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# Majorization effect on entropic functional: An application to a $\vee$-type
three-level atom-field interacting system

Debraj Nath  Department of Mathematics, Vivekananda College, Kolkata-700063,
WB, India<EMAIL_ADDRESS>
###### Abstract

Majorization effect on some entropic functional, such as von-Neumann, Shannon,
atomic Wehrl and Rényi entropies are investigated of a $\vee$-type three-level
atom, which interacts with a coherent field in a resonant cavity. Fidelity,
purity and linear entropy are investigated for two quantum systems, which
contain photon number distributions and Husimi $Q$ functions. A relation
between majorization and localization properties of continuous density
functional is established. The results are compared and verified for
continuous and discrete distributions of an atom-field interacting system.

Key words: von-Neumann entropy; Shannon entropy; atomic Wehrl entropy; Rényi
entropy; majorization effect; atom-field interacting Hamiltonian

##  I Introduction

The interaction between a two-level atom and a single-mode quantized
electromagnetic field in a lossless cavity is known as Jaynes-Cummings model
(JCM) [1]. It is exactly solvable under the rotating-wave approximation. It
has many applications in quantum optics, such as Rabi oscillations, atomic
inversion, sub-Poissonian statistics, squeezing and phase properties of
radiation field. Moreover, the interaction between three-level atom(s) and
field(s) have an important role in quantum optics, in the context of collapse-
revival phenomena, squeezing phenomena, entanglement, quantum information,
quantum computation and so on. A large number of theoretical and numerical
studies have been made on the subject of quantum information entropy of a
three-level atom of ladder, vee and lambda configurations, which interacts
with single-mode and multi-mode quantized field [2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In this connection, we have
noted that, different approaches have been used to determine the population of
the excited states of a three-level atom [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18], which is a measure of the two-photon absorption
probability [23]. These are associated with a two-level atom, driven by a
single-mode field and they are extension of Jaynes-Cummings model [1]. If an
atom collapses to the ground state with a probability $p$, then the
probability $p$ is called fidelity [24, 25]. In this paper, we have consider
the maximum value of $p$. A quantum state with density operator $\rho$, is
said to be pure, if $\rho^{2}=\rho$. The measure of purity $Tr[\rho^{2}]$ of a
moving three-level atom is investigated in ref. [6]. From the definition of
purity, one can define the linear entropy $1-Tr[\rho^{2}]$ as a complement
measure of purity of a state with normalized density functions $\rho$ [26,
27], which is obtained from the von-Neumann entropy in a near pure state by
Taylor series expansion. The linear entropy of tripartite two-level atom [28],
three-level atom [5] and four-level atom [29] were investigated earlier in
different context. Many authors focus their attention for measure the von-
Neumann entropy [30] of three-level atom(s) field interacting quantum system
[2, 3, 4, 5, 6, 7, 8, 19, 20, 21, 22]. It is an important measurement in the
field of quantum information and computation for mixed states. The von-Neumann
entropy of an atom and a field are investigated in a dissipative cavity in
ref. [4] and in the presence of non-linearity of a three-level atom of
different configurations, in ref. [14]. Another important entropy is Shannon
entropy [31] which is used for the measure of temporal evolution of a field in
an atom-field interacting system (see refs. [32, 33]). The Shannon entropy
investigated for three-level atom system (see refs. [4, 7]).

In this paper, a vee-type three-level atom have been considered, which
interacts with a field in a resonant cavity [9, 10, 11, 12, 13, 14, 15, 16,
17]. We would like to study the atomic coherent state [34] of a three-level
atom, whose evolution started with a coherent super position of its excited
states and it is interacting with a coherent field. The Husimi distribution is
the expectation of a quantum state and the expectation is defined by a basis
of coherent states [35], which are defined on a complex plane. The Boltzmann-
Gibbs entropy of the Husimi distribution is introduced by Wehrl and it is
called the atomic Wehrl entropy. It is minimum for coherent state and is
conjectured by Wehrl [36]. After that, it is proved by Lieb [37]. It is a
measure of coherence of a given state. The atomic Wehrl entropy of three-level
atom(s) is investigated in [4]. The Rényi entropy is a generalization of
Shannon entropy of arbitrary order [38]. It has many applications in the
fields of statistical mechanics, quantum information and computation.

Generally, majorization properties are used to compare the probability
distributions. Moreover, majorization techniques are used in quantum
information specially in quantum entanglement [42]. If two distributions can
compare, then their corresponding entropic uncertainty measurements can be
compared. In addition, the entropic uncertainty relations for any two
observables can be established by the majorization techniques [39, 40, 41, 42,
43]. The lower limit of the Rényi entropic sum for bipartite Hilbert space is
established in ref. [42]. In most of the published papers majorization
technique has been shown for discrete distributions.

In this paper, the majorization and the cumulative distribution of discrete
and continuous density functions will be described. Under this model, there
are two set of interacting quantum systems (i) first atom-field and (ii)
second atom-field. Both systems contain photon number distributions and quasi
probability distribution. The purity, linear entropy and the von-Neumann
entropy of reduced density matrices of atoms will be defined. Then, the
Shannon and Rényi entropies will be defined of photon number distributions
whereas, the atomic Wehrl and the Rényi entropies will be defined of quasi
probability distribution (Husimi $Q$-function). To understand the entropic
nature, the properties of majorization and cumulative distributions will be
investigated. Moreover, effective domain of continuous probability density
function will be defined. Then characterize its localization property with
respect to the effective domain [44, 45]. Next, the connection between
majorization and localization property of continuous density functions will be
investigated. The motivation of this paper is to find the majorization effect
on some entropic functional. As an example, we have considered an atom-field
interacting system. In this regard, the possibilities for reducing values of
information theoretic measures, such as purity, linear entropy [26, 27, 29],
von-Neumann entropy [30], Shannon entropy [31], atomic Wehrl entropy [36, 46,
47, 48] and Rényi entropy [38] will be examined by adding another identical
atom into the cavity, when the first atom collapses to the ground state, after
some time. The time will be taken, in such a way that, the atom collapses to
the ground state with maximum fidelity [24] and then the atom-field
interacting system will reduce to a single-mode field and thereafter the
second atom will be passed through the cavity. The detail calculations of
information measures depend on the initial conditions of the atom and the
field.

The article is organized as follows. In Section II, a theoretical model of an
atom-field interaction in a cavity of quantized electromagnetic field will be
defined. Then, exact solutions of this model for two quantum systems, such as
(i) first atom-field interaction and (ii) second atom-field interaction will
be calculated. In Section III, some definitions of entropies, such as von-
Neumann, Shannon, atomic Wehrl and Rényi will be reviewed. In addition, a
relation between majorization and localization property of density functions
will be defined. In Section IV, a application of entropic functional to this
model will be demonstrated. Numerical results of entropic functional will be
compared and verified of a vee type three-level atom in two quantum systems.
Finally, in Section V, some conclusions will be exhibited.

##  II Theoretical model

As an application, we have considered a three-level vee type atom [9, 10, 11,
12, 13, 14, 15, 16, 17] in a radiation field as shown in Fig. 1. The allowed
photon transitions are $\displaystyle|e\rangle\leftrightarrow|g\rangle$ and
$\displaystyle|i\rangle\leftrightarrow|g\rangle$, where $|e\rangle$,
$|i\rangle$, $|g\rangle$ are eigenstates of a three-level atom with
eigenenergies $\omega_{e}$, $\omega_{i}$, $\omega_{g}$. For, $\vee$-type three
level atom, there is no direct photon transition between levels $|i\rangle$
and $|e\rangle$. But it can be allowed for $\Lambda$ and $\varXi$ and $\Delta$
types three-level atom. For the sake of simplicity, a linear polarization
basis is taken and the corresponding polarization unit vector is taken as
real. Then the total Hamiltonian of an interacting system of a quantized
electromagnetic field and a vee type three-level atom is,

$\begin{array}[]{ll}H&=\sum\limits_{k}\hbar\left(\Omega_{1k}a_{k}^{{\dagger}}a_{k}+\Omega_{2k}b_{k}^{{\dagger}}b_{k}\right)+\sum\limits_{\mu=e,i,g}\omega_{\mu}|\mu\rangle\langle\mu|\\\
&+\hbar\sum\limits_{k}\left(\sum\limits_{l,m=e,g}g_{1k}^{lm}|l\rangle\langle
m|\left(a_{k}+a^{{\dagger}}_{k}\right)\right.\\\
&\left.+\sum\limits_{l,m=i,g}g_{2k}^{lm}|l\rangle\langle
m|\left(b_{k}+b^{{\dagger}}_{k}\right)\right)\\\ \end{array}$ (1)

where $a^{{\dagger}}$ and $b^{{\dagger}}$ (or $a$ and $b$) are creation (or
annihilation) operators of the quantized driving fields with frequencies
$\Omega_{1k}$ and $\Omega_{2k}$ respectively. The terms $g_{1k}^{lm}$, and
$g_{2k}^{lm}$ are the coupling constants. For the resonant cavity
$|e\rangle\langle g|a_{k}$, $|g\rangle\langle e|a_{k}^{{\dagger}}$,
$|i\rangle\langle g|b_{k}$ and $|g\rangle\langle i|b_{k}^{{\dagger}}$ are
energy conserving operators. The operators $|e\rangle\langle
g|a_{k}^{{\dagger}}$ and $|i\rangle\langle g|b_{k}^{{\dagger}}$ gain the
energies $\omega_{e}-\omega_{g}+\hbar\Omega_{1k}$ and
$\omega_{i}-\omega_{g}+\hbar\Omega_{2k}$ respectively. Similarly, the
operators $|g\rangle\langle e|a_{k}$ and $|g\rangle\langle i|b_{k}$ loss the
energies $\omega_{e}-\omega_{g}+\hbar\Omega_{1k}$ and
$\omega_{i}-\omega_{g}+\hbar\Omega_{2k}$ respectively. Therefore, the non
conserving operators have been neglected. In the rotating-wave approximation,
the total Hamiltonian is written by

