# Exciton transport in amorphous polymers and the role of morphology and
thermalisation

Francesco Campaioli<EMAIL_ADDRESS>Chemical and Quantum
Physics, and ARC Centre of Excellence in Exciton Science, School of Science,
RMIT University, Melbourne 3000, Australia  Jared H. Cole
<EMAIL_ADDRESS>Chemical and Quantum Physics, and ARC Centre of
Excellence in Exciton Science, School of Science, RMIT University, Melbourne
3000, Australia

###### Abstract

Understanding the transport mechanism of electronic excitations in conjugated
polymers is key to advancing organic optoelectronic applications, such as
solar cells, OLEDs and flexible electronics. While crystalline polymers can be
studied using solid-state techniques based on lattice periodicity, the
characterisation of amorphous polymers is hindered by an intermediate regime
of disorder and the associated lack of symmetries. To overcome these hurdles
we use a reduced state quantum master equation approach based on the
Merrifield exciton formalism. Using this model we study exciton transport in
conjugated polymers and its dependence on morphology and temperature. Exciton
dynamics consists of a thermalisation process, whose features depend on the
relative strength of thermal energy, electronic couplings and disorder,
resulting in remarkably different transport regimes. By applying this method
to representative systems based on poly(p-phenylene vinylene) (PPV) we obtain
insight into the role of temperature and disorder on localisation, charge
separation, non-equilibrium dynamics, and experimental accessibility of
thermal equilibrium states of excitons in amorphous polymers.

##  I Introduction

Organic semiconductors (OSCs) are at the forefront of the current research
efforts for the development and improvement of optoelectronic technology, such
as solar cells [1, 2, 3], organic light-emitting diodes (OLEDs) [4, 5], thin-
film transistors [6, 7, 8, 9], sensors and flexible electronics [10, 11, 12,
13]. In particular, conjugated polymers are a key class of OSCs for
photovoltaic applications, due to their ability to transport electronic
excitations, i.e., _excitons_ , over tens of nanometers, together with their
low productions cost, ease of fabrication and flexibility [14, 15]. Depending
on their chemical composition and fabrication conditions, polymeric
semiconductors can be found in a variety of different morphologies
characterised by specific exciton transport properties [16].

While the fundamental features of energy and charge transport across
crystalline, semicrystalline and amorphous polymers are qualitatively known,
the dependence of exciton dynamics on temperature and morphology is still an
important area of investigation [17, 18, 19, 20]. For example, the
introduction of amorphous polymers characterised by high electron mobility has
challenged the idea that crystalline ones would be optimal for charge
transport [21, 22, 23, 24, 25, 26]. The study of exciton transport in
amorphous polymers is however hindered by the complexity that arises from
their disordered nature, which prevents the use of solid-state techniques
based on lattice periodicity. In this intermediate transport regime between
band conduction and incoherent hopping, semiclassical techniques such as
Marcus theory often fail to reproduce the coherent quantum attributes of the
exciton dynamics [27, 28, 29, 30], while first-principles calculations like
multiconfiguration time-dependent Hartree method (MCTDH) become
computationally intractable due to the large size of the systems of interest
[31, 32, 33, 19, 34, 20].

An effective alternative to study exciton transport in disordered OSCs is
given by reduced state quantum master equations [15]. These allow for the
efficient description of the interaction between excitons and nuclear
vibrations, i.e., phonons, and account for both coherent and incoherent
dynamics. Quantum master equations have therefore been applied to a variety of
exciton transport problems in OSCs, such as natural photosynthetic complexes
[35, 36, 37, 38], molecular aggregates [39, 40, 41, 42], and disordered
systems [43, 34]. A key insight from this body of literature is that exciton-
phonon interactions have an essential role for exciton transport in OSCs. On
one hand, decoherence induced by local and uncorrelated phonon modes is known
to improve transport efficiency in disordered systems [44, 36, 45],
counteracting weak and strong localisation phenomena, such as Anderson
localisation [46, 47, 48, 49], that otherwise prevail when dynamics is
primarily coherent. On the other hand, strong spatial correlations between
bath modes can facilitate the emergence of decoherence-free subspaces,
improving transport of certain states in ordered systems [50].

Recently, quantum master equations have also been used to study exciton
transport in conjugated polymers, albeit only for the case of crystalline
morphologies. In Ref. [19], Lyskov _et al._ bridge first-principles
calculations to a Lindblad master equation [51, 52, 53, 54, 55], and obtain a
phenomenological model for triplet exciton dynamics in crystalline
poly(p-phenylene vinylene) (PPV). Their results show that, conversely to the
case of disordered OSCs, decoherence slows down exciton dynamics in
crystalline PPV, inducing a rapid transition from ballistic (coherent) to
diffusive (incoherent) transport. However, the model used in Ref. [19] cannot
be directly applied to amorphous polymers, whose difficult theoretical
characterisation presents several questions [56, 57, 58, 59, 60, 61, 62, 63,
64, 33, 65, 66, 67]. How do temperature and morphology affect exciton
transport? What are the features of the dynamical transition between localised
and delocalised states? Are thermal equilibrium states experimentally
accessible? How does disorder affect charge-separation dynamics?

Here we answer these questions generalising the master equation approach used
in Ref. [19] to conjugated polymers with arbitrary morphology. Firstly, since
excited electronic states of amorphous polymers are known to display charge
separation over a few monomers [24, 26], we formulate a master equation for
the dynamics of Merrifield excitons [68, 31], i.e., a strongly coupled
electron-hole pair. Accordingly, our approach can be seen as a direct
extension of the master equations used in Refs. [36, 37, 38, 19] for the
dynamics of charge-neutral Frenkel excitons [69], characteristic of OSCs with
low dielectric constant.

Furthermore, our model, introduced in Sec. II, provides a rigorous
thermodynamic description of exciton-phonon interactions, overcoming the
limitations of the phenomenological approach used in Ref. [19]. There, the
authors fit the master equation parameters to MCTDH calculations carried out
on small two-monomer subsystems over short time scales. Such approach,
affected by artificially short recurrence times, may fail to correctly
reproduce the dynamics for long time scales and returns decoherence rates that
do not necessarily obey thermodynamics in the long-time limit. Here, instead,
we use the Bloch-Redfield formalism to calculate temperature-dependent
decoherence rates from the phonons’ correlation functions [53, 70], to then
express the dynamics of the exciton’s reduced state using a master equation of
the Lindblad form [54, 35, 40, 41, 42]. This approach has the additional
advantage of providing ensemble average transport properties without
calculating many individual stochastic realisations of system-environment
dynamics.

In Sec. III, we use the model to study the exciton transport properties of
some representative oligo and poly(p-phenylene vinylene) (OPV and PPV)
systems, across different morphologies, from crystalline to highly disordered.
Instead of relying on material-specific modelling of polymers, we express our
calculations in terms of electronic couplings and thermal energy. This allows
us to explore the impact that disorder and temperature have on localisation,
charge-separation dynamics and equilibration time scales. Exciton transport is
therefore interpreted as a thermalisation process [71, 53, 54, 72, 70, 73],
whose features vary with the relative strength between thermal energy,
electronic couplings and disorder. We show how such differences in electronic
coupling determine whether the phonon bath cools or heats the system, with
dramatic effects on exciton transport properties. We conclude discussing our
results in Sec. IV, where we compare and contrast our findings with those of
other works, and outlining the use of our method to exciton transport problems
of great importance for optoelectronic applications.

##  II Methodology

The exciton transport model that we introduce in this article is based on the
following two premises. First, the characteristic times and energies
associated with exciton transport in OSCs at room temperature and natural
illumination imply that excitons do not interact with each other, as they are
sufficiently spatially and temporally separated [15]. Under these
circumstances, it is a standard approach to focus on the transport of a single
electronic excitation, formally defined by the single-exciton manifold [36,
37, 38, 31, 15, 19, 20]. Second, we consider polymeric materials whose
structural changes are far slower than the characteristic femtosecond to
nanosecond time scales of exciton dynamics, such as for the paradigmatic case
of PPV and its amorphous derivative MEH-PPV [24, 26]. This allows us to adopt
a snapshot approach for a medium’s morphology, which is assumed to be time-
independent. Additionally, we assume that the interaction between excitons and
rapidly oscillating conformational coordinates like bond-length alternations,
here modelled as a bath of independent harmonic oscillators, are well
described in the weak coupling approximation [35, 15].

Figure 1: Polymeric materials are here modelled as a network of $N$ nodes,
each of which represents a monomer. Monomers positions $\vec{r}_{k}$ and
conformational coordinates $\bm{Q}_{k}$ define the morphology of the medium
and the structure of the exciton’s Hamiltonian. Exciton transport on such
network is mediated by through-bond couplings between conjugated monomers that
are part of the same polymer, or by through-space couplings (gray line)
between pairs of monomers in proximity of each other.

Under these assumptions, the transport of a single electronic excitation in
OSCs is typically modelled as a _quantum stochastic walk_ (QSW) of a charge-
neutral Frenkel exciton over a network of $N$ nodes [74, 75], each of which
represents a monomer or a chromophore [35, 37, 15, 19]. However, motivated by
recent results on the electronic excitations of amorphous polymers [24, 26],
we here generalise this standard approach and extend the dynamics to charge-
separated exciton states using the Merrifield model [31]. In this formalism
the states of the single-exciton manifold are here given by the product of the
localised single-electron and single-hole bases,

$\mathcal{B}:=\\{|{jk}\rangle\\}_{j,h=1}^{N},$ (1)

where $|{jk}\rangle:=|{j}\rangle_{e}\otimes|{k}\rangle_{h}$, and where
$|{j}\rangle_{e}$ ($|{k}\rangle_{h}$) represents the state of the electron
(hole) localised on monomer $j$ ($k$) of a polymeric material. Every monomer
$k$ is also associated with its centre’s position
$\vec{r}_{k}:=(x_{k},y_{k},z_{k})$ and a set of local conformational
coordinates $\bm{Q}_{k}$, such as torsional angles and bond-length
alternations, as schematically represented in Fig. 1. These, together with the
bonds between pairs of monomers, define the morphology of the medium and
determine the structure of the exciton’s Hamiltonian, as described in the next
section.

