{\sigma}^{{\dagger}}(\sigma)\bigg{(}-\frac{\hbar
d^{2}}{2m_{a}d\sigma^{2}}+\hbar
U_{0}\hat{c}^{{\dagger}}\hat{c}\cos^{2}k\sigma\bigg{)}\hat{\boldsymbol{\psi}}_{\sigma}(\sigma)$
(1)
$\displaystyle+\hbar\Delta_{c}\hat{c}^{{\dagger}}\hat{c}-i\hbar\eta(\hat{c}-\hat{c}^{{\dagger}})-i\hbar\eta_{p}(\hat{c}e^{i\Delta_{p}}-\hat{c}^{{\dagger}}e^{-i\Delta_{p}}).$

Here
$\hat{\boldsymbol{\psi}}_{\sigma}(\sigma)=[\hat{\psi}_{x},\hat{\psi}_{y}]^{T}$
($\hat{\boldsymbol{\psi}}_{\sigma}^{{\dagger}}(\sigma)=[\hat{\psi}^{{\dagger}}_{x},\hat{\psi}^{{\dagger}}_{y}]^{T}$)
represents bosonic field operators for BECs oriented along $x$-axis and
$y$-axis, respectively, with equal atomic mass $m_{a}$. The second term in the
Hamiltonian corresponds to the one dimensional optical lattices along $x$-axis
and $y$-axis, where $\hat{c}$ ($\hat{c}^{\dagger}$) describes the annihilation
(creation) operator for the intra-cavity optical mode.
$U_{0}=g_{0}^{2}/\Delta_{a}$ accommodates the optical potential depth for the
atoms defined over the far off-resonant Rabi oscillation $g_{0}$ and atomic-
field detuning $\Delta_{a}$ Ref22 . $k=\omega_{E}/c$ is the wave number
corresponding to the lattice with speed of light $c$. In this study, we assume
that the both atomic degrees of freedom are equally interacting coupled to the
cavity mode in strong coupling regime $Ng_{0}^{2}/\Delta_{a}>>\kappa$ Ref31 ,
where $N$ is the number of atoms in each condensate. The third term
corresponds to the intra-cavity optical mode strength while the last two terms
accommodate the coupling strengths of intra-cavity field with the external
pump laser $|\eta|=\sqrt{P\times\kappa/\hbar\omega_{E}}$ and probe laser
$|\eta_{p}|=\sqrt{P_{p}\times\kappa/\hbar\omega_{p}}$, respectively.

The intra-cavity field interacting with trapped BECs generates photonic recoil
causing excitation of symmetrical momentum side $\pm 2l\hbar k$, with integer
$l$. By assuming weak optical field, i.e. $l=1$, one can consider maximum
atomic population saturated in $0^{th}$ and $1^{st}$ mode and can ignore
higher-order momentum side-modes. Under this consideration, the bosonic
operator for BECs can be formulated as Ref292 ,

$\hat{\boldsymbol{\psi}}_{\sigma}^{{\dagger}}(\sigma)\approx\frac{1}{\sqrt{L}}[\hat{a}_{\sigma
0}+\sqrt{2}\cos(k\sigma)\hat{a}_{\sigma 1}],(\sigma=x,y).$ (2)

Here $\hat{a}_{x0,y0}$ and $\hat{a}_{x1,y1}$ correspond to the bosonic field
operators for atomic distribution in $o^{th}$ and $1^{st}$ momentum side-modes
(along $x$ and $y$-axis), respectively. $L=L_{x}=L_{y}$ is the effective
length of the cavity. By considering modified bosonic operator, the
Hamiltonian (1) reads as Ref29 ; Ref30 ,

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{\sigma=x,y}\frac{\hbar
U_{0}}{2}\hat{c}^{{\dagger}}\hat{c}\big{(}\hat{a}^{{\dagger}}_{\sigma
0}\hat{a}_{\sigma 0}+\hat{a}^{{\dagger}}_{\sigma 1}\hat{a}_{\sigma
1}\big{)}+\frac{\sqrt{2}\hbar
U_{0}}{4}\hat{c}^{{\dagger}}\hat{c}\big{(}\hat{a}^{{\dagger}}_{\sigma
0}\hat{a}_{\sigma 1}$ (3) $\displaystyle+\hat{a}^{{\dagger}}_{\sigma
1}\hat{a}_{\sigma
0}\big{)}+\frac{2\hbar^{2}k^{2}}{m_{a}}\hat{a}^{{\dagger}}_{\sigma
1}\hat{a}_{\sigma
1}+\hbar\Delta_{c}\hat{c}^{{\dagger}}\hat{c}-i\hbar\eta(\hat{c}-\hat{c}^{{\dagger}})$
$\displaystyle-i\hbar\eta_{p}(\hat{c}e^{i\Delta_{p}}-\hat{c}^{{\dagger}}e^{-i\Delta_{p}}).$

