# Nonclassicality in a dispersive atom-cavity field interaction in presence of
an external driving field

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

###### Abstract

We investigate nonclassical properties of a state generated by the interaction
of a three-level atom with a quantized cavity field and an external classical
driving field. In this study, the fields being degenerate in frequency, are
highly detuned from the atom. The atom interacts with the quantized field in a
dispersive manner. The experimental set-up involves a three-level atom passing
through a cavity and interacting dispersively with the cavity field mode.
Simultaneously, the atom interacts with an external classical field that is in
resonance with the cavity field. The three-level atom can enter the cavity in
one of the bare states $\ket{e}$, $\ket{f}$ or $\ket{g}$ or in a superposition
of two of these states. In this paper, we consider superposition of $\ket{e}$
and $\ket{f}$. In our analysis, we focus on the statistical properties of the
cavity field after interacting with the atom. The state vector
$|\psi(t)\rangle$ describes the entire atom-field system but we analyze the
properties of the cavity field independently neglecting the atomic component
of the system. For this the atom part is traced out from $|\psi(t)\rangle$ to
acquire the cavity field state only, denoted by $\ket{\psi_{f}(t)}$. We
evaluate different nonclassical measures including photon number distribution,
Mandel’s $Q_{M}$ parameter, squeezing properties $S_{x}$ and $S_{p}$, Wigner
distribution, $Q_{f}$ function, second-order correlation function $g^{2}(0)$
etc. for the obtained cavity field state.

††journal: Int. J. Mod. Phys. B

##  1 Introduction

The well-known Jaynes-Cummings model (JCM) [1] is a theoretical framework that
describes the interaction between a two-level atom and a single-mode quantized
cavity electromagnetic field [2] with the rotating-wave approximation. The JCM
is considered to be the most fundamental model for studying the interaction
between matter and field in the field of quantum optics. Moreover, this model
has nonperturbative solutions that are exactly integrable. A large number of
multi-level and multi-mode extensions of the original JCM have been studied
over the years [3]. The driven Jaynes-Cummings model while the cavity and the
external driving field are close to or on resonance with the atom, has been
studied by several authors. For example, Alsing et. al. [4] studied the Stark
splitting in the quasienergies of the dressed states resulting from the
presence of the driving field subject to the condition that both driving and
cavity fields are resonant with the atom. Jyotsna and Agarwal [5] studied the
effect of the external field on the Rabi oscillations in a situation where the
cavity field is resonant with the atom and where the external field is both
resonant and non-resonant. Dutra et. al. [6] studied a similar model but where
the external field was taken to be quantized. Chough and Carmichael [7] have
studied the JCM with an external resonant driving field and have shown that
the collapses and revivals of the mean photon number occur over a much longer
time scale than the revival time of the Rabi oscillations for the atomic
inversion. Joshi [8], in a similar way, studied the driven two-photon JCM when
Nha et. al. [9] studied the preparation of a temporally stable single-photon
state in an atom-cavity field system with a driving classical field. To the
best of our knowledge, the dispersive interaction with an external driving
field has not considered substantially despite that it is a logical extension
of the previous work in this field.

In the context of original JCM, if the interaction occurs in limit of a large
(but not too large) detuning between the cavity field and the relevant atomic
transition frequency, then it is called dispersive which is proved to be of
great importance in various proposals and experiments for producing
superposition of macroscopically (or at least mesoscopically) distinguishable
quantum states, the so-called Schrödinger cat states in cavity QED system.
Dispersive atom-field interactions are utilized in quantum sensors, such as
atomic magnetometers and atomic clocks. These interactions enhance the
precision of measurements with reference to quantum metrology. They are used
to design and develop quantum-enhanced measurement devices with applications
in navigation, geophysics, and precision spectroscopy. Dispersive interactions
can be utilized to create quantum amplifiers, devices that amplify quantum
signals with minimal noise. These amplifiers are valuable in quantum
communication and quantum measurement. These interactions can lead to
interesting nonlinear optical effects in quantum systems which have
applications in the generation of nonclassical light states and quantum
nonlinear optical devices. In a nutshell, dispersive atom-field interactions
are central to a wide range of quantum technologies which primarily motivated
us to start this work. They enable precise control over atomic systems and
have applications in quantum computing, sensing, optics, and the study of
fundamental physics. In this article, we consider a situation wherein a strong
external coherent field (microwave or laser, depending on the type of the
cavity QED experiment) resonant with a cavity mode, interacts non resonantly,
i.e., dispersively, with a three-level atom passing through the cavity.

The study of nonclassical light continues to be an active and exciting field
of research in quantum optics with the potentiality for significant
technological advancements in the future. These states have numerous
applications in different regimes. For example, quantum cryptography, quantum
metrology, quantum computing, quantum imaging, and quantum sensing are many
areas where nonclassical states have been applied with great success. In
quantum cryptography [10], nonclassical light sources such as single-photon
sources [11] are crucial for implementing quantum key distribution protocols
[12]. These protocols enable secure communication by allowing the exchange of
cryptographic keys between two parties that cannot be intercepted without
being detected. Squeezed states can be used to improve the precision of
measurements beyond the standard quantum limit [13], which can be in fields
such as gravitational wave detection and atomic clocks. Nonclassical states
can be used as qubits in quantum computing [14, 15, 16, 17, 18, 19], where
they enable faster and more efficient processing of quantum information [20]
as compared to classical bits.

