# Coherent Perfect Absorption in Tavis-Cummings Models

Zibo Wang, Pawan Khatiwada, Dan Wang, and Imran M. Mirza  Macklin Quantum
Information Sciences,
Department of Physics, Miami University, Oxford, Ohio 45056, USA
<EMAIL_ADDRESS>
###### Abstract

We theoretically study the conditions under which two laser fields can undergo
Coherent Perfect Absorption (CPA) when shined on a single-mode bi-directional
optical cavity coupled with two two-level quantum emitters (natural atoms,
artificial atoms, quantum dots, qubits, etc.). In addition to being indirectly
coupled through the cavity-mediated field, in our Tavis-Cummings model the two
quantum emitters (QEs) are allowed to interact directly via the dipole-dipole
interaction (DDI). Under the mean-field approximation and low-excitation
assumption, in this work, we particularly focus on the impact of DDI on the
existence of CPA in the presence of decoherence mechanisms (spontaneous
emission from the QEs and the leakage of photons from the cavity walls). We
also present a dressed-state analysis of the problem to discuss the underlying
physics related to the allowed polariton state transitions in the Jaynes-
Tavis-Cummings ladder. As a key result, we find that in the strong-coupling
regime of cavity quantum electrodynamics, the strong DDI and the emitter-
cavity detuning can act together to achieve the CPA at two laser frequencies
tunable by the inter-atomic separation which are not possible to attain with a
single QE in the presence of detuning. Our CPA results are potentially
applicable in building quantum memories that are an essential component in
long-distance quantum networking.

##  I Introduction

Coherent Perfect Absorption or CPA, in short, is an interference phenomenon in
which the light fields incident on a lossy medium are neither reflected nor
transmitted but completely absorbed by the medium through the process of
destructive interference among all scattering amplitudes baranov2017coherent ;
baldacci2015coherent ; yan2018coherent . The captured light is then converted
to other forms of energy inside the medium such as electrical energy or heat
while the media in which CPA is achieved are called coherent perfect
absorbers. The CPA is a material independent phenomenon and typically hinges
on maintaining correct phases and amplitudes of the incident light beams. In
principle, CPA is not confined to optical domains of wave-matter interactions
and as a result, it has been reported to be attained with sound waves
meng2017acoustic  and in diverse types of materials such as graphene films
rao2014coherent , Bose-Einstein condensates mullers2018coherent , plasmonic
metasurfaces zhang2012controlling , and (broadly speaking) in the generic type
of complex scattering structures with and without hidden symmetries
chen2020perfect , etc.

Historically, Douglas Stone et al. chong2010coherent  were the first research
group to put forward the idea of CPA in 2010. In their celebrated work, they
introduced CPA as a time-reversed laser or anti-laser due to its ability to
completely absorb the incident coherent fields with the proper adjustment of
the phases and the intensities of the incoming light beams. In their
theoretical proposal, they used a slab of Si material and incident light in
the wavelength range $500-900~{}nm$. What they were able to show was that by
adding the right amount of dissipation and controlling the phase and amplitude
of the light beams accordingly, 99.9% of the incident light can be absorbed at
discrete frequency values. Within a year, in 2011 two important developments
were made. First was Stefano Longhi’s theoretical proposal of combining a
laser and an anti-laser in a single optical device while fulfilling the
condition of $\mathcal{P}\mathcal{T}$ symmetry longhi2010pt . Secondly, the
experimental demonstration of the CPA was reported by Wan et al. using a Ti-
sapphire light source shined onto a 110$\mu m$ thick silicon wafer. They were
able to achieve 99.4% of absorption experimentally in the infrared wavelength
regime wan2011time . However, the actual proposal involving the Si slab
suffered the limitation of only being workable in a narrowband regime. This
restriction was relaxed in 2012 with the work of Pu et al. where they
theoretically proposed perfect absorption in a broadband regime using heavily
doped ultrathin Si films pu2012ultrathin . Pu et al.’s work was experimentally
verified by Li et al. in 2015 li2015broadband . Advancing the possible
applications of the CPA, in 2019 Pichler et al. demonstrated the operation of
an anti-laser in random media pichler2019random  exhibiting the time-reversed
version of random lasers wiersma2008physics .

Fueled by the aforementioned notable developments, the CPA has shown several
important applications in the past decade. Examples include photodetection
instrumentation zheng2014broadband , solar cells villinger2021doubling ,
switches fang2014ultrafast , microwave & radiowave antennas grimm2021driving ,
sensors kravets2013singular , and logic operators papaioannou2016all , to name
a few. Entering the quantum domain, an immediate application of CPA turns out
to be in the storage of quantum information that is encoded in different
photonic degrees of freedoms lvovsky2009optical . Such type of information
storage is a vital step in performing typical protocols of quantum
communication where flying photons are utilized as information carriers while
QEs/qubits reside at the nodes of a network to store and manipulate the
information kimble2008quantum ; mirza2016multiqubit ; mirza2016two ;
mirza2013single ; mirza2015bi . For reliable information processing the
complete photon absorption by lossy qubits trapped inside leaky optical
cavities becomes an indispensable requirement that can be fulfilled by CPA. In
this regard, CPA has been investigated in cavity quantum electrodynamics
(CQED) setups with a single two-level atom wei2018coherent , single two-level
atom and a second-order nonlinear crystal embedded in an optical cavity
xiaong2020coherent , collection of two-level atoms in the linear regime of
CQED agarwal2015photon , a two-level atom collection in the nonlinear regime
of CQED with the possibility of optical bi-stability agarwal2016perfect ;
wang2017control , three-level atomic ensemble CQED wang2017interference , path
entangled photons in Fabry-Perót cavity with an absorbing medium
huang2014coherent , squeezed light beams incident upon absorbing beam
splitters hardal2019quantum , and in cavity quantum optomechanics
yan2014coherent ; zhang2016perfect .

Figure 1: (Color online) The setup for CPA in the Tavis-Cummings model. Two
identical QEs trapped inside a single-mode optical cavity are directly coupled
to each other through DDI. Two incoming laser beams pump the cavity mode and
we find the conditions under which the incoming fields are completely absorbed
even in the presence of spontaneous emission from the emitters and photon
losses from the imperfect cavity mirrors.

In all of the above-mentioned studies, the position of the QEs inside the
cavity is completely ignored. In particular, for many-atom scenarios, a single
atomic excitation operator is introduced which linearly sums up the excitation
operator for each atom. Such type of treatment (considering an atomic ensemble
as a single giant atom essentially) may work well in the standard Dicke regime
dicke1954coherence  where the wavelength $\lambda$ of the resonant field is
much larger than the size of the atomic ensemble $R$ i.e. $\lambda\gg R$.
However, in the opposite situations ( $\lambda\lesssim R$; the so-called
timed-Dicke regime scully2006directed ; svidzinsky2009evolution ) the
separation between the emitters cannot be neglected. Furthermore, it is known
that once the inter-emitter separation $R$ becomes smaller compared to
$\lambda$, quantum emitters tend to directly exchange the excitation through
the dipole-dipole interaction agarwal2012quantum ; mirza2020dimer .

Keeping in view this unexplored area of CPA in CQED, in this paper, we present
an in-depth study of CPA with two QEs trapped inside an optical cavity with
separation resolvable by the cavity mode field. Our system consists of two
laser beams shined on the mirrors of a bidirectional single-mode optical
cavity interacting with a pair of two-level QEs with dipole-dipole interaction
(DDI). By deriving the quantum Langevin equations and input-output relations
for atomic and field operators, we calculate the conditions under which the
transmission and reflection of laser beams can be completely suppressed. We
also performed a dressed state analysis to predict the frequencies at which
the incident beams are completely absorbed to excite the polariton state
transitions in the Jaynes-Tavis-Cummings ladder. As the main result of our
investigation, we conclude that for an in-resonance emitter-cavity situation
the strong DDI can diminish the CPA under the strong coupling regime of CQED.
However, by introducing large detunings while still working in the strong DDI
and strong CQED regimes, absorption can not only be enhanced by almost $40\%$
but CPA can be achieved at two distinct laser frequencies — a feature that is
not possible to realize with a single QE.

The paper is organized as follows. In Sec. II we outline the details of our
Tavis-Cummings model. In Sec. III we discuss the dissipative dynamics of our
setup through the quantum Langevin equations and input-output formalism. In
Sec. IV we report the conditions under which CPA can be observed. Therein we
also report a dressed state analysis for the interpretation of the results
which are discussed in Sec. V. Finally, in Sec. VI we close with a summary of
our findings and mention the future outlook.