$\begin{array}[]{ll}H&=\sum_{k}\hbar\left(\Omega_{1k}a_{k}^{{\dagger}}a_{k}+\Omega_{2k}b_{k}^{{\dagger}}b_{k}\right)+\sum_{\mu=e,i,g}\omega_{\mu}|\mu\rangle\langle\mu|\\\
&+\hbar\sum_{k}g_{1k}^{eg}\left(|e\rangle\langle g|a_{k}+|g\rangle\langle
e|a^{{\dagger}}_{k}\right)\\\ &+\hbar\sum_{k}g_{2k}^{ig}\left(|i\rangle\langle
g|b_{k}+|g\rangle\langle i|b^{{\dagger}}_{k}\right).\end{array}$ (2)

Figure 1:  Schematic representation of a vee-type three-level atom. $\omega$
is carried frequency of the driving field and $\Delta_{1}$, $\Delta_{2}$ are
the detuning parameters.

Therefore, for a single-mode quantized field the total Hamiltonian reduces to

$\displaystyle H=H_{0}+H_{1},$ (3)

where

$\begin{array}[]{rcl}\displaystyle H_{0}&=&\omega
a^{{\dagger}}a+\omega_{g}|g\rangle\langle g|+\omega_{i}|i\rangle\langle
i|+\omega_{e}|e\rangle\langle e|,\\\ \displaystyle
H_{1}&=&g_{1}\left(~{}a|e\rangle\langle g|+a^{{\dagger}}|g\rangle\langle
e|~{}\right)+g_{2}\left(~{}a|i\rangle\langle g|+a^{{\dagger}}|g\rangle\langle
i|~{}\right),\end{array}$ (4)

$a^{{\dagger}}$ (or $a$) is the creation (or annihilation) operator of the
quantized field with frequency $\omega$ and $g_{1}$, $g_{2}$ are the
corresponding atom-field coupling constants. For the sake of simplicity, real
coupling constant is considered and $\hbar=1$. Then the interaction
Hamiltonian becomes

$\begin{array}[]{rcl}\mathcal{H}&=&\displaystyle
g_{1}e^{i\Delta_{1}t}a|e\rangle\langle
g|+g_{2}e^{i\Delta_{2}t}a|i\rangle\langle g|+H.C,\end{array}$ (5)

where $\displaystyle\Delta_{1}=\omega_{e}-\omega_{g}-\omega$ and
$\displaystyle\Delta_{2}=\omega_{i}-\omega_{g}-\omega$.

###  II.1 Analytical solution of first atom-field interacting system

In the interacting picture, the state vector of an atom-field system at time
$t$ can be described by

$\displaystyle|\psi_{1}(t)\rangle=\sum_{n=0}^{\infty}C^{1}_{e,n}(t)~{}|e,n\rangle+C^{1}_{i,n}(t)~{}|i,n\rangle+C^{1}_{g,n}(t)~{}|g,n\rangle.$
(6)

Using (6) into the Schrödinger equation

$\displaystyle i\frac{\partial|\psi_{1}(t)\rangle}{\partial
t}=\mathcal{H}|\psi_{1}(t)\rangle,$ (7)

one obtains

$\begin{array}[]{l}\displaystyle
i\dot{C}^{1}_{g,n+1}=g_{1}e^{-i\Delta_{1}t}\sqrt{n+1}~{}C^{1}_{e,n}+g_{2}e^{-i\Delta_{2}t}\sqrt{n+1}~{}C^{1}_{i,n},\\\
\displaystyle
i\dot{C}^{1}_{i,n}=g_{2}e^{i\Delta_{2}t}\sqrt{n+1}~{}C^{1}_{g,n+1},\\\
\displaystyle
i\dot{C}^{1}_{e,n}=g_{1}e^{i\Delta_{2}t}\sqrt{n+1}~{}C^{1}_{g,n+1}.\end{array}$
(8)

In this paper, $\displaystyle\Delta_{1}=\Delta_{2}=\Delta$ and the atom at
initial time is a coherent superposition
$\left(\displaystyle|\psi_{1A}(0)\rangle=\cos\frac{\varphi}{2}|e\rangle\right.$
$\left.+\sin\frac{\varphi}{2}e^{i\psi}|i\rangle\right)$ of two excited states
$|e\rangle$ and $|i\rangle$. Moreover, the field at initial time is the
superposition of the photon number states
$\left(\displaystyle|\psi_{1F}(0)\rangle=\sum_{n=0}^{\infty}C_{n}|n\rangle,~{}\sum_{n=0}^{\infty}|C_{n}|^{2}=1\right)$.
Then, the initial state vector of the atom field interacting system can be
written as,

$\displaystyle|\psi_{1}(0)\rangle=\displaystyle\sum_{n=0}^{\infty}C_{n}\left(\cos\frac{\varphi}{2}~{}|e,n\rangle+\sin\frac{\varphi}{2}e^{i\psi}~{}|i,n\rangle\right).$
(9)

Using the initial condition (9), one obtains the solution of (8) as

$\begin{array}[]{ll}\displaystyle
C^{1}_{e,n}(t)&=\frac{A_{1}g_{1}}{(g_{1}^{2}+g_{2}^{2})\sqrt{n+1}}\left\\{\mu+\left(\frac{i}{2}\Delta\sin\mu
t-\mu\cos\mu t\right)e^{\frac{i}{2}\Delta t}\right\\}\\\
&+\sin\frac{\varphi}{2}e^{i\psi}~{}C_{n},\\\ \displaystyle
C^{1}_{i,n}(t)&=\frac{A_{1}g_{2}}{(g_{1}^{2}+g_{2}^{2})\sqrt{n+1}}\left\\{\mu+\left(\frac{i}{2}\Delta\sin\mu
t-\mu\cos\mu t\right)e^{\frac{i}{2}\Delta t}\right\\}\\\
&+\cos\frac{\varphi}{2}~{}C_{n},\\\ \displaystyle
C^{1}_{g,n+1}(t)&=A_{1}(n)ie^{-\frac{i}{2}\Delta t}\sin\mu t,\end{array}$ (10)

where

$\begin{array}[]{l}\displaystyle\mu=\frac{1}{2}\sqrt{\Delta^{2}+4(g_{1}^{2}+g_{2}^{2})(n+1)},\\\
\end{array}$ (11)

and $A_{1}$ is a function of $n$ and it is written as

$A_{1}(n)=-\frac{1}{\mu}\left(g_{1}\sin\frac{\varphi}{2}e^{i\psi}+g_{2}\cos\frac{\varphi}{2}\right)\sqrt{n+1}~{}C_{n}$
(12)

As an application, in section 4.1 we will considered
$C_{n}=e^{-\frac{|\alpha|^{2}}{2}}\frac{\alpha^{n}}{\sqrt{n!}}$, where
$\alpha=\sqrt{\bar{n}}~{}e^{i\zeta}$. From Eqs. (10) and (11), we obtain
$C^{1}_{g,0}=0$. Therefore, $|g,0\rangle$ will not be present in Eq. (6) for
any other calculations. The coefficients $C^{1}_{e,n}(t)$, $C^{1}_{i,n}(t)$
and $C^{1}_{g,n}(t)$ in solution (6) satify the condition
$\sum\limits_{n=0}^{\infty}\left[|C^{1}_{e,n}(t)|^{2}+|C^{1}_{i,n}(t)|^{2}+|C^{1}_{g,n}(t)|^{2}\right]=1$
and $C^{1}_{e,n}(t),C^{1}_{i,n}(t),C^{1}_{g,n}(t)\rightarrow 0$ as
$n\rightarrow\infty$. In this paper, we have checked that
$|C^{1}_{e,n}(t)|,|C^{1}_{i,n}(t)|,|C^{1}_{g,n}(t)|<10^{-9}$ for $n>30$ for
the driving field with mean photon number is $\bar{n}=8$. In this paper, we
have consider the driving field is a coherent laser field.

###  II.2 Fidelity

Let us assume that, the first atom collapses to the ground state at time $t$
with fidelity $f_{1}(t)$, which is defined by [24]

$f_{1}(t)=\frac{\displaystyle\sum\limits_{n=0}^{\infty}|C^{1}_{g,n+1}(t)|^{2}}{\displaystyle\sum\limits_{n=0}^{\infty}\left(|C^{1}_{e,n}(t)|^{2}+|C^{1}_{i,n}(t)|^{2}+|C^{1}_{g,n+1}(t)|^{2}\right)}.$
(13)

It satisfies the relation $0\leq f_{1}(t)\leq 1$, for $t\geq 0$. The
normalized time is denoted by $\tau$ and it is defined by $\tau=gt$. Let
$f_{1}$ be maximum at $\displaystyle\tau_{1}=gt_{1}$ and the atom-field
interacting system becomes

$\displaystyle|\psi_{1}(t_{1})\rangle=\eta_{1}\sum_{n=0}^{\infty}C^{1}_{g,n+1}(t_{1})~{}|n+1\rangle,$
(14)

where
$\displaystyle\eta_{1}(t_{1})=\left(\sum_{n=0}^{\infty}|C^{1}_{g,n+1}(t_{1})|^{2}\right)^{-\frac{1}{2}}$,
is normalization constant and $\displaystyle C^{1}_{g,n+1}(t_{1})$ is obtained
from Eq. (10) by setting $t=t_{1}$.