###  II.1 Exciton Hamiltonian

The exciton Hamiltonian $H_{\text{ex}}=H_{0}+H_{\text{F}}+H_{\text{CT}}$ is
given by the Coulomb term $H_{0}$, which defines the energetic landscape
associated with electron-hole configurations on the polymeric material, and
the Frenkel $H_{F}$ and charge $H_{CT}$ transfer terms, which model the
coherent transport of energy and charges, respectively. In the electron-hole
basis of Eq. (1), the Coulomb term $H_{0}$ reads

$\langle{eh|H_{0}|e^{\prime}h^{\prime}}\rangle=\delta_{ee^{\prime}}\delta_{hh^{\prime}}\bigg{(}E_{0}-\frac{1}{4\pi\epsilon_{0}\epsilon_{r}d_{eh}}\bigg{)},$
(2)

where $E_{0}^{-1}=(4\pi\epsilon_{0}\epsilon_{r}r_{0})$ is the exciton binding
energy of electron-hole pairs that are localised within the same monomers,
i.e., Frenkel excitons. The electron-hole distance,
$d_{eh}=r_{0}+\lVert\vec{r}_{e}-\vec{r}_{h}\rVert$, is obtained from the
distance between two monomers offset by the intrinsic electron-hole distance
$r_{0}$ for Frenkel excitons in the considered polymer [31]. The Coulomb term
is diagonal in the electron-hole basis, and vanishes for Frenkel excitons. The
morphology of the medium introduces static disorder in this term via the
inter-monomer distance $d_{eh}$.

The Frenkel term $H_{\text{F}}$ models the coherent transport of Frenkel
excitons and is given by

$\langle{eh|H_{\text{F}}|e^{\prime}h^{\prime}}\rangle=\delta_{eh}\delta_{e^{\prime}h^{\prime}}V_{\text{F}}(\vec{r}_{e},\bm{Q}_{e};\vec{r}_{e^{\prime}},\bm{Q}_{e^{\prime}}),$
(3)

where $V_{\text{F}}(\vec{r}_{j},\bm{Q}_{j};\vec{r}_{k},\bm{Q}_{k})$ is the
strength of the Frenkel coupling between a pair of monomers $j,k$, as a
function of the morphology of the medium. Frenkel transfer terms preserve
charge separation, and can only mediate the transport of Frenkel exciton
states, which form a $N$ dimensional subset of the full $N^{2}$ dimensional
Hilbert space associated with the single-exciton manifold of the Merrifield
model.

Similarly, the charge transfer (CT) term $H_{\text{CT}}$ models the coherent
transport of individual charges and is given by

$\begin{split}\langle{eh|H_{\text{CT}}|e^{\prime}h^{\prime}}\rangle=&\>\>\delta_{ee^{\prime}}V_{\text{CT}}(\vec{r}_{h},\bm{Q}_{h};\vec{r}_{h^{\prime}},\bm{Q}_{h^{\prime}})+\\\
&+\delta_{hh^{\prime}}V_{\text{CT}}(\vec{r}_{e},\bm{Q}_{e};\vec{r}_{e^{\prime}},\bm{Q}_{e^{\prime}}),\end{split}$
(4)

where $V_{\text{CT}}(\vec{r}_{j},\bm{Q}_{j};\vec{r}_{k},\bm{Q}_{k})$
represents the strength of the CT coupling between a pair $j,k$ of monomers,
as a function of the morphology of the medium. The CT term can map Frenkel
states into charge-separated states and vice versa, and its addition to the
Hamiltonian allows exciton states to explore the entirety of the single-
exciton manifold.

The dependence of the coupling strengths $V_{\text{F}}$ and $V_{\text{CT}}$ on
the morphology of the medium is described in terms of the relative arrangement
of pairs of monomers that are conjugated, or close enough to each other to
allow for weaker through-space couplings. Without loss of generality,
$V_{\text{F}}$ and $V_{\text{CT}}$ are assumed to be symmetric in the pairs’
indices and to vanish for monomers that are not connected by a bond, or out of
range for through-space interactions. An example of the dependence of
$V_{\text{F}}$ and $V_{\text{CT}}$ on the morphology for PPV-like conjugated
polymers is given by Eqs. (14) and (15) in Sec. III.

###  II.2 Exciton-phonon interaction

Exciton-phonon interactions induce decoherence and relax the exciton states
until a steady state is reached. The composite dynamics of exciton and phonons
is governed by the system-environment Hamiltonian
$H=H_{\text{ex}}+H_{\text{ex-ph}}+H_{\text{ph}}$. The vibrational modes
involved in this process, such as bond-length alternations and ring-breathing
modes [24, 26, 19], are modelled as a bath of local and independent harmonic
oscillators [15, 19]. Let us write the exciton-phonon interaction Hamiltonian
$H_{\text{ex-ph}}$ as

$H_{\text{ex-ph}}=\sum_{l}A_{l}\otimes B_{l},$ (5)

such that each coupling operator $A_{l}$ acting on the exciton space is
associated with a bath operator

$B_{l}=\sum_{m}g_{l,m}\Big{(}b_{l,m}^{\dagger}+b_{l,m}\Big{)}$ (6)

given by a sum over the displacement operators associated with the modes with
frequencies $\omega_{l,m}$, written in terms of creation and annihilation
operators. The tensor $g_{l,m}$ represents the coupling strengths between the
modes $\omega_{l,m}$ and the exciton coupling operators $A_{l}$. These are
assumed to be weak, with respect to the characteristic energies of
$H_{\text{ex}}$ and $H_{\text{ph}}$. In this notation the phonon Hamiltonian
reads

$H_{\text{ph}}=\sum_{l,m}\hbar\>\omega_{l,m}\bigg{(}b^{\dagger}_{l,m}b_{l,m}+\frac{\mathbb{1}}{2}\bigg{)}.$
(7)

The exciton coupling operators $A_{l}$ model the exchange of energy between
the exciton and the vibrational modes. These can either involve a single
monomer, or a bond between two monomers. For example, the interaction between
a Frenkel state $|{kk}\rangle\\!\langle{kk}|$ and ring-breathing modes
concerns an individual monomer, here labelled by $k$. Conversely, the
interaction between Frenkel transfer terms $|{jj}\rangle\\!\langle{kk}|+h.c.$
and bond-length alternation modes is associated with the bond between a pair
$j,k$ of monomers. The set of exciton coupling operators $A_{l}$ considered in
this model is given in Table 1.

###  II.3 Master equation

The dynamics of the reduced exciton state $\rho_{t}$ is governed by a Lindblad
master equation 111Also known as Gorini–Kossakowski–Sudarshan–Lindblad (GKSL)
master equation [55]. that is obtained from the system-environment Hamiltonian
$H$ by tracing over the phonon environment [53]

$\begin{split}\dot{\rho}_{t}=&-\frac{i}{\hbar}[H_{\text{ex}},\rho_{t}]+\sum_{\omega,l}\gamma_{l}(\omega)\bigg{(}A_{l}(\omega)\rho_{t}A_{l}^{\dagger}(\omega)+\\\
&+\frac{1}{2}\Big{\\{}A^{\dagger}_{l}(\omega)A_{l}(\omega),\rho_{t}\Big{\\}}\bigg{)}.\end{split}$
(8)

The Lindblad collapse operators $A_{l}(\omega)$ are here given by the spectral
representation of the coupling operators $A_{l}$,

$A_{l}(\omega)=\sum_{E_{\lambda^{\prime}}-E_{\lambda}=\omega}|{E_{\lambda}}\rangle\\!\langle{E_{\lambda}}|A_{l}|{E_{\lambda^{\prime}}}\rangle\\!\langle{E_{\lambda^{\prime}}}|,$
(9)

where $|{E_{\lambda}}\rangle$ are the eigenstates of the exciton Hamiltonian,
$H_{\text{ex}}|{E_{\lambda}}\rangle=E_{\lambda}|{E_{\lambda}}\rangle$, such
that $\omega$ matches the Bohr frequencies $E_{\lambda^{\prime}}-E_{\lambda}$.
The relaxation rates $\gamma_{l}(\omega)$ are obtained from the Fourier
transform of the bath correlation functions
$\langle{B_{l}(t)B_{l^{\prime}}(t^{\prime})}\rangle=\mathrm{Tr}[B_{l}(t)B_{l^{\prime}}(t^{\prime})\rho_{\text{ph}}]$
[53, 54]. Here, $\rho_{\text{ph}}=\exp(-\beta H_{\text{ph}})\mathcal{Z}^{-1}$
is the steady state of the phonon bath, in thermal equilibrium at inverse
temperature $\beta=(k_{B}T)^{-1}$, with partition function
$\mathcal{Z}=\mathrm{Tr}[\exp(-\beta H_{\text{ph}})]$ [54].

This general approach, also known as the Bloch-Redfield formalism, allows to
obtain relaxation rates from the energy and temperature dependent correlation
functions of the bath. The rates obtained this way satisfy thermodynamics and
detailed balance condition, leading to the correct thermal equilibrium
distribution for long time scales [70]. Here, we obtain a prescription for the
dynamics of exciton states in the Markovian approximation, under which
condition the master equation can be equivalently expressed in either Bloch-
Redfield or Lindblad formalism [70].