##  II System Model

As shown in Fig. 1, we consider a bidirectional optical cavity with a single-
mode resonant frequency $\omega_{c}$. The annihilation (creation) of a photon
in the quantized optical mode is performed through the operator
$\hat{a}(\hat{a}^{\dagger})$. Under realistic conditions, photons can leak out
from the imperfect cavity mirrors. We incorporate this process in our model by
introducing the left (right) cavity leakage rate $\kappa_{l}(\kappa_{r})$. We
also consider two two-level QEs (quantum dots, superconducting Josephson
junctions, naturally occurring Alkali atoms for instance Cs, etc.) trapped
inside the optical cavity. We describe the process of emitter excitation
(deexcitation) through the atomic raising (lowering) operators
$\hat{\sigma}^{\dagger}_{i}(\hat{\sigma}_{i})$ for the $i$th QE. In terms of
the energy eigenstates of the bare QEs these atomic operators are defined as
$\hat{\sigma}_{i}:=\left|g_{i}\right\rangle\left\langle e_{i}\right|$ and
$\hat{\sigma}^{\dagger}_{i}:=\left|e_{i}\right\rangle\left\langle
g_{i}\right|$ where $i=1,2$. Atomic transition frequency and spontaneous
emission rate for the $1^{st}(2^{nd})$ QE is given by $\omega_{eg_{1}}$ and
$\gamma_{1}$ ($\omega_{eg_{2}}$ and $\gamma_{2}$), respectively. Furthermore,
each QE interacts with the single-mode field through the standard Jaynes-
Cummings interaction with the coupling strength characterized by the parameter
$g_{ac}$. For simplicity, we set the coupling strength to be the same for both
QEs. Additionally, we suppose that the separation between the QEs is less than
a wavelength which opens the possibility of direct coupling between QEs
through the dipole-dipole interaction (DDI). The strength of the DDI $J_{12}$
sensitively depends on the inter-emitter separation and is defined as
zhang2014effects

$\displaystyle
J_{12}=\frac{3\Gamma_{0}c^{3}}{4\omega^{3}_{eg}r^{3}_{12}}\left(1-3\cos^{2}\phi\right),$
(1)

where $\Gamma_{0}$ is the atomic decay rate in the free space, $c$ is the
speed of light, $r_{12}$ is the inter-emitter separation, and $\phi$ is the
angle between the atomic dipole moment vector and the inter-emitter vector.
For simplicity, in the definition of $J_{12}$ we have assumed both QEs to be
identical i.e. $\omega_{eg_{1}}=\omega_{eg_{2}}=\omega_{eg}$, and
$\Gamma_{0_{1}}=\Gamma_{0_{2}}=\Gamma_{0}$. Furthermore, we assume atomic
dipole vectors to be perpendicular to the inter-emitter axis vector such that
$\cos\phi=0$.

To study coherent perfect absorption (CPA) we consider that two input lasers
with the same carrier frequency $\omega_{l}$ are being shined on our CQED
setup. One laser enters the cavity from the left mirror and the other from the
right mirror. Input operators $\hat{a}^{(l)}_{in}$ and $\hat{a}^{(r)}_{in}$
are used to describe the annihilation of photons in the left and right
incoming lasers, respectively, while the corresponding output operators are
given by $\hat{a}^{(l)}_{out}$ and $\hat{a}^{(r)}_{out}$. These operators are
defined in the next section.

Putting everything together, the system Hamiltonian $\hat{\mathcal{H}}_{sys}$,
under the rotating wave approximation and in a frame rotating with the laser
frequency $\omega_{l}$, is given by

$\displaystyle\hat{\mathcal{H}}_{sys}=$
$\displaystyle~{}\hbar\Delta_{c}\hat{a}^{\dagger}\hat{a}+\sum_{i=1,2}\hbar\Delta_{eg_{i}}\hat{\sigma}^{\dagger}_{i}\hat{\sigma}_{i}+\sum_{i=1,2}\hbar\Big{(}g_{i}\hat{a}\hat{\sigma}^{\dagger}_{i}$
$\displaystyle+g^{\ast}_{i}\hat{\sigma}_{i}\hat{a}^{\dagger}\Big{)}+\hbar
J_{12}\left(\hat{\sigma}^{\dagger}_{1}\hat{\sigma}_{2}+\hat{\sigma}^{\dagger}_{2}\hat{\sigma}_{1}\right),$
(2)

where $\Delta_{c}=\omega_{c}-\omega_{l}$,
$\Delta_{eg_{i}}=\omega_{eg_{i}}-\omega_{l}$ are the cavity and QEs detunings,
respectively. Non-vanishing commutation relations among different system
operators are given by

$\displaystyle\left[\hat{a},\hat{a}^{\dagger}\right]=1,\hskip
2.84526pt\text{and}\hskip
2.84526pt\\{\hat{\sigma}^{\dagger}_{j},\hat{\sigma}_{k}\\}=\delta_{jk},\hskip
2.84526pt\text{with}~{}~{}j=1,2;~{}k=1,2.$

The first term on the right-hand side of Eq. (II) describes the free
Hamiltonian of the optical cavity. The second term represents the free
Hamiltonian of the first and second QE. In the QE Hamiltonian, we have assumed
the ground state energies to be zero. The third and fourth terms show the
Jaynes-Cummings interaction between QEs and the single mode of the optical
cavity. The fifth and sixth terms represent the DDI between the QEs.

##  III Dissipative System Dynamics and Output Fields

Under the realistic treatment of the problem, we assume that the system is
open to interact with the environmental degrees of freedom. Consequently, we
decompose the total Hamiltonian $\hat{\mathcal{H}}$ of the global system
(including QEs, optical cavity, and environment) into three pieces as

$\displaystyle\hat{\mathcal{H}}=\hat{\mathcal{H}}_{sys}+\hat{\mathcal{H}}_{bath}+\hat{\mathcal{H}}_{int}.$
(3)

The system Hamiltonian $\hat{\mathcal{H}}_{sys}$ has already been specified in
Eq. (II). We focus on bath Hamiltonian $\hat{\mathcal{H}}_{bath}$ now. In
there, to describe the cavity mode leakage from the imperfect cavity mirrors,
we introduce two multimode quantum harmonic oscillators which model the
environment(/bath/reservoir) Hamiltonian $\hat{\mathcal{H}}_{bath}$ for the
intra-cavity field. One multimode environment for the left (or $l$) mirror and
the other for the right (or $r$) mirror. The annihilation of a photon in the
$\omega~{}(\nu)$th mode of the left (right) bath is carried out by the
operator $\hat{b}_{l}(\omega)\left(\hat{b}_{r}(\nu)\right)$. Moreover, to
incorporate spontaneous emission from both QEs we assume two independent
baths. We use $\hat{S}_{1}(\mu)$ and $\hat{S}_{2}(\xi)$ annihilation operators
to describe the photon destruction in the $\mu$th and $\xi$th modes,
respectively. With these considerations, the bath Hamiltonian takes the form

$\displaystyle\hat{\mathcal{H}}_{bath}=~{}\hbar\int^{\infty}_{-\infty}\omega\hat{b}^{\dagger}_{l}(\omega)\hat{b}_{l}(\omega)d\omega+\hbar\int^{\infty}_{-\infty}\nu\hat{b}^{\dagger}_{r}(\nu)\hat{b}_{r}(\nu)d\nu$
$\displaystyle+\hbar\int^{\infty}_{-\infty}\mu\hat{S}^{\dagger}_{1}(\mu)\hat{S}_{1}(\mu)d\mu+\hbar\int^{\infty}_{-\infty}\xi\hat{S}^{\dagger}_{2}(\xi)\hat{S}_{2}(\xi)d\xi,$
(4)

where the non-vanishing commutation relations among different bath operators
are given by

$\displaystyle\left[\hat{b}_{l}(\omega),\hat{b}^{\dagger}_{l}(\omega^{{}^{\prime}})\right]=\delta(\omega-\omega^{{}^{\prime}}),~{}\left[\hat{b}_{r}(\nu),\hat{b}^{\dagger}_{r}(\nu^{{}^{\prime}})\right]=\delta(\nu-\nu^{{}^{\prime}});$
$\displaystyle\left[\hat{S}_{1}(\mu),\hat{S}^{\dagger}_{1}(\mu^{{}^{\prime}})\right]=\delta(\mu-\mu^{{}^{\prime}}),~{}\left[\hat{S}_{2}(\xi),\hat{S}^{\dagger}_{2}(\xi^{{}^{\prime}})\right]=\delta(\xi-\xi^{{}^{\prime}}).$

Finally, the interaction Hamiltonian $\hat{\mathcal{H}}_{int}$, describing the
coupling between the optical mode and QEs with their respective environmental
degrees of freedom, is expressed as