As the maximum atomic population is saturated in $o^{th}$ and $1^{st}$ side-
modes of BECs, so we can consider $\hat{a}^{{\dagger}}_{\sigma
0}\hat{a}_{\sigma 0}+\hat{a}^{{\dagger}}_{\sigma 1}\hat{a}_{\sigma 1}\approx
N,(\sigma=x,y)$, where $N$ is the total number of atoms in each condensate.
Further, as the atomic population in $o^{th}$ mode is much higher than the
population in $1^{st}$ mode ($N_{0}>>N_{1}$), therefore, one can approximate
that $\hat{a}^{{\dagger}}_{\sigma 0}\hat{a}_{\sigma 0}\approx N$, leading to
$\hat{a}^{{\dagger}}_{\sigma 0}$ and $\hat{a}_{\sigma 0}\approx\sqrt{N}$
Ref292 . By considering these approximation and defining dimensionless
position $\hat{q}_{\sigma}=(1/\sqrt{2})(\hat{a}_{\sigma
1}+\hat{a}^{{\dagger}}_{\sigma 1}),(\sigma=x,y)$ and momentum
$\hat{p}_{\sigma}=(i/\sqrt{2})(\hat{a}_{\sigma 1}-\hat{a}^{{\dagger}}_{\sigma
1}),(\sigma=x,y)$ over the bosonic operator of $1^{st}$ side-modes, with with
canonical relation $[\hat{q}_{\sigma},\hat{p}_{\sigma}]=i$, we rewrite the
equation 3 as,

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\frac{\hbar
U_{0}N}{2}\hat{c}^{{\dagger}}\hat{c}+\sum_{\sigma=x,y}\bigg{(}\frac{\hbar\omega_{\sigma}}{2}\big{(}\hat{p}^{2}_{\sigma}+\hat{q}^{2}_{\sigma}\big{)}+\hbar
g_{\sigma}\hat{c}^{{\dagger}}\hat{c}\hat{q}_{\sigma}\bigg{)}$ (4)
$\displaystyle+$
$\displaystyle\hbar\Delta_{c}\hat{c}^{{\dagger}}\hat{c}-i\hbar\eta(\hat{c}-\hat{c}^{{\dagger}})-i\hbar\eta_{p}(\hat{c}e^{i\Delta_{p}}-\hat{c}^{{\dagger}}e^{-i\Delta_{p}}).$

Here first term corresponds to the influences of atomic degrees of freedom on
the intra-cavity field while the second term describes the motion of atomic
side-mode with recoil frequencies $\omega_{\sigma}=4\omega_{\sigma}=2\hbar
k^{2}/m_{a},(\sigma=x,y)$. The third term defines the coupling of cavity mode
with the atomic degrees of freedom, where
$g_{\sigma}=\omega_{c}\sqrt{\hbar/m_{bec}\Omega_{\sigma}}/L$ with effective
BECs masses $m_{bec}=\hbar\omega_{c}^{2}/(L^{2}NU^{2}_{0}\Omega_{\sigma})$.
The fourth, fifth and sixth terms corresponds to the strength of intra-cavity
field and its couplings with external pump and probe fields, as stated
previously.

In order to incorporate the effects of dissipation and damping along with the
depletion of BECs with standard quantum noise operators, we adopt the quantum
Langevin equations approach to govern the collective spatio-temporal dynamics
of the subsystems, given as,

$\displaystyle\frac{d\hat{c}}{dt}$ $\displaystyle=$
$\displaystyle(i\Delta+\sum_{\sigma=x,y}ig_{\sigma}\hat{q}_{\sigma}-\kappa)\hat{c}+\eta+\eta_{p}e^{-i\Delta_{p}t}$
(5) $\displaystyle+\sqrt{2\kappa}c_{in},$
$\displaystyle\frac{d\hat{p}_{\sigma}}{dt}$ $\displaystyle=$
$\displaystyle-4\omega_{\sigma}\hat{q}_{\sigma}-g_{\sigma}\hat{c}^{{\dagger}}\hat{c}-\gamma_{\sigma}\hat{p}_{\sigma}+\hat{F}_{\sigma},(\sigma=x,y),$
(6) $\displaystyle\frac{d\hat{q}_{\sigma}}{dt}$ $\displaystyle=$
$\displaystyle
4\omega_{\sigma}\hat{p}_{\sigma}-\gamma_{\sigma}\hat{q}_{\sigma}+\hat{F}_{\sigma},(\sigma=x,y).$
(7)

