# A study of the quasi-probability distributions of the Tavis-Cummings model
under different quantum channels

Devvrat Tiwari<EMAIL_ADDRESS>Subhashish Banerjee<EMAIL_ADDRESS>Indian Institute of Technology Jodhpur-342030, India

###### Abstract

We study the dynamics of the spin and cavity field of the Tavis-Cummings model
using quasi-probability distribution functions and the second-order coherence
function, respectively. The effects of (non)-Markovian noise are considered.
The relationship between the evolution of the cavity photon number and spin
excitation under different quantum channels is observed. The equal-time
second-order coherence function is used to study the sub-Poissonian behavior
of light and is compared with the two-time second-order coherence function in
order to highlight the (anti)-bunching properties of the cavity radiation.

##  I Introduction

Interactions between atoms and the cavity field play a critical role in
science and technology and provide a foundation to quantum optics [1, 2, 3,
4]. They are at the core of numerous developments in spectroscopy, quantum
information processing, sensing, and lasers, among others. The first exactly
solvable quantum mechanical model of a single two-state atom interacting with
a cavity mode of an electromagnetic field was given by Jaynes and Cummings [5,
6]. It was developed to examine the processes of spontaneous emission and
absorption of photons in a cavity as well as to detect the presence of Rabi
oscillations in atomic excitations. Experimental verification of the Jaynes-
Cummings model [7, 8] significantly increased its importance. The Tavis-
Cummings (TC) model [9, 10, 11, 12, 13, 3, 14], a multi-atom generalization of
the Jaynes-Cummings model, is of fundamental importance in the quest to
understand atom-field interactions. It appears with different variations in
quantum physics and has a resemblance with the Dicke model [15, 16] under
dipole and rotating wave (RW) approximations, modulo the different coupling
strengths between atoms and the cavity field and the inhomogeneous transition
frequencies of the individual atoms.

Cavity quantum electrodynamics (cavity-QED) studies the properties of atoms
interacting with photons in cavities [17, 18, 19, 20]. Many theoretical models
can now be realized in laboratories thanks to advances in cavity-QED
experiments during the previous few decades. It is now possible to control the
isolated evolution of a few atoms coupled to a single mode inside a cavity.
The TC model has been realized in a number of experiment [21, 22, 23, 24], and
their applications can be found in [23, 21, 25, 26, 27] and references
therein.

The notion of phase space is a very useful concept in analyzing the dynamics
of classical systems. However, a straightforward extension to the phase space
domain in quantum mechanics is hampered because of the uncertainty principle.
In spite of this, quasi-probability distribution functions (QDs) for quantum
mechanical systems analogous to their classical counterparts can be
constructed [28, 3, 29, 30, 31, 32, 33, 34, 13]. These QDs are extremely
helpful because they offer a quantum-classical relationship and make it easier
to calculate quantum mechanical averages that are analogous to classical phase
space averages. However, the QDs are not probability distributions since they
may also have negative values. The first such QD developed was the Wigner $W$
function [35, 36, 37, 38, 39, 40]. The $P$ function is a different, well-known
QD that acts as a witness to quantumness in the system. It can become singular
for some quantum states. The $W$ and $P$ functions, along with the $Q$
function [41, 42, 43], are used in the present study. The problem of operator
orderings is closely tied to these QDs. As a result, the $P$ and $Q$ functions
are related to normal and anti-normal orderings, respectively, while the $W$
function is connected to symmetric operator ordering [44].

The tremendous interest in these QDs can be attributed to a number of factors.
We can use them to identify a state’s non-classical characteristics
(quantumness in the system) [45]. Values of $P$ function that are non-positive
precisely characterize a non-classical state. $P$ function’s non-positivity is
a necessary and sufficient condition for non-classicality in a system,
although other QDs provide only sufficient criteria. Tasks that are impossible
in a classical state can be accomplished using a non-classical state. Numerous
investigations on non-classical states, such as those on squeezed,
antibunched, and entangled states, were motivated by this [46]. Interestingly,
many of these applications have been developed using spin-qubit systems.

A realistic quantum system is subjected to the influence of the environment.
These interactions significantly change the system’s dynamics and result in
the loss of information from the system to the environment. The theory of open
quantum systems (OQS) [47, 48, 49] provides a framework to study the impact of
the environment on a quantum system. Open quantum system ideas cater to a
broad spectrum of disciplines [44, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64]. In many circumstances, the dynamics of an OQS may be
characterized using a Markov approximation, which assumes that the environment
instantaneously recovers from its contact with the system, resulting in a
continuous flow of information from the system to the environment. However,
growing technical as well as technological advances are pushing the study into
regimes beyond Markovian approximation. A neat separation between system and
environment time scales can no longer be expected in many of these
circumstances, resulting in non-Markovian behavior [65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83].

With the motivation to understand the impact of noise, both Markovian and non-
Markovian, in the context of cavity-QED, specifically on the Tavis-Cummings
model, we will make use of the $W$, $P$, and $Q$ QDs to study the dynamics of
the spin system. For the characterization of the cavity field, the second-
order coherence function [1, 2] will be used.

This paper is organized as follows: in Sec. II, we present, briefly, the
Tavis-Cummings model and a master equation to understand the open system
dynamics of the TC model. A brief discussion on QDs is provided in Sec. III
followed by a study of the dynamics of the spin system using QDs in Sec. IV
and the dynamics of the cavity field in Sec. V, under ambient noisy
conditions. We then make our conclusions.