$\displaystyle\hat{\mathcal{H}}_{int}=~{}\hbar\sqrt{\frac{\kappa_{l}}{2\pi}}\int^{\infty}_{-\infty}\left(\hat{a}^{\dagger}\hat{b}_{l}(\omega)+\hat{b}^{\dagger}_{l}(\omega)\hat{a}\right)d\omega+\hbar\sqrt{\frac{\kappa_{r}}{2\pi}}\times$
$\displaystyle\int^{\infty}_{-\infty}\left(\hat{a}^{\dagger}\hat{b}_{r}(\nu)+\hat{b}^{\dagger}_{r}(\nu)\hat{a}\right)d\nu+\hbar\sqrt{\frac{\gamma_{1}}{\pi}}\int^{\infty}_{-\infty}\Big{(}\hat{\sigma}^{\dagger}_{1}\hat{S}_{1}(\mu)+$
$\displaystyle\hat{S}^{\dagger}_{1}(\mu)\hat{\sigma}_{1}\Big{)}d\mu+\hbar\sqrt{\frac{\gamma_{2}}{\pi}}\int^{\infty}_{-\infty}\left(\hat{\sigma}^{\dagger}_{2}\hat{S}_{2}(\xi)+\hat{S}^{\dagger}_{2}(\xi)\hat{\sigma}_{2}\right)d\xi.$
(5)

In the above equation, the terms with prefactor
$\sqrt{\kappa_{l}/2\pi}~{}(\text{or}~{}\sqrt{\kappa_{r}/2\pi})$ represent the
coupling of the cavity mode field with the left (or right) environment leading
to photon leakage from the cavity while the terms with the prefactor
$\sqrt{\gamma_{1}/2\pi}~{}(\text{or}~{}\sqrt{\gamma_{2}/2\pi})$ show the
interaction of first (or second) QE with its environment that will lead to
spontaneous emission. Note that in the description of system-environment
coupling, we have applied the rotating wave approximation such that within the
global system the excitation number is always conserved. Furthermore, we have
made use of the Markov approximation according to which all decay rates
($\kappa_{l},\kappa_{r},\gamma_{1}$ and $\gamma_{2}$) are assumed to be
independent of their respective bath mode frequency.

Next, to describe the dynamics of the open quantum system under study, we
formulate the problem in the Heisenberg picture. Therein (and as outlined in
Appendix A) the dissipative dynamics can be described in terms of Heisenberg-
Langevin equations (a set of coupled differential equations for the system
operators incorporating decay processes and quantum noise from the
environment). Since we are interested in the spectrum emitted by our CQED
system, we perform a trace operation on both sides of Eq. (29). As a result,
we obtain the following equations of motion for the expectation value of
operators

$\displaystyle\partial_{t}\left<\hat{a}\right>$
$\displaystyle=-\left(i\Delta_{c}+\frac{\kappa_{l}+\kappa_{r}}{2}\right)\left<\hat{a}\right>-i\sum\limits^{2}_{j=1}g^{\ast}_{j}\left<\hat{\sigma}_{j}\right>-\sqrt{\kappa_{l}}\left<\hat{a}^{(l)}_{in}\right>-\sqrt{\kappa_{r}}\left<\hat{a}^{(r)}_{in}\right>,$
(6a) $\displaystyle\partial_{t}\left<\hat{\sigma}_{j}\right>$
$\displaystyle=-\left(i\Delta_{eg_{j}}+\gamma_{j}\right)\left<\hat{\sigma}_{j}\right>+ig_{j}\left<\hat{\sigma}_{z,j}\hat{a}\right>+iJ_{jk}\left<\hat{\sigma}_{z,j}\hat{\sigma}_{k}\right>+\sqrt{2\gamma_{j}}\left<\hat{\sigma}_{z,j}\hat{\sigma}_{in,j}\right>,$
(6b) $\displaystyle\partial_{t}\left<\hat{\sigma}_{z,j}\right>=$
$\displaystyle-4\gamma_{j}\left(\frac{\left<\hat{\sigma}_{z,j}\right>+1}{2}\right)-2i\left(g_{j}\left<\hat{a}\hat{\sigma}^{\dagger}_{j}\right>-g^{\ast}_{j}\left<\hat{\sigma}_{j}\hat{a}^{\dagger}\right>\right)-2iJ_{jk}\left(\left<\hat{\sigma}^{\dagger}_{j}\hat{\sigma}_{k}\right>-\left<\hat{\sigma}^{\dagger}_{k}\hat{\sigma}_{j}\right>\right)$
$\displaystyle-2\sqrt{2\gamma_{j}}\left(\left<\hat{\sigma}^{\dagger}_{j}\hat{\sigma}_{in,j}\right>+\left<\hat{\sigma}^{\dagger}_{in,j}\hat{\sigma}_{j}\right>\right),$
(6c)

where $j=1,2$; $k=1,2$ and $j\neq k$ with $J_{12}=J_{21}=J$ and
$\hat{\sigma}_{z,j}\equiv[\hat{\sigma}^{\dagger}_{j},\hat{\sigma}_{j}]$. To
proceed with the analytic calculations we now apply three approximations that
are central to this work. The first approximation we employ is called the
“mean-field approximation” agarwal2010electromagnetically  according to which
we replace the mean value of product of operators with the product of the mean
value of each operator i.e. for any two arbitrary operators $\hat{o}_{1}$ and
$\hat{o}_{2}$ we write
$\left<\hat{o}_{1}\hat{o}_{2}\right>=\left<\hat{o}_{1}\right>\left<\hat{o}_{2}\right>$.
In the second approximation, we assume a symmetric cavity i.e. we set
$\kappa_{l}=\kappa_{r}=\kappa$. And in the third approximation we apply the
“weak-excitation assumption” li2014tunable ; hu2015extended  according to
which we assume that for the majority of the time (on the time scale set by
the inverse of the atomic decay rate) both QEs stay in their ground state
implying $\left<\hat{\sigma}_{z,j}\right>\approx-1$, $j=1,2$. With these
assumptions, we take the steady-state conditions and solve Eq. set (6) which
gives us the following form of the intracavity field

$\displaystyle\left<\hat{a}\right>=$
$\displaystyle\frac{-\sqrt{\kappa}\left(\left<\hat{a}^{(l)}_{in}\right>+\left<\hat{a}^{(r)}_{in}\right>\right)}{i\Delta_{c}+\kappa+\left(\frac{g^{\ast}_{1}g_{2}\left(i\Delta_{eg_{1}}+\gamma_{1}\right)+g_{1}g^{\ast}_{2}\left(i\Delta_{eg_{2}}+\gamma_{2}\right)-i\left(\left|g_{1}\right|^{2}+\left|g_{2}\right|^{2}\right)J_{12}}{\left(i\Delta_{eg_{1}}+\gamma_{1}\right)\left(i\Delta_{eg_{2}}+\gamma_{2}\right)+J^{2}_{12}}\right)},$
(7) $\displaystyle\implies\left<\hat{a}^{(l)}_{out}\right>=$
$\displaystyle\left<\hat{a}^{(l)}_{in}\right>-\frac{\kappa\left(\left<\hat{a}^{(l)}_{in}\right>+\left<\hat{a}^{(r)}_{in}\right>\right)}{i\Delta_{c}+\kappa+\left(\frac{g^{\ast}_{1}g_{2}\left(i\Delta_{eg_{1}}+\gamma_{1}\right)+g_{1}g^{\ast}_{2}\left(i\Delta_{eg_{2}}+\gamma_{2}\right)-i\left(\left|g_{1}\right|^{2}+\left|g_{2}\right|^{2}\right)J_{12}}{\left(i\Delta_{eg_{1}}+\gamma_{1}\right)\left(i\Delta_{eg_{2}}+\gamma_{2}\right)+J^{2}_{12}}\right)},$
(8)

while writing the expectation value of the output field we have taken the
trace of the input-output relations (as derived in Appendix B) i.e.

$\displaystyle\left<\hat{a}^{(l)}_{out}\right>=$
$\displaystyle\left<\hat{a}^{(l)}_{in}\right>+\sqrt{\kappa_{l}}\left<\hat{a}\right>,$
(9a) $\displaystyle\left<\hat{a}^{(r)}_{out}\right>=$
$\displaystyle\left<\hat{a}^{(r)}_{in}\right>+\sqrt{\kappa_{r}}\left<\hat{a}\right>,$
(9b)

and inserted Eq. (7) into Eq. (9). Note that in Eq. (8) we have neglected any
direct input to the QEs i.e. we have set $\left<\hat{\sigma}_{in,j}\right>=0$.
We notice that for the case of a single QE the above expression for
$\left<\hat{a}\right>$ simplifies to the following result which is already
reported by Agarwal et al. in Ref. agarwal2016perfect

$\displaystyle\left<\hat{a}\right>=\frac{-\sqrt{\kappa}\left(\left<\hat{a}^{(l)}_{in}\right>+\left<\hat{a}^{(r)}_{in}\right>\right)}{i\Delta_{c}+\kappa+\frac{|g|^{2}}{(i\Delta_{eg}+\gamma)}}.$
(10)

Since we have assumed identical inputs
$\left<\hat{a}^{(r)}_{in}\right>=\left<\hat{a}^{(l)}_{in}\right>$ and intra-
cavity parameters are identical for both left and right channels, therefore,
we conclude
$\left<\hat{a}^{(r)}_{out}\right>=\left<\hat{a}^{(l)}_{out}\right>$. Equipped
with these mathematical results, in the next subsection, we concentrate on the
conditions under which the CPA can be achieved.

