\section{Introduction}
\hspace{0.5cm}
In the recent years, the quantum entanglement and mixedness are two
biggest challenges of quantum information theory.
Quantum entanglement is a very important quantum characteristic that
has no classical counterpart.
Also, quantum entanglement is necessary for the realization of
quantum communication and the most important computational tasks,
such as quantum teleportation \cite{b1}, superdense coding \cite{b2}
and so on.
In addition, it is crucial to understand the fundamental aspects of
quantum mechanics, including quantum measurement and quantum
decoherence \cite{b3}.
Consequently, no system can be completely isolated from its
environment.
Thus in real experimental situations due to the coupling
environment, the entangled state inevitably loses its purity and
becomes mixed.
\\ \indent
Decoherence phenomenon is the most dangerous obstacle for all
entanglement manipulations.
However, the recent studies of decoherence effect for different
systems are increased due to the variety of applications in areas as
quantum optics and quantum information \cite{ad1,ad11,ad2,ad3}.
In fact, the interactions between a quantum system with the
environment are caused two types of irreversible effects, e.g.,
quantum dissipation and quantum decoherence \cite{b4,b5}.
The former will cause energy dissipation, and the latter will make
the system degenerate from coherent state to classical state.
Entanglement evolution under decoherence has attracted much interest
and many researchers have studied it extensively based on various
models in the view of environment-induced decoherence \cite{b5d,b5d1}.
\\ \indent
In the previous years, some researchers have found that state
entanglement can be decreased to zero abruptly and remained zero for
a finite time.
This phenomenon is called entanglement sudden death
\cite{b6,b7,b8,bb8,bbb8}, where entanglement of two-qubit decays to
zero in a finite time under the influence of pure vacuum noise
\cite{b9}.
Precisely, entanglement sudden death has been experimentally
observed in an implementation using twin photons \cite{b10} and
atomic ensembles \cite{b11}.
In many efforts have been devoted to study various types of
superconducting (SC) qubit which is considered as good candidate to
solve this problem.
\\ \indent
Nowadays, superconductors are viable elements and feasible within
current experimental technology.
SC qubit is represented a unique and its interesting approach to
quantum information and quantum computation because it naturally
allow strong coupling.
Some aspects which are discussed through these systems: entanglement
between a flux qubit and a superconducting quantum interference
device (SQUID) \cite{b12} have been realized experimentally.
In addition, there are other aspects: the strong coupling of a
single photon to a SC qubit has been experimentally demonstrated
\cite{b13} by using a one-dimensional transmission line resonator
\cite{b14}.
Recently, some quantum logic gates have proposed \cite{b14d1}
by using a system consisting of a number of SC-charge qubits
coupled to a resonator to study different quantum operations.
Also, a new physical scheme for implementing a discrete quantum
Fourier transform is proposed via SC qubits coupled to a single-mode
SC-cavity \cite{b14d2}.
However, the effect of environment cannot be neglected in these
situations \cite{b15}, because, the qubits are never isolated, but
under some environmental influence.
\\ \indent
Our purpose of this article is to give an analysis of the
entanglement dynamics of a SC phase qubit coupled to a torsional
resonator.
Consequently, some phenomena, such as the qubit inversion, purity
and negativity are theoretically investigated, mainly because such
states are more physical and experimentally relevant.
Thus we take into account decoherence and analyze its effect on
these phenomena.
Despite the complexity of the problem, the analytical descriptions
of the exact solution of the master equation is presented.
\\ \indent
This article is arranged as follows: The physical system of qubit
with torsional resonator and its dynamics  is provided in section 2.
The influence of  phase-damping on a SC qubit inversion and measures
of entanglement is presented in section 3.
Finally,the conclusions is demonstrated in section 4.
\section{The physical system of qubit with
torsional resonator and its dynamics }
\hspace{0.5cm}
We introduce a theoretical description of the superconducting phase
qubit coupled to a torsional resonator.
A phase qubit with $\kappa$ excess Cooper-pair charges consists of
two superconducting islands connected to a superconducting electrode
through two identical Josephson tunnel junctions of a  small size
with the same capacitance $C_{J}$, see Fig.\ref{w1}.
If the phase qubit is working in a regime with $k_{B}T\ll E_{C}\ll
E_{J}$, where $E_{J}$, $E_{C}$, $k_{B}$ and $T$ are Josephson
energy, charging energy, Boltzmann constant and temperature,
respectively.
The Hamiltonian of the superconducting phase qubit can be described
by the form \cite{b16}
\begin{eqnarray}\label{a}
\hat{H}_{qu}&=& \frac{(2e)^{2}}{2C_{J}}\kappa^{2}-\frac{\hbar
I_{0}}{2e}\Big(\cos(\pi f_{c})+\frac{
I}{I_{0}}\Theta\Big)\cos(\Theta-\pi f_{c}),
\end{eqnarray}
where $\Theta$, $I$, $I_{0}$, and $f_{c}$ are the phase difference
across the junction, bias current, junction critical current, and
the external flux which is generated by a classical applied magnetic
field, respectively.