Here $\Delta=\Delta_{c}-NU_{0}/2$ corresponds to the effective detuning of the
collective system while the $c_{in}$ represents Markovian noise operator
associated with cavity input field. The association of external harmonic trap
with BECs, which we have ignored in this study so-far, forces atomic momentum
side-modes to interact with the optical mode inside cavity, which eventually
results in the damping of atomic degrees of freedom Ref292 . $\gamma_{\sigma}$
(with $\sigma=x,y$) corresponds to such atomic damping factors. Further,
$F_{\sigma}$ (with $\sigma=x,y$) corresponds to the associated quantum noises
with the motion of atomic degrees of freedom, which are assume to be Markovian
Ref32 . Here it should be noted that, unlike macroscopic cavity optomechanics,
these damping and quantum noise operators are associated with both position
$q_{\sigma}$ as well as momentum $p_{\sigma}$ because of the microscopic
nature of atomic modes Ref29 ; Ref30 . However, in our study, by considering
strong cavity mode frequency $\hbar\omega_{c}>>k_{B}T$, where $k_{B}$ is the
Boltzmann constant while $T$ is the temperature of the external thermal
reservoir, we ignored the effects of associated quantum noises Ref33 .

In order to include the effects of associated first order quantum
fluctuations, we linearize quantum Langevin equations over the steady-states
of associated subsystems
$\mathcal{\hat{O}}(t)=\mathcal{O}_{s}+\mathcal{\delta\hat{O}}(t)$. Here
$\mathcal{\hat{O}}$ is a generic operator for any associated subsystems while
$\mathcal{O}_{s}$ corresponds to the steady-state of that operator, which can
be easily calculated from quantum Langevin equations by putting time
derivative equals to zero. The linearized Langevin equations will read as,

$\displaystyle\partial_{t}\delta\hat{c}(t)=$
$\displaystyle-(\kappa+i\Delta)\delta\hat{c}(t)+\sum_{\sigma=x,y}iG_{\sigma}\hat{q}_{\sigma}+\eta_{p}e^{-i\Delta_{p}t},$
(8) $\displaystyle\partial_{t}\delta\hat{c}^{{\dagger}}(t)=$
$\displaystyle-(\kappa-i\Delta)\delta\hat{c}(t)-\sum_{\sigma=x,y}iG_{\sigma}\hat{q}_{\sigma}+\eta_{p}e^{i\Delta_{p}t},$
(9) $\displaystyle\partial_{t}\delta\hat{q}_{\sigma}(t)=$ $\displaystyle
4\omega_{\sigma}\delta\hat{p}_{\sigma}(t)-\gamma_{\sigma}\delta\hat{q}_{\sigma}(t),(\sigma=x,y),$
(10) $\displaystyle\partial_{t}\delta\hat{p}_{\sigma}(t)=$
$\displaystyle-4\omega_{\sigma}\delta\hat{q}_{\sigma}(t)+G_{\sigma}(\delta\hat{c}(t)+\delta\hat{c}^{{\dagger}}(t))$
(12) $\displaystyle-\gamma_{\sigma}\delta\hat{p}_{\sigma}(t),(\sigma=x,y).$

Here $\partial_{t}$ represents time derivative and
$G_{\sigma}=g_{\sigma}\hat{q}|c_{s}|$ is the effective coupling between atomic
side-modes and the cavity mode, defined by the steady-state of intra-cavity
photons.

To measure the behavior of quantum interference in the form of EIT, we have to
compute the cavity transmission. In order to do so, we use linearized
subsystem quadratures (under condition $\eta>>\eta_{p}$) in
$\delta\mathcal{B}=\sum_{n\rightarrow\\{+,-\\}}\mathcal{B}_{n}e^{in\omega t}$,
where $\mathcal{B}$ is generic operator for any associated subsystem, and
compare the coefficients of probe exponential terms in linearized Langevin
equations. After solving these coefficients, the probe field terms appearing
with subscription minus $c_{-}$ contains the influences of quantum
interference,

$\displaystyle c_{-}(\Delta_{p})$ $\displaystyle=$
$\displaystyle\frac{\eta_{p}(\mathcal{X}_{x}(\Delta_{p}))}{(\kappa+i(\Delta-\Delta_{p}))(\mathcal{X}_{x}(\Delta_{p})+\mathcal{X}_{y}(\Delta_{p}))},$
(13) $\displaystyle\mathcal{X}_{x}(\Delta_{p})$ $\displaystyle=$
$\displaystyle
1+\frac{G_{x}^{2}\omega_{x}+G_{y}^{2}\gamma_{x}}{\kappa-i(\Delta_{p}+\Delta)},$
$\displaystyle\mathcal{X}_{y}(\Delta_{p})$ $\displaystyle=$
$\displaystyle\frac{G_{x}^{2}G_{y}^{2}\omega_{y}\gamma_{y}}{(\kappa-i(\Delta_{p}-\Delta))^{2}},$

where $\mathcal{X}_{x}(\Delta_{p})$ and $\mathcal{X}_{y}(\Delta_{p})$ can be
considered as modified susceptibilities for atomic degrees of freedom along
$x$-axis and $y$-axis, respectively, because they contain the corresponding
contributions of the condensates.