##  II The Model

Here we consider the Tavis-Cummings model with $N$ two-level atoms or spins
with inhomogeneous transition frequencies coupled to a single mode field
cavity with different coupling strengths. Under the dipole and rotating wave
approximations (RWA), we can write the Hamiltonian of the system (with
$\hbar=1$) as

$H=\frac{1}{2}\sum_{k=1}^{N}\omega_{k}\sigma^{z}_{k}+\omega_{c}a^{\dagger}a+\sum_{k=1}^{N}g_{k}(\sigma_{k}^{+}a+\sigma_{k}^{-}a^{\dagger}),$
(1)

where $\omega_{c}$ is the resonance frequency of the cavity field, and
$\omega_{k}$ and $g_{k}$ are the transition frequencies and coupling strength
for the $k^{th}$ spin. $\sigma^{z}_{k}$, $\sigma^{+}_{k}$ and $\sigma^{-}_{k}$
are the Pauli spin-$1/2$ operators.

To account for the losses in the spin and cavity system due to various
dissipative processes, such as spontaneous emission and cavity decay due to
imperfections in the cavity, we use the tools of open quantum systems. To this
end, the losses can be modeled by the following master equation

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i[H,\rho(t)]+\frac{1}{2}\sum_{k=1}^{N}\bigg{[}\gamma_{k}(N_{th,k}+1)(2\sigma_{k}^{-}\rho(t)\sigma_{k}^{+}$
$\displaystyle-\\{\sigma_{k}^{+}\sigma_{k}^{-},\rho(t)\\})+\gamma_{k}N_{th,k}(2\sigma_{k}^{+}\rho(t)\sigma_{k}^{-}-\\{\sigma_{k}^{-}\sigma_{k}^{+},\rho(t)\\})\bigg{]}$
$\displaystyle+\frac{1}{2}\left[\kappa(N_{th,k}+1)(2a\rho(t)a^{\dagger}-\\{a^{\dagger}a,\rho(t)\\})\right.$
$\displaystyle+\left.\kappa
N_{th,k}(2a^{\dagger}\rho(t)a-\\{aa^{\dagger},\rho(t)\\})\right].$ (2)

Here $\kappa$ corresponds to the cavity decay rate, $\gamma_{k}$ corresponds
to the spontaneous emission rate, and $N_{th}$ is the average number of
thermal photons. Constant values of these decay rates generate a semi-group
type of Lindblad equation, called the GKSL master equation [84, 85], whereas
their time dependence typically models non-Markovian scenarios. In the
subsequent sections, we will see the impact of various types of noise
(Markovian and non-Markovian) on the dynamics of the system.

##  III Quasi-probability Distribution Functions

Here we briefly discuss the Wigner $W$, $P$, and $Q$ quasi-probability
distribution functions (QDs) to be used subsequently.

###  III.1 The Wigner $W$ function

The Wigner $W$ function for a single spin-$j$ state, as a function of polar
and azimuthal angles expanded over special harmonics, can be given as

$W(\theta,\phi)=\bigg{(}\frac{2j+1}{4\pi}\bigg{)}\sum_{\mu,\eta}\rho_{\mu\eta}Y_{\mu\eta}(\theta,\phi),$
(3)

where $\mu=0,1,\dots,2j$ and $\eta=-\mu,-\mu+1,\dots,0,\dots,\mu-1,\mu$, and

$\rho_{\mu\eta}=\mathrm{Tr}\big{[}T_{\mu\eta}^{\dagger}\rho\big{]}.$ (4)

Further, $Y_{\mu\eta}$ are the spherical harmonics and $T_{\mu\eta}$ are the
multipole operators [86, 87] given by

$T_{\mu\eta}=\sum_{m,m^{\prime}}(-1)^{j-m}(2\mu+1)^{1/2}\begin{pmatrix}j&\mu&j\\\
-m&\eta&m^{\prime}\end{pmatrix}\ket{j,m}\bra{j,m^{\prime}},$ (5)

where $\begin{pmatrix}j_{1}&j_{2}&j\\\
m_{1}&m_{2}&m\end{pmatrix}=\frac{(-1)^{j_{1}-j_{2}-m}}{\sqrt{2j+1}}\bra{j_{1}m_{1}j_{2}m_{2}}{j-m}\rangle$
is the Wigner $3j$ symbol and $\bra{j_{1}m_{1}j_{2}m_{2}}{j-m}\rangle$ is the
Clebsch-Gordon coefficient. The multipole operators are orthogonal to each
other, and they form a complete set with
$T_{\mu\eta}^{\dagger}=(-1)^{\eta}T_{\mu,-\eta}$. The $W$ function satisfies
the normalization condition

$\int W(\theta,\phi)\sin\theta d\theta d\phi=1,$ (6)

and $W^{*}(\theta,\phi)=W(\theta,\phi)$. In a similar way, we can write the
$W$ function for an $N$ particle system, each with spin-$j$ as