Moreover, $\kappa$ and $\Theta$ are quantum mechanical conjugate
variables and satisfy $[\kappa,\Theta]=-i$.
\\ \indent
In this study, the bias current $I$ is assumed to be zero.
The Josephson energy, and the charging
energy for each junction also are assumed to be $E_{J}=\hbar
I_{0}/2e\sim6.8\times10^{-21}\,\,J$  and
$E_{C}=\frac{(2e)^{2}}{2C_{J}}\sim8.3\times10^{-27}\,\,J$  as estimated in \cite{c1}.
Therefore, the effective Josephson coupling $E_{J_{e}}=E_{J}\cos(\pi
f_{c})$ of phase qubit is controlled by the external flux $f_{c}$.
If the phase different $\Theta$ is confined to the period $\pi
f_{c}$, the states of a single qubit are restricted to a two
lowest-energy states $|g\rangle$ and $|e\rangle$.
\begin{figure}
\centering\includegraphics[height=10cm,width=14cm]{ph.pdf}
\caption{Circuit diagram of the phase qubit; a single Josephson
junction with Josephson coupling energy $E_{J}$ and Josephson
capacitance $C_{J}$ is driven with a bias current $I$. The rectangle
in this diagram refer to dissipative  part of the admittance of the
bias current $I$.}\label{w1}
\end{figure}
Hence, the Hamiltonian (\ref{a}) can be converted to a spin
representation as follows:
\begin{eqnarray}\label{b}
\hat{H}_{qu}&=& -\frac{\hbar\omega_{z}}{2}\hat{\sigma}_{z},
\end{eqnarray}
where $\omega_{z}\sim\frac{\sqrt{2E_{C}E_{J_{e}}}}{\hbar}$ is the
frequency of the phase qubit, and $\hat{\sigma}_{z}$ is the usual
Pauli spin operator.
\\ \indent
The torsional resonator mode is described by a harmonic
oscillator-like Hamiltonian as follows \cite{sss}
\begin{eqnarray}
\hat{H}_{mode}
&=&\frac{P^{2}_{\top}}{2I_{m}}+\frac{1}{2}I_{m}\omega^{2}_{v}
\top^{2},
\end{eqnarray}
where $P_{\top}$ is the angular momentum conjugate to $\top$ and
$\top\in[0,2\pi]$.
The fluctuations of the angle $\top$ can be described by the
fluctuation $\top_{0}=\sqrt{\hbar/I_{m}\omega_{v}}$ in the ground
state, where $I_{m}$, and $\omega_{v} $ are the rotational moment of
inertia of the torsional resonator, and vibration frequency,
respectively.
\\ \indent
If we put the SC qubit in the torsional resonator, the effective
flux is $f_{c}=f^{'}\sin{\top}$.
Hence, $\top$ is measured relative to the direction of $f_{c}$,
$E_{J}$ is the largest energy scale and coupling between the qubit
and torsional vibrational mode can be a large quantity.
Thus, we consider the qubit is coupled to resonator mode.
Also, the external magnetic field is parallely applied to the phase
qubit plane.
Then the external field in this case is given by:
$f_{c}=f^{'}(\top-\top_{e})$, where $\top_{e}$ is the angle at
equilibrium measured from the direction of $f_{c}$.
In this work, $f^{'}$ is the maximum value of magnetic flux when the
field is perpendicular to the qubit plane.
If $\top_{e}$ is non-zero, the coupling strength is decreased by
factor $\sin \top_{e}$, so consider the angle $\top_{e}=0$.
However, one concludes that the total Hamiltonian restricted to the
two states is given by:
\begin{eqnarray}\label{a1}
\hat{H}_{t}&=&
\hbar\omega\hat{a}^{\dag}\hat{a}-\frac{\hbar\omega_{z}}{2}\hat{\sigma}_{z}
+G(\hat{a}+\hat{a}^{\dag})\hat{\sigma}_{x},
\end{eqnarray}
where $(\hat{a}^{\dag})$ $\hat{a}$ and $(\hat{\sigma}^{\dag})$
$\hat{\sigma}$ are the (creation) annihilation operators for the
vibrational mode and phase qubit, respectively, $G=\pi
f^{'}\sqrt{\frac{E_{J}\omega_{z}}{2I_{m} \omega}}$ is the constant
coupling energy between the phase qubit and the torsional resonator,
and $\omega=\sqrt{\omega^{2}_{v} +2\pi^{2}f{^{'}}^{2}E_{J}/I_{m}}$
is the torsional oscillator frequency.
Because of the negative sign of the second term, the above
Hamiltonian (\ref{a1}) in the counter-rotating wave approximation
can be written as follows:
\begin{eqnarray}\label{a2}
\hat{H}&=&
\hbar\omega\hat{a}^{\dag}\hat{a}-\frac{\hbar\omega_{z}}{2}\hat{\sigma}_{z}
+G(\hat{a}\hat{\sigma}_{-}+\hat{a}^{\dag}\hat{\sigma}_{+}).