In this paper, we propose an extended driven Jaynes-Cummings model where the
cavity and external driving field are close to or on resonance with the atom.
The external field, resonant with the cavity mode, interacts with an
appropriately prepared atom as it passes through the cavity. The resulting
interaction is conditional on the state of the atom and effectively converts
the classical external field into a quantized cavity field with the same
frequency. It can be seen that when the injected atom is prepared in a
superposition of its bare atomic states, it becomes possible to generate
various types of Schrödinger cat states [21].

This paper is organized as follows. In Sect. 2, we discuss the driven Jaynes-
Cummings model in the regime where the atom is detuned with both the quantized
cavity field and driving external classical field such that the atom-cavity
field coupling is dispersive. We study nonclassical properties of the cavity
field state via a set of nonclassicality witnesses, namely zeros of $Q_{f}$
function, Mandel’s $Q_{M}$ parameter, negativity of Wigner function, squeezing
properties, second-order correlation function ($g^{2}(0)<1$) in Sect. 4.

##  2 State of interest

We consider an atom with three levels $\ket{e}$, $\ket{f}$, and $\ket{g}$
configured as in Fig. 1. We assume that only dipole type transitions can occur
consecutively: $\ket{e}\leftrightarrow\ket{f}\leftrightarrow\ket{g}$.
$\omega_{0}$ is the atomic transition frequency between the levels $\ket{e}$
and $\ket{f}$ and it is near resonance with a single-mode cavity field of
frequency $\omega_{c}$. We further assume that the transition
$\ket{f}\leftrightarrow\ket{g}$ is far out of resonance with the specific
cavity mode of interest or any other cavity mode. A strong, classical driving
field of frequency $\omega_{ex}$ interacts directly with the atom while
passing through the cavity.

Figure 1:  The system under consideration involves an atom interacting with
both a cavity field and an external driving field. The energy-level
configuration of the atom consists of three distinct levels denoted as
$\ket{e}$, $\ket{f}$, and $\ket{g}$. The transition frequency between levels
$\ket{e}$ and $\ket{f}$ is represented by $\omega_{0}$. In addition, there are
two distinct frequencies involved: $\omega_{c}$ corresponds to the cavity
field while $\omega_{ex}$ represents the frequency of the external driving
field. The cavity and external fields are positioned close to resonance but
not exactly with the atomic transition frequency $\omega_{0}$. This
arrangement allows for a dispersive interaction between the fields and the
atom. It is assumed that the condition $\omega_{c}=\omega_{ex}$ holds true.
Furthermore, the level $\ket{g}$ is significantly detuned from all the
frequencies involved in the system.

The Hamiltonian for a system of a three-level atom (represented by Pauli
matrices $\hat{\sigma}_{i}$) interacting with a quantized electromagnetic
field mode (represented by the annihilation and creation operators ${a}$ and
${a}^{\dagger}$, respectively) is given by [21]

$\displaystyle{H}$
$\displaystyle=\frac{1}{2}\hbar\omega_{0}\hat{\sigma}_{3}+\hbar\omega_{c}{a}^{\dagger}a+\hbar
g({a}\hat{\sigma}_{+}+{a}^{\dagger}\hat{\sigma}_{-})$
$\displaystyle+\hbar(Ee^{-i\omega_{ex}t}\hat{\sigma}_{+}+E^{*}e^{i\omega_{ex}t}\hat{\sigma}_{-})$
(1)

Here $\frac{1}{2}\hbar\omega_{0}\hat{\sigma}_{3}$ is the energy of the atom in
absence of any external fields where $\hat{\sigma}_{3}$ is the third Pauli
matrix representing the difference in population between the two considered
levels, $\hbar\omega_{c}{a}^{\dagger}a$ is the energy of the electromagnetic
field mode in absence of any atom, $a^{\dagger}$ and $a$ are the creation and
annihilation operators for the field mode, $\hbar
g({a}\hat{\sigma}_{+}+{a}^{\dagger}\hat{\sigma}_{-})$ is the interaction
Hamiltonian between the atom and the cavity field, where $\hbar g$ is the
coupling strength of the interaction and $\hat{\sigma}_{+}$ and
$\hat{\sigma}_{-}$ are the raising and lowering operators for the atom,
respectively,
$\hbar(Ee^{-i\omega_{ex}t}\hat{\sigma}_{+}+E^{*}e^{-i\omega_{ex}t}\hat{\sigma}_{-})$
is the external driving field that is applied to the atom, where $E$ is
proportional to the coupling constant between the atom and the amplitude of
the external classical field of frequency $\omega_{ex}$. This term can be used
to drive transitions between the two levels of the atom.