##  IV CPA conditions and the dressed State Analysis

###  IV.1 CPA conditions

The CPA requires all input fields to be completely absorbed resulting in no
output field detection in both left and right channels i.e.
$\left<\hat{a}^{(l)}_{out}\right>=\left<\hat{a}^{(r)}_{out}\right>=0$.
Therefore, we impose this condition on the output fields found in the last
section (Eq. (8)). After some rearrangement of terms, we arrive at the
following two conditions necessary to achieve CPA in the present setup

$\displaystyle\kappa\left(-\Delta^{2}_{eg}+\gamma^{2}+J^{2}\right)=-2\Delta_{c}\Delta_{eg}\gamma+2g^{2}\gamma,$
(11a) $\displaystyle
2\kappa\gamma\Delta_{eg}=~{}\Delta_{c}\left(-\Delta^{2}_{eg}+\gamma^{2}+J^{2}\right)+2g^{2}\left(\Delta_{eg}-J\right).\
$ (11b)

Here for the sake of simplicity we have assumed both QEs as well as both input
fields to be identical i.e. we have set
$\Delta_{eg_{1}}=\Delta_{eg_{2}}=\Delta_{eg}$, $\gamma_{1}=\gamma_{2}=\gamma$,
$g_{1}=g_{2}=g$ (with real-valued emitter-cavity coupling parameters), and
$\left<\hat{a}^{(l)}_{in}\right>=\left<\hat{a}^{(r)}_{in}\right>=\left<\hat{a}_{in}\right>$.
We now develop a physical understanding of these conditions. To this end, we
present two different descriptions. In the first description, we rely on the
mathematical results and analyze the phase relationship among the incident and
cavity-emitted light fields to study the destructive interference. In the
second approach, we will be paying attention to the polariton state formation
and inter-state transitions by performing a dressed-state analysis.

E$\left|e\right\rangle$$\left|g\right\rangle$$\hbar{J}$QE$\left|0\right\rangle$$\left|1\right\rangle$$\left|2\right\rangle$Field⋮$\left|g,0\right\rangle$$\frac{1}{\sqrt{2}}(\left|g,1\right\rangle+\left|e,0\right\rangle)$$\frac{1}{\sqrt{2}}(-\left|g,1\right\rangle+\left|e,0\right\rangle)$Dressed
States⋮$\sqrt{2}\hbar{g}$$\sqrt{2}\hbar{g}$$\frac{1}{\sqrt{2}}(\left|g,2\right\rangle+\left|e,1\right\rangle)$$\frac{1}{\sqrt{2}}(-\left|g,2\right\rangle+\left|e,1\right\rangle)$$2\hbar{g}$$2\hbar{g}$
Figure 2: (Color online) Energy-level diagram for two-QEs CQED architecture
with DDI between QEs. A polariton state manifold with up to two excitations is
drawn. In this diagram we have assumed an on-resonance case i.e.
$\omega_{eg}=\omega_{c}$.

####  IV.1.1 Interpretation based on the CPA conditions

To understand what is causing no filed detection at the two output channels,
we first consider a single input situation and assume there is no input field
present from the right channel of the cavity i.e.
$\left<\hat{a}^{(r)}_{in}\right>=0$. Under these conditions, Eq. (8), and Eq.
(11) yield

$\displaystyle\left<\hat{a}^{(r)}_{out}\right>=-\frac{1}{2}\left<\hat{a}^{(l)}_{in}\right>,\hskip
14.22636pt\text{and}\hskip
14.22636pt\left<\hat{a}^{(l)}_{out}\right>=+\frac{1}{2}\left<\hat{a}^{(l)}_{in}\right>.$
(12)

We notice that the expectation values of both outputs have the same magnitude
but the right (left) channel output is out (in) phase from the input field by
$\pi$ radian. Next, we assume the opposite situation and consider no input
field from the left channel i.e. $\left<\hat{a}^{(l)}_{in}\right>=0$. In this
case, we find

$\displaystyle\left<\hat{a}^{(r)}_{out}\right>=+\frac{1}{2}\left<\hat{a}^{(r)}_{in}\right>,\hskip
14.22636pt\text{and}\hskip
14.22636pt\left<\hat{a}^{(l)}_{out}\right>=-\frac{1}{2}\left<\hat{a}^{(r)}_{in}\right>.$
(13)

We notice in this case the left output field becomes out of phase from the
input field. Next, if we combine both of these situations and suppose
identical input fields
i.e.$\left<\hat{a}^{(l)}_{in}\right>=\left<\hat{a}^{(r)}_{in}\right>$ then we
notice that both output fields vanish. We observe that once the CPA conditions
are met there is a complete cancellation of the output fields in both left and
right channels due to the destructive interference between the incident laser
beams and the cavity-emitted field. It is also important to emphasize the need
of using two identical input fields for such a perfect cancellation of fields
at the output detection channels. We point out here that a similar
interpretation has also been presented in Ref. agarwal2015photon  in the
context of perfect photon absorption in a single QE CQED setting. This shows
that even though the DDI causes the intra-cavity field to change as compared
to the intra-cavity field found in the single QE scenario but as long as the
CPA conditions (which takes care of the DDI) are fulfilled the destructive
interference still occurs at the output channels.

####  IV.1.2 Dressed state analysis

In order to study the impact of DDI on the energy level configuration of our
Tavis-Cummings setup, in this subsection, we perform a dressed state analysis
of the problem. It turns out that with the presence of the second QE trapped
inside the optical cavity and directly interacting with the first QE through
the DDI, it becomes rather difficult to calculate the dressed states of the
combined atom-field system zhang2014effects . However, this issue can be
tackled under the weak-excitation approximation whereby following Ref.
nicolosi2004dissipation  we introduce a unitary transformation

$\displaystyle\hat{U}=\exp\left[-\frac{\pi}{4}\left(\hat{\sigma}^{\dagger}_{1}\hat{\sigma}_{2}-\hat{\sigma}^{\dagger}_{2}\hat{\sigma}_{1}\right)\right].$
(14)

Such a transformation can be used to reduce our two-QE CQED problem into an
effective single QE CQED problem, as we show below. To this end, we apply the
aforementioned $\hat{U}$ onto our original two QE system Hamiltonian (given in
Eq. (II)) and as a result, we obtain the following Hamiltonian

$\displaystyle\widetilde{H}_{sys}$
$\displaystyle\equiv\hat{U}^{\dagger}\hat{\mathcal{H}}_{sys}\hat{U}$
$\displaystyle=\hbar\Delta_{c}\hat{a}^{\dagger}\hat{a}+\hbar\left(\Delta_{eg}+J\right)\hat{\sigma}^{\dagger}_{1}\hat{\sigma}_{1}+\hbar\left(\Delta_{eg}-J\right)\hat{\sigma}^{\dagger}_{2}\hat{\sigma}_{2}$
$\displaystyle+\sqrt{2}\hbar
g\left(\hat{a}\hat{\sigma}^{\dagger}_{1}+\hat{\sigma}_{1}\hat{a}^{\dagger}\right),$
(15)

where we have adopted the simplification that both QEs have identical
transition frequencies. We notice that in the transformed frame only the first
QE (with a shifted frequency $\Delta_{eg}+J$) couples with the cavity field
mode with a modified coupling strength $\sqrt{2}g$. The second QE (with a
modified frequency $\Delta_{eg}-J)$) is completely decoupled from the field
and hence can be neglected for focusing on the interaction part of the
processes. Introducing a simplified notation in which
$\hat{\sigma}_{1}=\hat{\sigma}$,
$\hat{\sigma}^{\dagger}_{1}=\hat{\sigma}^{\dagger}$ and using the basis set
$\\{\left|e\right\rangle\otimes\left|n-1\right\rangle\equiv\left|e,n-1\right\rangle$,
$\left|g\right\rangle\otimes\left|n\right\rangle\equiv\left|g,n\right\rangle\\}$,
one can then diagonalize the transformed Hamiltonian Eq. (IV.1.2).
Accordingly, we obtain the following eigensystem (with eigenvalues
$\lambda_{\pm}$ and corresponding eigenvectors
$\left|\lambda_{\pm}\right\rangle$)