###  II.3 Analytical solution of second atom-field interacting system

Now, another identical atom which passes through the cavity, when the first
atom leaves the cavity [49, 50]. Then, the entangled state of the second atom-
field interacting system, at time $t$, is written by,

$\displaystyle|\psi_{2}(t)\rangle=\sum_{n=0}^{\infty}C^{2}_{e,n}(t)~{}|e,n\rangle+C^{2}_{i,n}(t)~{}|i,n\rangle+C^{2}_{g,n}(t)~{}|g,n\rangle.$
(15)

Now, substituting (15) into the Schrödinger equation (7), we obtain

$\begin{array}[]{l}\displaystyle
i\dot{C}^{2}_{g,n+1}=g_{1}e^{-i\Delta_{1}t}\sqrt{n+1}~{}C^{2}_{e,n}+g_{2}e^{-i\Delta_{2}t}\sqrt{n+1}~{}C^{2}_{i,n},\\\
\\\ \displaystyle
i\dot{C}^{2}_{i,n}=g_{2}e^{i\Delta_{2}t}\sqrt{n+1}~{}C^{2}_{g,n+1},\\\ \\\
i\dot{C}^{2}_{e,n}=g_{1}e^{i\Delta_{2}t}\sqrt{n+1}~{}C^{2}_{g,n+1}.\end{array}$
(16)

In a similar manner,
$\displaystyle|\psi_{2A}(0)\rangle=\cos\frac{\varphi^{{}^{\prime}}}{2}|e\rangle+\sin\frac{\varphi^{{}^{\prime}}}{2}e^{i\psi^{{}^{\prime}}}|i\rangle$.
Then the state vector for the second atom-field interacting system at $t=0$,
can be written as,

$\displaystyle|\psi_{2}(0)\rangle=\eta_{1}\sum_{n=0}^{\infty}C^{1}_{g,n+1}(t_{1})\Big{(}\cos\frac{\varphi^{{}^{\prime}}}{2}|e,n+1\rangle+\sin\frac{\varphi^{{}^{\prime}}}{2}e^{i\psi^{{}^{\prime}}}|i,n+1\rangle\Big{)}.$
(17)

To avoid the tedious calculations, $\displaystyle\Delta_{1}=\Delta_{2}=\Delta$
and using the initial condition (17), one obtains the solution of (16) as,

$\begin{array}[]{ll}\displaystyle
C^{2}_{e,n}(t)&=\frac{A_{2}g_{1}}{(g_{1}^{2}+g_{2}^{2})\sqrt{n+1}}\left\\{\mu+\left(\frac{i}{2}\Delta\sin\mu
t-\mu\cos\mu t\right)e^{\frac{i}{2}\Delta t}\right\\}\\\
&+\sin\frac{\varphi^{{}^{\prime}}}{2}e^{i\psi^{{}^{\prime}}}\eta_{1}~{}C_{g,n}(t_{1}),\\\
\displaystyle
C^{2}_{i,n}(t)&=\frac{A_{2}g_{2}}{(g_{1}^{2}+g_{2}^{2})\sqrt{n+1}}\left\\{\mu+\left(\frac{i}{2}\Delta\sin\mu
t-\mu\cos\mu t\right)e^{\frac{i}{2}\Delta t}\right\\}\\\
&+\cos\frac{\varphi^{{}^{\prime}}}{2}\eta_{1}~{}C_{g,n}(t_{1}),\\\
C^{2}_{g,n+1}(t)&=A_{2}ie^{-\frac{i}{2}\Delta t}\sin\mu t,\end{array}$ (18)

where

$\begin{array}[]{l}\displaystyle\mu=\frac{1}{2}\sqrt{\Delta^{2}+4(g_{1}^{2}+g_{2}^{2})(n+1)},\\\
\displaystyle
A_{2}(n)=-\frac{1}{\mu}\left(g_{1}\sin\frac{\varphi^{{}^{\prime}}}{2}e^{i\psi^{{}^{\prime}}}+g_{2}\cos\frac{\varphi^{{}^{\prime}}}{2}\right)\sqrt{n+1}~{}\eta_{1}~{}C_{g,n}^{1}(t_{1}).\end{array}$
(19)

From Eqs. (18) and (19), we obtain
$C^{2}_{e,0}=C^{2}_{i,0}=C^{2}_{g,0}=C^{2}_{g,1}=0$. Therefore, $|e,0\rangle$,
$|i,0\rangle$, $|g,0\rangle$ and $|g,1\rangle=0$ will not be present in Eq.
(15) for next calculations. Using the definition of fidelity in (13),
similarly, one can find the fidelity $f_{2}(t)$ for which the second atom
collapses to the ground state by the relation

$f_{2}(t)=\frac{\displaystyle\sum\limits_{n=1}^{\infty}|C^{2}_{g,n+1}(t)|^{2}}{\displaystyle\sum\limits_{n=1}^{\infty}\left(|C^{2}_{e,n}(t)|^{2}+|C^{2}_{i,n}(t)|^{2}+|C^{2}_{g,n+1}(t)|^{2}\right)}.$
(20)

###  II.4 Total and reduced density matrices

The total density matrix of the $j$ system is defined by
$\rho=|\psi_{j}\rangle\langle\psi_{j}|$. Let $\rho_{A_{j}}$ and $\rho_{F_{j}}$
be the reduced density matrices of the atom and the field. Then one obtains,

$\rho_{A_{j}}=tr_{F_{j}}\left(|\psi_{j}\rangle\langle\psi_{j}|\right)=\left(\begin{array}[]{ccc}\rho_{A_{j}}^{ee}&\rho_{A_{j}}^{ei}&\rho_{A_{j}}^{eg}\\\
\rho_{A_{j}}^{ie}&\rho_{A_{j}}^{ii}&\rho_{A_{j}}^{ig}\\\
\rho_{A_{j}}^{ge}&\rho_{A_{j}}^{gi}&\rho_{A_{j}}^{gg}\end{array}\right),$ (21)

and

$\rho_{F_{j}}=tr_{A_{j}}\left(|\psi_{j}\rangle\langle\psi_{j}|\right),$ (22)

where

$\begin{array}[]{l}\rho_{A_{j}}^{\mu\nu}(t)=\sum\limits_{n=0}^{\infty}C_{\mu,n}^{j}(t)C_{\nu,n}^{j*}(t),~{}\mu,\nu=g,i,e;\\\
\rho_{A_{j}}^{\mu\nu}(t)=\left(\rho_{A_{j}}^{\nu\mu}(t)\right)^{*},~{}\sum\limits_{\mu=g,i,e}\rho_{A_{j}}^{\mu\mu}(t)=1.\end{array}$
(23)

Therefore, the photon number distribution $\displaystyle p_{j}(n,t)$ is
defined by [51]

$\displaystyle p_{j}(n,t)=\langle
n|\rho_{F_{j}}|n\rangle,~{}j=1,2,~{}n=0,1,2,....~{}.$ (24)

##  III Preliminaries of some entropic functionals

###  III.1 Purity and linear entropy

For a pure state, measure of entanglement is widely used whereas, for mixed
state measure of entanglement, distillable entanglement and relative entropy
are accepted for uncertainty measurements [52]. The measure of mixedness of a
quantum state with density matrix $\rho(t)$ is defined by
$\mathcal{L}(t)=1-P(t)=1-Tr\left[\rho(t)^{2}\right]$. The quantity,
$\mathcal{L}(t)=0$ for pure states and $\mathcal{L}=\frac{N-1}{N}$, for a
normalized $N$-qubit quantum system with idensitity $1/N$ [53]. In this case,
the state is called fully mixed. In particular, for a maximally entangled
bipartite state [54, 55], $\mathcal{L}(t)=0.5$. The quantity $\mathcal{L}(t)$
is known as linear entropy [28, 54, 55, 56]. It can be written as a function
of the Bloch sphere radius $\eta_{R}(t)$ and it is defined by
$\mathcal{L}=\frac{1}{2}\left(1-\eta_{R}^{2}(t)\right)$. For the JCM, the
linear entropy has been defined in the text book [56]. Moreover, $P(t)$
describes the purity of a quantum system [57]. If $P(t)=1$, then, the system
is a pure state and if $P(t)<1$, then the system is a mixed state. In this
paper, we will find the purity and the linear entropy of the reduced density
matrices of two quantum systems. The purity of the reduced density matrix
$\rho_{A}$ is defined by

$P(t)=\sum\limits_{\mu=g,i,e}\left(\rho_{A}^{\mu\mu}\right)^{2}+2\left(|\rho_{A}^{ei}|^{2}+|\rho_{A}^{eg}|^{2}+|\rho_{A}^{ig}|^{2}\right).$
(25)

The quantity $P(t)$ is also used as a measure of disequilibrium of a quantum
system and it is defined as the average position of the density function
$\rho_{A}(t)$. This $\mathcal{L}$ defines the measure of impurity of a quantum
system. Moreover, it has been used as a measure of decoherence [58],
entanglement [54, 55], complexity [59] and mixedness [60, 53].