We are currently assuming that $\omega_{0}$, $\omega_{c}$ and $\omega_{ex}$
are distinct. To remove the time dependence in the Hamiltonian $H$, the system
is rotating at the frequency $\omega_{ex}$. The modified Hamiltonian in the
rotating frame is obtained as

$\displaystyle H_{R}$
$\displaystyle=i\hbar\dot{R}(t)R^{\dagger}(t)+R(t)H(t)R^{\dagger}(t)$
$\displaystyle=\frac{1}{2}\hbar\Delta\hat{\sigma}_{3}+\hbar(w_{c}-w_{ex})a^{\dagger}a$
$\displaystyle+\hbar g\left(e^{-i\Delta
t}\left({a}+\lambda\right)\hat{\sigma}_{+}+e^{i\Delta
t}\left(a^{\dagger}+\lambda^{*}\right)\hat{\sigma}_{-}\right)$

where $R(t)=\exp{-i\omega_{ex}t(\hat{\sigma}_{3}+a^{\dagger}a)}$ is the
rotation operator, $\lambda=\frac{E}{g}$, $\Delta=\omega_{0}-\omega_{ex}$ is
the detuning between the atomic transition frequency and the external cavity
field frequency. Assuming the resonance condition between the external field
and the cavity field and introducing new auxiliary Bose operators
${b}={a}+\lambda$ and ${b}^{\dagger}={a}^{\dagger}+\lambda^{*}$ which satisfy
$[{b},{b}^{\dagger}]=1$, the Hamiltonian can be written as

$\displaystyle H_{R}$
$\displaystyle=\frac{1}{2}\hbar\Delta\hat{\sigma}_{3}+\hbar g\left(e^{-i\Delta
t}{b}\hat{\sigma}_{+}+e^{i\Delta t}{b^{\dagger}}\hat{\sigma}_{-}\right)$ (2)

which superficially looks like the interaction picture Hamiltonian of the
usual detuned JCM.

If the detuning between the atom and the fields is sufficiently large, one can
use the standard techniques [22] to derive the effective atom-field
interaction Hamiltonian as

$\displaystyle\int_{0}^{t}H_{R}(t^{\prime})dt^{\prime}\int_{0}^{t^{\prime}}H_{R}(t^{\prime\prime})dt^{\prime\prime}\approx\frac{ih^{2}}{\Delta}[b^{\dagger}b\hat{\sigma}_{3}+\frac{1}{2}[b,b^{\dagger}]\,(\hat{\sigma}_{3}+I)]t$

While performing the above integration, the second-order terms in $1/\Delta$
arriving from the exponential of $\Delta$ and an additional factor of $\Delta$
in the denominator are neglected. Here
$\hat{\sigma}_{3}=\hat{\sigma}_{+}\hat{\sigma}_{-}-\hat{\sigma}_{-}\hat{\sigma}_{+}=\ket{e}\bra{e}-\ket{f}\bra{f}$,
$I=\ket{e}\bra{e}+\ket{f}\bra{f}$. Hence using
$\hat{\sigma}_{+}\hat{\sigma}_{-}=\frac{1}{2}(\hat{\sigma}_{3}+I)$, the
effective Hamiltonian arrives at

$\displaystyle H_{\mathrm{eff}}$
$\displaystyle=h\chi[\hat{\sigma}_{+}\hat{\sigma}_{-}+b^{\dagger}b\hat{\sigma}_{3}]$
$\displaystyle=h\chi[\hat{\sigma}_{+}\hat{\sigma}_{-}+D^{\dagger}(\lambda)a^{\dagger}aD(\lambda)\hat{\sigma}_{3}]$
$\displaystyle=h\chi[\hat{\sigma}_{+}\hat{\sigma}_{-}+(a^{\dagger}a+\lambda
a^{\dagger}+\lambda^{*}a+|\lambda|^{2})\hat{\sigma}_{3}]$ (3)

where $\chi=\frac{g^{2}}{\Delta}$. In the limit $\lambda\rightarrow 0$ (in
case of no external driving field), the usual dispersive interaction
Hamiltonian
$H_{\mathrm{eff}}=h\chi[\hat{\sigma}_{+}\hat{\sigma}_{-}+a^{\dagger}a\hat{\sigma}_{3}]$
can be recovered. But with $\lambda\neq 0$, the interaction is no longer
purely dispersive as it contains terms of the form that creates or destroys
photons in the cavity conditional on the state of the atom. If the atom is
prepared in the far off-resonance state $\ket{g}$, the cavity field remains
unaffected because there is no interaction between the atom and the field. But
when the atom is prepared in either the excited state $\ket{e}$ or the
intermediate state $\ket{f}$, and if the cavity field is initially in a vacuum
state $\ket{0}$, the external classical driving field can generate a coherent
state of the quantized field.