In the Markovian limit that characterises Eq. (8), the bath correlation
functions only depend on the time interval $\tau$ between two time steps
$t-t^{\prime}=\tau$ [53, 54]. Moreover, in virtue of the mutual independence
of the vibrational modes for each pair of coupling operators $A_{l}$,
$A_{l^{\prime}}$, the bath correlation functions reduce to

$\begin{split}\langle{B_{l}(\tau)B_{l^{\prime}}(0)}\rangle&=\delta_{ll^{\prime}}\sum_{m}g^{2}_{l,m}\bigg{(}\Big{(}1+\nu_{\beta}\big{(}\omega_{l,m}\big{)}\Big{)}e^{-i\omega_{l,m}\tau}+\\\
+&\nu_{\beta}\big{(}\omega_{l,m}\big{)}e^{i\omega_{l,m}\tau}\bigg{)},\end{split}$
(10)

where $\nu_{\beta}(\omega)=(\exp(\hbar\beta\omega)-1)^{-1}$ is the bosonic
distribution function at inverse temperature $\beta$ [53, 54].

To calculate the rates $\gamma_{l}(\omega)$, the sum over the couplings
$g_{l,m}$ of Eq. (10) is replaced with an integral in the frequency domain,
$\sum_{m}g_{l,m}^{2}\to\int J_{l}(\omega)d\omega$, where $J_{l}(\omega)$ is a
spectral density that can be used to sample the coupling strengths $g_{l,m}$.
To do so we use an Ohmic spectral density $J_{l}(\omega)$

$J_{l}(\omega)=\frac{\Lambda_{l}}{\hbar}\frac{\omega}{\Omega_{l}}\exp\bigg{(}-\frac{\omega}{\Omega_{l}}\bigg{)},$
(11)

where $\Lambda_{l}=\hbar\int_{0}^{\infty}d\omega J_{l}(\omega)/\omega$ is the
reorganisation energy associated with coupling operator $A_{l}$ [53, 35]. In
this way the relaxation rates read

$\gamma_{l}(\omega)=2\pi\bigg{(}J_{l}(\omega)(1+\nu_{\beta}(\omega))+J_{l}(-\omega)\nu_{\beta}(-\omega)\bigg{)}.$
(12)

Exciton coupling operators ($A_{l}$) | Site $k$ | Bonds $j,k$
---|---|---
Frenkel | $|{kk}\rangle\\!\langle{kk}|$ | $|{jj}\rangle\\!\langle{kk}|+h.c.$
Electron | $|{k}\rangle\\!\langle{k}|_{e}\otimes\mathbb{1}_{h}$ | $(|{j}\rangle\\!\langle{k}|_{e}+h.c.)\otimes\mathbb{1}_{h}$
Hole |  $\mathbb{1}_{e}\otimes|{k}\rangle\\!\langle{k}|_{h}$ | $\mathbb{1}_{e}\otimes(|{j}\rangle\\!\langle{k}|_{e}+h.c.)$
Table 1: Exciton coupling operators $A_{l}$ classified by their action on
Frenkel, electron, and hole states and transfer terms.

Exciton-phonon interactions also introduce further disorder, which is captured
by energy fluctuations $\delta E_{l}$ in the exciton Hamiltonian,
$H_{\text{ex}}\to H_{\text{ex}}+\sum_{l}\delta E_{l}\>A_{l}$ [43, 34]. These
are sampled from a normal distribution with zero average and standard
deviation $\sigma_{l}(\beta)$, given by

$\sigma^{2}_{l}(\beta)=\int_{0}^{\infty}\>d\omega\>\hbar\>J_{l}(\omega)\coth\bigg{(}\frac{\hbar\beta\omega}{2}\bigg{)},$
(13)

as done in Ref. [77, 42]. The addition of such energy fluctuations affects
site energies and coherent transfer terms, leading to asymmetry in the charge
separation landscape and promoting localisation in the centre of mass dyamics
[49, 78, 79].

In the next section we use Eq. (8) to study exciton transport in conjugated
polymers across different morphologies. Before discussing the results, let us
make a few remarks on the approximations that characterise Eq. (8). First of
all, it is worth pointing out that, in Eq. (8) we have neglected the Lamb-
Shift Hamiltonian term. This term accounts for the contributions of the
coupling operators $A_{l}$ to the coherent dynamics of the exciton. Lamb-Shift
corrections have often been neglected in exciton transport problems,
especially for the case of local and uncorrelated phonon baths [35, 37, 38].
When applying Eq. (8) in Sec. III we will assume the exciton Hamiltonian to
already include energy shift arising from the weak coupling with the phonon
environment. There, Lamb-Shift corrections are expected to be well below the
precision limit of a qualitative study focused on order-of-magnitude
differences in exciton energies and electronic couplings.

Another important observation is that we have assumed the vibrational modes to
be uncoupled with each other, which is the reason why Eq. (8) does not feature
combinations of $A_{l}(\omega)$ and $A_{l^{\prime}}(\omega)$ Lindblad
operators. It is reasonable to imagine that this approximation would not be
justified for some combinations of coupling operators, such as for those
vibrational modes coupled with the same monomer. When spatial correlations
become important, e.g., when phonon modes are delocalised over several
monomers, they can be simply accounted for and included into Eq. (8) via the
addition of terms in $A_{l}$ and $A_{l^{\prime}}$, and the associated rates
arising from the spatial dependence of the correlation functions $\langle
B_{l}(t),B_{l^{\prime}}(0)\rangle$ [80]. The presence of strong and long-range
spatial correlations allows for the formation of decoherence free subspaces
and gives rise to super and subradiance phenomena [81], which have been
studied for biological light-harvesting complexes [70], but remain rather
unexplored for the case of organic polymers.

Finally, the weak coupling approximation for the exciton-phonon interaction,
necessary to obtain Eq. (8), is arguably the main source of deviations from a
rigorous description of the exciton transport. While some vibrational modes,
such as those associated with torsional angles and foldings of a polymer
chain, are very likely to be well described by a weak coupling approximation,
others modes, such as bond-length alternations, can couple strongly with the
excitons. In such cases it is possible to overcome the limitations imposed by
Eq. (8) by performing a polaron-transformation [82, 83, 84, 85], which models
the dynamics of an exciton followed by nuclear deformations, i.e., a polaron
(or _dressed-exciton_), using master equations similar to Eq. (8), such as the
secular polaron-transformed Redfield equation (sPTRE) [86, 87, 88, 89, 43, 90,
33, 91, 34].

Exciton Hamiltonian parameters — OPV, PPV Parameter |  | Hartree AU
---|---|---
Frenkel electron-hole distance | $r_{0}$ | $12.7$
Exciton biding energy |  $E_{0}$ | $7.90\times 10^{-2}$
Frenkel transfer parameters | $j_{0}$ | $5.03\times 10^{-3}$
| $j_{2}$ | $1.20\times 10^{-2}$
| $j_{4}$ | $3.23\times 10^{-2}$
Charge transfer parameter | $t_{0}$ | $9.78\times 10^{-2}$
Relative permittivity | $\epsilon_{0}$ | $1$
Table 2: Exciton Hamiltonian parameters used in Eqs. (2), (14), and (15) for
the OPV and PPV-like materials considered in Secs. III.1 and III.2, expressed
in Hartree atomic units [31]. Frenkel transfer parameters have been scaled in
Sec. III.3 in order to study the dependence of transport properties on the
strength of the electronic couplings.

##  III Results

We now use Eq. (8) to study room-temperature exciton transport in some
representative conjugated oligomers and polymers, exploring the transition
from ordered to disordered morphologies. To compare our findings with previous
results, we base our calculation upon the electronic properties of OPV and
PPV, used in Refs. [31, 19]. In particular, we focus on the following systems.
First, we consider a short OPV hexamer to explore centre of mass (CoM) and
charge separation (CS) dynamics with a full Merrifield exciton approach, in
Sec. III.1. Then, in Sec. III.2, we model a long PPV polymer with 51 repeated
unit to explore intrachain Frenkel exciton dynamics across different
morphological regimes. Finally, we examine the role of intrachain and
interchain transport and its dependence on electronic coupling strength in
Sec. III.3, by looking at Frenkel exciton transport in OPV oligomers.

The different regimes of disorder, from crystalline to amorphous, are here
achieved sampling torsional angles and bond length-alternations from a normal
distribution with zero average and tunable standard deviation. The functional
dependence of Frenkel and CT terms on the relative arrangement between two
conjugated monomers, given by
$V_{\text{F}}(d,\theta)=f(d)\Theta_{\text{F}}(\theta)$ and
$V_{\text{CT}}(d,\theta)=f(d)\Theta_{\text{CT}}(\theta)$, respectively, is
calculated upon their distance $d$ and the relative angle $\theta$ between the
two monomers’ planes. The angular dependence $\Theta_{\text{F}}(\theta)$ and
$\Theta_{\text{CT}}(\theta)$ is modelled as in Ref. [31],

$\displaystyle\Theta_{F}(\theta)=(j_{4}-j_{0})\cos^{4}\theta+(j_{2}-j_{0})\cos^{2}\theta-2j_{0},$
(14) $\displaystyle\Theta_{\text{CT}}(\theta)=t_{0}\cos\theta,$ (15)

while the dependence on the intra-monomer distance is heuristically modelled
with a Gaussian factor $f(d)=\exp[(d-r_{0})^{2}/4r_{0}^{2}]$.