$\displaystyle\lambda_{\pm}=n\hbar\Delta_{c}+\frac{\hbar}{2}\widetilde{\Delta}_{ac}\pm\frac{\hbar}{2}\widetilde{\Omega}_{n}\left(\widetilde{\Delta}_{ac}\right),$
(16)
$\displaystyle\left|\lambda_{+}\right\rangle=\cos\left(\frac{\Phi_{n}}{2}\right)\left|e,n-1\right\rangle+\sin\left(\frac{\Phi_{n}}{2}\right)\left|g,n\right\rangle,$
(17)
$\displaystyle\left|\lambda_{-}\right\rangle=-\cos\left(\frac{\Phi_{n}}{2}\right)\left|g,n\right\rangle+\sin\left(\frac{\Phi_{n}}{2}\right)\left|e,n-1\right\rangle,$
(18)

where $\widetilde{\Delta}_{ac}:=\Delta_{eg}-\Delta_{c}+J$,
$\widetilde{\Omega}_{n}\left(\widetilde{\Delta}_{ac}\right):=\sqrt{\widetilde{\Delta}^{2}_{ac}+8g^{2}n}$,
and
$\tan\left(\frac{\Phi_{n}}{2}\right):=\frac{2g\sqrt{2n}}{\widetilde{\Delta}_{ac}}$.
Note that the above results readily represent the single QE case as well if we
set $J=0$ and replace $\sqrt{2}g\rightarrow g$. In Fig. 2, we have plotted the
energy-level diagram of our Tavis-Cummings model. In the bare state situation
(blue and green colored energy level) we have $2n$ number of uncoupled states
possible (where $n$ being the cavity photon number). When the QE and the
cavity field are allowed to interact, atom-cavity entanglement causes the
polariton state formation (red-colored states shown in Fig. 2). When we
compare two-QEs dressed states to the single QE problem, we observe that one
of the main changes is that the excited dressed states are now split by an
amount $2g\sqrt{2n}$. Furthermore, the ground state of the field-coupled
single-QE is shifted by an amount $\hbar J$.

##  V Results and Discussion

To set the stage of results, in Fig. 3, we plot the output spectra
$\left|\left<{\hat{a}_{out}}\right>\right|^{2}/\left|\left<{\hat{a}_{in}}\right>\right|^{2}$
(also called the normal-mode spectra) in the presence of two identical inputs
(and consequently identical outputs) as a function of the detuning
$\Delta\equiv\omega_{c}-\omega_{l}$ with $\omega_{eg_{j}}=\omega_{c},\forall
j=1,2$. The main focus of this plot is to analyze the impact of the DDI on the
spectral properties of our Tavis-Cummings setup. The results focusing on CPA
conditions are presented in the next figure onward. For comparison, we have
plotted the single QE ($N=1$) case (grey dotted curve) in Fig. 3 as well.
Using Eq.(10), for $N=1$ case, we obtain

Figure 3: The normalized output spectra as a function of detunning
$\Delta=\omega_{c}-\omega_{l}$ for the case of two identical input fields
(i.e. $\hat{a}^{(l)}_{in}=\hat{a}^{(r)}_{in}$ implying
$\hat{a}^{(l)}_{out}$=$\hat{a}^{(r)}_{out}=\hat{a}_{out}$). In this figure we
have assumed an on-resonance condition i.e.
$\omega_{eg_{1}}=\omega_{eg_{2}}=\omega_{c}$ and varied the strength of the
DDI $J_{12}=J$ between the two QEs ($N=2$ cases). For comparison $N=1$ (single
QE) case is also displayed. The rest of the parameters are
$\gamma_{1}=\gamma_{2}=\gamma$, $\kappa=\gamma$, $g_{1}=g_{2}=10\gamma$.

$\displaystyle\left<\hat{a}_{out}\right>=\left<\hat{a}_{in}\right>-\frac{2\kappa\left<\hat{a}_{in}\right>}{i\Delta+\kappa+\frac{|g|^{2}}{(i\Delta_{eg}+\gamma)}}.$
(19)

For the single QE case, we note that the spectrum is a frequency doublet with
equal depths of both resonances. Moreover, the position of the resonances is
symmetric around $\Delta=0$. Ignoring the spontaneous emission and cavity
losses (which have been neglected while performing the dressed-state
analysis), the separation between the two dips is given by $2g$. This is the
typical Rabi splitting encountered in the Jaynes-Cummings model and it is
regarded as the hallmark of the strong coupling regime of light-matter
interaction in the CQED and circuit QED platforms thompson1998nonlinear ;
cui2006emission ; wallraff2004strong .

Figure 4: The output field intensities (plot (a)), the intracavity field (plot
(b)), and the collective emitter excitation probability (plot (c)) for the
case of a single input drive entering the cavity from the left hand side i.e.
$\left<\hat{a}^{(r)}_{in}\right>=0$. In plot (a), the lighter (darker) colored
curves are the output field intensities from the left (right) channel. The
parameters are the same as used in the previous figure.

Figure 5: (a) Output intensities, (b) intracavity field intensity, and (c)
collective emitter excitation probability plotted as a function of detuning
$\Delta/\gamma$. Compared to Fig. 4 the difference now is that we have assumed
two identical input fields pumping the cavity mode from the left and right
mirrors. The rest of the parameters and the curve types are the same as used
in Fig. 4.

Moving on to the two identical QE problem, we notice that the expectation
value of the output operator takes the value

$\displaystyle\left<\hat{a}_{out}\right>=\left<\hat{a}_{in}\right>-\frac{2\kappa\left<\hat{a}_{in}\right>}{i\Delta+\kappa+\left(\frac{2|g|^{2}\left(i\Delta+\gamma\right)-2i\left|g\right|^{2}J}{\left(i\Delta+\gamma\right)^{2}+J^{2}}\right)}.$
(20)

As shown in Fig. 3, we note that for all cases of two QEs, the spectrum
remains a frequency doublet. However, compared to the single QE case, the
separation between the two dips, for $J=0$ case, takes the value $2\sqrt{2}g$.
See the red thick solid curve with $g=10\gamma$, $J=0$ and we are observing a
splitting of $\sim 28.2\gamma$. This observation is consistent with the
prediction of the dressed state analysis where the peak separation (for either
$J=0$ or $J\neq 0$ scenario) is expected to take the value
$\Delta\lambda=\lambda_{+}-\lambda_{-}=\hbar\widetilde{\Omega}_{n}\left(\widetilde{\Delta}_{ac}\right)$.
Next, we notice that the separation between the resonances tends to increase
as DDI $J$ is introduced and consequently enhanced (see blue dashed, green
dotted dashed, and brown solid curves). This trend can again be explained
using the dressed-state picture, where we expect the peak separation to
increases monotonically with $J$ (i.e.
$\Delta\lambda\propto\sqrt{8g^{2}+J^{2}}$). Additionally, we remark that the
minima in the output spectra take a higher value as we increase $J$. This
trend can be explained by calculating the transition matrix element
$\sqrt{\kappa}\left\langle\lambda_{+}\right|\hat{a}\left|g,0\right\rangle=\sqrt{\kappa}\sin\left(\Phi_{0}/2\right)$
and
$\sqrt{\kappa}\left\langle\lambda_{-}\right|\hat{a}\left|g,0\right\rangle=-\sqrt{\kappa}\cos\left(\Phi_{0}/2\right)$.
When $J=0$, it turns out that the QEs and cavity mode equally contribute such
that the peak depths take the same value, where when $J\neq 0$ is introduced
this equal contribution is disturbed. Overall in Fig. 3, we observe that the
inclusion of two QEs allows us better control of the spectral properties by
tuning the inter-emitter separation (which is equivalent to altering the DDI
as from Eq. (1) the inter-emitter separation $r_{12}$ is linked to the DDI $J$
through
$r^{3}_{12}=\frac{3\Gamma_{0}c^{3}}{4\pi^{3}\omega^{3}_{eg}}\frac{1}{J}$).