$\displaystyle W(\theta_{1},\phi_{1},\dots,\theta_{N},\phi_{N})$
$\displaystyle=\bigg{(}\frac{2j+1}{4\pi}\bigg{)}\sum_{\mu_{1},\eta_{1}}\sum_{\mu_{2},\eta_{2}}...\sum_{\mu_{N},\eta_{N}}$
$\displaystyle\rho_{\mu_{1}\eta_{1}\mu_{2}\eta_{2},\dots,\mu_{N}\eta_{N}}Y_{\mu_{1}\eta_{1}}(\theta_{1},\phi_{1})$
$\displaystyle\times Y_{\mu_{2},\eta_{2}}(\theta_{2},\phi_{2})\dots
Y_{\mu_{N}\eta_{N}}(\theta_{N},\phi_{N}),$ (7)

where
$\rho_{\mu_{1}\eta_{1}\mu_{2}\eta_{2}\dots\mu_{N}\eta_{N}}=\mathrm{Tr}\big{\\{}\rho
T^{\dagger}_{\mu_{1}\eta_{1}}T^{\dagger}_{\mu_{2}\eta_{2}}\dots
T^{\dagger}_{\mu_{N}\eta_{N}}\big{\\}}$, satisfying the normalization
condition

$\displaystyle\int
W(\theta_{1},\phi_{1},\dots,\theta_{N},\phi_{N})\sin\theta_{1}\sin\theta_{2}\dots\sin\theta_{N}d\theta_{1}d\phi_{1}$
$\displaystyle\times d\theta_{2}d\phi_{2}\dots d\theta_{N}d\phi_{N}=1.$ (8)

###  III.2 The $P$ function

The $P$ function for a single spin-$j$ particle is defined as

$\rho=\int d\theta d\phi
P(\theta,\phi)|\theta,\phi\rangle\langle\theta,\phi|,$ (9)

and can be shown to be

$\displaystyle P(\theta,\phi)$
$\displaystyle=\sum_{\mu,\eta}\rho_{\mu\eta}Y_{\mu\eta}(\theta,\phi)\left(\frac{1}{4\pi}\right)^{1/2}(-1)^{\mu-\eta}$
$\displaystyle\times\Bigg{(}\frac{(2j-\mu)!(2j+\mu+1)!}{(2j)!(2j)!}\Bigg{)}^{1/2}.$
(10)

Here $\ket{\theta,\phi}$ is the atomic coherent state [88] which in terms of
Wigner-Dicke states $\ket{j,m}$ can be expressed as

$\displaystyle\ket{\theta,\phi}=\sum_{m=-j}^{j}\begin{pmatrix}2j\\\
m+j\end{pmatrix}^{1/2}\sin^{j+m}\left(\frac{\theta}{2}\right)\cos^{j-m}\left(\frac{\theta}{2}\right)e^{-i(j+m)\phi}\ket{j,m}.$
(11)

Moreover, the $P$ function for $N$ spin-$j$ particles is

$\displaystyle P(\theta_{1},\phi_{1},\dots\theta_{N},\phi_{N})$
$\displaystyle=\sum_{\mu_{1},\eta_{1}}\sum_{\mu_{2},\eta_{2}}...\sum_{\mu_{N},\eta_{N}}\rho_{\mu_{1}\eta_{1}\mu_{2}\eta_{2},\dots,\mu_{N}\eta_{N}}Y_{\mu_{1}\eta_{1}}(\theta_{1},\phi_{1})Y_{\mu_{2},\eta_{2}}(\theta_{2},\phi_{2})\dots
Y_{\mu_{N}\eta_{N}}(\theta_{N},\phi_{N})(-1)^{\mu_{1}-\eta_{1}+\mu_{2}-\eta_{2}+\dots\mu_{N}-\eta_{N}}$
$\displaystyle\times\left(\frac{1}{4\pi}\right)^{N/2}\left(\frac{\sqrt{(2j-\mu_{1})!(2j-\mu_{2})!\dots(2j-\mu_{N})!(2j+\mu_{1}+1)!(2j+\mu_{2}+1)!\dots(2j+\mu_{N}+1)!}}{[(2j)!(2j)!]^{N/2}}\right).$
(12)

###  III.3  The $Q$ function

The $Q$ function for a single spin-$j$ state is defined as

$Q(\theta,\phi)=\frac{2j+1}{4\pi}\bra{\theta,\phi}\rho\ket{\theta,\phi},$ (13)

and can be shown to be

$\displaystyle Q(\theta,\phi)$
$\displaystyle=\bigg{(}\frac{1}{4\pi}\bigg{)}^{1/2}\sum_{\mu,\eta}\rho_{\mu\eta}Y_{\mu\eta}(\theta,\phi)(-1)^{\mu-\eta}(2j+1)$
$\displaystyle\times\bigg{(}\frac{(2j)!(2j)!}{(2j-\mu)!(2j+\mu+1)!}\bigg{)}^{1/2}.$
(14)

For $N$ spin-$j$ particles, the normalized $Q$ function is

$\displaystyle Q(\theta_{1},\phi_{1},\dots\theta_{N},\phi_{N})$
$\displaystyle=\sum_{\mu_{1},\eta_{1}}\sum_{\mu_{2},\eta_{2}}...\sum_{\mu_{N},\eta_{N}}\rho_{\mu_{1}\eta_{1}\mu_{2}\eta_{2},\dots,\mu_{N}\eta_{N}}Y_{\mu_{1}\eta_{1}}(\theta_{1},\phi_{1})Y_{\mu_{2},\eta_{2}}(\theta_{2},\phi_{2})\dots
Y_{\mu_{N}\eta_{N}}(\theta_{N},\phi_{N})(-1)^{\mu_{1}-\eta_{1}+\mu_{2}-\eta_{2}+\dots\mu_{N}-\eta_{N}}$
$\displaystyle\times\left(\frac{(2j+1)^{N}}{(4\pi)^{N/2}}\right)\left(\frac{[(2j)!(2j)!]^{N/2}}{\sqrt{(2j-\mu_{1})!(2j-\mu_{2})!\dots(2j-\mu_{N})!(2j+\mu_{1}+1)!(2j+\mu_{2}+1)!\dots(2j+\mu_{N}+1)!}}\right).$
(15)