The parameters used in Eqs. (2), (14), and (15) are given in Tab. 2 and
expressed in Hartree atomic units, while the relative permittivity of the
considered materials is assumed to be $\epsilon_{r}=1$ for simplicity. The
characteristic electronic couplings associated with Frenkel and charge-
transfer terms can be as high as 2 eV, and, thus, much larger than the 26 meV
associated with the thermal energy of the phonon bath at room temperature of
300K. When required, Frenkel through-space couplings are included for non-
conjugated monomers that are separated by $\approx r_{0}$, with characteristic
energies around 10 meV [92, 19].

Figure 2: Exciton transport for different instances of OPV hexamer
morphologies at room temperature $T=300K$. For each case the initial state
$\rho_{0}$ is a Frenkel exciton localised on the first monomer. Expectation
value (solid lines) and standard deviation (shaded areas) of CoM (blue) and CS
(red) are expressed in the sites basis. The energy required for charge
dissociation (dotted-dashed magenta line) is shown in eV (on the same $y$-axis
as CS). The histograms insets schematically represent the populations (not in
scale) of the partial states of electron (black bars) and hole (light gray
bars) on each site at a given time. Figure 3: (_Left_) Exciton transport for
crystalline (_top-left_) and amorphous (_bottom-left_) PPV polymers with 51
monomers, at room temperature $T=300K$. For each case the initial state
$\rho_{0}$ is a Frenkel exciton localised on the central monomer ($k=26)$.
Expectation value (solid lines) and standard deviation (shaded areas) of CoM
(blue) are expressed in the sites basis. Delocalisation is monitored by purity
(dashed light blue lines) and average IPR in the sites basis (dashed-dotted
red line). The histograms insets schematically represent the populations (not
in scale) of the exciton on each site at a given time. (_Right_) The
dependence of steady state delocalisation on the morphology is studied varying
the standard deviation $\sigma_{\theta}$ of random torsional angles, and
sampling 100 PPV polymers with 51 repeated units for each value of
$\sigma_{\theta}$.

Reorganisation energies $\Lambda_{l}$ and cut-off frequencies $\Omega_{l}$
associated with vibrational modes define relaxation rates $\gamma_{l}(\omega)$
and energy fluctuations $\sigma_{l}(\beta)$, as prescribed by Eq. (11) [53,
54]. For the considered systems we have chosen reorganisation energies of 500
meV and 50 meV for monomer-local and bond-local vibrational modes,
respectively. The chosen cut-off frequencies are 1500 $\text{cm}^{-1}$ for
modes that couple with Frenkel states, 1000 $\text{cm}^{-1}$ for modes that
couple with individual charges and Frankel transfer terms, and 500
$\text{cm}^{-1}$ for modes that couple with CT terms [93, 94, 32, 19].

To propagate an initial exciton state $\rho_{0}$ we exactly solve the system
of linear differential equations associated with Eq. (8) written in Liouville
space [95]. In this way we obtain a map $\Lambda_{t}$ for the dynamics of the
state $\rho_{t}=\Lambda_{t}[\rho_{0}]$. Exciton transport properties are
studied evaluating time-dependent expectation value and standard deviation of
CoM and CS operators. These can be expressed in terms of monomer indices
$\langle{eh|X_{\text{CoM}}|e^{\prime}h^{\prime}}\rangle=\delta_{ee^{\prime}}\delta_{hh^{\prime}}(e+h)/2$,
and
$\langle{eh|X_{\text{CS}}|e^{\prime}h^{\prime}}\rangle=\delta_{ee^{\prime}}\delta_{hh^{\prime}}(e-h)$,
as done in Ref. [31]. Alternatively, they can be expressed in terms of monomer
coordinates $\vec{r}_{k}=(x_{k},y_{k},z_{k})$, by replacing $e$ ($h$) with a
given Cartesian coordinate, such as $x_{e}$ ($x_{h}$).

Localisation properties of Frenkel states are inferred from purity,
$\mathcal{P}[\rho_{t}]=\mathrm{Tr}[\rho_{t}^{2}]$, and average inverse
participation ratio (IPR) [96, 94, 29, 34, 66]. Let
$\rho=\sum_{k}p_{k}|{r_{k}}\rangle\\!\langle{r_{k}}|$ be expressed in its
eigenbasis $\\{|{r_{k}}\rangle\\}$, and let $\\{|{n}\rangle\\}$ be the sites
basis then

$\overline{\texttt{IPR}}(\rho)=\sum_{k}p_{k}\bigg{(}\sum_{n}|\langle{n|r_{k}}\rangle|^{4}\bigg{)}^{-1},$
(16)

i.e., the weighted sum of the
$\texttt{IPR}=1/\sum_{n}|\langle{n|\psi}\rangle|^{4}$ for each pure state
$|{r_{k}}\rangle$ making up the density operator $\rho$ [97, 34]. From the
IPR, delocalisation length can be estimated using
$l=(\overline{\texttt{IPR}}(\rho))^{1/d}$ for a $d$-dimensional system [34].
Since the considered systems are either one-dimensional or ensembles of
coupled one-dimensional systems we estimate the delocalisation length simply
using $\overline{\texttt{IPR}}(\rho)$.

###  III.1 OPV hexamer

This system consists of six conjugated monomers and it is studied with a full
Merrifield exciton approach, i.e., including all the possible CS states. For
each morphology, we initialise the exciton in a Frenkel state localised on the
first monomer. For comparison, a similar system was studied in Ref. [31] with
a MCTDH method, requiring around $10^{6}$ configurations for its numerical
implementation, as opposed to the 36 required for the solution of Eq. (8)
222As discussed in Sec. II this advantage in terms of computational cost comes
at the expense of several approximations that reduce the accuracy of the
master equation.

The crystalline morphology, shown in Fig. 2 (_top_), is obtained for a
perfectly planar arrangement of the monomers with constant bond-length
alternation and no static disorder. It is characterised by trivial CS
dynamics, which remains constant and equal to zero for all times. The exciton
undergoes a initial ultrafast ballistic transport regime (1-10 fs), followed
by a diffusive transport regime that lasts for the first 100 fs. During the
evolution the partial states of electron and hole are perfectly symmetric, as
shown in the histograms insets of Fig. 2 representing the population (not in
scale) of the partial states of electron (black bars) and hole (light gray
bars) on each site at a given time.

The semicrystalline morphology, an instance of which is shown in Fig. 2
(_middle_), is obtained for torsional angles normally distributed with
standard deviation of 0.5 rad. The presence of disorder allows for a non
trivial CS dynamics, thus reducing the energy barrier required for charge
dissociation (CD) of the exciton. Despite the CoM dynamics has similar
features to those of the crystalline morphology, asymmetry in the partial
states of electron and hole is generally present both during the evolution and
in the steady states.

The amorphous phase, an instance of which is shown in Fig. 2 (_bottom_), is
obtained for torsional disorder with standard deviation of 1 rad. It is
characterised by noticeable CS dynamics, with electron-hole separation over
more than 2 monomers, lasting for hundreds of femtoseconds. Such CS dynamics
can considerably reduce the CD energy for relatively long time spans of
hundreds of femtoseconds. Remarkably, high-disorder allows for the formation
of long-lived non-equilibrium states, before thermal equilibrium is reached
around 100 ps. Exciton lifetimes may be shorter than the time required to
reach thermal equilibrium in such systems. This implies that thermal
equilibrium states of the excitons are generally not experimentally
accessible.

Figure 4: (_Left_) Exciton transport for an arrangement of three
semicrystalline OPV octamers with weak (10 – 100 meV) (_top-left_) and strong
(0.1 – 1 eV) (_bottom-left_) electronic couplings at room temperature
$T=300K$. For each case the initial state $\rho_{0}$ is a Frenkel exciton
localised on first monomer of the first octamer. Expectation value (solid
lines) and standard deviation (shaded areas) of CoM (blue) are expressed in
the sites basis. Delocalisation is monitored by purity (dashed light blue
lines) and average IPR in the sites basis (dashed-dotted red line). The insets
schematically represent the populations of the exciton on each site at a given
time. (_Right_) The dependence of exciton delocalisation on the electronic
couplings is studied varying the characteristic strength of Frenkel couplings
from 10 meV to 1 eV. For each coupling regime we sampled 100 OPV oligomers
with 10 repeated units and semicrystalline morphology ($\sigma_{\theta}=0.5$
rad). Delocalisation is studied by evaluating IPR, purity and standard
deviation of CoM for steady states and for states at 100 fs, with $\rho_{0}$
initialised in a Frenkel state localised on the first monomer.

###  III.2 PPV polymer

We now consider a PPV polymer with 51 repeated units. We restrict the dynamics
to the Frenkel manifold to study the transport regimes of the CoM while
limiting the computational cost. Excitons are here initialised in a Frenkel
state localised on the central monomer ($k=26$) of each considered polymer.

The crystalline morphology, shown in Fig. 3 (_top-left_), is characterised by
a transition from ballistic to diffusive exciton transport during the first
100 fs, in analogy with the results of Ref. [19]. The remaining part of the
dynamics consists in a relaxation process that guides the exciton towards
thermal equilibrium. The absence of disorder in the sites energies and
electronic couplings allows for the formation of highly delocalised steady
states, with $\overline{\texttt{IPR}}>30$.

However, in contrast with the results of Ref. [19], steady state populations
are not evenly spread across the polymer, and thus only partially mixed. This
is because the decoherence model used in Ref. [19] does not account for the
spectral response between exciton coupling operators and the phonon bath. The
results of Ref. [19] can be qualitatively reproduced using Eq. (8) by
replacing the Lindblad operators $A_{l}(\omega)$ with the coupling operators
$A_{l}$, and the associated relaxation rates $\gamma_{l}(\omega)$ with
frequency independent rates $\gamma_{l}$. In such simplified limit, steady
states become diagonal in the sites’ basis. Here, instead, we use a Bloch-
Redfield approach for rates and dissipators and obtain the correct thermal
equilibrium states, which are diagonal in the exciton Hamiltonian basis.