Further, to compute probe field components in cavity transmission, we use
standard input-output relation $c_{out}=\sqrt{2\kappa}c-c_{in}$, which leads
to,

$\displaystyle\mathcal{E}_{p}(\omega_{p})=$
$\displaystyle(\eta_{p}-\sqrt{2\kappa}c_{-}(\Delta_{p}))/\eta_{p}$
$\displaystyle=$ $\displaystyle 1-\sqrt{2\kappa}c_{-}(\Delta_{p})/\eta_{p}.$
(14)

Here the amplitude
$\mathcal{E}_{out}(\Delta_{p})=\frac{\sqrt{2\kappa}c_{-}(\Delta_{p})}{\eta_{p}}$
defines the absorption (in-phase behavior) and dispersion (out-of-phase
behavior) of the cavity probe transmission with its real and imaginary parts,
respectively. Here one can note that the
$\mathcal{X}_{y}(\Delta_{p})\rightarrow 0$ when any of the condensates will be
uncoupled to the system (i.e. $G_{x}\rightarrow 0$ or $G_{y}\rightarrow 0$).
In this situation, the probe transmission amplitude will read as,

$\displaystyle\mathcal{E}_{out}(\Delta_{p})=$
$\displaystyle\frac{\sqrt{2\kappa}}{\kappa+i(\Delta-\Delta_{p})},$ (15)

which is the case of total absorption with no quantum interference (or no EIT)
and it can also be seen in the following results as well.

##  III Electromagnetically Induced Transparencies

The strongly driven cavity mode and the probe laser create double excitation
for both transverse condensates (oriented along $x$-axis and $y$-axis). The
quantum interference at these double excitation yields to the possibilities
for the dark states and leads to the transparencies for probe light Ref7 . In
another analogy, when the intra-cavity optical mode interacts with the
condensates, it exerts radiation pressure force. If the radiation pressure
force is resonant (or near resonant) with the mechanical frequencies of both
condensates, then it generates Stokes and anti-Stokes scattering of probe
light from the edges of atomic oscillators. However, under the condition
$Ng_{0}^{2}/\Delta_{a}>>\kappa$, the Stokes scattering will be suppressed
because the collective system will be in resolved-sideband regime, where the
Stokes scattering is off-resonant with cavity mode Ref7 ; Ref8 ; Ref9 . Thus,
only anti-Stokes scatterings will survive, which eventually result in the
appearance of dark states in probe transmission, as can be seen in absorption
($Re[\mathcal{E}_{out}]$) and dispersion ($Im[\mathcal{E}_{out}]$) of probe
transmission illustrated in Figs. 2(a) and 2(b).

Figure 2: (a) Absorption $Re[\mathcal{E}_{out}]$ and dispersion
$Im[\mathcal{E}_{out}]$ of cavity probe transmission as function of normalized
pump-probe detuning $\Delta_{p}/\kappa$, for various atom-cavity couplings
$G_{x}/\omega_{x}$ and $G_{y}/\omega_{y}$. For black, blue and red curves
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=0$, $0.6$, and $0.9$, respectively. The
other parameters are considered as, $\Delta=0$,
$\gamma_{x}=\gamma_{y}=0.1\kappa$, $\omega_{x}=\omega_{y}=0.1\Delta_{c}$, and
$\kappa\approx 0.1\times 2\pi$kHz.

In absence of coupling between BECs and cavity
($G_{x}/\omega_{x}=G_{y}/\omega_{y}=0$), the probe light will be completely
absorb by the cavity without facing any quantum interference, as illustrated
by the black curves in Figs. 2(a) and 2(b). However, when we couple atomic
states with cavity mode, the quantum interference at atomic transitional
pathways yields two EIT windows, as can be seen by two dips in blue curves of
Figs. 2(a) and 2(b). Further, when we increase the coupling strengths for both
condensates, the EIT windows get robustly enhanced, as illustrated by red
curves in Figs. 2(a) and 2(b).

The probe light after, getting split from BS, interacts with condensates
trapped in two arms of the cavity (along $x$ and $y$-axis). The quantum
interaction of probe light at double excitation levels (created by probe light
in presence of cavity excitation) of each condensates in each arm of the
cavity engineers the feature of EIT for probe light, likewise as it happens in
conventional optomechanical systems in the case of OMIT Ref29 ; Ref27 ; Ref28
. After getting reflected from high-$Q$ cavity mirrors in both arms, the probe
light with quantum nonlinear feature of EIT merges again at BS, from where it
will leave the cavity. Here it should be noted that the quantum nonlinear
features of EITs only exist when both atomic condensates are coupled to the
cavity mode. If any of the both condensates gets failed to couple with the
cavity mode then EIT windows will not appear in the probe transmission. It is
because when any of the condensates becomes uncoupled to the cavity field, the
probe light from the arm of uncoupled condensate will suppress (dominate) the
EIT effects of the probe light coming from other arm with coupled condensate.
It eventually results in the absence of EIT windows in probe transmission
unlike the case of conventional hybrid optomechanical system, where if any of
the oscillators is coupled to the system then there will be an EIT window
Ref29 ; Ref26 ; Ref27 ; Ref28 .