##  IV Dynamics of the spin system: Quasi-probability Distributions

We consider the dynamics of the spin system inside the cavity under the
influence of various quantum channels and study their $W$, $P$, and $Q$ quasi-
probability distributions. In order to bring out the impact of non-Markovian
effects on the evolution of the spin, we make a comparison with the
corresponding analysis under the influence of the GKSL master equation. We
consider $N=4$ and take the initial state of the spin-cavity system to be
$\rho=\rho_{a}(0)\otimes|\alpha\rangle\langle\alpha|$, where
$\rho_{a}(0)=|g\rangle\langle g|$ ($\ket{g}$ denotes the ground state of the
atomic system), and $\ket{\alpha}$ ($\alpha=|\alpha|e^{i\zeta}$) is the field
coherent state, such that
$\ket{\alpha}=e^{-\braket{n}/2}\sum_{n=0}^{\infty}\frac{(\sqrt{\braket{n}})^{n}}{\sqrt{n!}}e^{in\zeta}\ket{n}$,
where $n$ is the photon number in the Fock state $\ket{n}$. In the numerics
below, we choose $\braket{n}=6$ and $\zeta=\pi/2$.

###  IV.1 Impact of noisy channels

We now study the impact of a number of noisy channels, both Markovian as well
as non-Markovian, on the dynamics of the spin system inside the cavity.

####  IV.1.1 Squeezed Generalized Amplitude Damping Channel

We consider the dynamics of the spin system impacted by the squeezed
generalized amplitude damping (SGAD) channel [63, 89], which is a GKSL type of
semi-group evolution (Markovian in nature). It is worth noting that here and
in all the master equations treated below, the cavity loss is accounted for by
the term corresponding to $\kappa$. The master equation for the evolution of
the spin-cavity system is

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i[H,\rho(t)]+\frac{1}{2}\sum_{k=1}^{N}\sum_{j=1}^{2}(2R_{jk}\rho(t)R^{\dagger}_{jk}-R_{jk}^{\dagger}R_{jk}\rho(t)$
$\displaystyle-\rho(t)R_{jk}^{\dagger}R_{jk})+\frac{1}{2}\kappa(2a\rho(t)a^{\dagger}-a^{\dagger}a\rho(t)-\rho(t)a^{\dagger}a),$
(16)

where $R_{1k}=(\gamma_{k}(N_{th,k}+1))^{1/2}R$ and
$R_{2k}=(\gamma_{k}N_{th,k})^{1/2}R$ with
$R=\sigma_{k}^{-}\cosh(r)+e^{i\Phi}\sigma_{k}^{+}\sinh(r)$ and
$N_{th,k}=\frac{1}{e^{h\omega_{k}/k_{B}T}-1}$. Also, $r$ and $\Phi$ are the
bath squeezing parameters.

Figure 1: Variation of the $W$, $P$, and $Q$ functions for the spin system
with time under the squeezed generalized amplitude damping channel. Here we
have chosen $\omega_{1}=\omega_{4}=1.11$, $\omega_{2}=\omega_{3}=1.15$,
$\omega_{c}=1.15$, $\Phi=\pi/4$, $g_{1}=g_{4}=0.55$, $g_{2}=0.52$,
$g_{3}=0.5$, $\gamma_{k}=0.01g_{k}$ and $\kappa=0.01$. The subplots (a), (b),
(c), and (d) have different values of the squeezing parameter $r$ and
temperature $T$. Also, $\theta_{i}=\\{\pi/4,3\pi/5,2\pi/3,3\pi/4\\}$ and
$\phi_{i}=\\{3\pi/4,\pi/3,\pi/4,\pi/6\\}$.

After the evolution of the system $\rho$ through the SGAD channel, we trace
out the cavity degrees of freedom and use Eqs. (7), (12) and (15) on the spin
system to calculate their $W$, $P$ and $Q$ functions, respectively. The QDs
for the case of temperature $T$ and squeezing parameter $r$ equal to zero are
plotted in Fig. 1(a). It is observed that the $P$ and $W$ functions frequently
take negative values, indicating quantumness in the system. Further, with an
increase in the value of $r$ and $T$ in Figs. 1(b), 1(c), 1(d), it is observed
that the negativity of QDs decreases. This indicates depletion of non-
classicality in the system with an increase in the squeezing parameter $r$ or
temperature $T$.