As disorder is increased, localisation becomes more evident, undermining the
ultrafast ballistic transport regime that would otherwise characterise a
crystalline morphology. The amorphous morphology, an instance of which is
shown in Fig. 3 (_bottom-left_), is characterised by sub-diffusive exciton
transport within the first 100 fs, followed by thermal relaxation. States are
remarkably less delocalised both during the dynamics and at equilibrium, with
$\overline{\texttt{IPR}}\approx 6$ and CoM standard deviations much lower than
for the crystalline morphology.

To further examine the relation between morphology and exciton delocalisation
we study the IPR of steady states at room temperature for different
morphological regimes. To do so we vary the standard deviation
$\sigma_{\theta}$ for the random torsional angles between 0 (crystalline) and
1 rad (amorphous), sampling 100 different PPV polymers with 51 monomers for
each morphology. The results, presented in Fig. 3 (_right_), show the high
sensitivity of exciton delocalisation to conformational defects.
Delocalisation rapidly drops from $\overline{\texttt{IPR}}\approx 35$ for
crystalline polymers to $\overline{\texttt{IPR}}\approx 4$ for amorphous ones.

###  III.3 Dependence on electronic couplings

We now study the dynamics of Frenkel excitons for fixed morphology while
varying the strength of the electronic couplings. This allows us to explore
the different regimes of exciton transport and thermal equilibrium that are
associated with weak (10 – 100 meV) or strong (0.1 – 1 eV) electronic
couplings. First, we illustrate such difference by looking at Frenkel exciton
dynamics for a system given by three closely arranged OPV octamers that
interact via through-space couplings, shown in Fig. 4 (_left_).

Systems characterised by weak electronic couplings rapidly lose their
coherence, which vanishes within the first 10 – 20 fs. As shown in Fig. 4
(_top-left_), exciton transport is incoherent both during an initial
intrachain transport over one oligomer and during the slower interchain
transport across the different oligomers. The thermal energy is sufficient to
populate several eigenstates of the exciton Hamiltonian. Thermal equilibrium
is therefore characterised by low-purity and large CoM standard deviation.

Strong electronic couplings ($\approx 1$ eV) also display ultrafast intrachain
dynamics, followed by a slower interchain transport. However, the thermal
energy is not high enough to populate several states of the exciton
Hamiltonian, therefore the steady states are rather pure and localised around
the most energetically favourable clusters of conjugated monomers, as shown in
Fig. 4 (_bottom-left_).

To systematically explore the dependence of exciton delocalisation on the
strength of the electronic couplings (and thus on the thermal energy) we study
IPR, purity and CoM standard deviation of transient (100 fs) and steady states
of semicrystalline oligomers. The characteristic strength of Frenkel couplings
is varied from 10 meV to 1 eV. For each coupling regime we sample 100 OPV
oligomers with 10 repeated units and fixed semicrystalline morphology
($\sigma_{\theta}=0.5$ rad). As shown in Fig. 4 (_right_), the average IPR
decreases only slightly ($\approx 10\%$) over two orders of magnitude of
electronic couplings strength. However, transport properties change
dramatically, with CoM standard deviation and purity varying remarkably for
both transient and steady states.

###  III.4 Final remarks

Using the model introduced in this article the dynamics of excitons in
conjugated polymers is understood as a quantum thermalisation process, whose
features strongly depend on the amount of disorder (i.e., morphology) and on
the relative magnitude of thermal energy and electronic couplings (i.e.,
temperature). The non-equilibrium dynamics is characterised by an ultrafast
and ballistic transport transient, followed by an intermediate diffusive or
sub-diffusive process that occurs both through-bond and through-space. The
dynamics culminates with thermal relaxation. Increasing disorder enables
localisation, thus decreasing delocalisation lengths and IPR and hindering the
efficacy of exciton transport. The amorphous phase is characterised by the
presence of long-lived non-equilibrium states and prominent charge-separation
dynamics, which however is limited to a few monomers, as previously reported
in Ref. [24].

In polymers characterised by weak electronic couplings, of the order 10 – 100
meV, the environment heats the system, driving the excitons low-purity thermal
states, with non-negligible populations across the whole spectrum of energy
eigenstates of the exciton Hamiltonian. Even though the intrinsic localisation
lengths are low due to the presence of disorder, the thermal energy of the
phonon bath (around 26 meV at room temperature) is sufficient to populate
monomers that are far apart from the initial exciton location. In contrast, in
materials characterised by strong Frenkel and charge-transfer couplings
($\approx 1$ eV), the phonon environment cools the system, leading the
excitons to high-purity thermal state, with low-energy eigenstates of the
exciton Hamiltonian being the only few populated ones. For this reason,
despite the slower intrachain transport, systems with weak electronic
couplings can display a more efficient transport mechanism over the picosecond
time-scale at room temperature. This becomes particularly important for
triplet excitons, which typically have lower mobility and longer lifetime than
singlet excitons.

##  IV Conclusions

In this article we have introduced a general master equation to study the
dynamics of Merrifield excitons in conjugated polymers as a function of
temperature and morphology of the medium. Using this method we have explored
the general features of energy and charge transport in some representative
systems based on PPV’s electronic properties, confirming well known paradigms
of quantum transport in disordered systems, and revealing non-equilibrium
features such as long-lived states and charge-separation dynamics.

Beyond the qualitative understanding of exciton dynamics in amorphous
polymers, we expect our method to be applicable for the quantitative study of
energy and charge transport properties of specific materials. This can be done
using a multi-scale approach based on molecular dynamics and first-principle
calculations, as demonstrated in Refs. [41, 42, 19]. We also anticipate that
this approach could be used to study disorder-dependent effects in OSCs, such
as trapped-charge induced photoluminescence peak displacement [99, 100] and
individual charge-carrier pathways in amorphous polymers [101].

###### Acknowledgements.

This research was supported by the Australian Research Council under grant
number CE170100026. FC thanks Igor Lyskov and Ivan Kassal for insightful
discussions. This research was undertaken with the assistance of resources
from the National Computational Infrastructure (NCI), which is supported by
the Australian Government. The authors acknowledge the people of the Woi
wurrung and Boon wurrung language groups of the eastern Kulin Nations on whose
unceded lands we work. We respectfully acknowledge the Traditional Custodians
of the lands and waters across Australia and their Elders: past, present, and
emerging.

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# Nonclassical properties of a deformed atom-cavity field state

Naveen Kumar<EMAIL_ADDRESS>Deepak<EMAIL_ADDRESS>Arpita
Chatterjee<EMAIL_ADDRESS>Department of Mathematics, J. C. Bose
University of Science and Technology,
YMCA, Faridabad 121006, India

###### Abstract

We analyze here a nonclassical state produced by an atom-cavity field
interaction. The two-level atom is passed through the single-mode
electromagnetic cavity field. By deforming the field operators and introducing
nonlinearity to the classic Jaynes–Cummings model, we explore the system in
respect of a nonlinear Hamiltonian. Assuming that the atom is in an excited
state and the field is in a coherent state initially, the analytic expression
for the state vector of the entire system state vector is obtained. With the
help of the derived state vector, we calculate the photon number distribution,
Mandel’s $Q_{M}$ parameter, Wigner function, anti-bunching, squeezing
properties and $Q$ function etc.

##  1 Introduction

The conventional Jaynes–Cummings model (JCM) [1] describes the atom-cavity
interaction as a two-level atom is colliding with a single mode of the
electromagnetic field in the matter–radiation coupling. This model is proposed
as a basic design to investigate the semiclassical behaviour of quantum
radiation field [2]. The generalized Jaynes-Cummings model, illustrating the
nonlinear interaction of a two or multi-level atom with a cavity field,
results a deformed JCM structure. In quantum optics and quantum information
processing [3], the nonclassical light field is of major interest for a number
of reasons. In an all-optical quantum information processing device [4], the
single-photon Fock state, a nonclassical state, is an essential resource.
Controlling the emission of a single radiator, such as a molecule or a quanta
[5], can be used to create these states. Fock state can also be prepared using
cavity QED experiments in which atoms interact one at a time with a high $Q$
resonator. A $\pi$ quantum Rabi pulse in a microwave cavity [6] or an
adiabatic passage sequence in an optical cavity[7] can produce a one-photon
Fock state in this way. The study of these states yields a fundamental
understanding of quantum fluctuations and a new method of quantum
communication or imaging that surpasses the standard quantum noise limit.
Nonclassical states have a wide range of real-world applications. For example,
squeezed states are used to reduce the noise level in one of the phase-space
quadratures below the quantum limit [8], entangled states are employed to
realize a quantum computer and to transfer quantum information [9]. Here under
the rotating-wave approximation (RWA), we investigate the dynamics of two-
photon correlations generated by the interaction of a semiclassical two-level
atom with a single-mode cavity field.

The following is a breakdown of the paper’s structure. We begin by describing
the interaction between a two-level atom and a single-mode field and solve the
time-dependent Schrödinger equation [8], and derive the definite form of the
atom-cavity state vector. To find the field’s nonclassical properties, we
first trace out the atom part from the atom-field system. We then check the
nonclassicality criteria for distribution of photon numbers that means the
probability of detecting $n$ photons in the state $\hat{\rho}$. It is a smooth
function of $n$ for classical states like coherent and thermal states and an
oscillating function of $n$ for nonclassical states like squeezed states, as
described in Schleich and Wheeler’s [10] seminal work on interference in phase
space. We further discuss the nonclassicality of the cavity field described by
Mandel’s $Q_{M}$ parameter, Wigner function, anti-bunching, squeezing
parameter, and $Q$ function.