###  III.2 von-Neumann entropy

A standard measure of mixedness of a quantum state is the von-Neumann entropy
and it is defined by [30]

$\displaystyle\mathcal{N}=-Tr[~{}\rho\ln\rho],$ (26)

where $\rho$ is the density operator. Then, $\mathcal{N}=0$ for a pure state
and $\mathcal{N}>0$ for a mixed state. As the entropy is a measure of purity
of states, it does not differentiate between pure states. When the state
initially is a pure state, the system remains in a pure state at any time and
its entropy is always zero. The von-Neumann entropies $\mathcal{N}_{A}$ and
$\mathcal{N}_{F}$ of an atom and a field are defined by

$\begin{array}[]{l}\displaystyle\mathcal{N}_{A}=-Tr_{A}[~{}\rho_{A}\ln\rho_{A}],~{}\displaystyle\mathcal{N}_{F}=-Tr_{F}[~{}\rho_{F}\ln\rho_{F}],\end{array}$
(27)

where $\rho_{A}$ and $\rho_{F}$ are the reduced density matrices of an atom
$A$ and a field $F$ respectively. From, the Araki and Lieb theorem [37], one
obtains

$\displaystyle|\mathcal{N}_{A}-\mathcal{N}_{F}|\leq\mathcal{N}\leq|\mathcal{N}_{A}+\mathcal{N}_{F}|,$
(28)

where $\mathcal{N}$ is the total entropy of the complete atom-field system. If
$\mathcal{N}=0$, then one obtains
$\displaystyle\mathcal{N}_{A}=\mathcal{N}_{F}$. If $\lambda_{A,i}$, $i=1,2,3$
are the eigenvalues of the density operator $\rho_{A}$, then von-Neumann
entropy can be written as

$\mathcal{N}_{A}=-\sum\limits_{i}\lambda_{A,i}\ln\lambda_{A,i}$ (29)

###  III.3 Shannon entropy

The temporal evolution of the information about dynamics of a field is
measured by the Shannon entropy [31]. It depends on the diagonal elements of
the reduced field density matrix. We can not measure the information of the
phase. The Shannon entropy is associated with the classical probability
distribution. Therefore, the Shannon entropy of the photon number distribution
$\displaystyle p_{j}(n,t)$ is defined by [32, 33]

$\displaystyle\mathcal{H}(t)=-\sum_{n=0}^{\infty}~{}p_{j}(n,t)\ln p_{j}(n,t).$
(30)

###  III.4 Atomic Wehrl entropy

A different definition, which clearly exhibits the unique character of
coherent states, has been introduced by Wehrl [36]. In this section, we will
define the atomic Wehrl entropy. To do, let us first define the atomic
coherent state of a three-level atom [34, 61, 62],

$\displaystyle|\theta,\Phi\rangle=\sum\limits_{\mu=e,i,g}k_{\mu}|\mu\rangle,$
(31)

where

$\displaystyle k_{e}=\cos^{2}\frac{\theta}{2},~{}\displaystyle
k_{i}=\sqrt{2}\cos\frac{\theta}{2}\sin\frac{\theta}{2}e^{i\Phi},~{}\displaystyle
k_{g}=\sin^{2}\frac{\theta}{2}e^{2i\Phi}.$ (32)

It has the following properties

$\begin{array}[]{l}\langle\theta,\Phi|\theta,\Phi\rangle=1,~{}\displaystyle\frac{3}{4\pi}\displaystyle\int_{\phi=0}^{2\pi}d\phi\int_{\theta=0}^{\pi}\sin\theta
d\theta|\theta,\Phi\rangle\langle\theta,\Phi|=I_{3},\end{array}$ (33)

where $I_{3}$ is the identity matrix of order 3. Then the Husimi $Q$-function
of an atom is defined by

$\displaystyle
Q(t,\theta,\Phi)=\langle\theta,\Phi|\rho_{A}|\theta,\Phi\rangle=\sum_{\mu\nu}~{}k_{\mu}^{*}k_{\nu}~{}\rho_{A}^{\mu\nu},~{}\displaystyle
k_{\mu\nu}=k_{\mu}^{*}k_{\nu}.$ (34)

The function $Q$ is also called the atomic quasiprobability distribution
function and the normalization condition is given by

$\displaystyle\frac{3}{4\pi}\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}Q(\tau,\theta,\Phi)\sin\theta~{}d\theta~{}d\Phi=1.$
(35)

In this case, the atomic Wehrl entropy of the Husimi $Q$ function is defined
by [62, 63, 64, 65, 66, 67]

$\mathcal{W}=-\displaystyle\frac{3}{4\pi}\displaystyle\int_{0}^{2\pi}d\Phi\int_{0}^{\pi}\sin\theta~{}d\theta\,Q(\tau,\theta,\Phi)\ln
Q(\tau,\theta,\Phi).$ (36)

The Wehrl entropy is minimum for coherent state and maximum for the
delocalized states [62] and it corresponds to the maximally mixed state.

###  III.5 Rényi entropy

The one-parameter Rényi entropy of order $\bar{\alpha}>0,~{}\neq 1$ of
$\mathcal{P}$ is defined by [38]

$\begin{array}[]{ll}\mathcal{R}_{\mathcal{P}}^{(\bar{\alpha})}&=\frac{1}{1-\bar{\alpha}}\ln\left[\displaystyle\sum\limits_{i=1}^{n}p_{i}^{\bar{\alpha}}\right],\mbox{if~{}}\mathcal{P}=(p_{1},p_{2},...,p_{n}),\mbox{is~{}discrete},\\\
\\\
&=\frac{1}{1-\bar{\alpha}}\ln\left[\displaystyle\int_{\Omega^{D}}\left[p(x)\right]^{\bar{\alpha}}dx\right],\mbox{if~{}}\mathcal{P}=(p(x):x\in
M^{D}),\\\ &~{}\mbox{is~{}continuous},\end{array}$ (37)

where $\Omega^{D}\subset M^{D}$ is the effective domain of $\mathcal{P}$ [44,
45]. The Rényi entropy (37) can be redefined by

$\begin{array}[]{ll}\mathcal{R}_{\mathcal{P}}^{(\bar{\alpha})}&=\frac{1}{1-\bar{\alpha}}\ln\sum\limits_{i}p_{i}^{\bar{\alpha}},~{}\mathcal{P}=(p_{1},p_{2},...,p_{n}),\\\
\\\
&=\frac{1}{1-\bar{\alpha}}\ln\sum\limits_{i}p_{i}^{\bar{\alpha}}\mathbb{L}(\Omega_{i}),~{}\mathcal{P}=(p(x):x\in
M^{D}),\end{array}$ (38)

where

$\begin{array}[]{l}\displaystyle\sum\limits_{i}p_{i}\mathbb{L}(\Omega_{i})=1,~{}0\leq
p_{i}\leq 1,\mathbb{L}(\Omega_{i})\geq 0,\\\
p(x)=\displaystyle\sum\limits_{i}p_{i}\,\chi_{\Omega_{i}}(x),\\\
\begin{array}[]{ll}\chi_{\Omega_{i}}(x)&=1,x\in\Omega_{i},\\\
&=0,x\notin\Omega_{i},\end{array}\end{array}$ (39)

$\Omega^{D}=\cup_{i}\Omega_{i}$ and $\mathbb{L}(\Omega_{i})$ is the Lebesgue
measure of the set $\Omega_{i}$. The Rényi entropy of a continuous random
variable can expressed as a limit of a sum (37) and the corresponding
integation can be considred as an example of Lebesgue integration. Then the
sum (38) is a good approximation of (37), if the partition point set
$\Omega_{i}$ has infinitesimal Lebesgue measure, that is
$\mathbb{L}(\Omega_{i})\rightarrow 0$, for continuous distribution. The Rényi
entropy of a finite dimensional discrete distribution is always positive and
it may be negative for continuous distribution. Another generalized entropy,
of order $\bar{\alpha}>0,~{}\neq 1$ is defined by

$\mathcal{T}_{\mathcal{P}}^{(\bar{\alpha})}=\frac{1}{1-\bar{\alpha}}\left(e^{(1-\bar{\alpha})\mathcal{R}_{\mathcal{P}}^{(\bar{\alpha})}}-1\right),~{}\bar{\alpha}>0,~{}\neq
1.$ (40)

The entropy $\mathcal{T}_{\mathcal{P}}^{(\bar{\alpha})}$ is called the Tsallis
entropy [68]. If $\mathcal{P}$ is a delta function, then
$\mathcal{R}_{\mathcal{P}}^{(\bar{\alpha})}=\mathcal{T}_{\mathcal{P}}^{(\bar{\alpha})}=0$.
In the limiting case, $\bar{\alpha}\rightarrow 1$, the Rényi and Tsallis
entopies reduce to the Shannon entropy. Moreover, if $\mathcal{P}$ is a Husimi
Q function and $\bar{\alpha}\rightarrow 1$, then, the Rényi entropy reduces to
an atomic Wehrl entropy. For two special cases, $\bar{\alpha}=0$ and
$\bar{\alpha}=\infty$, we obtain [44, 45, 69]