\end{eqnarray}
\\ \indent
Under the previous  hypothesis, the non-unitary time evolution of
present interaction system with taking into account the dissipative
effect is given through the density operator.
The density operator is described by a master equation of the form:
\begin{eqnarray}\label{a3}
\hbar\hat{\dot{\rho}} &=& i[\hat{\rho},\hat{H}]+
\frac{\hbar\gamma}{2}\Big(\hat{\sigma}_{z}\hat{\rho}\hat{\sigma}_{z}-\hat{\rho}\Big),
\end{eqnarray}
where $\gamma$ is a positive parameter that represents a dispersive
reservoir (or phase damping of the qubit system).
It is one of the variables that does not interchange energy with the
qubit system.
Dispersive reservoir of the qubit may occur due to dephasing
interactions that might arise.
For instance, the dispersive reservoir may occur from the elastic
collisions in a qubit vapor.
The exact solution for Eq.\ref{a3} in the case of a high-$Q$
torsional resonator ($\gamma/G\ll1$) is obtained by analytic method.
The used analytic method is presented by the dressed-states
representation \cite{c3,c4,c5}.
This representation is consisting of the complete set of Hamiltonian
eigenfunctions.
Here, the states $|g\rangle$ and $|e\rangle$ are corresponding to
the higher energy and the lower energy, respectively.
In the invariant subspace spanned by $|e,n+1\rangle$ and
$|g,n\rangle$, the eigenvalues and eigenfunctions of the Hamiltonian
(\ref{a2}) are given by:
\begin{eqnarray}
\hat{H}|\Psi^{\pm}_{n}\rangle&=&
E^{\pm}_{n}|\Psi^{\pm}_{n}\rangle,\qquad\qquad \hat{H}|e,0\rangle=
-\frac{\hbar\omega_{z}}{2}|e,0\rangle,
\end{eqnarray}
with the abbreviations
$ E_{n}^{{\pm}} =\hbar \omega(n+\frac{1}{2})\pm R_{n}$,
$|\Psi_{n}^{\pm}\rangle =
\beta_{n}^{\mp}|g,n\rangle\pm\beta_{n}^{\pm}|e,n+1\rangle$,
$\beta_{n}^{\pm}=\frac{1}{\sqrt{2}}\sqrt{1\mp\frac{\delta}{R_{n}}}$,
$R_{n}=G\sqrt{(n+1)+(\frac{\delta}{G})^{2}}$,
and $\delta=\frac{\hbar(\omega_{z}-\omega)}{2}$ is the detuning
parameter which represents the difference between transition energy
of qubit  and torsional energy.
\\ \indent
To obtain the analytical solution for Eq.\ref{a3}, we suppose that
the qubit is initially in the general superposition of the excited
state and ground state while the torsional resonator is assumed to
be initially in a coherent state, that is
\begin{eqnarray}
\hat{\rho}(0) &=&
\Big(\cos\theta|e\rangle+e^{i\varphi}\sin\theta|g\rangle\Big)
\Big(\cos\theta|e\rangle+e^{i\varphi}\sin\theta|g\rangle\Big)^{\dagger}
\otimes|\alpha\rangle\langle\alpha|.
\end{eqnarray}
Consequently, the time-dependent analytical solution for the density
matrix with the initial previous state is given by:
\begin{eqnarray}\label{a4}
\hat{\rho}(t) &=&\rho^{00}_{ee}|e,0\rangle\langle
e,0|+\sum^{\infty}_{n,m=0}\Big\{\rho^{n,m}_{gg}(t)|g,n\rangle\langle
g,m|+\rho^{n,m+1}_{ge}(t)|g,n\rangle\langle e,m+1| \nonumber\\&&
+\rho^{n+1,m}_{eg}(t)|e,n+1\rangle\langle
g,m|+\rho^{n+1,m+1}_{ee}(t)|e,n+1\rangle\langle e,m+1|\Big\},
\end{eqnarray}
where all elements of the density matrix (\ref{a4}) are also
functions in $\theta,\varphi,\delta$.