It is worth mentioning that the negativity of $W$ and $P$ functions also
depend on the choice of the system parameters $\theta_{i}$ and $\phi_{i}$ for
each atom. Here we have run simulations on many values of $\theta_{i}$ and
$\phi_{i}$ and have chosen the values of $\theta_{i}$ and $\phi_{i}$ showing
greater negativity. Given that there are two parameters ($\theta_{i}$ and
$\phi_{i}$) for each atom, therefore, for the case considered here ($N=4$), we
have a set of 8 parameters. It is not possible to show the variation of $W$ or
$P$ function for all the 8 parameters here. However, to illustrate the
dependence of negativity of $W$ or $P$ function over the parameters
$\theta_{i}$ and $\phi_{i}$, we have chosen
$\theta_{1}=\frac{\theta}{4},\theta_{2}=\frac{3\theta}{5},\theta_{3}=\frac{2\theta}{3},\theta_{4}=\frac{3\theta}{4}$
and
$\phi_{1}=\frac{3\phi}{4},\phi_{2}=\frac{\phi}{3},\phi_{3}=\frac{\phi}{4},\phi_{4}=\frac{\phi}{6}$
and evolved the $P$ function with $\theta$ and $\phi$ for a given time. This
is depicted in Fig. 2, where we can see the negativity of the $P$ function
varies as we change $\theta$ and $\phi$.

Figure 2: Variation of the $P$ function for the spin system with parameters
$\theta$ and $\phi$ for a given time under the squeezed generalized amplitude
damping channel. The parameters have the following values:
$\omega_{1}=\omega_{4}=1.11$, $\omega_{2}=\omega_{3}=1.15$, $\omega_{c}=1.15$,
$g_{1}=g_{4}=0.55$, $g_{2}=0.52$, $g_{3}=0.5$, $\gamma_{i}=0.01g_{i}$, $r=0$
and $T=0$, and $\kappa=0.01$.

Next, we consider the impact of non-Markovian noise on the spin system inside
the cavity. To this effect, we consider the following non-Markovian channels.

####  IV.1.2  Phase covariant eternal non-Markovian channel

We now discuss the phase covariant eternal CP-indivisible dynamics [82] of the
spin system using the master equation

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i[H,\rho(t)]+\sum_{k=1}^{N}\left[\gamma_{1k}(t)\mathcal{L}_{1k}(\rho(t))+\gamma_{2k}(t)\mathcal{L}_{2k}(\rho(t))\right.$
$\displaystyle+\left.\gamma_{3k}(t)\mathcal{L}_{3k}(\rho(t))\right]+\frac{1}{2}\kappa(2a\rho(t)a^{\dagger}-a^{\dagger}a\rho(t)-\rho(t)a^{\dagger}a),$
(17)

where

$\displaystyle\mathcal{L}_{1k}$
$\displaystyle=\sigma_{k}^{+}\rho(t)\sigma_{k}^{-}-\frac{1}{2}\\{\rho(t),\sigma_{k}^{-}\sigma_{k}^{+}\\},$
(18) $\displaystyle\mathcal{L}_{2k}$
$\displaystyle=\sigma_{k}^{-}\rho(t)\sigma_{k}^{+}-\frac{1}{2}\\{\rho(t),\sigma_{k}^{+}\sigma_{k}^{-}\\},$
$\displaystyle\mathcal{L}_{3k}$
$\displaystyle=\sigma_{k}^{z}\rho(t)\sigma_{k}^{z}-\rho(t).$

Further,

$\displaystyle\gamma_{1k}(t)$ $\displaystyle=\nu_{k}(1+q),$ (19)
$\displaystyle\gamma_{2k}(t)$ $\displaystyle=\nu_{k}(1-q),$
$\displaystyle\gamma_{3k}(t)$
$\displaystyle=-\frac{\nu_{k}(1-q^{2})\sinh(2\nu_{k}t)}{2[1+q^{2}+(1-q^{2})\cosh(2\nu_{k}t)]},$

such that $\nu_{k}>0$ and $|q|<1$. Here $\gamma_{1k}$, $\gamma_{2k}$, and
$\gamma_{3k}$ denote the energy gain, energy loss, and pure dephasing rates,
respectively. The cavity degrees of freedom are now traced out from the
solution of Eq. (17) and subsequently used to calculate the QDs of the spin
system using Eqs. (7), (12) and (15).

Figure 3: Variation of the $W$, $P$, and $Q$ function for the spin system with
time after evolution through phase covariant eternal non-Markovian channel
given in Eq. (17). The green dashed curve denotes the variation of the $P$
function when the pure dephasing rate in Eq. (19) $\gamma_{3k}(t)$ is chosen
to be in the limiting range of $t\to\infty$, which becomes $-\nu_{k}/2$. We
have chosen the parameters to be $\omega_{1}=\omega_{4}=1.11$,
$\omega_{2}=\omega_{3}=1.15$, $\omega_{c}=1.15$, $g_{1}=g_{4}=0.55$,
$g_{2}=0.52$, $g_{3}=0.5$, $\nu_{1}=\nu_{3}=2.2$, $\nu_{2}=\nu_{4}=2.4$ and
$q=0.75$. For the QDs, we have taken
$\theta_{i}=\\{\pi/4,3\pi/5,2\pi/3,3\pi/4\\}$ and
$\phi_{i}=\\{3\pi/4,\pi/3,\pi/4,\pi/6\\}$.