Assuming that the atom is initially prepared in state $\ket{f}$ and is
injected through a cavity in the vacuum state $\ket{0}$, the atom-field system
inside the cavity evolves as

$\displaystyle\ket{\psi_{f}(t)}$
$\displaystyle=\exp[-iH_{\mathrm{eff}}t/\hbar]\ket{0}\ket{f}$
$\displaystyle=\exp[i\chi t(a^{\dagger}a+\lambda
a^{\dagger}+\lambda^{*}a+|\lambda|^{2})]\ket{0}\ket{f}$
$\displaystyle=\exp[i|\lambda|^{2}\sin(\chi t)]\ket{-\lambda(1-e^{i\chi
t})}\ket{f}$

where $\ket{-\lambda(1-e^{i\chi t})}$ is a coherent state of the cavity field.
Again if the atom is initially in the state $\ket{e}$, the state vector while
the atom is inside the cavity becomes

$\displaystyle\ket{\psi_{e}(t)}$
$\displaystyle=\exp[-iH_{\mathrm{eff}}t/\hbar]\ket{0}\ket{e}]$
$\displaystyle=\exp[-i\chi t-i\chi t(a^{\dagger}a+\lambda
a^{\dagger}+\lambda^{*}a+|\lambda|^{2})]\ket{0}\ket{e}$
$\displaystyle=e^{-i\chi t}\exp[-i|\lambda|^{2}\sin(\chi
t)]\ket{-\lambda(1-e^{-i\chi t})}\ket{e}$

where $\ket{-\lambda(1-e^{-i\chi t})}$ is a coherent state of the cavity
field. If the atom is prepared in the general superposition state
$\sin\theta\ket{e}+e^{i\phi}\cos\theta\ket{f}$ and the cavity field in the
vacuum state, then at time $t\geq 0$, the following entangled state is
obtained

$\ket{\psi(t)}=\sin(\theta)\ket{\psi_{e}(t)}\ket{e}+e^{i\phi}\cos(\theta)\ket{\psi_{f}(t)}\ket{f}$

For the simple most case $\theta={\pi}/{4}$ and $\phi=0$, the state vector
becomes

$\displaystyle\ket{\psi(t)}=\frac{1}{\sqrt{2}}\left(\ket{\psi_{e}(t)}\ket{e}+\ket{\psi_{f}(t)}\ket{f}\right)$
(4)

The state vector for the cavity field after the atom-field interaction is
obtained by tracing out the atom part from $\ket{\psi(t)}\bra{\psi(t)}$ as
following:

$Tr_{\mathrm{atom}}(\ket{\psi}\bra{\psi})=\ket{\psi_{\mathrm{field}}}\bra{\psi_{\mathrm{field}}}$

$\ket{\psi_{\mathrm{field}}}$ is the state of interest in rest of the article.

##  3 Generalised expectation

The generalised expectation value with respect to the cavity field is obtained
as

$\displaystyle\bra{\psi_{\mathrm{field}}}a^{{\dagger}p}a^{q}\ket{\psi_{\mathrm{field}}}=\frac{1}{2}\sum_{n=0}^{\infty}\frac{\lambda^{n}\lambda^{*(p+n-q)}}{(n-q)!}$
$\displaystyle\times\bigg{[}\exp{-2|\lambda|^{2}(1-\cos(\chi t)}(1-e^{(i\chi
t)})^{p+n-q}(1-e^{(-i\chi t)})^{n}$ $\displaystyle+\exp(i\chi
t)\exp{-2|\lambda|^{2}(1-e^{(i\chi t)})}(1-e^{(i\chi t)})^{p+2n-q}$
$\displaystyle+\exp(-i\chi t)\exp{-2|\lambda|^{2}(1-e^{(-i\chi
t)})}(1-e^{(-i\chi t)})^{p+2n-q}$
$\displaystyle+\exp{-2|\lambda|^{2}(1-\cos(\chi t)}(1-e^{(i\chi
t)})^{n}(1-e^{(-i\chi t)})^{p+n-q}\bigg{]}$ (5)

##  4 Nonclassical Properties

In this section, we derive some criteria to witness the nonclassicality of the
considered quantum state.

###  4.1 Photon number distribution

Photon number distribution is the probability distribution for finding $l$
photons in a given cavity field state and it can be obtained as the
expectation of the density field in Fock state basis as

$\displaystyle P(l)$ $\displaystyle=\bra{l}\rho\ket{l}$
$\displaystyle=\frac{1}{2}\sum_{n=0}^{\infty}\frac{\lambda^{n}\lambda^{*m}}{\sqrt{n!m!}}$
$\displaystyle\times\bigg{[}\exp{-2|\lambda|^{2}(1-\cos(\chi t)}(1-e^{(-i\chi
t)})^{n}(1-e^{(i\chi t)})^{m}$ $\displaystyle+\exp{-2|\lambda|^{2}(1-\cos(\chi
t)}(1-e^{(i\chi t)})^{n}(1-e^{(-i\chi t)})^{m}\bigg{]}$
$\displaystyle\times\bra{l}\ket{n}\bra{m}\ket{l}$

For $m=n=l$, the photon number distribution for the cavity field is obtained
as

$\displaystyle P(l)$
$\displaystyle=\frac{|\lambda|^{n}}{n!}\bigg{[}\exp{-|\lambda|^{2}(2-2\cos(\chi
t)}(2-2\cos(\chi t))^{l}\bigg{]}$ (6)

(6) provides the probability of observing $l$ number of photons in a given
cavity field state. The simplified form of the photon number distribution is
illustrated in Fig. 2. It is clear that $P(l)$ behaves in a wave-like manner
as a function of $\lambda$ and $\chi t$. Initially $P(l)$ increases with
$\lambda$ and then decreases. $P(l)$ attains a maximum value of 0.5 for
$\lambda$, and 1.08 for $\chi t$. When plotting $P(l)$ for different photon
numbers, the highest value of $P(l)$ is 0.83 for $l=2$.