##  2 State of interest

In a typical Jaynes-Cummings model, a two-level atom interacts with a single-
mode quantized electromagnetic field oscillating with frequency $\Omega$. We
consider $\sigma_{ij}=\ket{i}\bra{j},\,(i,j=1,2)\,$ with $\ket{2}$ ($\ket{1}$)
as the excited (ground) state of the atom. With these notations, the resonant
Jaynes-Cummings Hamiltonian can be written as (with units such that $\hbar=1$)
[11, 12]

$\displaystyle H=\Omega
N+\omega\sigma_{3}+g(a\sigma_{21}+a^{\dagger}\sigma_{12})$ (1)

where $N=a^{\dagger}a$ is the number operator for the electromagnetic field
mode, $\sigma_{3}=\ket{2}\bra{2}-\ket{1}\bra{1}$ is the atomic inversion
operator, $\omega=\omega_{2}-\omega_{1}$ is the atomic transition frequency,
and the coupling constant $g$ is effectively the product of the dipole moment
and the mode function of the cavity mode at the position of the atom. The
first two terms of (1) are the free Hamiltonians of a single-mode
electromagnetic field and a two-level atom. When the atom is interacting with
the cavity field, it may drop (jump) from the excited (ground) to ground
(excited) state by emitting (absorbing) a photon. The Hamiltonian (1)
describes one of the few exactly solvable models in quantum optics. The
deformation can be added to the atom-field system in two different ways.
First, the field properties can be modified by an intensity-dependent coupling
between the atom and the radiation field, i.e. the coupling is no longer
linear in the field variables. Secondly, a system is considered in which the
electromagnetic field mode is excited in a Kerr medium. This system may be
realized by the experiment with a Rydberg atom in a nonlinear Kerr-like
cavity. We consider here the first kind of nonlinearity only. The Hamiltonian
describing the dynamics of this quantum system in the rotating wave
approximation (RWA) is obtained from (1) by deforming the bosonic operators as

$H=\Omega{\hat{A}}^{\dagger}\hat{A}+(\omega_{1}\sigma_{11}+\omega_{2}\sigma_{22})+g(\hat{A}\sigma_{21}+{\hat{A}}^{\dagger}\sigma_{12}).$
(2)

The $f$-deformed annihilation ($\hat{A}$) and creation (${\hat{A}}^{\dagger}$)
operators are constructed from the usual bosonic operators $\hat{a}$,
${\hat{a}}^{\dagger}$ and the number operator $\hat{n}=a^{\dagger}a$ such as
${\hat{A}}^{\dagger}=f(\hat{n})a^{\dagger}$ and $\hat{A}=\hat{a}f(\hat{n})$,
where $f(\hat{n})$ is an arbitrary real function of the number operator
$\hat{n}$ [13, 14]. One way of realizing nonlinear Jaynes-Cummings Hamiltonian
is to consider atomic systems with vibrational sidebands [15]. The spatial
structure of the cavity field determines the form for the function
$f(a^{\dagger}a)$. By proper choice of cavity mode structure, it is possible
to design arbitrary atom–field couplings. Another way to arrive at atom-field
coupling, which has polynomial dependence on the photon number, is to use many
lasers with different phases and Rabi frequencies to interact with a trapped
ion with vibrational sidebands [11]. The number of lasers required is same as
the order of the polynomial. Once the specific form of the function $f(n)$,
considered to be a polynomial, is fixed, the required phases and the Rabi
frequencies of the external lasers can be determined. With the technological
advancement in trapping and availability of lasers, it is possible to engineer
the atom–environment coupling by varying laser frequencies and their
intensities. Some $f$-deformed bosonic commutations relations are

$\displaystyle[\hat{A},{\hat{A}}^{\dagger}]$
$\displaystyle=(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n})=g(\hat{n}),\,\textrm{say}$
$\displaystyle[{\hat{A}}^{\dagger}\hat{A},\hat{A}]$
$\displaystyle=-g(\hat{n})\hat{A},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\hat{A}}^{\dagger}\hat{A},{\hat{A}}^{\dagger}]={\hat{A}}^{\dagger}g(\hat{n})$
$\displaystyle[{\hat{A}}^{\dagger}\hat{A},{\hat{A}}^{\dagger}g(\hat{n})]$
$\displaystyle={\hat{A}}^{\dagger}g^{2}(\hat{n}),~{}~{}~{}~{}~{}[{\hat{A}}^{\dagger}\hat{A},{\hat{A}}^{\dagger}g^{p}(\hat{n})]={\hat{A}}^{\dagger}g^{p+1}(\hat{n})$
$\displaystyle[{\hat{A}}^{\dagger}\hat{A},-g(\hat{n})\hat{A}]$
$\displaystyle=g^{2}(\hat{n})\hat{A},~{}[{\hat{A}}^{\dagger}\hat{A},(-1)^{p}g^{p}(\hat{n})\hat{A}]=(-1)^{p+1}g^{p+1}(\hat{n})\hat{A}$

The Hamiltonian in the interaction picture is

$\displaystyle H_{r}$ $\displaystyle=$ $\displaystyle
e^{iH_{r_{0}}t}H_{r_{1}}e^{-iH_{r_{0}}t}$

where

$\displaystyle H_{r_{0}}$
$\displaystyle=\Omega{\hat{A}}^{\dagger}\hat{A}+w_{1}\sigma_{11}+w_{2}\sigma_{22}$
$\displaystyle H_{r_{1}}$
$\displaystyle=g(\hat{A}\sigma_{21}+{\hat{A}}^{\dagger}\sigma_{12})$

Applying the Baker-Campbell-Hausdorff formula, $H_{r}$ can be calculated as

$\displaystyle H_{r}$ $\displaystyle=$ $\displaystyle
H_{r_{1}}+(it)[H_{r_{0}},H_{r_{1}}]+\frac{(it)^{2}}{2!}[H_{r_{0}},[H_{r_{0}},H_{r_{1}}]]+\frac{(it)^{3}}{3!}[H_{r_{0}},[H_{r_{0}},[H_{r_{0}},H_{r_{1}}]]]+\ldots$
(3) $\displaystyle=$
$\displaystyle\sum_{r=0}^{\infty}\frac{(it)^{r}}{r!}[H_{0},[H_{0},[H_{0},...H_{1}]]]_{r\text{
times}}$

and thus the effective Hamiltonian as

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle
g[e^{-ith(\hat{n})}A\sigma_{21}+A^{\dagger}e^{ith(\hat{n})}\sigma_{12}]$ (4)

where $h(\hat{n})=\Omega f^{2}(\hat{n})+(\omega_{2}-\omega_{1})$.
Solving the Schrödinger equation of motion [8]

$\displaystyle i\frac{\partial|\psi\rangle}{\partial t}=H|\psi\rangle,$ (5)

for any arbitrary state
$|\psi\rangle=\sum{C_{1,n+1}(t)|1,n+1\rangle+C_{2,n}(t)|2,n\rangle}$,
$\ket{2}$ and $\ket{1}$ are the excited and ground states of atom,
respectively, the equations of motion in terms of the probability amplitudes
of the state vector $\ket{\psi}$ are

$\displaystyle\dot{C}_{2,n+1}(t)$ $\displaystyle=$ $\displaystyle
e^{ih(\hat{n})t}\sqrt{n+1}\,C_{1,n}(t),$ $\displaystyle\dot{C}_{1,n}(t)$
$\displaystyle=$ $\displaystyle e^{-ih(\hat{n})t}\sqrt{n+1}\,C_{2,n+1}(t).$

Assuming that the atom enters the cavity in its excited state $\ket{2}$ and
the radiation field is initially in a coherent state, that means
$C_{1,n}(0)=0$ and
$C_{2,n+1}(0)=C_{n}(0)=e^{-\frac{|\beta|^{2}}{2}}\frac{\beta^{n}}{\sqrt{n!}}$,
the above equations can be solved as

$\displaystyle C_{2,n+1}(t)$
$\displaystyle=\frac{e^{-\frac{|\beta|^{2}}{2}}\beta^{n}}{{\sqrt{n!}}(m_{2}-m_{1})}\left\\{m_{2}e^{m_{1}t}-m_{1}e^{m_{2}t}\right\\}$
(6) $\displaystyle C_{1s,n}(t)$
$\displaystyle=\frac{e^{-ih(n)t}m_{1}m_{2}e^{-\frac{|\beta|^{2}}{2}}\beta^{n}}{\sqrt{n!}\sqrt{n+1}gf(n+1)(m_{2}-m_{1})}\left\\{e^{m_{1}t}-e^{m_{2}t}\right\\}$
(7)

where

$\displaystyle 2m_{1,2}=ih(n)\pm i\sqrt{4g^{2}(n+1)f^{2}(n+1)+h^{2}(n)}$

The state vector $|\psi(t)\rangle$ for an arbitrary state describes the time
evolution of the whole atom-field system. Here we focus on some analytical
properties of the single-mode cavity field, which is obtained from
$|\psi(t)\rangle$ by tracing out the atom part

${\rho(t)}_{f}=Tr_{a}[\rho(t)],$ (8)

where the subscript $a\,(f)$ denotes the atom (field). We consider
${\rho(t)}_{f}$ throughout the paper for determining the properties of the
field left into the cavity. The general form of the expectation $\langle
a^{\dagger p}a^{q}\rangle$ with respect to the state vector $\ket{\psi}$ can
be calculated as follows:

$\displaystyle\langle a^{\dagger p}a^{q}\rangle$ $\displaystyle=$
$\displaystyle\bra{\psi}a^{\dagger p}a^{q}\ket{\psi}$ (9) $\displaystyle=$
$\displaystyle\sum_{n}\bigg{[}C^{*}_{2,n+p-q+1}(t)C_{2,n+1}(t)\frac{\sqrt{(n+p-q+1)!(n+1)!}}{(n+1-q)!}$
$\displaystyle+C^{*}_{1,n+p-q}(t)C_{1,n}(t)\frac{\sqrt{(n+p-q)!n!}}{(n-q)!}\bigg{]}$

##  3 Nonclassicality criteria under study

In this section, we derive some criteria to witness the nonclassicality of the
considered quantum state $\ket{\psi}$. While plotting different distribution
functions, the numerical values of the atomic transition frequencies and the
coupling constant are chosen in such a way that they are in a weak coupling
regime $(g\ll\omega_{i},\,i=1,2)$, which is the primary assumption for
applying RWA. For simplicity, three different forms of $f(\hat{n})$, being an
arbitrary polynomial function of $\hat{n}$, are considered. The following
plots correspond to the cases $f(n)=\sin({n}),\,1/\sin({n})$, and $\ln({n})$.
Here, $\sin({n})$ is a bounded and periodic polynomial function, $1/\sin({n})$
is an unbounded, periodic polynomial function and $\ln({n})$ is an unbounded,
non-periodic, monotonically increasing polynomial function. In the limiting
case $f(\hat{n})=1$, the Hamiltonian reduces to the basic Jaynes-Cummings
model.