In Fig. 3, we have plotted the dynamics of the spin system through phase
covariant eternal non-Markovian (PCEnM) master equation, Eq. (17). It is
observed that under the influence of the PCEnM channel, which is an eternal
CP-indivisible non-Markovian channel for $t>0$ (except when $t\to\infty$), the
$P$ function takes negative values for longer periods of time. This indicates
that quantumness in the system is retained for a longer time. Further, in Fig.
3, variation of the $P$ function for a limiting value of the pure dephasing
rate $\gamma_{3k}(t\to\infty)=-\nu_{k}/2$ is plotted. It can be observed that
at shorter times, the $P$ functions for time-dependent $\gamma_{3k}(t)$ (black
dot-dashed curve) and constant $\gamma_{3k}$ (green dashed curve) are slightly
different. However, as the time-dependent $\gamma_{3k}(t)$ converges to
$-\nu_{k}/2$, the $P$ functions coincide. The difference in the evolution of
the $P$ function is due to the structure of the PCEnM channel, which is
eternally non-Markovian when $\gamma_{3k}(t)$ is time-dependent and Markovian
when it becomes time-independent in the limit $t\to\infty$.

####  IV.1.3 Non-Markovian amplitude damping channel

Here we consider the non-Markovian amplitude damping (NMAD) channel [81, 90]
for the evolution of the spin system. To this end, the master equation for the
evolution of the system is given by

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i[H,\rho]+\sum_{k=1}^{N}\gamma_{k}^{\prime}(t)[\sigma_{k}^{-}\rho\sigma_{k}^{+}-\frac{1}{2}\\{\sigma_{k}^{+}\sigma_{k}^{-},\rho(t)\\}]$
$\displaystyle+\frac{1}{2}\kappa(2a\rho(t)a^{\dagger}-a^{\dagger}a\rho(t)-\rho(t)a^{\dagger}a),$
(20)

where $\gamma_{k}^{\prime}(t)=-2\Re\left(\frac{\dot{F}(t)}{F(t)}\right)$ is
the time dependent decoherence rate. $F$ is the decoherence function given by

$F(t)=e^{\frac{-q^{\prime}t}{2}}\left(\frac{q^{\prime}}{l}\sinh\left[\frac{lt}{2}\right]+\cosh\left[\frac{lt}{2}\right]\right),$
(21)

with $l=\sqrt{q^{{}^{\prime}2}-2\gamma^{\prime}_{k}q^{\prime}}$. Here
$\gamma^{\prime}_{k}$ and $q^{\prime}$ parameterize the bath spectral density
[81]. Now

$\displaystyle\gamma_{k}^{\prime}(t)$
$\displaystyle=-\frac{2}{|F(t)|}\frac{d|F(t)|}{dt}$
$\displaystyle=2\Re\left(\frac{\gamma^{\prime}_{k}}{\sqrt{1-\frac{2\gamma^{\prime}_{k}}{q^{\prime}}}\coth\left(\frac{1}{2}q^{\prime}t\sqrt{1-\frac{2\gamma^{\prime}_{k}}{q^{\prime}}}\right)+1}\right).$
(22)

Figure 4: Variation of the $W$, $P$, and $Q$ function for the spin system with
time. Subplot (a) shows the QDs for the spin system after the GKSL type of
evolution, modeling an AD channel, and (b) shows the QDs for the spin system
after evolution through the NMAD channel given in Eq. (20). The parameters
have the following values: $\omega_{1}=\omega_{4}=1.11$,
$\omega_{2}=\omega_{3}=1.15$, $\omega_{c}=1.15$, $g_{1}=g_{4}=0.55$,
$g_{2}=0.52$, $g_{3}=0.5$, $\gamma_{i}=0.5+g_{i}$,
$\gamma^{\prime}_{i}=\gamma_{i}/2$, $q^{\prime}=0.05$ and $\kappa=0.01$. For
the QDs, we have taken $\theta_{i}=\\{\pi/4,3\pi/5,2\pi/3,3\pi/4\\}$ and
$\phi_{i}=\\{3\pi/4,\pi/3,\pi/4,\pi/6\\}$.

The evolution, in this case, is non-Markovian for
$q^{\prime}<2\gamma^{\prime}_{k}$ as that regime leads to damped oscillations.
Here $\Re$ represents the real part of the quantity inside the bracket. In
Fig. 4, we have plotted the evolution of the spin system through the GKSL
master equation (modeling an AD channel), Fig. 4 (a), and through the NMAD
channel, Fig. 4 (b). We observe that under the influence of the NMAD channel,
the $P$ function becomes negative before the corresponding case under the GKSL
master equation.

####  IV.1.4 Semi-Markov dephasing channel

We now discuss the evolution of the spin system through the semi-Markov
dephasing channel [83] using the master equation

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i[H,\rho]+\sum_{k=1}^{N}{\tilde{\gamma}}_{k}(t)[\sigma_{k}^{z}\rho\sigma_{k}^{z}-\rho]$
$\displaystyle+\frac{1}{2}\kappa(2a\rho(t)a^{\dagger}-a^{\dagger}a\rho(t)-\rho(t)a^{\dagger}a),$
(23)

where

$\displaystyle{\tilde{\gamma}_{k}(t)}=\frac{2\tilde{\gamma}_{k}}{s\sqrt{1-\frac{8\tilde{\gamma}_{k}}{s^{2}}}\coth\left(\frac{1}{2}st\sqrt{1-\frac{8\tilde{\gamma}_{k}}{s^{2}}}\right)+s}.$
(24)