Figure 2: Variation of photon number distribution $P(l)$ as a function of (a)
$\lambda$ with $\chi t=2,l=2$, (b) $\chi t$ with $\lambda=2,l=2$, (c) $l$ with
$\lambda=1$ and $\chi t=4$.

Here the shape of the photon number distribution provides insights into
nonclassical behaviour of the considered quantum state as a consequence that
nonclassical states may have unique features in their photon number
distributions, such as multiple peaks or oscillatory behaviour. Experimental
techniques and tools, such as photon-number-resolving detectors and homodyne
or heterodyne detectors that measure photon number distributions with high
precision are used in various experiments in quantum optics and quantum
information science to verify the nonclassical nature of the quantum states.

###  4.2 Mandel’s $Q_{M}$ parameter

Next to determine the photon statistics of a single-mode radiation field, we
consider the Mandel’s $Q_{M}$ parameter defined by [23]

$Q_{M}=\frac{\langle a^{{\dagger}^{2}}a^{2}\rangle}{\langle
a^{\dagger}a\rangle}-\langle a^{\dagger}a\rangle$ (7)

The parameter $Q_{M}$ is used to quantify the deviation of the variance of the
photon number distribution for a given state from the Poissonian distribution
associated with a coherent state. When $Q_{M}=0$, it indicates a Poissonian
distribution, whereas for $-1\leq Q_{M}<0~{}(Q_{M}>0)$, the field follows sub-
(super-) Poissonian photon statistics. However, it is important to note that
the negativity of $Q_{M}$ is not a necessary criterion to differentiate
quantum states into classical and nonclassical regimes. It is merely a
sufficient condition. There are cases where a state can exhibit nonclassical
behaviour even if $Q_{M}$ is positive [24]. The expectation values can be
found as following:

$\displaystyle\langle a^{{\dagger}^{2}}a^{2}\rangle$ $\displaystyle=$
$\displaystyle\langle\psi_{\mathrm{field}}|a^{{\dagger}^{2}}a^{2}|\psi_{\mathrm{field}}\rangle$
$\displaystyle\langle a^{\dagger}a\rangle$ $\displaystyle=$
$\displaystyle\langle\psi_{\mathrm{field}}|a^{\dagger}a|\psi_{\mathrm{field}}\rangle$

Substituting these in (7), the expression for Mandel’s $Q_{M}$ parameter for
the given cavity field is obtained and plotted as a function of $\lambda$ and
$\chi t$ in Fig. 3. Here we can see that $Q_{M}$ exhibits wave nature with
varying amplitudes. The negativity of $Q_{M}$ ascertains nonclassical nature
of the considered cavity field state. Also the nonclassicality decreases while
$\lambda$ increases. The state portrays sub-Poissonian light $(Q_{M}<0)$ for
$1.8<\lambda<3.5$ and for specific values of $\chi t$ ranging from 0.70 to
11.7. Mandel’s $Q_{M}$ parameter can be used in a variety of applications,
including the characterization of nonclassical states of light, the
measurement of photon number correlations in quantum states, and the
implementation of quantum communication protocols. It is a useful tool for
quantifying the degree of photon number correlations in a quantum state, and
has played an important role in the development of quantum optics and quantum
information science.

Figure 3: Variation of Mandel’s $Q_{M}$ parameter as a function of (a)
$\lambda$ with $\chi t=0.2$, (b) $\chi t$ with $\lambda=0.35$.

###  4.3 Squeezing properties

To analyze the quantum fluctuations of the field quadratures, we consider two
Hermitian operators which are combinations of photon creation and annihilation
operators as following:

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

with the commutation relation $\left[\hat{x},\hat{p}\right]=i/2.$ They obey
the Heisenberg uncertainty principle of the form
$\langle(\Delta\hat{x})^{2}\rangle\langle(\Delta\hat{p})^{2}\rangle\geq 1/16$,
and thus the quadrature squeezing occurs whenever
$\langle(\Delta\hat{x})^{2}\rangle<\frac{1}{4}$ or
$\langle(\Delta\hat{p})^{2}\rangle<\frac{1}{4}$. It is convenient to introduce
the squeezing parameters as [25]

$\displaystyle S_{x}$ $\displaystyle=2\langle a^{\dagger}a\rangle+\langle
a^{2}\rangle+\langle a^{{\dagger}2}\rangle-{\langle a\rangle}^{2}-{\langle
a^{\dagger}\rangle}^{2}-2\langle a\rangle\langle a^{\dagger}\rangle$
$\displaystyle S_{p}$ $\displaystyle=2\langle a^{\dagger}a\rangle-\langle
a^{2}\rangle-\langle a^{{\dagger}2}\rangle+{\langle a\rangle}^{2}+{\langle
a^{\dagger}\rangle}^{2}-2\langle a\rangle\langle a^{\dagger}\rangle$

Squeezing occurs in the $x$ or $p$ quadrature when $-1<S_{x}<0$ or
$-1<S_{p}<0$, respectively. The negativity of squeezing serves as a sufficient
condition rather than a necessary one. The expectations can be computed using
(3). We have plotted the squeezing values $S=(S_{x},S_{p})$ as a function of
$\lambda$ and $\chi t$ in Fig. 4. Here $S_{x}$ becomes negative as a function
of $\lambda$ as well as $\chi t$, indicating the nonclassical nature of the
given cavity field state.