Figure 3: (a) Absorption $Re[\mathcal{E}_{out}]$ versus normalized
$\Delta_{p}/\kappa$, for $G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.5$ (black),
$G_{x}/\omega_{x}=0.5,G_{y}/\omega_{y}=2$ (blue),
$G_{x}/\omega_{x}=2,G_{y}/\omega_{y}=0.5$ (red), and
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=2$ (brown), at constant
$\omega_{x}=\omega_{y}=0.1\Delta_{c}$. (b) Absorption $Re[\mathcal{E}_{out}]$
versus normalized $\Delta_{p}/\kappa$, for
$\omega_{x}=\omega_{y}=0.1\Delta_{c}$ (black),
$\omega_{x}=0.4\Delta_{c},\omega_{y}=0.1\Delta_{c}$ (blue),
$\omega_{x}=0.1\Delta_{c},\omega_{y}=0.4\Delta_{c}$ (red), and
$\omega_{x}=\omega_{y}=0.4\Delta_{c}$ (black), with constant
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.5$. The rest of the parameters are the
same as mentioned in Fig. 2.

Further, as stated previously, one can mathematically understand this from
equation (13). When any of the condensate is uncoupled
$\mathcal{X}_{y}(\Delta_{p})\rightarrow 0$, which leads to the empty cavity
configuration as described in equation (15). Furthermore, one can also note
from the equation (13) that even in the case stationary $\omega_{y}=0$
transverse condensate (along $y$-axis) or in the case where it is performing
undamped motion $\gamma_{y}=0$, the EIT effects will also disappear from the
probe transmission, because of the suppression of EIT effects coming from
longitudinal arm.

In Fig. 2, we discussed the EITs effects when both condensates are equally
coupled to the cavity mode. However, in the case where both condensates are
not equally coupled to the system, the EITs behavior gets interestingly
modified, as illustrated in Fig. 3 (a). The decrease in coupling strength of
any condensate ($G_{x}$ or $G_{y}$) reduces the strength of quantum
interference happening in EITs. But, as the both condensates equally
contribute to EITs, so if we reduce the strength of coupling for one
condensate while keeping the coupling of second condensate constant, or keep
the coupling of first one same and reduce the coupling for second one (equal
to first one in first case), their influences on EITs will be same. It can be
seen in blue ($G_{x}/\omega_{x}=0.5,G_{y}/\omega_{y}=2$) and red
($G_{x}/\omega_{x}=2,G_{y}/\omega_{y}=0.5$) curves of Fig. 3 (a).

The increase or decrease in the frequencies ($\omega_{x}$ and $\omega_{y}$) of
the condensate also equally alter the EITs behavior, as illustrated in Fig. 3
(b). By increasing the frequencies of the condensates, the edges of EITs
windows move away from the resonant state of pump-probe detuning
($\Delta_{p}=0$) to the off-resonant domain and mimic like a Fano resonances.
It is because higher mechanical frequencies of condensate will prolong the
quantum interference by increase the gap between central modes and side-modes
of the condensates Ref29 . Further, in the case of unequal frequencies, the
EITs response will be same as it is in the case of unequally coupled
condensates, as illustrated by the blue
($\omega_{x}=0.4\Delta_{c},\omega_{y}=0.1\Delta_{c}$) and red
($\omega_{x}=0.1\Delta_{c},\omega_{y}=0.4\Delta_{c}$) curves of Fig. 3 (b).

##  IV Fano Resonances

Figure 4: Fano resonance in absorption $Re[\mathcal{E}_{out}]$ versus
normalized $\Delta_{p}/\kappa$ at $G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.8$. In
(a), black, blue, red and brown curves correspond to
$\Delta=0\kappa,0.2\kappa,0.5\kappa$ and $0.8\kappa$, respectively. In (b),
black, blue, red and brown curves correspond to
$\Delta=0\kappa,-0.2\kappa,-0.5\kappa$ and $-0.8\kappa$, respectively.
Remaining parameters are the same as mentioned in Fig. 2.