Figure 5: Variation of the $W$, $P$, and $Q$ function for the spin system with
time. Subplot (a) shows the QDs for the spin system after Lindblad type
evolution, modeling an AD channel, and (b) shows the QDs for the spin system
after evolution through semi-Markov dephasing channel given in Eq. (23). The
parameters are chosen to be: $\omega_{1}=\omega_{4}=1.11$,
$\omega_{2}=\omega_{3}=1.15$, $\omega_{c}=1.15$, $g_{1}=g_{4}=0.55$,
$g_{2}=0.52$, $g_{3}=0.5$, $\gamma_{1}=\gamma_{3}=0.31$,
$\gamma_{2}=\gamma_{4}=0.32$, $\tilde{\gamma}_{i}=\gamma_{i}/2$, $s=0.1$ and
$\kappa=0.01$. Also, for the QDs, we have taken
$\theta_{i}=\\{\pi/4,3\pi/5,2\pi/3,3\pi/4\\}$ and
$\phi_{i}=\\{3\pi/4,\pi/3,\pi/4,\pi/6\\}$. Subplot (c) shows the variation of
QDs for pure Hamiltonian evolution of the system with
$\tilde{\gamma}_{k}=\kappa=0$.

In this case, the evolution is CP-indivisible and non-Markovian for
$p>\frac{s^{2}}{8}$. From Fig. 5, we observe that the evolution of the QDs for
the spin system is nearly the same under the evolution of the system through
the semi-Markov dephasing channel and through the GKSL master equation,
modeling an AD channel. Further, we also plotted the variation of QDs when the
system’s evolution is through the pure Hamiltonian
($\tilde{\gamma}_{k}=\kappa=0$) in Fig. 5(c). It can be observed that the $P$
function is the most negative when the system is evolved under pure
Hamiltonian, and the negativity of the $P$ function reduces slightly when
evolution is under the semi-Markov dephasing channel. The $P$ function is the
least negative when the AD channel is used for the system’s evolution.

Figure 6: Variation of the cavity photon number $\langle a^{\dagger}a\rangle$
and the spin excitation $\langle\sigma^{+}_{k}\sigma^{-}_{k}\rangle$ with time
where the evolution of the spin system is under different quantum channels:
(a) through GKSL master equation (Eq. (2)) with parameters:
$\gamma_{i}=0.01g_{i}$, (b) under PCEnM channel (Eq. (17)) (with parameters:
$\nu_{i}=0.02g_{i}$), (c) through NMAD channel (Eq. (20)) with parameters:
$\gamma^{\prime}_{i}=0.5+g_{i}/2$, $q^{\prime}=0.05$, and (d) semi-Markov
dephasing channel (Eq. (23)) with parameters:
$\tilde{\gamma}_{1}=\tilde{\gamma}_{3}=0.31/2$,
$\tilde{\gamma}_{2}=\tilde{\gamma}_{4}=0.32/2$, $s=0.1$. Values of
$g_{1}=g_{4}=0.55$, $g_{2}=0.52$, $g_{3}=0.5$, $\omega_{1}=\omega_{4}=1.11$,
$\omega_{2}=\omega_{3}=1.15$, $\omega_{c}=1.15$ and $\kappa=0.01$ are same for
all the channels.

We will next examine the dynamics of the cavity field. As a connection to the
spin dynamics studied till now, we observe the effect of the evolution of the
spin-cavity system due to the impact of various quantum channels using the
spin excitation $\langle\sigma^{+}_{k}\sigma^{-}_{k}\rangle$ (where $k$ runs
from 1 to $N$) and cavity photon number $\langle a^{\dagger}a\rangle$, and is
depicted in Fig. 6. The peaks and dips in the spin excitation are in contrast
with those in the cavity photon number, indicating a transfer of excitation
from the spins to the cavity. This could be attributed to the fact that total
excitation number $\sum_{k=1}^{4}\sigma^{+}_{k}\sigma^{-}_{k}+a^{\dagger}a$ is
conserved in the TC model (for negligible dissipation, as for the parameters
chosen in the present case) and serves as a consistency check. This was also
observed in the context of a mesoscopic spin-cavity system [14].

##  V Dynamics of cavity field

In this section, we study the dynamics of the cavity field. To this end, we
use the second-order coherence function to characterize the evolution of the
field. The second-order coherence function ($g^{(2)}$) is one of the most
important characterizers of a light source into classical or non-classical and
bunched or anti-bunched. It is defined as

$g^{(2)}(\tau)=\frac{\langle\hat{a}^{\dagger}(t)\hat{a}^{\dagger}(t+\tau)\hat{a}(t+\tau)\hat{a}(t)\rangle}{\langle\hat{a}^{\dagger}(t)\hat{a}(t)\rangle^{2}},$
(25)

where $\hat{a}$ and $\hat{a}^{\dagger}$ are the bosonic annihilation and
creation operators, respectively, representing the cavity field. Here we
calculate the $g^{(2)}(0)$ function after the evolution of the cavity field
through the GKSL master equation (AD channel) and make a comparison with the
AD channel being replaced by the NMAD channel, using the following master
equation

$\displaystyle\frac{d\rho(t)}{dt}$
$\displaystyle=-i[H,\rho]+\sum_{k=1}^{N}\gamma_{k}(2\sigma_{k}^{-}\rho(t)\sigma_{k}^{+}-\\{\sigma_{k}^{+}\sigma_{k}^{-},\rho(t)\\})$
$\displaystyle+\kappa^{\prime}(t)[\hat{a}\rho(t)\hat{a}^{\dagger}-\frac{1}{2}\\{\hat{a}^{\dagger}\hat{a},\rho(t)\\}],$
(26)

where

$\kappa^{\prime}(t)=2\Re\left(\frac{\kappa^{\prime}}{\sqrt{1-\frac{2\kappa^{\prime}}{b}}\coth\left(\frac{1}{2}bt\sqrt{1-\frac{2\kappa^{\prime}}{b}}\right)+1}\right).$
(27)

Here, we have chosen $b<2\kappa^{\prime}$ for non-Markovian evolution, and
$\Re$ denotes the real part of the quantity inside the bracket.