Figure 4: Variation of squeezing parameters $S=(S_{x},S_{p})$ as a function of
(a) $\lambda$ with $\chi t=1$, (b) $\chi t$ with $\lambda=1$.

Squeezed states of light has the potential to enhance the capabilities of
quantum devices. They are crucial in the generation of entangled photon pairs,
which are essential for various quantum technologies, including quantum
teleportation and quantum cryptography. This light is proposed to improve the
sensitivity of gravitational wave detectors in today’s most referenced LIGO
(Laser Interferometer Gravitational-Wave Observatory) experiments. It can
enhance the precision of interferometric measurements of space-time
distortions caused by gravitational waves.

###  4.4 Wigner distribution

The nonclassical characteristic of a quantum state can be detected by
examining its phase-space distribution, namely the Wigner function. This
function establishes a connection between the density operator and a
distribution in phase space. It contains comprehensive information about the
state of a particular physical system [26]. This distribution is often
referred to as a quasiprobability distribution because it has the ability to
take on negative values. These negative values serve as indicators of the
system’s departure from classical behavior.

Given a quantum state $\hat{\rho}$, the Wigner function of the system is
defined using the coherent state basis as follows [13, 27, 28]

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

where $|\gamma\rangle=\exp(-|\gamma|^{2}/2+\gamma\hat{a}^{\dagger})|0\rangle$
is a coherent state. By using the relation [29]

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

the Wigner function can be expressed in a series form as follows [30],

$\displaystyle
W(\beta,\beta^{*})=\frac{2}{\pi}\sum_{k=0}^{\infty}(-1)^{k}\langle\beta,k|\hat{\rho}|\beta,k\rangle,$
(8)

where $|\beta,k\rangle$ is the standard displaced number state. The presence
of negativity in the Wigner function indicates that the associated state is
nonclassical [31]. But observing positive values for the entire Wigner
function cannot lead to the conclusion that the state is classical. For
example, squeezed state owning a Gaussian Wigner function that is positive
everywhere, is a widely recognized nonclassical state. In the case of a
nonclassical state, the negativity of the Wigner function is a necessary
condition. Therefore, a state exhibiting a negative region in its phase-space
distribution is inherently nonclassical. The displaced number state
$|\beta,k\rangle$ can be alternatively represented in the number state basis
as follows,

$\displaystyle|\beta,k\rangle$ $\displaystyle=$ $\displaystyle
D(\beta)|k\rangle$ (9) $\displaystyle=$ $\displaystyle
e^{-\frac{|\beta|^{2}}{2}}\sum_{l=0}^{k}\frac{\beta^{*{l}}}{l!}\sqrt{\frac{k!}{(k-l)!}}$
$\displaystyle\times\sum_{p=0}^{\infty}\frac{\beta^{p}}{p!}\sqrt{\frac{(k-l+p)!}{(k-l)!}}|k-l+p\rangle$

Thus

$\displaystyle\langle\beta,k|n\rangle=e^{-\frac{|\beta|^{2}}{2}}\sum_{l=0}^{k}\frac{(-\beta)^{l}}{l!}\sqrt{\frac{k!}{(k-l)!}}$
(10)
$\displaystyle\times\sum_{p=0}^{\infty}\frac{\beta^{*{p}}}{p!}\sqrt{\frac{(k-l+p)!}{(k-l)!}}\langle
k-l+p|n\rangle$ $\displaystyle=$ $\displaystyle
e^{-\frac{|\beta|^{2}}{2}}\sum_{l=0}^{k}\frac{(-\beta)^{l}}{l!}\sqrt{\frac{k!}{n!}}\frac{(\beta^{*})^{n-k+l}n!}{(k-l)!(n-k+l)!}$
$\displaystyle=$ $\displaystyle
e^{-\frac{|\beta|^{2}}{2}}\sqrt{\frac{k!}{n!}}(\beta^{*})^{n-k}\sum_{l=0}^{k}\frac{(-|\beta|^{2})^{l}n!}{(k-l)!l!(n-k+l)!}$
$\displaystyle=$ $\displaystyle
e^{-\frac{|\beta|^{2}}{2}}\sqrt{\frac{k!}{n!}}(\beta^{*})^{n-k}L_{k}^{(n-k)}\left(|\beta|^{2}\right),$

where $L_{l}^{(k)}(x)=\sum_{n=0}^{l}\frac{(-x)^{n}(l+k)!}{(l-n)!n!(l+n)!}$ is
the associated Laguerre polynomial [32]. Now