$\begin{array}[]{ll}\mathcal{R}_{\mathcal{P}}^{(0)}&=\ln
n,~{}\mathcal{P}=(p_{1},p_{2},...,p_{n}),\\\
\mathcal{R}_{\mathcal{P}}^{(0)}&=\ln\mathbb{L}(\Omega^{D}),\mathcal{P}=(p(x):x\in
M^{D}),\end{array}$ (41)

and

$\begin{array}[]{ll}\lim\limits_{\bar{\alpha}\rightarrow\infty}\mathcal{R}_{\mathcal{P}}^{(\bar{\alpha})}&=-\ln\max\limits_{i}p_{i},~{}\mathcal{P}=(p_{1},p_{2},...,p_{n}),\\\
&=-\ln\sup\limits_{{x\in\Omega^{D}}}p(x),~{}\mathcal{P}=(p(x):x\in
M^{D}).\end{array}$ (42)

###  III.6 Majorization

Let $\mathcal{P}=(p(1),p(2),\dots,p(n))$ be a finite discrete probability
distribution (FDD). Then the cumulative distribution of $\mathcal{P}$ is
defined by

$\displaystyle\begin{array}[]{ll}cp(k)=\sum\limits_{j=1}^{k}p(j),~{}k=1,2,\dots,<n,~{}cp(n)=1.\end{array}$

If $\mathcal{P}=(p(1),p(2),\dots,p(n),\dots)$ is an infinite discrete
probability distribution, then we can write

$\displaystyle\begin{array}[]{l}cp(k)=\sum\limits_{j=1}^{k}p(j),~{}k=1,2,\dots,<\infty,~{}cp(\infty)=1.\end{array}$

Let $\mathcal{P}\left(p:I\rightarrow[0,1]\right)$ be a continuous probability
density function defined on $I\subset\mathbb{R}^{r}$. Then the cumulative
distribution of $\mathcal{P}$ is defined by

$\displaystyle\begin{array}[]{ll}cp(I_{x})=\displaystyle\int_{I_{x}}p(y)\,d\mu,~{}I_{x}\subset
I,~{}cp(I)=1.\end{array}$

The measure $d\mu$ depends on the domain of definition of $\mathcal{P}$. Let
$\mathcal{P}_{1}=(p_{1}(1),p_{1}(2),\dots,p_{1}(n))$ and
$\mathcal{P}_{2}=(p_{2}(1),p_{2}(2),\dots,p_{2}(n))$ be two finite discrete
distributions. Then $\mathcal{P}_{1}$ majorizes $\mathcal{P}_{2}$
$(\mathcal{P}_{1}\succ\mathcal{P}_{2})$, if they satisfy the relations [43]

$\displaystyle\begin{array}[]{l}cp_{1}^{\downarrow}([k])\geq
cp_{2}^{\downarrow}([k]),~{}1\leq
k<n,~{}cp_{1}^{\downarrow}([n])=cp_{2}^{\downarrow}([n])=1,\end{array}$

where
$p_{j}[1]=\max\left\\{p_{j}(i):i=1,2,\dots,n\right\\},~{}p_{j}[n]=\min\left\\{p_{j}(i):i=1,2,\dots,n\right\\},~{}p_{j}[1]\geq
p_{j}[2]\geq...\geq p_{j}[n]$ for $j=1,2$.

If $\mathcal{S}$ is a convex entropic functional of density functions, then
$\mathcal{P}\succ\mathcal{Q}\implies\mathcal{S}(\mathcal{P})\geq\mathcal{S}(\mathcal{Q})$
holds [36, 41, 42]. If $\mathcal{S}$ is a concave entropic functional of
density functions, then
$\mathcal{P}\succ\mathcal{Q}\implies\mathcal{S}(\mathcal{P})\leq\mathcal{S}(\mathcal{Q})$
holds [36, 41, 42].
As we know that the Shannon entropy $\mathcal{H}$ is a concave entropic
functional, therefore,
$\mathcal{P}\succ\mathcal{Q}\implies\mathcal{H}(\mathcal{P})\leq\mathcal{H}(\mathcal{Q})$.
The entropic moment
$\mathcal{E}_{\bar{\alpha}}(\mathcal{P})=\displaystyle\sum\limits_{i=1}^{n}p_{i}^{\bar{\alpha}}$
is a concave or convex funcational, if $0<\bar{\alpha}<1$ or $\bar{\alpha}>1$,
therefore,

$\displaystyle\mathcal{P}\succ\mathcal{Q}\implies\left\\{\begin{array}[]{ll}\mathcal{E}_{\bar{\alpha}}(\mathcal{P})\leq\mathcal{E}_{\bar{\alpha}}(\mathcal{Q})&\mbox{if~{}}0<\alpha<1\\\
\mathcal{E}_{\bar{\alpha}}(\mathcal{P})\geq\mathcal{E}_{\bar{\alpha}}(\mathcal{Q})&\mbox{if~{}}\bar{\alpha}>1\end{array}\right..$

Consquently, the Rényi entropy,
$\mathcal{R}_{\mathcal{P}}^{(\bar{\alpha})}=\displaystyle\frac{1}{1-\bar{\alpha}}\ln\left[\mathcal{E}_{\bar{\alpha}}(\mathcal{P})\right]$
is a concave functional for any $\bar{\alpha}>0$. Hence,
$\mathcal{P}\succ\mathcal{Q}\implies\mathcal{R}_{\mathcal{P}}^{(\bar{\alpha})}\leq\mathcal{R}_{\mathcal{Q}}^{(\bar{\alpha})}$
for all $\bar{\alpha}>0$.
Similarly, the Rényi and atomic Wehrl entropies are concave entropic
functional of Husimi $Q$-functions. Therefore, for two Husimi $Q$-functions
$\mathcal{Q}_{1}$, $\mathcal{Q}_{2}$, such that
$\mathcal{Q}_{1}\succ\mathcal{Q}_{2}$, then
$\mathcal{R}_{\mathcal{Q}_{1}}^{(\bar{\alpha})}\leq\mathcal{R}_{\mathcal{Q}_{2}}^{(\bar{\alpha})}$,
for all $\bar{\alpha}>0$ and
$\mathcal{W}({\mathcal{Q}_{1}})\leq\mathcal{W}({\mathcal{Q}_{2}})$.
Contrapositively, if $\mathbb{S}$ is a set of convex functionals of discrete
distributions and $\mathcal{P}(p)$ and $\mathcal{Q}(q)$ are two discrete
distributions which satisfy the relation
$\sum\limits_{i}\mathcal{S}(p(i))\geq\sum\limits_{i}\mathcal{S}(q(i))$, for
all $\mathcal{S}\in\mathbb{S}$, then $\mathcal{P}\succ\mathcal{Q}$.
Similarly, if $\mathbb{S}$ is a set of concave functionals of discrete
distributions and $\mathcal{P}(p)$ and $\mathcal{Q}(q)$ are two discrete
distributions which satisfy the relation
$\sum\limits_{i}\mathcal{S}(p(i))\leq\sum\limits_{i}\mathcal{S}(q(i))$ for all
$\mathcal{S}\in\mathbb{S}$, then $\mathcal{P}\succ\mathcal{Q}$.
The localization effect on Rényi entropy and the generalized Rényi complexity
has been shown in our previous studies [44, 45].

Let $\mathcal{P}_{1}\left(p_{1}:I\rightarrow[0,1]\right)$ and
$\mathcal{P}_{2}\left(p_{2}:I\rightarrow[0,1]\right)$ be two continuous
probability density functions defined on $I\subset\mathbb{R}^{r}$. Then
$\mathcal{P}_{1}$ majorizes $\mathcal{P}_{2}$
$\left(\mathcal{P}_{1}\succ\mathcal{P}_{2}\right)$, if [70]

$\displaystyle\begin{array}[]{l}\displaystyle\int\left[p_{1}(x)-r\right]^{+}d\mu\geq\displaystyle\int\left[p_{2}(x)-r\right]^{+}d\mu,~{}\mbox{holds,~{}for~{}all~{}}r\geq
0,\end{array}$

where $\left[y\right]^{+}=\max\left\\{y,0\right\\}$.

Let $f$ and $g$ be two continuous density functions, then
$\displaystyle\int\left[f-t\right]^{+}d\mu\leq\int\left[f-t\right]^{+}d\mu$
holds for all $t\geq 0$ if and only if
$\displaystyle\int\phi(f)\,d\mu\leq\int\phi(g)\,d\mu$, holds for all real
valued convex continuous function $\phi$, such that $\phi(0)=0$ and the
integral exist, where $\left[y\right]^{+}=\max\\{y,0\\}$ [70].

Let $f$ and $g$ be two continuous density functions, then
$\displaystyle\int\left[f-t\right]^{+}d\mu\leq\int\left[f-t\right]^{+}d\mu$
holds for all $t\geq 0$ if and only if
$\displaystyle\int\phi(f)\,d\mu\geq\int\phi(g)\,d\mu$, holds for all real
valued concave continuous function $\phi$, such that $\phi(0)=0$ and the
integral exist, where $\left[y\right]^{+}=\max\\{y,0\\}$ [70].

A region $\Omega\subset I\subset\mathbb{R}^{r}$ is called the effective domain
of the density function $\mathcal{P}(p:I\rightarrow[0,1])$, if [44, 45]
$\displaystyle\int_{\Omega}p(x)d\mu=1$ and for $\Omega_{1}\subset I$, such
that $\displaystyle\int_{\Omega_{1}}p(x)d\mu=1$, then
$\mathbb{L}(\Omega)\leq\mathbb{L}(\Omega_{1})$. Let
$\mathcal{P}_{1}\left(p_{1}:I\rightarrow[0,1]\right)$ and
$\mathcal{P}_{2}\left(p_{2}:I\rightarrow[0,1]\right)$ be two continuous
probability density functions and $\Omega_{1}$, and $\Omega_{2}$ be their
respective effective domains. Then, $\mathcal{P}_{1}$ is called localized from
$\mathcal{P}_{2}$, if [44, 45]
$\mathbb{L}(\Omega_{1})\leq\mathbb{L}(\Omega_{2})$. It is to be noted that, if
the number of zeros of $p_{2}$ is greater than the number of zeros of $p_{1}$,
then $\mathbb{L}(\Omega_{1})\leq\mathbb{L}(\Omega_{2})$. Now, we have
investigated a relation between majorization and localization properties for
continuous distributions.