###  3.1 Photon number distribution

Photon number distribution is the probability distribution for finding $n$
photons in a given cavity field. It can be found as the expectation of the
density field with respect to the number state $n$ as

$\displaystyle p(n)=\langle n|\hat{\rho}|n\rangle$ (10)

After tracing out the atom part, the density operator $\hat{\rho}_{f}$ is
given by

$\displaystyle\hat{\rho}_{f}$ $\displaystyle=$
$\displaystyle|\psi\rangle_{f}\,{}_{f}\langle\psi|$ (11) $\displaystyle=$
$\displaystyle\sum_{n}\Big{[}|C_{2,n+1}(t)|^{2}|n+1\rangle\langle
n+1|+C_{2,n+1}(t){C^{*}_{1,n}(t)}|n+1\rangle\langle n|$
$\displaystyle+{C^{*}_{2,n+1}(t)}C_{1,n}(t)|n\rangle\langle
n+1|+|C_{1,n}(t)|^{2}|n\rangle\langle n|\Big{]}$

Substituting (11) into (10), the photon number distribution for the cavity
field is obtained as

$\displaystyle p(n)$ $\displaystyle=$ $\displaystyle\langle
n|\hat{\rho}|n\rangle$ (12) $\displaystyle=$
$\displaystyle|C_{2,n+1}(t)|^{2}+|C_{1,n}(t)|^{2}$

Here (12) gives the probability of finding $n$ photons in a given cavity field
state. The simplified form of photon number distribution can be plotted in
Fig. 1 for three different functional values of $f(n)$, namely
$\sin(n),\,1/\sin(n)$ and $\ln(n)$.

Figure 1: Variation of $p(n)$ with respect to (a) $t$ and for
$g=0.5,\,\beta=2,\,\Omega=1\mbox{GHz},\,\omega_{1}=\omega_{2}=100\mbox{J},\,n=5$,
(b) $\beta$ and for
$t=1\mbox{hr},\,g=0.5,\,\Omega=1\mbox{GHz},\,\omega_{1}=\omega_{2}=100\mbox{J},\,n=5$,
(c) $\Omega$ and for
$t=1\mbox{hr},\,g=0.5,\,\beta=2,\,\omega_{1}=\omega_{2}=100\mbox{J},\,n=5$,
(d) $n$ and for
$t=1\mbox{hr},\,g=0.5,\,\beta=2,\,\Omega=1\mbox{GHz},\,\omega_{1}=\omega_{2}=100\mbox{J}$.
Red, green, and blue lines represent $f({n})=\sin({n})$, $1/\sin({n})$, and
$\ln({{n}})$ respectively.

We can conclude from Fig. 1 that the variations of $p(n)$ with respect to
$t,\,\Omega$ follow wave nature and the amplitude of waves is maximum for
$\ln(n)$ and minimum for $1/\sin(n)$. The variation of $p(n)$ with respect to
$\beta$ is first increasing and then decreasing and maximum value attained is
0.25. It can be further observed that the peak of $p(n)$ is highest (lowest)
for $f(n)=\ln(n)$ $(1/\sin(n))$. While plotting $p(n)$ for different number of
photons $n$, all the three functions almost coincide for $n=1$ and have
distinct nature for $n>1$.

###  3.2 Mandel’s $Q_{M}$ parameter

Sub-Poissonian photon statistics are one of the most striking nonclassical
characteristics of a quantum system. Mandel’s $Q_{M}$ parameter, which
measures how far the photon number distribution deviates from the Poissonian
statistics, is used to determine such characteristics [16]. $Q_{M}$ is zero
for a coherent field, and the minimal value $Q_{M}=-1$ is obtained for Fock
states, having a well-defined number of photons by definition. The field
statistics is sub-Poissonian if $-1\leq Q_{M}<0$, and thus the phase space
distribution cannot be interpreted as a classical probability function. As a
result, the negative values of $Q_{M}$ is sufficient for a field to be
nonclassical. But the negativity of $Q_{M}$ is not a necessary condition to
distinguish the quantum states into nonclassical and classical regimes. For
example, a state may be nonclassical even though $Q_{M}$ is positive. If
$Q_{M}>0$, the state’s statistics is super-Poissonian but no conclusions can
be drawn about their character, for example the $Q_{M}$ parameter for a well-
known nonclassical thermal field is always positive. Next to determine the
photon statistics of a single-mode radiation field, we consider the Mandel’s
$Q_{M}$ parameter defined by

$Q_{M}=\frac{\langle a^{{\dagger}^{2}}a^{2}\rangle}{\langle
a^{\dagger}a\rangle}-\langle a^{\dagger}a\rangle$ (13)

Mandel’s $Q_{M}$ function can be plotted for different $f(n)$ such as
$\sin{n}$, $1/\sin(n)$, and $\ln(n)$ with respect to different parameters in
Fig. 2.

Figure 2: Variation of $Q_{M}$ with respect to (a) $t$ and for
$g=0.5,\,\beta=2,\,\Omega=1\mbox{GHz},\,\omega_{1}=\omega_{2}=100\mbox{J}$,
(b) $\beta$ and for
$t=1\mbox{hr},\,g=0.5,\,\Omega=1\mbox{GHz},\,\omega_{1}=\omega_{2}=100\mbox{J}$,
(c) $\Omega$ and for
$t=1\mbox{hr},\,g=0.5,\,\beta=2,\,\omega_{1}=\omega_{2}=100\mbox{J}$. Red,
green, and blue lines represent $f({n})=\sin({n})$, $1/\sin({n})$, and
$\ln({{n}})$ respectively.

Here we can observe that $Q_{M}$ follows wavy nature with respect to $t$ with
maximum amplitude when $f(n)=\ln(n)$. Further $Q_{M}$ shows nonclassicality
for all these three functions with respect to all the parameters along with
$\ln(n)$ has minimum nonclassicality whereas $1/\sin(n)$ has maximum
nonclassicality. Thus, in general we can conclude that the considered quantum
state is nonclassical in nature for different polynomial functional values of
$f(n)$ and order of nonclassicality may increase or decrease according to the
choice of $f(n)$. Also the nonclassicality decreases with increasing of
$\beta$, $\Omega$ and wavy in nature while increasing $t$.

###  3.3 Wigner function

The nonclassicality of a quantum state can be studied in terms of its phase-
space distribution characterized by the Wigner function. For a quantum state
$\hat{\rho}$, the Wigner function of the system is defined in terms of the
coherent state basis as [8, 17, 18]

$\displaystyle W(\gamma,\gamma^{*})=\frac{2}{\pi^{2}}e^{2|\gamma|^{2}}\int
d^{2}\delta{\langle-\delta|\hat{\rho}|\delta\rangle
e^{-2(\beta^{*}\delta-\beta\delta^{*})}},$

where $|\delta\rangle=\exp(-|\delta|^{2}/2+\delta\hat{a}^{\dagger})|0\rangle$
is a coherent state. By using the relation [19]

$\displaystyle\sum_{n=k}^{\infty}n_{C_{k}}\,y^{n-k}=(1-y)^{-k-1},$

the Wigner function can be expressed in series form as [20]

$\displaystyle
W(\gamma,\gamma^{*})=\frac{2}{\pi}\sum_{k=0}^{\infty}(-1)^{k}\langle\gamma,k|\hat{\rho}|\gamma,k\rangle,$
(14)

where $|\gamma,k\rangle$ is the usual displaced number state. The partial
negative value of the Wigner function is a one-sided condition for the
nonclassicality of the related state [21], in the sense that one cannot
conclude the state is classical when the Wigner function is positive
everywhere. For example, the Wigner function of the squeezed state is Gaussian
and positive everywhere but it is a well-known nonclassical state. For a
classical state, a necessary but not sufficient condition is the positivity of
the Wigner function. Hence, a state with a negative region in the phase-space
quadrature is essentially nonclassical. Now the displaced number state
$|\gamma,k\rangle$ can be expressed in a number state basis as

$\displaystyle|\gamma,k\rangle$ $\displaystyle=$ $\displaystyle
D(\gamma)|k\rangle$ $\displaystyle=$ $\displaystyle
e^{-\frac{|\gamma|^{2}}{2}}\sum_{m=0}^{k}\frac{\gamma^{*{m}}}{m!}\sqrt{\frac{k!}{(k-m)!}}\sum_{p=0}^{\infty}\frac{\gamma^{p}}{p!}\sqrt{\frac{(k-m+p)!}{(k-m)!}}|k-m+p\rangle$