The elements of the previous density matrix are given by the
following formulas:
\begin{eqnarray*}
\hspace{-0.63cm}\rho^{n+1,m+1}_{ee}(t)=\left\{%
\begin{array}{ll}
x_{n}x^{*}_{m}e^{-iA_{n,m}t}\{2\beta^{+}_{n}\beta^{-}_{n}
\beta^{+}_{m}\beta^{-}_{m}\sin^{2}\theta [\mathbb{E}^{+-}_{n,m}\cos
(R^{+-}_{n,m}t)-\mathbb{K}^{++}_{n,m}\cos (R^{++}_{n,m}t)]\\
+\xi_{n,m}\cos^{2}\theta[({\beta_{n}^{+}}^{2}{\beta_{m}^{+}}^{2}e^{-iR^{+-}_{n,m}t}
+{\beta_{n}^{-}}^{2}{\beta_{m}^{-}}^{2}e^{iR^{+-}_{n,m}t})\mathbb{E}^{+-}_{n,m}
\\+({\beta_{n}^{+}}^{2}{\beta_{m}^{-}}^{2}e^{-iR^{++}_{n,m}t}
+{\beta_{n}^{-}}^{2}{\beta_{m}^{+}}^{2}e^{iR^{++}_{n,m}t})
\mathbb{K}^{++}_{n,m}]
\\+\frac{\tilde{\alpha}\beta^{+}_{m}\beta^{-}_{m}\sin2\theta}{2\sqrt{n+1}}
[({\beta_{n}^{+}}^{2}e^{-iR^{+-}_{n,m}t}
-{\beta_{n}^{-}}^{2}e^{iR^{+-}_{n,m}t})
\mathbb{E}^{+-}_{n,m}+({\beta_{n}^{-}}^{2}e^{iR^{++}_{n,m}t}
-{\beta_{n}^{+}}^{2}e^{-iR^{++}_{n,m}t}) \mathbb{K}^{++}_{n,m}]
\\+\frac{\tilde{\alpha}^{*}\beta^{+}_{n}
\beta^{-}_{n}\sin2\theta}{2\sqrt{m+1}}
[({\beta_{m}^{+}}^{2}e^{-iR^{+-}_{n,m}t}
-{\beta_{m}^{-}}^{2}e^{iR^{+-}_{n,m}t})
\mathbb{E}^{+-}_{n,m}+({\beta_{m}^{-}}^{2}e^{-iR^{++}_{n,m}t}
-{\beta_{m}^{+}}^{2}e^{iR^{++}_{n,m}t})\mathbb{K}^{++}_{n,m}]
\},&\\
\qquad\qquad\qquad\qquad\quad\quad
\hbox{$\forall$ n$\neq$ m;} \\
|x_{n}|^{2}\{\sin^{2}\theta[\Pi^{+,-}-v^{+,-}]
+\xi_{n,n}\cos^{2}\theta [\varpi^{+,-}+v^{+,-}]
+\Omega^{+,-}+\Gamma^{+,-}
\},&\\
\qquad\qquad\qquad\qquad\quad\quad
\hbox{$\forall$ n=m.} \\
\end{array}%
\right.
\end{eqnarray*}
\begin{eqnarray*}
\hspace{-0.63cm}\rho^{n,m}_{gg}(t)=\left\{%
\begin{array}{ll}
x_{n}x^{*}_{m}e^{-iA_{n,m}t}\{
\sin^{2}\theta[({\beta_{n}^{-}}^{2}{\beta_{m}^{-}}^{2}e^{-iR^{+-}_{n,m}t}
+{\beta_{n}^{+}}^{2}{\beta_{m}^{+}}^{2} e^{iR^{+-}_{n,m}t})
\mathbb{E}^{+-}_{n,m}\\+({\beta_{n}^{+}}^{2}{\beta_{m}^{-}}^{2}
e^{iR^{++}_{n,m}t} +{\beta_{n}^{-}}^{2}{\beta_{m}^{+}}^{2}
e^{-iR^{++}_{n,m}t})\mathbb{K}^{++}_{n,m}]\\
+2\beta^{+}_{n}\beta^{-}_{n}\beta^{+}_{m} \beta^{-}_{m}\xi_{n,m}
\cos^{2}\theta [\mathbb{E}^{+-}_{n,m}\cos
(R^{+-}_{n,m}t)-\mathbb{K}^{++}_{n,m} \cos (R^{++}_{n,m}t)]
\\+\frac{\tilde{\alpha}\beta^{+}_{n}\beta^{-}_{n}\sin2\theta}{2\sqrt{n+1}}
[({\beta_{m}^{-}}^{2}e^{-iR^{+-}_{n,m}t}
-{\beta_{m}^{+}}^{2}e^{iR^{+-}_{n,m}t})\mathbb{E}^{+-}_{n,m}
+({\beta_{m}^{+}}^{2}e^{-iR^{++}_{n,m}t}
-{\beta_{m}^{-}}^{2}e^{iR^{++}_{n,m}t})\mathbb{K}^{++}_{n,m}]
\\+\frac{\tilde{\alpha}^{*}\beta^{+}_{m}
\beta^{-}_{m}\sin2\theta}{2\sqrt{m+1}}
[({\beta_{n}^{-}}^{2}e^{-iR^{+-}_{n,m}t}
-{\beta_{n}^{+}}^{2}e^{iR^{+-}_{n,m}t})\mathbb{E}^{+-}_{n,m}
+({\beta_{n}^{+}}^{2}e^{iR^{++}_{n,m}t}
-{\beta_{n}^{-}}^{2}e^{-iR^{++}_{n,m}t})
\mathbb{K}^{++}_{n,m}]\},&\\
\qquad\qquad\qquad\qquad\quad\quad
\hbox{$\forall$ n$\neq$ m;} \\
|x_{n}|^{2}\{\sin^{2}\theta[\varpi^{+,-}+v^{+,-}]
+\xi_{n,n}\cos^{2}\theta[\Pi^{+,-}-v^{+,-}]
-\Omega^{+,-}-\Gamma^{+,-}\},\\
\qquad\qquad\qquad\qquad\quad\quad
\hbox{$\forall$ n=m.} \\
\end{array}%
\right.