$\displaystyle\langle\beta,k|\hat{\rho}|\beta,k\rangle=\frac{1}{2}\sum_{m,n=0}^{\infty}\frac{\lambda^{n}\lambda^{*m}}{\sqrt{m!n!}}$
$\displaystyle\times\bigg{[}\exp{-2|\lambda|^{2}(1-\cos(\chi t)}(1-e^{(-i\chi
t)})^{n}(1-e^{(i\chi t)})^{m}$ $\displaystyle+\exp{-2|\lambda|^{2}(-\cos(\chi
t)}(1-e^{(i\chi t)})^{n}(1-e^{(-i\chi t)})^{m}\bigg{]}$
$\displaystyle\times\langle\beta,k\ket{n}\bra{m}\beta,k\rangle$ (11)

Substituting (4.4) into (8), we get the final expression of the Wigner
function. The surface plot of the Wigner function $W$ with respect to
$\lambda$ and $\beta$ is given in Fig. 5. It is evident that $W$ exhibits
negative regions as well as peaks, indicating the nonclassical and non-
Gaussian characteristics of the cavity field. In addition, the Wigner function
demonstrates wavy nature and goes negative with respect to the parameter $\chi
t$. It is important to note that the identification of negative values within
the Wigner function is a clear indicator of non-Gaussian and nonclassical
characters. These type of states hold great significance in the field of
quantum information applications, mainly due to their resistance to effective
simulation by classical computers [33, 34].

Figure 5: Variation of Wigner function $W$ with respect to (a) $\lambda$ with
$\chi t=2$ and $\beta=1$, (b) $\beta$ with $\chi t=1$ and $\lambda=1$, (c)
$\chi t$ with $\beta=\lambda=1$.

###  4.5 $Q_{f}$ Function

The uncertainty principle imposes limitation on directly describing a quantum
mechanical system in phase-space. Consequently, quasiprobability distributions
have been introduced, which are highly valuable in quantum mechanics as they
establish a correspondence between the quantum and classical realms. They also
facilitate the calculation of quantum mechanical averages in a manner similar
to classical phase-space averages [35]. One notable quasiprobability
distribution is the $Q_{f}$ function, where the presence of zeros serves as an
indicator of nonclassical behavior [36]. The $Q_{f}$ function can be computed
using the following expression

$Q_{f}=\bra{\alpha}\hat{\rho}\ket{\alpha}$ (12)

where $\ket{\alpha}$ is the usual coherent state. Substituting
$\ket{\alpha}=e^{-\frac{|\alpha|^{2}}{2}}\sum_{p=0}^{\infty}\frac{\alpha^{p}}{\sqrt{p!}}\ket{p}$
in (12), we obtain

$\displaystyle Q_{f}$
$\displaystyle=\frac{1}{2}\sum_{m,n=0}^{\infty}\frac{\lambda^{n}\lambda^{*m}}{\sqrt{m!n!}}$
$\displaystyle\times\bigg{[}\exp{-|\lambda|^{2}(2-2\cos(\chi t)}(1-e^{(-i\chi
t)})^{n}(1-e^{(i\chi t)})^{m}$ $\displaystyle+\exp{-|\lambda|^{2}(2-2\cos(\chi
t)}(1-e^{(i\chi t)})^{n}(1-e^{(-i\chi t)})^{m}\bigg{]}$
$\displaystyle\times\bra{\alpha}\ket{n}\bra{m}\ket{\alpha}$

where

$\displaystyle\bra{\alpha}\ket{n}\bra{m}\ket{\alpha}$
$\displaystyle=\sum_{p=0}^{\infty}\frac{\alpha^{*p}}{\sqrt{p!}}e^{-\frac{|\alpha|^{2}}{2}}\sum_{q=0}^{\infty}\frac{\alpha^{q}}{\sqrt{q!}}e^{-\frac{|\alpha|^{2}}{2}}\bra{p}\ket{n}\bra{m}\ket{q}$
$\displaystyle=e^{-{|\alpha|^{2}}}\frac{\alpha^{*n}}{\sqrt{n!}}\frac{\alpha^{m}}{\sqrt{m!}}$

$Q_{f}$ is a complex-valued function that can be plotted in the complex plane,
and it provides a complete description of the quantum state of a system. The
surface plot of $Q_{f}$ as a function of $\lambda$ and $\alpha$ is shown in
Fig. 6. It is clearly seen that $Q_{f}$ almost overlaps with zero for certain
values of the parameters, which indicates the nonclassical behaviour of the
cavity field state. This function has a number of important applications in
quantum optics, quantum information processing, and quantum computing. For
example, it can be used to calculate the probability of detecting a photon in
a particular mode of a quantum field, or to describe the entanglement of two
or more quantum systems.

Figure 6: Variation of $Q_{f}$ function with respect to (a) $\lambda$ with
$\chi t=1$ and $\alpha=1$, (b) $\alpha$ with $\chi t=1$ and $\lambda=1$, (c)
$\chi t$ with $\alpha=1$ and $\lambda=1$.