Figure 7: Variation of the equal-time second-order coherence function
$g^{(2)}(0)$ with time. The black solid line refers to the evolution of the
cavity mode through the GKSL master equation (AD channel), and the dot-dashed
red line refers to the corresponding evolution under the NMAD channel. The
parameters are: $\omega_{1}=\omega_{4}=1.11$, $\omega_{2}=\omega_{3}=1.15$,
$\omega_{c}=1.15$, $g_{1}=g_{4}=0.55$, $g_{2}=0.52$, $g_{3}=0.5$,
$\gamma_{i}=0.1g_{i}$, $\kappa=0.5$, $\kappa^{\prime}=\kappa/2$, and $b=0.05$.

In Fig. 7, we draw a comparison between the evolution of the cavity field
impacted by the AD and NMAD channels. In both cases, we observe a similar
pattern of dips and blips in the $g^{(2)}(0)$ function. The $g^{(2)}(0)$ is
observed to be less than $1$ a number of times, indicating the sub-Poissonian
(non-classical) nature of light. A decay in the $g^{(2)}(0)$ function sets in
earlier for evolution under the NMAD channel as compared to that under the AD
channel. An important parameter characterizing the light source is the Mandel
Q parameter. It is defined as

$\text{Q}=\langle\hat{n}\rangle(g^{(2)}(0)-1),$ (28)

where $\hat{n}$ is the photon number operator. In Fig. 8, the negative values
of the Mandel Q parameter depict the sub-Poissonian behavior of light.

We finally calculate the $g^{(2)}(\tau)$ function, shown in Fig. 8. Light is
said to be anti-bunched if $g^{(2)}(0)<g^{(2)}(\tau)$ and bunched if
$g^{(2)}(0)>g^{(2)}(\tau)$ [1].

Figure 8: Variation of Mandel Q parameter and the second-order coherence
function $g^{(2)}(\tau)$ (for $\tau=0$ and $\tau=3$) and the difference
$g^{(2)}(0)-g^{(2)}(\tau)$ with time. In this case, the cavity’s evolution is
considered through the GKSL master equation. The parameters are:
$\omega_{1}=\omega_{4}=1.11$, $\omega_{2}=\omega_{3}=1.15$, $\omega_{c}=1.15$,
$g_{1}=g_{4}=0.55$, $g_{2}=0.52$, $g_{3}=0.5$, $\gamma_{i}=0.01g_{i}$,
$\kappa=0.1$.

In Fig. 8, we observe that the light is anti-bunched a number of times. Also,
at around $t=5$, we observe that the light becomes anti-bunched
($g^{(2)}(0)-g^{(2)}(\tau)<0$) even when the $g^{(2)}(0)$ function is not less
than 1. This brings out the difference between the anti-bunched and sub-
Poissonian behavior of light [91] in the present context.

##  VI Conclusion

We have discussed the Tavis-Cummings model in a noisy environment. The impact
of both Markovian and non-Markovian noise on the dynamics of the Tavis-
Cummings model was observed. The dynamics of the spin system and the cavity
field were studied using the $W$, $P$, and $Q$ quasi-probability distributions
and second-order coherence function ($g^{(2)}$ function), respectively. The
effect of squeezing and temperature on the quasi-probability distributions was
analyzed. We observed that, in general, an increase in the squeezing
parameters and temperature diminishes non-classicality in the system for a
given set of atomic parameters. We have studied the impact of PCEnM, NMAD, and
semi-Markov dephasing channels, in their non-Markovian limits, over the
evolution of the atomic system using the $W$, $P$, and $Q$ functions. We have
compared the impact of NMAD and semi-Markov dephasing channels with the
variation of these quasi-probability distribution functions when the atomic
system is evolved through the GKSL master equation modeling an AD channel,
which is semi-group and models a Markovian evolution. This brings out the
impact of memory effects on the quasiprobability distribution functions. From
the study of the influence of noise on the spin-cavity system, specifically on
the spin excitation and the cavity photon number, it was observed that, in
general, the spin excitation and the cavity photon number have a complementary
behavior. The dynamics of the Mandel Q parameter revealed the cavity field to
be sub-Poissonian for different evolution times for the parameters considered.
The $g^{(2)}(\tau)$ function was also computed and compared with the
$g^{(2)}(0)$ function in order to bring out bunching and anti-bunching in the
light. Interestingly, a number of instances were observed where sub-Poissonian
and anti-bunching behavior of light were not in tandem.

## Acknowledgements

The authors acknowledge useful discussions with Himadri Shekhar Dhar during
the preliminary stage of the work. SB acknowledges support from the
Interdisciplinary Cyber Physical Systems (ICPS) programme of the Department of
Science and Technology (DST), India, Grant No.: DST/ICPS/QuST/Theme-1/2019/6.
SB also acknowledges support from the Interdisciplinary Research Platform
(IDRP) on Quantum Information and Computation (QIC) at IIT Jodhpur.

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