Fano resonances – a fascinating implication of quantum nonlinear interactions
of light that emerge in off-resonant configuration of EIT – possess crucial
importance in hybrid and complex quantum systems Ref280 ; Ref2800 , especially
where multiple optical modes are needed to be transmitted through one channel
or through one photonic pathway. In optomechanical system the phenomenon of
Fano resonances has been extensively studied Ref29 ; Ref27 ; Ref28 , even in
the four mirror cavity optomechanical system Ref281 ; Ref282 . However, these
Fano resonances are worth exploring in our system where two transversely
coupled BECs are placed inside a four mirror cavity system. The dynamics of
Fano resonances can observed in our system by measuring the probe transmission
in off-resonant cavity detuning (pump-caviy) $\Delta$ configuration with
respect to the EITs spectrum, as illustrated in Fig. 4.

When we shift the cavity detuning towards positive ($\Delta>0$) from the
resonant (or EITs) regime ($\Delta=0$), the peak of probe transmission starts
moving towards the positive pump-probe detuning $\Delta_{p}>0$, as can be seen
in Fig. 4 (a). But the EIT dip occurring around $\Delta_{p}\approx+0.5\kappa$
remain at same position for each increase in cavity detuning $\Delta$ (or for
each Fano line shape), yielding in a resonance formation (known as Fano
resonance) over the pump-probe detuning $\Delta_{p}$. It should be noted here,
in this configuration, a weak resonance is also appearing around
$\Delta_{p}\approx-0.5\kappa$, but it can be comparatively ignored (or
suppressed) with respect to other resonance. Similarly, if we decrease the
strength of cavity detuning below zero $\Delta<0$, the probe transmission
spectrum (or Fano lines) starts moving towards the left of the pump-probe
detuning ($\Delta_{p}<0$) and the Fano resonance gets sifted to the other EIT
dip occurring around $\Delta_{p}\approx-0.5\kappa$, as illustrated in Fig. 4
(b). Here one observe that the width of absorption $Re[\mathcal{E}_{out}]$
increases in off-resonant cavity domain but the Fano resonance will again
remain at the same position. The interesting thing that augments the
significance of current model is that the both transparencies windows remain
at the same position unaffected by the direction system detuning. It is unlike
the previous studies Ref29 ; Ref28  where the saturation of Fano lines to any
EIT window effects the position of second window and presents more appealing
argument for the Fano resonance applications.

Figure 5: Fano resonance in Absorption $Re[\mathcal{E}_{out}]$ versus
normalized pump-probe detuning $\Delta_{p}/\kappa$ and normalized cavity
detuning $\Delta/\kappa$ at $G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.6$ (a) and
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=1$ (b). Remaining parameters are the same
as mentioned in Fig. 2.

In order to further explain the behavior of Fano resonances, in Fig. 5, we
illustrate the probe absorption $Re[\mathcal{E}_{out}]$ as a function of
normalized cavity detuning $\Delta/\kappa$ and normalized pump-probe detuning
$\Delta_{p}/\kappa$. In the absence of BECs-cavity detuning (i.e.
$G_{x}=G_{y}=0$), the probe transmission will form a diagonal band (bright
strip) versus $\Delta/\kappa$ and $\Delta_{p}/\kappa$ without showing any
break or resonance, which is obvious and is not illustrated here. But when we
coupled transversely BECs with the cavity arms, two Fano resonances appear in
the form of two breaks around $\Delta_{p}\approx\pm 0.5$ in the probe
transmission band, as shown in Fig. 5 (a). If we further increase the strength
of coupling between BECs and cavity, the gap between the beaks will be
increased resulting in enhanced Fano resonances, as illustrated in Fig. 5 (b).
Similarly, like EITs, these dips produced by Fano resonances only exist when
both atomic states are coupled to the system and if any of the condensates
becomes uncoupled, then there will be no Fano resonance.

##  V Fast and Slow Light Dynamics

Figure 6: The slow light behavior with the probe group delay $\tau_{g}$ as a
function of external pump laser power $P/P_{p}$. The black, blue solid, and
blue dashed curves represent $G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.3$, $0.8$,
and $2$, respectively. Remaining parameters are the same as mentioned in Fig.
2.

The dynamics of fast and slow light possess great significance in order to
take quantum nonlinear optical interactions towards practical quantum
computation Ref4 ; Ref5 ; Ref6 . In our case, to govern the behavior of slow
as well as fast light, the phase $\Phi_{p}$ of total probe transmission
$\mathcal{E}_{p}$ possesses crucial importance, because with its robust
diffusion, one can calculate probe transmission delay (or group delay)
$\tau_{g}$, reading as.