\end{eqnarray*}
\begin{eqnarray*}
\hspace{-1.01cm}\rho^{n+1,m}_{eg}(t)=\left\{%
\begin{array}{ll}
x_{n}x^{*}_{m}e^{-iA_{n,m}t}\{
\beta^{+}_{n}\beta^{-}_{n}\sin^{2}\theta
[({\beta_{m}^{-}}^{2}e^{-iR^{+-}_{n,m}t}-{\beta_{m}^{+}}^{2}
e^{iR^{+-}_{n,m}t})
\mathbb{E}^{+-}_{n,m}+({\beta_{m}^{+}}^{2}e^{-iR^{++}_{n,m}t}
-{\beta_{m}^{-}}^{2}e^{iR^{++}_{n,m}t}) \mathbb{K}^{++}_{n,m}]\\
+\beta^{+}_{m}\beta^{-}_{m}\xi_{n,m} \cos^{2}\theta
[({\beta_{n}^{+}}^{2}e^{-iR^{+-}_{n,m}t}
-{\beta_{n}^{-}}^{2}e^{iR^{+-}_{n,m}t})\mathbb{E}^{+-}_{n,m}
+({\beta_{n}^{-}}^{2}e^{iR^{++}_{n,m}t}
-{\beta_{n}^{+}}^{2}e^{-iR^{++}_{n,m}t})\mathbb{K}^{++}_{n,m}]
\\+\frac{\tilde{\alpha}\sin2\theta}{2\sqrt{n+1}}
[({\beta_{n}^{+}}^{2}{\beta_{m}^{-}}^{2}e^{-iR^{+-}_{n,m}t}
+{\beta_{n}^{-}}^{2}{\beta_{m}^{+}}^{2}e^{iR^{+-}_{n,m}t})\mathbb{E}^{+-}_{n,m}
+({\beta_{n}^{+}}^{2}{\beta_{m}^{+}}^{2}e^{-iR^{++}_{n,m}t}
+{\beta_{n}^{-}}^{2}{\beta_{m}^{-}}^{2}e^{iR^{++}_{n,m}t})\mathbb{K}^{++}_{n,m}]
\\+\frac{\tilde{\alpha}^{*}\sin2\theta}{\sqrt{m+1}}
\beta^{+}_{n}\beta^{-}_{n}\beta^{+}_{m}\beta^{-}_{m}[\mathbb{E}^{+-}_{n,m}
\cos(R^{+-}_{n,m}t)-\mathbb{K}^{++}_{n,m}\cos (R^{++}_{n,m}t)]\},&\\
\qquad\qquad\qquad\qquad\quad\quad
\hbox{$\forall$ n$\neq$ m;} \\
|x_{n}|^{2}\{\beta_{n}^{+}\beta_{n}^{-}
\sin^{2}\theta[({\beta_{n}^{-}}^{2}
-{\beta_{n}^{+}}^{2})e^{\frac{-\gamma
b_{n}t}{2}}\mathbb{E}^{+-}_{n,n}+(
{\beta_{n}^{+}}^{2}e^{-iR^{++}_{n,n}t}-
{\beta_{n}^{-}}^{2}e^{iR^{++}_{n,n}t})\mathbb{K}^{++}_{n,n}]
\\+\beta_{n}^{+}\beta_{n}^{-}
\xi_{n,n}\cos^{2}\theta[({\beta_{n}^{+}}^{2} -{\beta_{n}^{-}}^{2})
e^{\frac{-\gamma b_{n}t}{2}}\mathbb{E}^{+-}_{n,n}+(
{\beta_{n}^{-}}^{2}e^{iR^{++}_{n,n}t}-
{\beta_{n}^{+}}^{2}e^{-iR^{++}_{n,n}t})\mathbb{K}^{++}_{n,n}]
\\+\frac{{\beta_{n}^{+}}^{2}{\beta_{n}^{-}}^{2}
\sin2\theta(\tilde{\alpha}+\tilde{\alpha}^{*})}{\sqrt{n+1}}
e^{\frac{-\gamma b_{n}t}{2}}\mathbb{E}^{+-}_{n,n}
\\+\frac{\sin2\theta}{\sqrt{n+1}}[\frac{\tilde{\alpha}}{2}(
{\beta_{n}^{+}}^{4}e^{-iR^{++}_{n,n}t}+
{\beta_{n}^{-}}^{4}e^{iR^{++}_{n,n}t})
-\tilde{\alpha}^{*}{\beta_{n}^{+}}^{2}{\beta_{n}^{-}}^{2}
\cos(R^{++}_{n,n}t)]\mathbb{K}^{++}_{n,n}\},&\\
\qquad\qquad\qquad\qquad\quad\quad
\hbox{$\forall$ n=m.} \\
\end{array}%
\right.