Theorem: Let $\mathcal{P}_{1}\left(p_{1}:I\rightarrow[0,1]\right)$ and
$\mathcal{P}_{2}\left(p_{2}:I\rightarrow[0,1]\right)$ be two density
functions, such that $\mathcal{P}_{1}\succ\mathcal{P}_{2}$, then
$\mathcal{P}_{1}$ is localized from $\mathcal{P}_{2}$.

Proof. Since, $\mathcal{P}_{1}\succ\mathcal{P}_{2}$, therefore,

$\displaystyle\int\left[p_{1}(x)-r\right]^{+}d\mu\geq\displaystyle\int\left[p_{2}(x)-r\right]^{+}d\mu,~{}\mbox{for~{}
all}~{}r\geq 0.$ (49)

Then for each $r\geq 0$, there exists $I_{1}(r),~{}I_{2}(r)\subset I$, such
that,
$\displaystyle\int\left[p_{1}(x)-r\right]^{+}d\mu=\displaystyle\int_{I_{1}}(p_{1}(x)-r)d\mu=\displaystyle\int_{I_{1}}\,p_{1}(x)\,d\mu-r\mathbb{L}(I_{1})\leq
1$, and
$\displaystyle\int\left[p_{2}(x)-r\right]^{+}d\mu=\displaystyle\int_{I_{2}}(p_{2}(x)-r)d\mu=\displaystyle\int_{I_{2}}\,p_{2}(x)\,d\mu-r\mathbb{L}(I_{2})\leq
1$.
In the limiting case $(r\rightarrow 0+)$, we obtain

$\begin{array}[]{l}\displaystyle\lim\limits_{r\rightarrow
0+}\int\left[p_{1}(x)-r\right]^{+}d\mu=1,~{}\displaystyle\lim\limits_{r\rightarrow
0+}I_{1}(r)=\Omega_{1},\\\ \displaystyle\lim\limits_{r\rightarrow
0+}\int\left[p_{2}(x)-r\right]^{+}d\mu=1,~{}\displaystyle\lim\limits_{r\rightarrow
0+}I_{2}(r)=\Omega_{2},\end{array}$ (50)

where $\Omega_{1}$ and $\Omega_{2}$ are respectively effective domain of
$p_{1}$ and $p_{2}$.
If $r>1$, then
$\displaystyle\int\left[p_{1}(x)-r\right]^{+}d\mu=\displaystyle\int\left[p_{2}(x)-r\right]^{+}d\mu=0$.
Therefore,
$\displaystyle\int_{I_{1}(r)}\,p_{1}(x)\,d\mu-r\mathbb{L}(I_{1}(r))\geq\displaystyle\int_{I_{2}(r)}\,p_{2}(x)\,d\mu-r\mathbb{L}(I_{2}(r))$,
for $0\leq r\leq 1$,
$\displaystyle\int_{I_{1}(r)}\,p_{1}(x)\,d\mu-\displaystyle\int_{I_{2}(r)}\,p_{2}(x)\,d\mu+r\left[\mathbb{L}(I_{2}(r))-\mathbb{L}(I_{1}(r))\right]\geq
0$, for $0\leq r\leq 1$.
Taking $\lim\limits_{r\rightarrow 0+}$ on both sides and using (50) we obtain
$\lim\limits_{r\rightarrow
0+}r\left[\mathbb{L}(I_{2}(r))-\mathbb{L}(I_{1}(r))\right]\geq 0$.
If possible let $\mathbb{L}(\Omega_{2})-\mathbb{L}(\Omega_{1})<0$, then there
exists $\delta>0$ such that $\mathbb{L}(I_{2}(r))-\mathbb{L}(I_{1}(r))<0$ for
all $r\in(0,\delta)$. Therefore,
$r\left[\mathbb{L}(I_{2}(r))-\mathbb{L}(I_{1}(r))\right]<0$ for all
$r\in(0,\delta)$. This implies that $\lim\limits_{r\rightarrow
0+}r\left[\mathbb{L}(I_{2}(r))-\mathbb{L}(I_{1}(r))\right]<0$ and it will
contradict that $\mathcal{P}_{1}\succ\mathcal{P}_{2}$.
Hence $\mathbb{L}(\Omega_{1})\leq\mathbb{L}(\Omega_{2})$, that is,
$\mathcal{P}_{1}$ is localized from $\mathcal{P}_{2}$.
Note: We can not conclude about the converse of this theorem. For example, in
one dimensional particle in a box of length $d$, we can say that each density
function $p_{n}(x)=\sqrt{\frac{d}{2}}\,\sin\left(\frac{(n+1)\pi x}{d}\right)$,
$0\leq x\leq d$, $(n=0,1,2,,\cdots)$ has effective domain $[0,~{}d]$ whose
Lebesgue measure is $d$. In particular, for noncentral Kratzer potential we
have been prove that the narrowest confined and widest spread radial wave
functions dominate the localization property of rotational wave functions for
the optimum measure of Rényi entropy. In addition, we have been found that,
the minimum and the maximum values of the Rényi entropy for the narrowest
confined and widest spread radial wave functions, respectively [44].

##  IV Applications of entropic functionals to an atom-field interaction
system

###  IV.1 Fidelity, purity and linear entropy

For numerical calculations, the driving field is a coherent electromagnetic
field [51] have been considered. Therefore, coefficients of number state
$|n\rangle$ are defined by $\displaystyle
C_{n}=e^{-\frac{|\alpha|^{2}}{2}}\frac{\alpha^{n}}{\sqrt{n!}}$,
$\displaystyle\alpha=\sqrt{\bar{n}}~{}e^{i\zeta}$, $n=0,1,2,\cdots$. The
quantity $\bar{n}$, is called mean photon number of coherent field and $\zeta$
is phase of $\alpha$. In addition,

$\begin{array}[]{l}\Delta=\psi=\psi^{{}^{\prime}}=0,~{}\varphi=\varphi^{\prime}=\frac{\pi}{2},g_{1}=g_{2}=g,\end{array}$
(51)

have been considered. Then, first atom collapses to the ground state at
$\displaystyle\tau_{1}=0.137503$, with fidelity $f_{1}(\tau_{1})=0.990682$,
which is the maximum value of $f_{1}$, for a coherent field having mean photon
number $\bar{n}=8$. With the help of Eqs. (18) and (19), we observe that, the
second atom-field interacting system is same as that of first atom-field
interacting system, if $f_{1}$ is equal to one. But in the real system the
fidelity is always less than one. Therefore, the maximum fidelity is
considered in this paper. The fidelity’s for which the first and the second
atoms collapse to the ground state are plotted in Fig. 2 (A). From this, it is
observed that, the first atom is stable to the ground state but the second
atom is a superposition of $|e\rangle$, $|i\rangle$ and $g\rangle$. The purity
and the linear entropy of states of first-atom and second-atom field
interacting systems are displayed in Figs. 2 (B) and 2 (C), under the
conditions (51). The second atom field interacting system is always a mixed
state. From, this figure, it is observed that, purity of first system is
greater than that of the second system.

Figure 2:  Plot of (A) the values of the fidelity’s for which the first (dot
dashed) and second (solid) atom collapse to the ground state. (B) The purity
of the reduced density matrices of the first (dot dashed) and second (solid)
atoms. (C) Linear entropy of the first (dot dashed) and the second (solid)
atoms. The parameters are $g_{1}=g_{2}=g$, $\Delta=0$, $\psi=0$,
$\varphi=\frac{\pi}{2}$, $\psi^{{}^{\prime}}=0$,
$\varphi^{{}^{\prime}}=\frac{\pi}{2}$, $\tau_{1}=0.137503$, $\zeta=0$ and
$\bar{n}=8$.