Afterward, we focus on the perfect absorption of the input fields. In Fig. 4
we consider the case of a single input drive from the left cavity channel
(i.e. we set $\left<\hat{a}^{(r)}_{in}\right>=0$). In all plots of this
figure, we are working in the strong coupling regime of CQED (with
cooperativity $\mathcal{C}\equiv g^{2}/(2\kappa\gamma)=50$) where the single
input drive is on resonance with the polariton states frequencies
$\lambda_{\pm}\propto\widetilde{\Omega}_{n}/2$. As we have seen in Sec. IV A,
for this case, the single input field fails to undergo complete destructive
interference at the output channels
$\Big{(}\left<\hat{a}^{(r)}_{out}\right>=-1/2\left<\hat{a}^{(l)}_{in}\right>=$
with
$\left<\hat{a}^{(l)}_{out}\right>=+1/2\left<\hat{a}^{(l)}_{in}\right>\Big{)}$
and hence no CPA is achieved. Fig. 4(a) confirms this result for both single
and double QE scenarios. For two QE cases, we notice as the DDI is enhanced
the left output field intensities show a decreasing tendency on the positive
$\Delta$-axis such that for $J=15\gamma$ case the left output field shows an
almost complete suppression at $\Delta\approx 7.8\gamma$. However, CPA is
still not achieved as at the same frequency, we notice the right output field
reaches more than $60\%$ intensity value. Furthermore, on the negative
$\Delta$-axis, the output field intensities show a trend opposite to their
positive $\Delta$-axis behavior as $J$ is increased. In Fig. 4(b) and 4(c) we
present the intracavity field (photonic excitation) and total emitter
excitation normalized to the input field
$\hat{a}^{(l)}_{in}\equiv\hat{a}_{in}$, respectively. In both of these plots,
we again observe a frequency doublet profile due to the strong coupling regime
($\mathcal{C}>1)$ of CQED. As a prominent feature, we observe that as
$J>\kappa$ and eventually when $J>g$, the peak on the positive $\Delta$-axis
of Fig. 4 takes a much higher value as compared to the peak on the negative
$\Delta$-axis. This asymmetry arises due to the fact that as $J>g$,
$\sin\left(\Phi_{n}/2\right)\approx 0$ while
$\cos\left(\Phi_{n}/2\right)\approx 1$ which favors the
$\left|\lambda_{-}\right\rangle\longrightarrow\left|g,0\right\rangle$
transition over the
$\left|\lambda_{+}\right\rangle\longrightarrow\left|g,0\right\rangle$
transition (see Fig. 2). In the atomic excitation probability (Fig. 4(c)) we
don’t observe such an asymmetry as our setup is pumped with a light field
directly shined onto the cavity mirrors that causes more impact on the cavity
bare states.

Next, we turn our attention to the case when both cavity mirrors are shined
with laser fields of equal phase and amplitude
($\hat{a}^{(l)}_{in}=\hat{a}^{(r)}_{in}$). As we have already discussed in
Sec. IV A if the CPA conditions given in Eq. (11) are met, perfect photon
trapping would occur. In Fig. 5(a), we numerically solve and plot the
normalized output field intensities
$\left|\left<\hat{a}_{out}\right>\right|^{2}/\left|\left<\hat{a}_{in}\right>\right|^{2}$
as a function of detuning $\Delta$. We confirmed the existence of CPA for both
$N=1$ and $N=2$ with $J=0$ cases (gray dotted and thick solid red curves)
where Eq. (11) conditions are satisfied. In other words at the two frequencies
where the photons are perfectly absorbed are the frequencies where the
incoming laser is tuned with the polariton resonance $\pm\sqrt{2g^{2}n}$.
Furthermore, the intracavity field intensity
$\left|\left<\hat{a}\right>\right|^{2}/\left|\left<\hat{a}_{in}\right>\right|^{2}$
and collective atomic excitation probabilities
$\left|\left<\hat{\sigma}_{1}+\hat{\sigma}_{2}\right>\right|^{2}/\left|\left<\hat{a}_{in}\right>\right|^{2}$
plotted in Fig. 5(b) and Fig. 5(c) shows that at the same frequencies the
photon energy is completely absorbed within the CQED system. As a key finding,
we notice that the enhancement in the DDI destroys the CPA (see blue dotted
and brown thin solid curves in Fig. 5(a)) by creating a mismatch between the
left and right-hand sides of conditions described in Eq. (11). Particularly,
when the value of DDI exceeds the atom-cavity coupling g i.e. when $J>g$ this
behavior, as evident from the brown thin solid curve, becomes most pronounced
such that the CPA diminishes to almost $35\%$ with $J/g=3/2$.

|  |
---|---|---
|  |
Figure 6: Outfield intensities and collective atomic excitation probabilities
for an off-resonance case. In the upper and lower rows of plots, we have
considered $\Delta_{ac}=15\gamma$ and $\Delta_{ac}=-15\gamma$, respectively.
Brown solid curves represent cases with $J=20\gamma$. The rest of the
parameters are the same as used in Fig. 4. Yellow highlighted regions in the
bottom row of plots show the frequency points of perfect photon absorption.
Note that, due to the selection of a far-off resonance scenario, in some of
the plots we have extended the range of $\Delta/\gamma$ values on the
horizontal axis to capture all spectral features.

In the results, up till now, we have assumed on-resonance conditions i.e. we
have set $\omega_{c}=\omega_{eg}$. An important question in this context is in
what ways an off-resonant situation can impact the CPA in the presence of DDI?
We now address this question. From the CPA conditions mentioned in Eq. (11),
we find four sets of possible values for detunings $\Delta_{eg}$ and
$\Delta_{c}$ out of which two sets can take real values and are given by

$\displaystyle\Delta_{eg}=-J\pm\sqrt{\frac{2g^{2}\gamma}{\kappa}-\gamma^{2}},~{}~{}~{}\Delta_{c}=\pm\sqrt{\frac{2g^{2}\kappa}{\gamma}-\kappa^{2}},$
$\displaystyle\implies\Delta_{ac}=-J\pm\sqrt{\frac{2g^{2}\gamma}{\kappa}-\gamma^{2}}\mp\sqrt{\frac{2g^{2}\kappa}{\gamma}-\kappa^{2}},$
(21)

where the DDI introduces a shift in the atom-laser (and consequently atom-
cavity) detuning frequency. Additionally, from the dressed state analysis as
shown in Eq. (16), we have found that the polariton state frequency is given
by
$\lambda_{\pm}/\hbar=\frac{1}{2}\widetilde{\Delta}_{ac}\pm\frac{1}{2}\widetilde{\Omega}_{n}\left(\widetilde{\Delta}_{ac}\right)$.
We observe that for the perfect photon trapping to occur both conditions
needed to be satisfied simultaneously. The second condition on $\lambda_{\pm}$
would ensure the excitation of polariton states while the first condition (as
mentioned in Eq. (V)) would then fulfill the requirement for a perfect
absorption of photons interacting with the atom-cavity setup. In Fig. 6 we
plot the output and inter-cavity field intensities along with the collective
atomic excitation probabilities subjected to an off-resonant situation (i.e
when $\omega_{c}\neq\omega_{eg}$). In the upper (lower) row of plots, we take
$\Delta=15\gamma$ ($\Delta=-15\gamma$) while in all plots a strong coupling
regime of CQED has been assumed. In the $\Delta=15\gamma$ case we notice in
Fig. 6(a) as DDI is elevated, the DDI and atom-cavity detuning act together to
suppress the CPA. On the contrary, in Fig. 6(d) the negative detuning combined
with the strong DDI tends to promote the CPA. As a result, for the specially
chosen numerical parameters of Fig. 6, we observe the perfect photon trapping
at $\Delta=-3\gamma$ and $\Delta=-32\gamma$. From Fig. 6(a), (d) and as well
from reference agarwal2015photon , we emphasize that the presence of CPA
provides a better control (through changing the inter-atomic separation) and
allowed us to achieve CPA at two tunable frequencies which are not possible to
attain with a single QE case in the presence of detuning (see grey-dotted
curves in Fig. 6(a) and (d)). As a last noticeable point in this figure, we
note from Fig. 6(e) and Fig. 6(f) that at the CPA frequencies the contribution
in the polariton state is dominated by the atomic excited state as compared to
the cavity field contribution which is consistent with the arguments presented
in Ref. zanotto2014perfect ; agarwal2014nanomechanical .

---
Figure 7: Plots showing the dependence of normalized optical properties on the
relative phase $\Delta\Phi=\varphi_{l}-\varphi_{r}$ between the two input
fields for the cases of a single QE, two QEs without any DDI, and two QEs with
strong DDI of $10\gamma$. (a) output field intensity with $d=l,r$, (b) intra-
cavity field intensity, and (c) atomic excitation probability. In plot (a),
due to same choice of parameters for both atoms, the output intensities in
left and right channel follow an identical pattern. In plot (b) and (c) we
have used i.e. $\hat{a}^{(l)}_{in}=\hat{a}^{(r)}_{in}\equiv\hat{a}_{in}$. As
before, the same parameters of strong cavity QED have been considered except
that now we have set $\Delta_{ac}=10\gamma$.