\end{eqnarray*}
Where
\begin{eqnarray*}
\begin{array}{ll}
\rho^{n,m+1}_{ge}= (\rho^{n+1,m}_{eg})^{\dag},\:\:
\rho^{00}_{ee}=e^{-|\alpha|^{2}}\cos^{2}\theta,\:\:
\tilde{\alpha}=\alpha e^{-i\varphi},
\:\:\xi_{n,m}=\frac{|\alpha|^{2}}{\sqrt{(n+1)(m+1)}},\\
x_{n}=e^{(-\frac{|\alpha|^{2}}{2})}\frac{\alpha^{n}}{\sqrt{n!}},\:\:
R^{+-}_{n,m}=R_{n}-R_{m},\:\:R^{++}_{n,m}=R_{n}+R_{m},\\
\Delta^{+-}_{n,m}=\frac{1}{2}[1
-({\beta^{+}_{n}}^{2}{\beta^{+}_{m}}^{2}
+{\beta^{-}_{n}}^{2}{\beta^{-}_{m}}^{2}
-{\beta^{+}_{n}}^{2}{\beta^{-}_{m}}^{2}
-{\beta^{-}_{n}}^{2}{\beta^{+}_{m}}^{2})],\\
\Delta^{++}_{n,m}=\frac{1}{2}[1
+({\beta^{+}_{n}}^{2}{\beta^{+}_{m}}^{2}
+{\beta^{-}_{n}}^{2}{\beta^{-}_{m}}^{2}
-{\beta^{+}_{n}}^{2}{\beta^{-}_{m}}^{2}
-{\beta^{-}_{n}}^{2}{\beta^{+}_{m}}^{2})],\\
\mathbb{E}^{+-}_{n,m}=e^{-\gamma
t\Delta^{+-}_{n,m}},\:\:\mathbb{K}^{++}_{n,m}=e^{-\gamma
t\Delta^{++}_{n,m}},\:\:
b_{n}=4{\beta_{n}^{+}}^{2}{\beta^{-}_{n}}^{2},\:\:A_{n,m}=\hbar\omega(n-m),\\
\Gamma^{+,-}=\frac{\beta_{n}^{+}\beta_{n}^{-}
\sin2\theta}{2\sqrt{n+1}}[{\beta_{n}^{-}}^{2} (\tilde{\alpha}
e^{iR^{++}_{n,n}t}+\tilde{\alpha}^{*}
e^{-iR^{++}_{n,n}t})-{\beta_{n}^{+}}^{2} (\tilde{\alpha}
e^{-iR^{++}_{n,n}t}+\tilde{\alpha}^{*}
e^{iR^{++}_{n,n}t})]\mathbb{K}^{++}_{n,n},\\
\Omega^{+,-}=\frac{\beta_{n}^{+}\beta_{n}^{-}\sin2\theta(\tilde{\alpha}
+\tilde{\alpha}^{*})}{2\sqrt{n+1}}
({\beta_{n}^{+}}^{2}-{\beta_{n}^{-}}^{2})e^{\frac{-\gamma
b_{n}t}{2}} \mathbb{E}^{+-}_{n,n},\\
\Pi^{+,-}=[({\beta_{n}^{+}}^{4}
+{\beta_{n}^{-}}^{4})\sinh(\frac{\gamma
b_{n}t}{2})+2{\beta_{n}^{+}}^{2}{\beta_{n}^{-}}^{2}\cosh(\frac{\gamma
b_{n}t}{2})] \mathbb{E}^{+-}_{n,n},\\
\varpi^{+,-}=[({\beta_{n}^{+}}^{4} +{\beta_{n}^{-}}^{4})
\cosh(\frac{\gamma
b_{n}t}{2})+2{\beta_{n}^{+}}^{2}{\beta_{n}^{-}}^{2}\sinh(\frac{\gamma
b_{n}t}{2})]\mathbb{E}^{+-}_{n,n}\:\: \textsf{and}\\
v^{+,-}=2{\beta_{n}^{+}}^{2}{\beta_{n}^{-}}^{2}
\cos(R^{++}_{n,n}t)\mathbb{K}^{++}_{n,n}.
\end{array}
\end{eqnarray*}
\indent
To calculate the asymptotic behavior of the density matrix, we set
$Gt\rightarrow\infty$, as follows:
\begin{eqnarray*}
\hat{\rho}(Gt\rightarrow\infty) &=&\rho^{00}_{ee}|e,0\rangle\langle
e,0|+\frac{1}{2} \sum_{n=0}^{\infty}
|x_{n}|^{2}\Big(\sin^{2}\theta+\xi_{n,n}\cos^{2}\theta\Big)
\nonumber\\&&\times \Big(|e,n+1\rangle\langle
e,n+1|+|g,n\rangle\langle g,n|\Big),
\end{eqnarray*}
which is a statistically mixed state.
\section{The influence of dispersive reservoir on some physical phenomena}
In the current section, we study the existence of some phenomena of
the previous system.
In an effort to present a numerical characterization, we have
performed some calculations for the properties quantity for a
particular set of parameters, some of which can be considered as
realistic, while some other parameters look perhaps too optimistic.
\subsection{SC qubit inversion phenomenon}
\hspace{0.5cm}
The qubit inversion is one of the important dynamic variables of the
properties of quantum mechanics.