###  4.6 Second-order correlation

The second-order correlation function $g^{2}(0)$ at zero time delay for a
single-mode radiation field is defined as [37]

$g^{2}(0)=\frac{\bra{\psi_{\mathrm{field}}}a^{{\dagger}2}a^{2}\ket{\psi_{\mathrm{field}}}}{\bra{\psi_{\mathrm{field}}}a^{{\dagger}}a\ket{\psi_{\mathrm{field}}}^{2}}$
(13)

If for a field state $g^{2}(0)>1$, the field is said to be bunched with super-
Poissonian field statistics, whereas if $g^{2}(0)<1$, the field is said to be
antibunched with sub-Poissonian field statistics.

Figure 7: Variation of second-order correlation $g^{2}(0)$ as a function of
(a) $\lambda$ with $\chi t=1$, (b) $\chi t$ with $\lambda=2$.

We have seen from Fig. 7 that $g^{2}(0)$ is less then one for some particular
values of $\lambda$ as well as $\chi t$ which imparts that the cavity field
state is antibunched.

###  4.7 Lower order antibunching

The theory of majorization by [38] and [39] gives us the expression of
antibunching as

$d_{1}=\langle a^{{\dagger}^{2}}a^{2}\rangle-{\langle
a^{\dagger}a\rangle}^{2}$

For a quantum state to be nonclassical, $d_{1}$ should be negative. The
negativity of $d_{1}$ implies that the probability of independent photons is
more than the clustered photons. Here $d_{1}$ can simply be calculated by
substituting the expectations $\langle a^{{\dagger}^{2}}a^{2}\rangle$ and
$\langle a^{\dagger}a\rangle$ from (3). Fig. 8 clearly shows that $d_{1}$
becomes negative with respect to $\lambda$ and $\chi t$, indicating that the
cavity field state is antibunched. We have observed that the cavity QED field
exhibits antibunching behaviour for a short period of time that means the
system presents signature of temporary antibunching. In many experiments, it
is important to stabilize the antibunching effect over time to ensure that the
nonclassical properties of the light source are maintained. Temporary
antibunching can also be used to generate entangled photon pairs in a random
sequence which is necessary for the success of teleportation protocol. Thus
temporary antibunching present for a limited time period is expected to play
an important role in the study of cavity QED system.

Figure 8: The variation of the lower order antibunching $d_{1}$ as a function
of (a) $\lambda$ with $\chi t=1$, (b) $\chi t$ with $\lambda=2$.

##  5 Conclusion

In this paper, we have investigated the generation of a state by the
interaction of an atom with both a quantized cavity field and an external
classical driving field. In this study, we have considered an atom with three
levels $\ket{e}$, $\ket{f}$ and $\ket{g}$ passes through a cavity and
interacts dispersively with the cavity field mode. At the same time it
interacts with an external classical field tuned into resonant with the cavity
field. We have further assumed that the transition
$\ket{f}\leftrightarrow\ket{g}$ is significantly far from being in resonance
and the transition frequency for $\ket{e}\leftrightarrow\ket{f}$ is in close
proximity to resonance with the frequency of a single-mode cavity field. A
coherent state of quantized field is generated, when the atom prepared in
either states $\ket{e}$ or $\ket{f}$, and the cavity field is initially in
vacuum state $\ket{0}.$ We have considered that the atom is initially in the
superposition of these two states and the cavity is in vacuum state. After the
atom-cavity interaction, the cavity field state is obtained by tracing out the
atom part from the generalized state vector. Different statistical properties
like photon number distribution, Wigner function, Mandel’s $Q_{M}$ parameter,
squeezing properties $S_{x}$ and $S_{p}$, $Q_{f}$ function, second-order
correlation $g^{2}(0)$ etc. are investigated. It is observed that Wigner
function $W$, Mandel’s $Q_{M}$, squeezing ($S_{x}$, $S_{p}$) are negative for
some specific parametric values that admits the nonclassical nature of the
cavity field state. Also the zeroes of $Q_{f}$ function suggests that the
behaviour of the cavity state is nonclassical. The second-order correlation
function $g^{2}(0)$ with respect to $\lambda$ is less than 1 that also imparts
the nonclassical behaviour of the cavity field.

In particular, we have investigated photon number distribution, quadrature
squeezing, Mandel’s $Q_{f}$ etc. and provided analytical expressions for these
in dependence on different state parameters. Furthermore, the Wigner function
of the cavity-mode is calculated and we could infer the state’s nonclassical
character from its negativities. For the preparation of states with different
nonclassical features optimal parameter regions are identified. Summing up,
the presented approach can easily be implemented in current cavity QED
experiments and provides a versatile method for the engineering of
nonclassical states of cavity fields. Dispersive interactions can be used to
create quantum networks where information is stored and processed using atoms
trapped in optical cavities. Future research may focus on scaling up these
networks and improving their efficiency for secure quantum communication.

ACKNOWLEDGEMENT

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

Data Availability Statement

Data generated or analyzed during this study are provided in full within the
article.

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