###  IV.2 von-Neumann entropy

The density matrix $\rho_{A_{1}}$ depends on $t$ and the density matrix
$\rho_{A_{2}}$ depends on time $t_{1}$ and $t$. The von-Neumann entropy of the
$j$th atom is defined by

$\displaystyle\mathcal{N}_{j}=-\sum_{i=1}^{3}\lambda_{i}^{j}\ln\lambda_{i}^{j},~{}j=1,2,$
(52)

where $\lambda_{i}^{j}~{}(i=1,2,3)$ are eigenvalues of $\rho_{A_{j}}$.
Therefore, $\lambda_{i}^{j}(i=1,2,3)$ are the roots of the equation

$\begin{array}[]{l}x^{3}-x^{2}+x\left[\rho_{A_{j}}^{ee}\rho_{A_{j}}^{ii}+\rho_{A_{j}}^{ee}\rho_{A_{j}}^{gg}+\rho_{A_{j}}^{ii}\rho_{A_{j}}^{gg}-|\rho_{A_{j}}^{ei}|^{2}\right.\\\
\left.-|\rho_{A_{j}}^{eg}|^{2}-|\rho_{A_{j}}^{ig}|^{2}\right]-det(\rho_{A_{j}})=0~{}\mbox{for~{}}j=1,2.\end{array}$
(53)

The explicit form of the von-Neumann entropy of the $j$th system is obtained
by

$\displaystyle\mathcal{N}_{j}=-\lambda_{+}^{j}\ln\lambda_{+}^{j}-\lambda_{-}^{j}\ln\lambda_{-}^{j},~{}j=1,2,$
(54)

where two sets of eigen values
$\mathcal{P}_{1}^{\downarrow}=(\lambda_{+}^{1},\lambda_{-}^{1},0)$ and
$\mathcal{P}_{2}^{\downarrow}=(\lambda_{+}^{2},\lambda_{-}^{2},0)$ in
increasing order are defined by

$\lambda_{\pm}^{j}=\frac{1}{2}\left[1\pm\sqrt{1-8\rho_{A_{j}}^{ee}+16(\rho_{A_{j}}^{ee})^{2}+8|\rho_{A_{j}}^{eg}|^{2}}\right].$
(55)

The von-Neumann entropies of two atoms are binary entropies
$\mathcal{N}_{j}=\mathcal{N}_{j}(\lambda_{+}^{j})=\mathcal{N}_{j}(\lambda_{-}^{j})$,
$j=1,2$. The cumulative distributions can be written as

$\begin{array}[]{l}cp_{1}^{\downarrow}(1)=\lambda_{+}^{1},\,cp_{1}^{\downarrow}(2)=1,\,cp_{1}^{\downarrow}(3)=1,\\\
cp_{2}^{\downarrow}(1)=\lambda_{+}^{2},\,cp_{2}^{\downarrow}(2)=1,\,cp_{2}^{\downarrow}(3)=1.\end{array}$
(56)

Figure 3:  (Color online) Plot of the eigen values (A) $\lambda_{+}^{1}$,
$\lambda_{+}^{2}$ and (B) $\lambda_{-}^{1}$, $\lambda_{-}^{2}$. The parameters
are as taken in Fig. 2.

Figs. 3, display eigen values $\lambda_{+}^{1,2}$ and $\lambda_{-}^{1,2}$.
From this figure, it is observed, that $\lambda_{+}^{1}-\lambda_{+}^{2}$ is
indefinite in nature with respect to $\tau$.

Figure 4:  Plot of the von-Neumann entropy of the (A) 1st atom, (B) 2nd atom
and (C) the difference between the von-Neumann entropies of the 2nd and the
1st atoms. The parameters are as taken in Fig. 2.

The von-Neumann entropies $\mathcal{N}_{1}(\tau)$ and $\mathcal{N}_{2}(\tau)$
are plotted Figs. 4 (A) and 4 (B) respectively, with respect to $\tau$ and
difference $\displaystyle\mathcal{N}_{2}(\tau)-\mathcal{N}_{1}(\tau)$ is
plotted in Fig. 4 (C). From Fig. 4 (C), it can be conclude that,
$\mathcal{N}_{2}>\mathcal{N}_{1}$ for some values of $\tau$ and
$\mathcal{N}_{2}<\mathcal{N}_{1}$ for other values $\tau$. Therefore, von-
Neumann entropy can not be controlled by passing atom(s) into the cavity under
the conditions (51).

###  IV.3 Shannon entropy

The photon number distribution $p_{j}(n,t)$ of the field when the $j$th atom
passes through the cavity is given by,

$\begin{array}[]{lll}p_{1}(n,\tau)&=e^{-|a|^{2}}\cos^{2}(\sqrt{2}\,\tau),n=0,\\\
&=\frac{|\alpha|^{2n}e^{-|\alpha|^{2}}}{n!}\Big{(}\cos^{2}(\sqrt{2n+2}\,\tau)+\frac{n}{|\alpha|^{2}}\sin^{2}(\sqrt{2n}\,\tau)\Big{)},\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n>0,\\\
p_{2}(n,\tau)&=0,n=0\\\
&=\eta_{1}^{2}e^{-|\alpha|^{2}}\sin^{2}(\sqrt{2}\tau_{1})\cos^{2}(\sqrt{4}\tau),n=1,\\\
&=\eta_{1}^{2}e^{-|\alpha|^{2}}\left(\frac{|\alpha|^{2n}}{n!}\sin^{2}(\sqrt{2n+2}\,\tau_{1})\cos^{2}(\tau\sqrt{2n+4})\right.\\\
&\left.+\frac{|\alpha|^{2n-2}}{(n-1)!}\sin^{2}(\sqrt{2n}\,\tau_{1})\sin^{2}(\tau\sqrt{2n+2})\right),~{}n>1.\\\
\end{array}$ (57)

Figure 5:  (Color online) Plot of the photon number distributions of (A) 1st
atom-field interaction, (B) 2nd atom-field interaction and (C) difference
between $p_{1}$ and $p_{2}$. Cumulative photon number distributions of (D) 1st
atom-field interaction, (E) 2nd atom-field interaction and (F) difference
between $cp_{2}$ and $cp_{1}$. Cumulative photon number distributions in
decreasing order of (G) 1st atom-field interaction, (H) 2nd atom-field
interaction and (I) difference between $cp_{2}^{\downarrow}$ and
$cp_{1}^{\downarrow}$. The parameters are as taken in Fig. 2.

Figs. 5 (A) and Fig. 5 (B), show the photon number distributions of two fields
and their difference is plotted in Fig. 5 (C). From this figure it is clear
that, $p_{1}(n,\tau)-p_{2}(n,\tau)$ is neither strictly positive nor strictly
negative. The cumulative distributions $cp_{1}(n,\tau)$, $cp_{2}(n,\tau)$ of
photon numbers are shown in Fig. 5 (D), (E) respectively and the difference
between them is shown in Fig. 5 (F), which is strictly positive. The
cumulative distributions of decreasing orders $cp_{1}^{\downarrow}(n,\tau)$
and $cp_{2}^{\downarrow}(n,\tau)$ are plotted in Figs. 5 (G), (H) and their
difference in Fig. 5 (I). From Fig. 5 (I), it can be observed that, photon
number distributions of two systems satisfy the majorization condition
$p_{2}^{\downarrow}\succ p_{1}^{\downarrow}$.

Now, using $0\ln 0=0$, and photon number distributions defined in Eq.(57), one
obtains the Shannon entropy $\mathcal{H}_{j}(t)$ of the field $F_{j}$. The
Shannon entropy $(\mathcal{H}_{1}(t))$ of first atom-field interaction is a
function of $t$ and $(\mathcal{H}_{2}(t))$ of second system is a function of
$t_{1}$ and $t$.

Figure 6:  Plot of the Shannon entropy of 1st atom-field interaction (black
line) and 2nd atom-field interaction (red line). The parameters are as taken
in Fig. 2.

Then, Shannon entropies $\mathcal{H}_{1}(t)$ and $\mathcal{H}_{2}(t)$ are
plotted in Fig. 6 as functions of $\tau$.

From Fig. 6, one can observe that, the Shannon entropy $\mathcal{H}_{2}(t)$ of
the field of second system is always less than that of $\mathcal{H}_{1}(t)$ of
the field of first system. Therefore, Shannon entropy can be reduced by
passing another atom(s) into the cavity in the same process.

###  IV.4 Atomic Wehrl entropy

The Husimi $Q$ function of the $j$th atom is obtained as

$Q_{j}(\tau,\theta,\phi)=\sum\limits_{i=1}^{3}Q_{j}^{(i)}(\tau,\theta,\phi),$
(58)

where

$\begin{array}[]{ll}Q_{1}^{(1)}(\tau,\theta,\phi)&=\frac{e^{-|\alpha|^{2}}}{16}\Big{(}5+4\cos\theta-\cos
2\theta\\\ &+2\sqrt{2}\cos\phi(2\sin\theta+\sin 2\theta)\Big{)}\\\
&\times\sum\limits_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\cos^{2}(\sqrt{2n+2}\,\tau),\\\
Q_{1}^{(2)}(\tau,\theta,\phi)&=\frac{e^{-|\alpha|^{2}}}{4\sqrt{2}}\Large((1-\cos
2\theta)\sin 2\phi\\\ &+\sqrt{2}(2\sin\theta-\sin 2\theta)\sin\phi\Large)\\\
&\times\sum\limits_{n=1}^{\infty}\frac{|\alpha|^{2n-1}}{\sqrt{n!(n-1)!}}\cos(\sqrt{2n+2}\,\tau)\sin(\sqrt{2n}\,\tau),\\\
Q_{1}^{(3)}(\tau,\theta,\phi)&=\frac{e^{-|\alpha|^{2}}}{8}\left(3-4\cos\theta+\cos
2\theta\right)\\\
&\sum\limits_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\sin^{2}(\sqrt{2n+2}\,\tau),\end{array}$
(59)

and

$\begin{array}[]{ll}Q_{2}^{(1)}(\tau,\theta,\phi)&=\frac{\eta_{1}^{2}e^{-|\alpha|^{2}}}{16}\Big{(}5+4\cos\theta-\cos
2\theta\\\ &+2\sqrt{2}\cos\phi(2\sin\theta+\sin 2\theta)\Big{)}\\\
&\times\sum\limits_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\sin^{2}(\sqrt{2n+2}\,\tau_{1})\cos^{2}(\tau\sqrt{2n+4}),\\\
Q_{2}^{(2)}(\tau,\theta,\phi)&=\frac{\eta_{1}^{2}e^{-|\alpha|^{2}}}{4\sqrt{2}}\Big{(}(1-\cos
2\theta)\sin 2\phi\\\ &+\sqrt{2}(2\sin\theta-\sin 2\theta)\sin\phi\Big{)},\\\
&\times\sum\limits_{n=1}^{\infty}\Big{[}\frac{|\alpha|^{2n-1}}{\sqrt{n!(n-1)!}}\cos(\sqrt{2n+4}\,\tau)\\\
&\sin(\sqrt{2n+2}\,\tau)\sin(\sqrt{2n+2}\,\tau_{1})\sin(\sqrt{2n}\,\tau_{1})\Big{]}\\\
Q_{2}^{(3)}(\tau,\theta,\phi)&=\frac{\eta_{1}^{2}e^{-|\alpha|^{2}}}{8}\left(3-4\cos\theta+\cos
2\theta\right)\\\
&\sum\limits_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\sin^{2}(\sqrt{2n+2}\,\tau_{1})\sin^{2}(\tau\sqrt{2n+4}).\end{array}$
(60)

Let us consider

$\begin{array}[]{ll}\left[Q_{j}(\tau,r)\right]^{+}=\displaystyle\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}\left[Q_{j}(\tau,\theta,\phi)-r\right]^{+}\sin\theta\,d\theta
d\phi,r\geq 0\end{array}$ (61)

for $,j=1,2$.