It gives information about the behavior of the qubit during the
interaction period.
Therefore, by using the analytic solution of the master equation and
keeping in mind that the state $|g\rangle$ is the one with higher
energy, we discuss the effect of dispersive reservoir on qubit
inversion, which is given by:
\begin{eqnarray}
W_{qu}(t)&=& \rho^{00}_{ee}+\sum_{i}\Big(\rho^{ii}_{ee}(t)
-\rho^{ii}_{gg}(t)\Big).
\end{eqnarray}
It is generally accepted that the revivals of inversion appears as a
consequence of quantum coherence which are built up during the
interaction between the resonator and the qubit.
\\ \indent
Numerical calculations which indicate the influence of dispersive
reservoir $\frac{\gamma}{G}$ on qubit inversion is exhibited in
Fig.\ref{w2}a, Fig.\ref{w2}b and Fig.\ref{w2}c.
The time evolution of $W_{qu}$ is plotted  as a function of the
scaled time $\frac{Gt}{\pi}$ and $\frac{\gamma}{G}$, in the
resonance case, for a different values of  $\theta$ and $\alpha$.
In the absence of detuning effect and $\frac{\gamma}{G}$, see
Fig.\ref{w2}a and Fig.\ref{w2}b, one finds that $W_{qu}$ fluctuates
around zero, and the collapse and revival phenomena start to appear.
It is seen from Fig.\ref{w2}a and Fig.\ref{w2}b a short period of
collapse after onset of the interaction followed immediately by a
reasonable revival period.
After this period of collapse, one can not see another collapse
period because of existence of interference in the fluctuation, so
in these cases of $\theta$ the revival period is not apparent.
The reason for this is due to the value of coherence parameter
$\alpha$.
If $\alpha=6$, the revival and collapse phenomena are more
pronounced than the other case for different values of $\theta$, say
$\frac{\pi}{2}$. This result is observed in Fig.\ref{w2}c.
Furthermore, for many different values of $\theta$, the revival and
collapse phenomena are more pronounced.
Also, it is observed from Fig.\ref{w2} that the periodic
oscillations of $W_{qu}$ between $1$ as a maximum value and $-1$ as
a minimum value around $0$.
\\ \indent
If the dispersive reservoir parameter is regarded
$\frac{\gamma}{G}\neq0$, the time evolution of $W_{qu}$ is shown in
Figs.\ref{w2}a, Fig.\ref{w2}b and Fig.\ref{w2}c.
At a weak dispersive reservoir, the collapses and revivals
phenomenon is still observed for a short time.
The increase of collapses region slowly due to the influence of the
damping.
But with large values of dispersive reservoir, the amplitude of
revivals is greatly decreased.
With $\frac{\gamma}{G}>0.1$, all revivals are completely washed out,
and the system is collapsed completely.
This means that the probability of existence for the qubit in the
state $|g\rangle $ and $|e\rangle$ are equal.
\begin{figure}
\hspace{-1cm}\includegraphics[height=9cm,width=10cm]{a.pdf}\includegraphics[height=9cm,width=10cm]{b.pdf}\\
\hspace{-1cm}\includegraphics[height=9cm,width=10cm]{d.pdf}\includegraphics[height=9cm,width=10cm]{c.pdf}
\caption{The evolution of SC qubit inversion plotted against the
scaled time $\frac{Gt}{\pi}$ and dispersive reservoir
$\frac{\gamma}{G}$ in the resonance case for $\theta=\frac{\pi}{10}$
in (a), $\theta=\frac{\pi}{3}$ in (b) and $\theta=\frac{\pi}{2}$ in
(c), and off-resonance case for $\theta=\frac{\pi}{3}$ and
$\frac{\delta}{G}=6$ in (d), with the coherence parameter $\alpha=3$
in (a, b, d) and $\alpha=6$ in (c).}\label{w2}
\end{figure}
\\ \indent
To investigate the influence of detuning parameter
$\frac{\delta}{G}$ and $\frac{\gamma}{G}$ parameter on the behavior
of qubit inversion, we display in Fig.\ref{w2}d the behavior of
inversion in off-resonance case with $\theta=\frac{\pi}{3}$,
$\frac{\delta}{G}=6$ and $\alpha=3$.
Firstly, for $\frac{\gamma}{G}=0$, one can observe from
Fig.\ref{w2}d, the revivals and collapses phenomena are more
pronounced than the case of $\frac{\delta}{G}=0$, and the curve of
qubit inversion is shifted down.
Also, one can note that from  Fig.\ref{w2}d the revival period is
increased, and it can distinguish other periods of revival in that
case, that is in the case of off-resonance for this case of $\theta$
the revival period is more apparent.
In fact the revival time is given by: \Big($\tau_{R}=2\pi
G\sqrt{\delta^{2}+|\alpha|^{2}}$\Big) which increases by increasing
$\delta$.
However, we can see from Fig.\ref{w2}d, that the detuning parameter
leads to the state of qubit to tend to $|g\rangle$ more than
$|e\rangle$.