Thus

$\displaystyle\langle\gamma,k|n\rangle$ (16) $\displaystyle=$ $\displaystyle
e^{-\frac{|\gamma|^{2}}{2}}\sum_{m=0}^{k}\frac{(-\gamma)^{m}}{m!}\sqrt{\frac{k!}{(k-m)!}}\sum_{p=0}^{\infty}\frac{\gamma^{*{p}}}{p!}\sqrt{\frac{(k-m+p)!}{(k-m)!}}\langle
k-m+p|n\rangle$ $\displaystyle=$ $\displaystyle
e^{-\frac{|\gamma|^{2}}{2}}\sqrt{\frac{k!}{n!}}(\gamma^{*})^{n-k}L_{k}^{(n-k)}\left(|\gamma|^{2}\right)$

where $L_{m}^{(k)}(x)=\sum_{n=0}^{m}\frac{(-x)^{n}(m+k)!}{(m-n)!n!(k+n)!}$ is
the associated Laguerre polynomial [22]. Substituting (LABEL:eqne) into (14)
we get

$\displaystyle W(\gamma,\gamma^{*})$ $\displaystyle=$
$\displaystyle\frac{2e^{-|\gamma|^{2}}}{\pi}\sum_{k=0}^{\infty}(-1)^{k}k!\sum_{n=0}^{\infty}\bigg{[}|c_{a,n}|^{2}\frac{|\gamma|^{2(n-k)}}{n!}\left\\{L_{k}^{(n-k)}\left(|\gamma|^{2}\right)\right\\}^{2}$
(17)
$\displaystyle+|c_{b,n+1}|^{2}\frac{|\gamma|^{2(n+1-k)}}{(n+1)!}\left\\{L_{k}^{(n+1-k)}\left(|\gamma|^{2}\right)\right\\}^{2}\bigg{]}$

Figure 3: Variation of $W$ with respect to $\beta$ ($\gamma$) in the first
(second) row and for $f(n)=\sin(n)$ in (a) and (d), $1/\sin{n}$ in (b) and
(e), $\ln{n}$ in (c) and (f). $W$ is plotted with respect to $t$ in (g). Other
parameters are the same as in Fig. 1.

Here it is clear that the Wigner function shows the non-gaussian [23]
behaviour with respect to real and imaginary $\beta$ and $\gamma$. As the
Wigner function has no negative region, we can conclude that this function is
not showing nonclassicality. Also the variation of Wigner function with
respect to the parameter $t$ shows wavy nature.

###  3.4 Anti-bunching

The theory of majorization in [24, 25] gives us the expression of anti-
bunching in a quantum state which satisfies the condition for nonclassicality
as

$d_{(1)}=\langle a^{{\dagger}^{2}}a^{2}\rangle-{\langle
a^{\dagger}a\rangle}^{2}<0$

The negativity of $d_{(1)}$ implies that the probability of detecting
independent photons is more than the clustered photons. We can find the
expression of $d_{(1)}$ by using the expectation values of
$a^{{\dagger}^{2}}a^{2}$ and $a^{\dagger}a$ from (9). Also the anti-bunching
criteria can be plotted in Fig. 4 with respect to different parameters. It is
seen that anti-bunching has the similar behaviour as $Q_{M}$.

Figure 4: Variation of $d_{(1)}$ for different $f(n)$ and with respect to (a)
$t$, (b) $\beta$ and (c) $\Omega$. Other parameters are the same as in Fig. 1.

###  3.5 Squeezing properties

In order to analyze the quantum fluctuations of the field quadratures, we
consider two Hermitian operators which are combinations of photon creation and
annihilation operators as

${x}=\frac{{a}+{a}^{\dagger}}{2},\,\,\,\,\,\,{p}=\frac{{a}-{a}^{\dagger}}{2i}$

with the commutation relation $\left[{x},{p}\right]=i/2.$ They obey the
Heisenberg uncertainty relation of the form
$\langle(\Delta{x})^{2}\rangle\langle(\Delta{p})^{2}\rangle\geq\frac{1}{16}$,
and thus the quadrature squeezing occurs whenever
$\langle(\Delta{x})^{2}\rangle<\frac{1}{4}$ or
$\langle(\Delta{p})^{2}\rangle<\frac{1}{4}$. It is convenient to introduce the
squeezing parameters as [26]

$\displaystyle s_{x}$ $\displaystyle=$ $\displaystyle
4\langle(\Delta{x})^{2}\rangle-1$ $\displaystyle=$ $\displaystyle
2\langle{a}^{\dagger}{a}\rangle+\langle{a}^{2}\rangle+\langle{a}^{{\dagger}2}\rangle-\langle{a}\rangle^{2}-\langle{a}^{\dagger}\rangle^{2}-2\langle{a}\rangle\langle{a}^{\dagger}\rangle,$

and

$\displaystyle s_{p}$ $\displaystyle=$ $\displaystyle
4\langle(\Delta{p})^{2}\rangle-1$ $\displaystyle=$ $\displaystyle
2\langle{a}^{\dagger}{a}\rangle-\langle{a}^{2}\rangle-\langle{a}^{{\dagger}2}\rangle+\langle{a}\rangle^{2}+\langle{a}^{\dagger}\rangle^{2}-2\langle{a}\rangle\langle{a}^{\dagger}\rangle,$

such that the squeezing exists in ${x}$ or ${p}$ quadrature if $-1<s_{x}<0$ or
$-1<s_{p}<0$, respectively. The negativity of squeezing is not a necessary
condition but a sufficient one. The expectations can be calculated using (9).

Figure 5: Variation of Squeezing parameters $S=s_{x}$ (thick line) and $s_{p}$
(thin line) with respect to $t$ in (a), $\beta$ in (b) and $\Omega$ in (c).
Other parameters are the same as in Fig. 1.

In Fig. 5 we observe that the squeezing parameters $S=s_{x}$, $s_{p}$ are
positive for all parametric values of $t$, $\beta$, and $\Omega$ and thus they
fail to detect the nonclassical nature of the cavity field.

###  3.6 $Q$ Function

The uncertainty principle prevents a direct phase-space description of a
quantum mechanical system. This fact leads to the creation of quasiprobability
distributions, which are very useful in quantum mechanics because they provide
a quantum-classical correspondence and make it easier to calculate quantum
mechanical averages in a way that is similar to the calculation of classical
phase-space averages [27]. The $Q$ function is one such quasiprobability
distribution, and its zeros are a sign of nonclassicality [28]. The $Q$
function is calculated as follows:

$\displaystyle Q$ $\displaystyle=\frac{1}{\pi}\bra{\alpha}{\rho}\ket{\alpha}$
$\displaystyle=\bra{\alpha}\sum_{n,p}[C_{1,n+1}C^{*}_{1,p+1}|n+1\rangle\bra{p+1}+C_{2,n}C^{*}_{2,p}|n\rangle\bra{p}]\ket{\alpha}$
$\displaystyle=\sum_{n,p}[C_{1,n+1}C^{*}_{1,p+1}e^{\frac{-|\alpha|^{2}}{2}}\frac{\alpha^{*(n+1)}}{\sqrt{(n+1)!}}e^{\frac{-|\alpha|^{2}}{2}}\frac{\alpha^{p+1}}{\sqrt{(p+1)!}}+C_{2,n}C^{*}_{2,p}e^{\frac{-|\alpha|^{2}}{2}}\frac{\alpha^{*(n)}}{\sqrt{(n)!}}e^{\frac{-|\alpha|^{2}}{2}}\frac{\alpha^{p}}{\sqrt{p!}}$
$\displaystyle=e^{-|\alpha|^{2}}\sum_{n,p}\left[\frac{\alpha^{*n}\alpha^{p}}{\sqrt{p!n!}}\left\\{C_{1,n+1}C^{*}_{1,p+1}\frac{|\alpha|^{2}}{\sqrt{(p+1)(n+1)}}+C_{2,n}C^{*}_{2,p}\right\\}\right]$

Figure 6: Variation of $Q$ with respect to $\beta$ ($\alpha$) in the first
(second) row and for $f(n)=\sin(n)$ in (a) and (d), $1/\sin{n}$ in (b) and
(e), $\ln{n}$ in (c) and (f). $Q$ is plotted with respect to $t$ in (g) and
$\Omega$ in (h). Other parameters are the same as in Fig. 1.

In Fig. 6, we observe that the $Q$ distribution shows the wavy and non-
Gaussian nature and remains positive everywhere.

##  4 Conclusion

In this article, we have considered a deformed model of interaction between a
two-level atom and a single-mode electromagnetic cavity field. Assuming that
the atom enters the cavity in its excited state $\ket{2}$ and the field is
initially in a coherent state, we have derived the cavity field state. For
this cavity field state, we have found some statistical properties like photon
number distribution, Wigner function, Mandel’s $Q_{M}$ parameter, anti-
bunching, squeezing parameter, and $Q$-function. All of these criteria have
specific conditions for witnessing nonclassicality such as the negativity of
the lower-order antibunching and the Poissonian statistics of Mandel’s $Q_{M}$
parameter are used to reveal the nonclassicality of the quantum field state.
Here we have observed that the $Q$-function, Wigner function, and squeezing
parameter failed to show nonclassicality.

ACKNOWLEDGEMENT

Naveen and Deepak acknowledge the financial support from the Council of
Scientific and Industrial Research, Govt. of India (Grant no.
09/1256(0004)/2019-EMR-I and 09/1256(0006)/2019-EMR-I, respectively).

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