$\displaystyle\tau_{g}=$
$\displaystyle\frac{\partial}{\partial\omega_{p}}\Phi_{p}(\omega_{p})=\frac{\partial}{\partial\omega_{p}}\bigg{(}arg\big{(}\mathcal{E}_{p}(\omega_{p})\big{)}\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\omega_{p}}\bigg{(}arg\big{(}1-\sqrt{2\kappa}c_{-}(\Delta_{p})/\eta_{p}\big{)}\bigg{)}.$
(16)

By illustrating the transmission group delay $\tau_{g}$ versus external pump
field power $P/P_{p}$, we measure the dynamics of fast and slow probe light,
as shown in Fig. 6. With weak BECs-cavity couplings
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.3$, the group delay decreases for weak
external pump laser power $P/P_{p}$. But the group delay again starts increase
with increase in power $P/P_{p}$ and gets saturated around
$\tau_{g}\approx-0.2\mu s$ at higher values of $P/P_{p}$, as can be seen by
the black curve of Fig. 6.

Figure 7: The group delay $\tau_{g}$ versus normalized cavity detuning
$\Delta/\kappa$ and normalized cavity decay $\kappa/\gamma$, where
$\gamma=\gamma_{x}=\gamma_{y}$, with $G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.3$
(a) and $G_{x}/\omega_{x}=G_{y}/\omega_{y}=1$ (b). The shads of blue and red
colors corresponds to the distribution of fast and slow light, respectively.
Remaining parameters are the same as mentioned in Fig. 2.

However, if we increase the coupling strengths of BECs to
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.8$, the group delay $\tau_{g}$ decreases
rapidly for initial values of $P/P_{p}$. But after $P\approx 25P_{p}$, it
starts increasing again and this time it saturates at much higher values,
around $\tau_{g}\approx-0.01\mu s$, for the higher values of $P/P_{p}$,
yielding to slow probe transmission, as illustrated with blue solid curve of
Fig. 6. If we further increase the strength of BECs-cavity coupling, the group
delay $\tau_{g}$ will more rapidly decay initially but will get saturated to
further higher values with $P/P_{p}$, as can be seen by the blue dotted curve
of Fig. 6, where the BECs couplings have been increase to
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.8$. It is happening because, at higher
strengths of BECs coupling with cavity, the quantum interference resulting in
enhance dark states will increase the widths EITs windows, which eventually
yields to slow transmission of probe light. The advantage of current system
over the previous studies on fast and slow in atom-cavity systems Ref28  is
the simultaneous effects of both condensates that enhances the group delay of
probe transmission.

To further enhance the understanding of the fast and slow light dynamics, we
measured the probe transmission delay versus normalized cavity detuning
$\Delta/\kappa$ as well as cavity decay rate $\kappa$, as shown in Fig. 7. It
is crucially important to know how much the cavity leakage $\kappa$ can alter
the dynamics of fast and slow light, and at which configuration, we can get
maximum slow light in our system. At weak BECs-cavity couplings
$G_{x}/\omega_{x}=G_{y}/\omega_{y}=0.3$, the group delay $\tau_{g}$, between a
particular interval of cavity detuning $\Delta/\kappa$, decreases with
increase in cavity decay $\kappa$ and reaches at minimum value (maximum fast
light) around $\kappa\approx 0.2\gamma$ (here $\gamma=\gamma_{x}=\gamma_{y}$).
However, if we further increase the cavity decay $\kappa$, the $\tau_{g}$
jumps to its maximum value (maximum slow light) and start decreasing from its
maximum value with increase in $\kappa$. But again after
$\kappa\approx\gamma$, it starts to increase and reach to the maximum value at
$\kappa\approx 2.1\gamma$, from where it suddenly jumps to its minimum values,
as shown in Fig. 7 (a). The higher values of $\tau_{g}$ or the slow light is
almost forming a oval like shape during the cavity decay interval
$0.2\gamma<\kappa<2.1\gamma$, where the maxima of slow light are occurring at
the edges with the maxima of fast light. However, if we increase the couplings
between BECs and cavity, that oval shape will shift towards higher values of
cavity decay $\kappa$. Because, at higher couplings, the strength of EIT
windows will increase allowing probe transmission to overcome (or handle)
higher values of $\kappa$. Although, these novel dynamics of fast and slow
light in our system are critically dependent on the various system parameters,
but it still magnifies the understanding of the behavior of slow as well as
fast light.

##  VI Conclusion

We investigate electromagnetically induced transparencies in a four mirror
cavity with two Bose-Einstein condensates, trapped along the transverse arms
of the cavity, and driven by a strong pump laser and a weak probe laser. The
cavity mode, strongly driven by the external pump laser and a weak probe
laser, excites atomic states to the double excitation configuration with an
intermediate level. We show that the quantum interference occurring at these
double excitation levels leads to the two novel transparency windows, which
only exist when both Bose-Einstein condensates are coupled to the system. The
strength of these electromagnetically induced transparencies can not only be
increased with an increase in atom-cavity couplings, but the frequencies of
condensates also equally contribute to the transparency windows. Further, by
demonstrating the behavior of Fano resonances, we illustrate the occurrence of
two Fano resonances corresponding to the two transparency windows and the
strength of these windows can be increased with an increase in atom-cavity
coupling. Furthermore, we illustrate the dynamics of fast and slow light and
conclude that the increase in atom-cavity couplings can significantly slow
down the transmission of probe light. These results enhance the understanding
of quantum nonlinear optics with hybrid complex systems, especially containing
ultra-cold atomic states.