Figure 7:  (Color online) In (A) $\left[Q_{1}(\tau,r)\right]^{+}$, (B)
$\left[Q_{2}(\tau,r)\right]^{+}$, their difference (C)
$\left[Q_{1}(\tau,r)\right]^{+}-\left[Q_{2}(\tau,r)\right]^{+}$ with respect
to $\tau$. Similarly, they are plotted with respect to $r$ in (D), (E) and
(F). The parameters are as taken in Fig. 2.

Now, Figs. 7 (A), (B) and (C), show $\left[Q_{1}(\tau,r)\right]^{+}$,
$\left[Q_{2}(\tau,r)\right]^{+}$ and their differences with respect to $\tau$
for different $r$. From Fig. 7 (C), it is clear that,
$\left[Q_{1}(\tau,r)\right]^{+}-\left[Q_{2}(\tau,r)\right]^{+}$ is oscillating
with respect to $\tau$ for different values of $r$.

Similarly, Figs. 7 (D), (E) and (F), display $\left[Q_{1}(\tau,r)\right]^{+}$,
$\left[Q_{2}(\tau,r)\right]^{+}$ and differences between them with respect to
$r$, for different normalized time $\tau$.

We obtain, $\lim\limits_{r\rightarrow 0}\left[Q_{j}(\tau,r)\right]^{+}=1$, and
$\lim\limits_{r\rightarrow 1}\left[Q_{j}(\tau,r)\right]^{+}=0$, for $j=1,2$.
Moreover, it is verified that,
$\left[Q_{1}(\tau,r)\right]^{+}\gtrless\left[Q_{2}(\tau,r)\right]^{+}$, for
$0\leq r\leq 1$ which implies $Q_{1}(\tau,r)\nsucc Q_{1}(\tau,r)$,
$Q_{1}(\tau,r)\nprec Q_{1}(\tau,r)$ and therefore,
$\implies\mathcal{W}_{1}(\tau)\lessgtr\mathcal{W}_{2}(\tau)$.

Figure 8:  Plot of the atomic Wehrl entropy of (A) the 1st atom, (B) the atom
and (C) the difference between them. The parameters are as taken in Fig. 2.

The atomic Wehrl entropies $\mathcal{W}_{1}(\tau)$ and $\mathcal{W}_{2}(\tau)$
are plotted with respect to $\tau$ in Fig. 8 (A) and 8 (B) respectively. Fig.
8 (C) shows the difference between atomic Wehrl entropies,
$\mathcal{W}_{2}(\tau)-\mathcal{W}_{1}(\tau)$.

From Fig. 8 (C), it is observed that, difference between atomic Wehrl
entropies is oscillating as
$\left[Q_{j}(\tau,r)\right]^{+}-\left[Q_{j}(\tau,r)\right]^{+}$ is indefinite
(7 (F)) with respect to $r$ for different values of $\tau$. Therefore, atomic
Wehrl entropy cannot be controlled by adding atom(s) into the cavity.

###  IV.5 Rényi entropy

Using the definition (37), the Rényi entropy of the photon number
distributions (57) is defined by

$\mathcal{R}_{\mathcal{P}_{j}}^{(\bar{\alpha})}(\tau)=-\frac{1}{1-\bar{\alpha}}\ln\sum\limits_{n=n_{0}}^{n_{max}}\left[p_{j}(n,\tau)\right]^{\bar{\alpha}},j=1,2$
(62)

and for the Husimi Q function it is defined by

$\mathcal{R}_{Q_{j}}^{(\bar{\alpha})}(\tau)=-\frac{1}{1-\bar{\alpha}}\ln\displaystyle\left[\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}\left(Q_{j}(\tau,\theta,\phi)\right)^{\bar{\alpha}}\sin\theta
d\theta d\phi\right],$ (63)

for $j=1,2$. The Rényi entropy of photon number distributions $p_{j}(n,\tau)$
is independent of its order $p^{\downarrow}_{j}(n,\tau)$, i.e.,
$\mathcal{R}_{\mathcal{P}_{j}}^{\downarrow(\bar{\alpha})}(\tau)=\mathcal{R}_{\mathcal{P}_{j}}^{(\bar{\alpha})}(\tau)$.

Figure 9:  Plot of the Rényi entropy of the photon number distributions in (A)
of order $\bar{\alpha}=2$, (B) of order $\bar{\alpha}=1/2$ for $p_{1}$ (black
lines) and of $p_{2}$ (blue lines). The parameters are as taken in Fig. 2.
Figure 10:  Plot of the Rényi entropy of the Husimi function in (A)-(C) of
order $\bar{\alpha}=2$, (D)-(F) of order $\bar{\alpha}=1/2$ for $Q_{1}$ and
$Q_{2}$. The parameters are as taken in Fig. 2.

In Fig. 9, we have plotted the Rényi entropies
$\mathcal{R}_{\mathcal{P}_{1}}^{\downarrow(\bar{\alpha})}$ and
$\mathcal{R}_{\mathcal{P}_{2}}^{\downarrow(\bar{\alpha})}$ of the photon
number distributions $p_{1}(n,\tau)$ and $p_{2}(n,\tau)$ for order
$\bar{\alpha}=2$, and $1/2$. From this figure, it is clear that,
$\mathcal{R}_{\mathcal{P}_{2}}^{\downarrow(2)}(\tau)<\mathcal{R}_{\mathcal{P}_{1}}^{\downarrow(2)}(\tau)$
and
$\mathcal{R}_{\mathcal{P}_{2}}^{\downarrow(1/2)}(\tau)<\mathcal{R}_{\mathcal{P}_{1}}^{\downarrow(1/2)}(\tau)$.
Similarly, Rényi entropies $\mathcal{R}_{\mathcal{Q}_{1}}^{(\bar{\alpha})}$
and $\mathcal{R}_{\mathcal{Q}_{2}}^{(\bar{\alpha})}$ of the Husimi functions
$Q_{1}$ and $Q_{2}$ are plotted in Fig. 10. From this figure one can observe
that,
$\mathcal{R}_{\mathcal{Q}_{1}}^{(\bar{\alpha})}-\mathcal{R}_{\mathcal{Q}_{2}}^{(\bar{\alpha})}$
is oscillating. Since
$\left[Q_{1}(\tau,r)\right]^{+}-\left[Q_{2}(\tau,r)\right]^{+}$ is oscillating
hence, $Q_{1}(\tau,r)\nsucc Q_{1}(\tau,r)$, $Q_{1}(\tau,r)\nprec
Q_{1}(\tau,r)$. Therefore, the Rényi entropy of Husimi $Q$ function cannot be
controlled by adding atom(s) into the cavity.

##  V Conclusions

In this paper, we have investigated of some entropy functionals of a vee type
three-level atom which interacts with a coherent field in a resonant cavity.
Two set of quantum systems (i) first atom-field interaction (ii) second atom-
field interaction have been considered. Both systems have discrete (photon
number distributions) and continuous probability distribution (Husimi $Q$
function). To understand the majorization effect of continuous probability
density functions on entropy functionals, localization property [44, 45] is
defined. It is observed that, if $\mathcal{P}\succ\mathcal{Q}$, then
$\mathcal{P}$ is localized from $\mathcal{Q}$, but the converse is not true
(please see theorem). Entropy functionals of second atom-field interacting
system is same as that of first atom-field interacting system, if the first
atom collapses to the ground state with fidelity is equal to one. The maximum
fidelity and the linear entropy of two quantum systems are investigated. The
first quantum system is more pure than the second quantum system. The von-
Neumann, Shannon, atomic Wehrl and Rényi entropies are investigated for two
quantum systems of photon number distributions and for quasi probability
distributions. The Shannon and Rényi entropies of photon number distributions
can be reduced by passing atom(s) into the cavity one by one. The von-Neumann
entropy of atoms, atomic Wehrl and Rényi entropies of Husimi $Q$ functions
cannot be controlled by adding atom(s) into the cavity. A similar behavior
will be obtained for ladder and lambda configurations of three-level atom-
field interacting system. It may be remarked that, all results are obtained
for solutions of a vee-type three-level atom and single-mode quantized field.
This majorization technique can be applied for solutions for Hamiltonian which
is defined in Eq. (1).

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