The output spectral properties of our Tavis-Cummings model also depend on the
relative phase between the two input fields, which up to this point has been
set equal to zero in all results for simplicity’s sake. We now introduced this
relative phase $\Delta\Phi$ through the relations

$\displaystyle\left<\hat{a}^{(l)}_{in}\right>=\left|\left<\hat{a}^{(l)}_{in}\right>\right|e^{i\varphi_{l}},~{}~{}~{}\text{and}~{}~{}~{}~{}\left|\left<\hat{a}^{(r)}_{in}\right>\right|e^{i\varphi_{r}}.$
(22)

In Fig. 7, we report the spectral properties of our setup as we vary the
relative phase $\Delta\Phi$ between $-2\pi$ to $2\pi$. For comparisons, we
present the cases of a single QE (grey dotted curve), two QEs with no DDI
(solid red curve), and two QEs with a strong DDI of $10\gamma$ (green dotted
dashed curve). First of all, in all plots, we notice a symmetric profile
around $\Delta\Phi=0$, while the output field intensities increases from $0$
to their maximum value of $1$ as $\Delta\Phi$ is varied from $0$ to $\pi$.
Whereas, the intra-cavity field and atomic excitation probability change from
their respective maximum values to the minimum value of $0$. This pattern
extends down to all three cases of $N=1$; $N=2,J=0$; and $N=2,J=10\gamma$. We
further point out that we observe a similar trend in our spectra for the on-
resonance i.e. $\Delta_{ac}$ case as well (results not shown here). However,
in the presence of a positive detuning of $10\gamma$, the differences between
the $N=2,J\neq 0$ case (green dotted dashed curve), and other cases are most
pronounced in the plot (b) and (c). In both plots (b) and (c), we observe that
the positive detuning in some sense works in conjunction with the strong DDI
to produce considerably elevated maximum values of atomic excitation
probability and thereby intra-cavity field intensity as compared to the
corresponding $N=2,J=0$ case, and thus can be utilized to distribute the
incoming photon energy between the optical mode and the atomic degrees of
freedom.

##  VI Summary and Conclusions

In this paper, we have studied the presence or absence of CPA of coherent
light fields in the Tavis-Cummings models of CQED. We derived the quantum
Langevin equations and input-output relations and studied the optical
properties of the system under the low-excitation limit, mean-field
approximation, and symmetric cavity assumption. We mathematically predicted
the conditions required to achieve CPA in the presence of DDI and developed
further physical understanding of the process by performing the dressed-state
analysis involving the polariton states manifold.

In our main findings, we concluded that a single input is not enough for the
perfect destructive interference between the incoming laser field and the
field emitted by our CQED system. And even in the presence of the two input
fields, strong DDI tends to destroy the CPA when the QEs and cavity mode are
in resonance. This behavior extended down to the scenario when atoms and
fields were positively detuned. However, as the main result of this work, we
found that some values of negative detuning (for instance $-15\gamma$) can
cancel the impact of strong DDI on the CPA allowing the perfect absorption of
the photons at two controllable laser frequencies under the strong coupling
regime of CQED. We compared this behavior to a single emitter CQED case in the
presence of detuning and found that this behavior is special to the Tavis-
Cummings models (without the possibility to be achieved with the single QE
CQED studies conducted earlier).

Finally, we discussed the dependence of the CPA on the relative phase between
the incoming light fields. We observed a periodic pattern of outfield
intensities as a function of relative phase and concluded that a strong DDI
and a positive detuning can generate considerable enhancement in the intra-
cavity and atomic excitation peak heights even in the presence of incoherent
dissipation from the polariton states. With the current progress in the CQED
and circuit QED platforms, our results may find applications in the
development of quantum photonic storage devices such as quantum memories with
usage in building quantum communication networks lvovsky2009optical ;
kimble2008quantum .

###### Acknowledgements.

IMM would like to thank Michael J. Crescimanno for bringing the subject of CPA
to our attention. This work is supported by the Miami University College of
Arts and Science and Physics Department start-up funding.

##  Appendix A Quantum Langevin Equations

In this Appendix we outline the derivation of the quantum Lengevin equations
steck2007quantum  which lay the foundations of our open quantum system
treatment in the Heisenberg picture. To this end, we first notice that the
relevant system operators in the present problem are: $\hat{a}(t)$,
$\hat{\sigma}_{j}(t)$, $\hat{\sigma}_{z,j}(t)$; while baths are represented in
terms of the operators $\hat{b}_{l}(\omega;t)$, $\hat{b}_{r}(\nu;t)$, and
$\hat{S}_{j}(\chi_{j};t)$ with $j=1,2$, $k=1,2$, $J_{jk}=J_{kj}$, and
$\chi_{1}=\mu$, $\chi_{2}=\xi$. Note that in this notation $\omega,\nu,$ and
$\chi_{j}$ are not frequencies conjugate to the time variable $t$ in the
Fourier space sense, rather these are the frequencies labeling the bath modes
for the cavity field and QEs. Using the Heisenberg equation of motion we find
that these operators obey the following set of coupled differential equations

$\displaystyle\partial_{t}\hat{a}(t)=-i\Delta_{c}\hat{a}(t)-i\sum\limits^{2}_{j=1}g^{\ast}_{j}\hat{\sigma}_{j}(t)-i\sqrt{\frac{\kappa_{l}}{2\pi}}\int^{\infty}_{-\infty}d\omega\hat{b}_{l}(\omega;t)-i\sqrt{\frac{\kappa_{r}}{2\pi}}\int^{\infty}_{-\infty}d\nu\hat{b}_{r}(\nu;t),$
(23a)
$\displaystyle\partial_{t}\hat{\sigma}_{j}(t)=-i\Delta_{eg_{j}}\hat{\sigma}_{j}(t)+ig_{j}\hat{\sigma}_{z,j}(t)\hat{a}(t)+iJ_{jk}\hat{\sigma}_{z,j}(t)\hat{\sigma}_{k}(t)+i\sqrt{\frac{\gamma_{j}}{\pi}}\hat{\sigma}_{z,j}(t)\int^{\infty}_{-\infty}d\mu\hat{S}_{j}(\mu;t),$
(23b)
$\displaystyle\partial_{t}\hat{\sigma}_{z,j}(t)=-2i\left(g_{j}\hat{a}(t)\hat{\sigma}^{\dagger}_{j}(t)-g^{\ast}_{j}\hat{\sigma}_{j}(t)\hat{a}^{\dagger}(t)\right)-2iJ_{jk}\left(\hat{\sigma}^{\dagger}_{j}(t)\hat{\sigma}_{k}(t)-\hat{\sigma}^{\dagger}_{k}(t)\hat{\sigma}_{j}(t)\right)$
$\displaystyle\hskip
52.63759pt-2i\sqrt{\frac{\gamma_{j}}{\pi}}\left(\hat{\sigma}^{\dagger}_{j}(t)\int^{\infty}_{-\infty}d\mu\hat{S}(\mu;t)-\int^{\infty}_{-\infty}d\mu\hat{S}^{\dagger}(\mu;t)\hat{\sigma}_{j}(t)\right),$
(23c)
$\displaystyle\partial_{t}\hat{b}_{l}(\omega;t)=-i\omega\hat{b}_{l}(\omega;t)-i\sqrt{\frac{\kappa_{l}}{2\pi}}\hat{a}(t),$
(23d)
$\displaystyle\partial_{t}\hat{b}_{r}(\nu;t)=-i\omega\hat{b}_{r}(\nu;t)-i\sqrt{\frac{\kappa_{r}}{2\pi}}\hat{a}(t),$
(23e)
$\displaystyle\partial_{t}\hat{S}_{j}(\chi_{j};t)=-i\chi_{j}\hat{S}_{j}(\chi_{j};t)-i\sqrt{\frac{\gamma_{j}}{\pi}}\hat{\sigma}_{j}(t).$
(23f)

We remark that the structure of the above equation set allows us to
straightforwardly extend the problem to $N$ number of QEs, in which case one
would need to solve a set of $3(N+1)$ coupled differential equations. For the
present case, proceeding with the derivation as the next step, we eliminate
the bath operators from Eq. (23a), (23b), and (23c) and solve for the system
dynamics. For this purpose, we take Eq. (23d) and apply the transformation:
$\hat{b}_{l}(\omega;t)\longrightarrow\hat{b}_{l}(\omega;t)e^{i\omega t}$ under
which we find

$\partial_{t}\left[\hat{b}_{l}(\omega;t)e^{i\omega
t}\right]=-i\sqrt{\frac{\kappa_{l}}{2\pi}}\hat{a}(t)e^{i\omega t}.$ (24)

Next, we integrate the above equation from some arbitrary past time $t_{0}$ to
some present time $t$ and obtain the solution

$\hat{b}_{l}(\omega;t)=\hat{b}_{l}(\omega;t_{0})e^{-i\omega(t-t_{0})}-i\sqrt{\frac{\kappa_{l}}{2\pi}}\int^{t}_{t_{0}}dt^{{}^{\prime}}\hat{a}(t^{{}^{\prime}})e^{-i\omega(t-t^{{}^{\prime}})}.$
(25)

Following the same line of calculations, we find that the other bath operators
obey