###### Acknowledgements.

L.Z.X. is supported by Zhejiang Provincial Natural Science Foundation of China
under Grant No. Z21A040009 and the National Natural Science Foundation of
China under Grant No. 12074344. G.X.L. acknowledges the support of National
Natural Science Foundation of China under Grant No. 11774316. W.M.L.
acknowledges the support of National Key R&D Program of China under grants No.
2016YFA0301500, NSFC under grants Nos. 61835013 and 61775242, Strategic
Priority Research Program of the Chinese Academy of Sciences under grants Nos.
XDB01020300 and XDB21030300.

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# Training on Art Composition Attributes to Influence CycleGAN Art Generation

Holly Grimm
Independent Software Developer and Artist
<EMAIL_ADDRESS>

###### Abstract

I consider how to influence CycleGAN, image-to-image translation, by using
additional constraints from a neural network trained on art composition
attributes. I show how I trained the the Art Composition Attributes Network
(ACAN) by incorporating domain knowledge based on the rules of art evaluation
and the result of applying each art composition attribute to apple2orange
image translation.

##  1 Introduction

The standard adversarial and cyclical losses of a CycleGAN [1] were augmented
with additional loss terms from a convolutional neural network trained with
art composition attributes. During training of the CycleGAN, the user
specifies values for each of the art composition attributes. For instance, if
a target contrast value of 10 is specified, the generator should output images
with more contrast than if the target contrast value is 1.

###  1.1 Art Composition Attributes

Eight art composition attributes were selected: variety of texture, variety of
shape, variety of size, variety of color, contrast, repetition, primary color,
and color harmony. Five hundred images from the WikiArt dataset [2] were
labeled with these attributes. Figures 1, 2, 3 and 4 are examples of low and
high values for variety of texture and contrast.

Figure 1: low texture
Figure 2: high texture

Figure 3: low contrast
Figure 4: high contrast

##  2 ACAN

Training consisted of fine-tuning a ResNet50 [3] pretrained on the ImageNet
dataset. ResNet50 is a fifty-layer deep residual network with 16 residual
blocks. Global Average Pooling (GAP) is applied to the ReLU output from each
of the sixteen ResNet block activations, called rectified convolution maps
[4]. The sixteen GAP outputs were concatenated and L2 normalization was
applied to create a merge layer. From the merge layer, there are eight
outputs, one for each of the attributes.

##  3 CycleGAN and ACAN

In addition to the standard CycleGAN losses (Adversarial, Cycle-Consistency,
and Identity) the ACAN losses are a series of eight losses generated when the
translated image is passed through the ACAN with eight target values. The
difference between these target values and the values output by the network
are the attribute losses.

##  4 Results

Below is a sampling of the results of running the CycleGAN training with ACAN
on the apple2orange dataset. Even with a small training set size of 500
images, the ACAN is able to learn and generate apples with the eight art
compositional attributes.

Figure 5: original apple image
Figure 6: color harmony analogous

Figure 7: color harmony complementary
Figure 8: color harmony monochromatic

Figure 9: low contrast
Figure 10: high contrast

Figure 11: low color
Figure 12: high color

Figure 13: low texture
Figure 14: high texture

Figure 15: low repetition
Figure 16: high repetit.

Figure 17: original

Figure 18: translated

Figure 19: reconstructed

The most surprising result of this project is the painterly effect that the
ACAN was able to inject into the CycleGAN generated images as seen in Figures
18 and 19.

#### Acknowledgments

This project was initially developed as part of the 2018 OpenAI Scholars
program. I would like to thank my mentor, Christy Dennison from OpenAI, for
her helpful comments along with support from Larissa Schiavo, Joshua Achiam,
Jack Clark, and Greg Brockman from OpenAI.

## References

[1] Zhu, J. & Park, T. & Isola, P. & Efros, A.A. (2017) Unpaired Image-to-
Image Translation using Cycle-Consistent Adversarial Networks.
arXiv:1703.10593.

[2] Nichol, K. (2016) Kaggle dataset: Painter by numbers.
https://www.kaggle.com/c/painter-by-numbers.

[3] He, K. & Zhang, X. & Ren, S. & Sun, J. (2015) Deep Residual Learning for
Image Recognition arXiv:1512.03385.

[4] Malu, G. & Bapi, R.S. & Indurkhya, B. (2017) Learning Photography
Aesthetics with Deep CNNs. arXiv:1707.03981.