$\displaystyle\hat{b}_{r}(\nu;t)$
$\displaystyle=\hat{b}_{r}(\nu;t_{0})e^{-i\nu(t-t_{0})}-i\sqrt{\frac{\kappa_{r}}{2\pi}}\int^{t}_{t_{0}}dt^{{}^{\prime}}\hat{a}(t^{{}^{\prime}})e^{-i\nu(t-t^{{}^{\prime}})},$
(26) $\displaystyle\hat{S}_{j}(\chi_{j};t)$
$\displaystyle=\hat{S}_{j}(\chi_{j};t_{0})e^{-i\chi_{j}(t-t_{0})}-i\sqrt{\frac{\gamma_{j}}{\pi}}\int^{t}_{t_{0}}dt^{{}^{\prime}}\hat{\sigma}_{j}(t^{{}^{\prime}})$
$\displaystyle\times e^{-i\mu(t-t^{{}^{\prime}})}.$ (27)

Inserting Eq. (25), (26), and (27) into Eq. (23a), (23b), and (23c) (with
$\hat{b}_{l}(\omega;t_{0})$, $\hat{b}_{r}(\nu;t_{0})$, and
$\hat{S}_{j}(\chi_{j};t_{0})$ essentially entering as the initial conditions
for the left cavity channel, right cavity channel and QEs) and defining the
set of input operators gardiner1985input  as

$\displaystyle\int^{\infty}_{-\infty}d\omega\hat{b}_{l}(\omega;t_{0})e^{-i\omega(t-t_{0})}d\omega$
$\displaystyle:=-i\sqrt{2\pi}\hat{a}^{(l)}_{in}(t),$ (28a)
$\displaystyle\int^{\infty}_{-\infty}d\nu\hat{b}_{r}(\nu;t_{0})e^{-i\nu(t-t_{0})}$
$\displaystyle:=-i\sqrt{2\pi}\hat{a}^{(r)}_{in}(t),$ (28b)
$\displaystyle\int^{\infty}_{-\infty}d\chi_{j}\hat{S}_{j}(\chi_{j};t_{0})e^{-i\chi_{j}(t-t_{0})}$
$\displaystyle:=-i\sqrt{2\pi}\hat{\sigma}_{in,j}(t),$ (28c)

we arrive at the following set of quantum Langevin equations describing the
time evolution of system operators $\hat{a}$, $\hat{\sigma}_{j}$ and
$\hat{\sigma}_{z,j}$

$\displaystyle\partial_{t}\hat{a}(t)=$
$\displaystyle-\Big{(}i\Delta_{c}+\frac{\kappa_{l}+\kappa_{r}}{2}\Big{)}\hat{a}(t)-i\sum\limits^{2}_{j=1}g^{\ast}_{j}\hat{\sigma}_{j}(t)$
$\displaystyle-\sqrt{\kappa_{l}}\hat{a}^{(l)}_{in}(t)-\sqrt{\kappa_{r}}\hat{a}^{(r)}_{in}(t),$
(29a) $\displaystyle\partial_{t}\hat{\sigma}_{j}(t)=$
$\displaystyle-\left(i\Delta_{eg_{j}}+\gamma_{j}\right)\hat{\sigma}_{j}(t)+ig_{j}\hat{\sigma}_{z,j}(t)\hat{a}(t)$
$\displaystyle+iJ_{jk}\hat{\sigma}_{z,j}(t)\hat{\sigma}_{k}(t)+\sqrt{2\gamma_{j}}\hat{\sigma}_{z,j}(t)\hat{\sigma}_{in,j}(t),$
(29b) $\displaystyle\partial_{t}\hat{\sigma}_{z,j}(t)=$
$\displaystyle-4\gamma_{j}\Big{(}\frac{\hat{\sigma}_{z,j}(t)+\hat{1}}{2}\Big{)}-2i\Big{(}g_{j}\hat{a}(t)\hat{\sigma}^{\dagger}_{j}(t)$
$\displaystyle-g^{\ast}_{j}\hat{\sigma}_{j}(t)\hat{a}^{\dagger}(t)\Big{)}-2iJ_{jk}\Big{(}\hat{\sigma}^{\dagger}_{j}(t)\hat{\sigma}_{k}(t)$
$\displaystyle-\hat{\sigma}^{\dagger}_{k}(t)\hat{\sigma}_{k}(t)\Big{)}-2\sqrt{2\gamma_{j}}\Big{(}\hat{\sigma}^{\dagger}_{j}(t)\hat{\sigma}_{in,j}(t)$
$\displaystyle+\hat{\sigma}^{\dagger}_{in,j}(t)\hat{\sigma}_{j}(t)\big{)}.$
(29c)

We notice that there are three types of terms that are arising in the above
set of Langevin equations. The terms with detunings and atom-cavity coupling
rates describe the internal dynamics of the system. The terms with prefactors
$\kappa_{l},\kappa_{r}$, and $\gamma_{j}$ are the decay terms representing the
processes of energy loss from the system to the environment. Finally, the
terms with prefactors $\sqrt{\kappa_{l}}$, $\sqrt{\kappa_{r}}$, and
$\sqrt{\gamma_{j}}$ are the quantum noise/input terms describing the impact of
baths on the system at some past time $t_{0}$.

##  Appendix B Input-Output relations

We begin from Eq. (24) and (as before) we integrate it from some past time
$t_{0}$ to some present time $t$. We then integrate the resultant equation
over all possible environment modes (using $\int^{+\infty}_{-\infty}d\omega
e^{i\omega(t-t^{{}^{\prime}})}=2\pi\delta(t-t^{{}^{\prime}})$) and by
recognizing the input operator we arrive at the following equation for the
left channel

$\displaystyle\int^{\infty}_{-\infty}d\omega\hat{b}_{l}\left(\omega;t\right)=-i\sqrt{2\pi}\hat{a}^{(l)}_{in}(t)-i\sqrt{\frac{\pi\kappa_{l}}{2}}\hat{a}(t).$
(30)

Next, we specify an arbitrary future time $t_{1}$ and now integrate the Eq.
(24) from present time $t$ to $t_{1}$. After integrating over all possible
$\omega$ values we find

$\displaystyle\int^{\infty}_{-\infty}d\omega\hat{b}_{l}\left(\omega;t\right)=-i\sqrt{2\pi}\hat{a}^{(l)}_{out}(t)+i\sqrt{\frac{\pi\kappa_{l}}{2}}\hat{a}(t),$
(31)

where we identify the output operator for the field propagating in the left
channel $\hat{a}^{(l)}_{out}(t)$ as

$\displaystyle-i\sqrt{2\pi}\hat{a}^{(l)}_{out}(t):=\int^{\infty}_{-\infty}d\omega\hat{b}_{l}\left(\omega;t_{1}\right)e^{-i\omega(t_{1}-t)}d\omega.$
(32)

Comparing Eq. (30) and Eq. (31) we arrive at the following expression that
relates the output field and input fields (commonly known as the Collett and
Gardiner input-output relation gardiner1985input ) for the left channel

$\displaystyle\hat{a}^{(l)}_{out}(t)=\hat{a}^{(l)}_{in}(t)+\sqrt{\kappa_{l}}~{}\hat{a}(t).$
(33)

Following similar lines of calculations, we find the following set of input-
output relations for the right cavity channel and QEs

$\displaystyle\hat{a}^{(r)}_{out}(t)$
$\displaystyle=\hat{a}^{(r)}_{in}(t)+\sqrt{\kappa_{r}}~{}\hat{a}(t),$ (34a)
$\displaystyle\hat{\sigma}_{out,j}(t)$
$\displaystyle=\hat{\sigma}_{in,j}(t)+\sqrt{\gamma_{j}}~{}\hat{\sigma}_{j}(t).$
(34b)

Here $\hat{a}^{(r)}_{out}(t)$ and $\hat{a}^{(r)}_{in}(t)$ are the output and
input field operators for the right channel, respectively while
$\hat{\sigma}_{out,j}(t)$ and $\hat{\sigma}_{in,j}(t)$ are the respective
output and input operators for the $j$th QE. These output operators are
defined as

$\displaystyle-i\sqrt{2\pi}\hat{a}^{(r)}_{out}(t):$
$\displaystyle=\int^{\infty}_{-\infty}d\nu\hat{b}_{r}\left(\nu;t_{1}\right)e^{-i\nu(t_{1}-t)}d\nu,$
(35a) $\displaystyle-i\sqrt{2\pi}\hat{\sigma}_{out,j}(t):$
$\displaystyle=\int^{\infty}_{-\infty}d\chi_{j}\hat{S}_{j}\left(\chi_{j};t_{1}\right)e^{-i\chi_{j}(t_{1}-t)}d\chi_{j}.$
(35b)

We make this remark that the generic structure of the aforementioned input-
output relations essentially works as boundary conditions relating the output
field with the input fields while incorporating the system response. We also
note that the construction of the input, output, and system operators in the
input-output formalism is such that the condition of causality is always
satisfied gardiner2004quantum .

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