Besides that, if $\frac{\gamma}{G}\neq0$, the collapses and revivals
phenomenon is still observed in a very short time.
This phenomenon disappears slowly by increasing time due to the
influence of $\frac{\gamma}{G}$.
One can observe rapid deterioration for the revivals of $W_{qu}$ at
the large values of $\frac{\gamma}{G}$.
From Fig.\ref{w2}d, we observe increasing in the behavior of
$W_{qu}$ curve, after that the value of $W_{qu}$ is almost to zero
at time $Gt\simeq50\pi$.
Moreover, the collapse region has becomes very large.
However, if we take large values of $\frac{\delta}{G}$ for the
different values of $\theta$, the amplitudes of the fluctuations are
diminished, the collapse periods are elongated, and a large decrease
is happened in the amplitudes of periodic oscillations of maximum
value and minimum value for the inversion.
\subsection{Dynamical properties of coherence and entanglement phenomena}
\hspace{0.5cm}
In order to understand the influence of dispersive reservoir of
qubit on some dynamical properties, such as coherence and
entanglement, we use in the following the solution in Eq.\ref{a4}.
Then we investigate the effect of dispersive reservoir on the
dynamical properties.
\subsubsection{Dynamical properties of coherence phenomenon}
\hspace{0.5cm}
In this part, we study the behavior of purity as a measure of the
amount of coherence loss caused by the dispersive reservoir of qubit
and the unitary qubit-vibrational mode interaction.
The purity of this state is measured by the von Neumann entropy
\cite{b19} as follows:
\begin{eqnarray}
S_{qu(r)}(t)&=& -\sum_{i=1}^{\infty}\lambda^{i}_{qu(r)}(t)\ln
\lambda^{i}_{qu(r)}(t),
\end{eqnarray}
where $\lambda^{i}_{qu(r)}$ are the instantaneous eigenvalues of the
two subsystems (qubit and torsional resonator).
In a bipartite quantum system, the system and subsystem entropies,
at any time $t$, are limited by the following Araki-Lieb
inequality
$ |S_{qu}-S_{r}|<S<|S_{qu}+S_{r}|. $
As a result, if the density operator $\rho$ describes a pure state,
the entropy $S$ becomes $0$ and remains constant.
This means that, if the system is initially prepared in a pure state
at any time $t>0$, the reduced entropies of the two subsystems are
identical, that is, $S_{qu}(t) = S_{r} (t)$.
Otherwise $S>0$, that is, $S_{qu}(t) \neq S_{r} (t)$.
\\ \indent
Here, we begin from pure state, therefore, we either use $S_{r}$ or
$S_{qu}$ in the absence of dispersive reservoir, to measure the
amount of entanglement between the two subsystems, whether in the
resonance or off-resonance cases.
Exactly, $S_{qu}(0)=S_{r}(0)=0$.
In this work, we devote our study to the von Neumann entropy for the
reduced density operator of the qubit that can be expressed as
follows:
\begin{eqnarray}
S_{qu}(t)&=&-\sum_{i=\pm} \lambda^{i}_{qu}(t)\ln
\lambda^{i}_{qu}(t),
\end{eqnarray}
where $\lambda^{\pm}_{qu}(t)
=\frac{1}{2}\Big(\rho^{00}_{ee}+\sum_{n=0}^{\infty}
[\rho^{n,n}_{ee}(t)+\rho^{n,n}_{gg}(t)]\Big)
\pm\sqrt{\Lambda^{2}_{n}(t) +\Upsilon^{2}_{n}(t)}$, with $
\Lambda_{n}(t)=\frac{1}{2}\Big(\rho^{00}_{ee}+\sum_{n=0}^{\infty}
[\rho^{n,n}_{ee}(t)-\rho^{n,n}_{gg}(t)]\Big)$\,\,\, and \,\,\,
$\Upsilon_{n}(t)=|\sum_{n=0}^{\infty}\rho^{n,n+1}_{ge}(t)|$.
\begin{figure}
\hspace{-1cm}\includegraphics[height=9cm,width=10cm]{u.pdf}\includegraphics[height=9cm,width=10cm]{v.pdf}\\
\hspace{-1cm}\includegraphics[height=9cm,width=10cm]{w.pdf}\includegraphics[height=9cm,width=10cm]{r.pdf}
\caption{The evolution of purity plotted against the scaled time
$\frac{Gt}{\pi}$ and dispersive reservoir $\frac{\gamma}{G}$ in the
resonance case for $\theta=\frac{\pi}{3}$ in (a), and off-resonance
case for $\theta=\frac{\pi}{3}$ and $\frac{\delta}{G}=4$ in (b),
$\theta=\frac{\pi}{4}$ and $\frac{\delta}{G}=10$ in (c) and
$\theta=\frac{\pi}{8}$ and $\frac{\delta}{G}=25$ in (d), with
$\alpha=3$.}\label{w3}
\end{figure}
 \indent
In our discussion for the influence of dispersive reservoir of qubit
on the purity, the time evolution of the purity against
$\frac{Gt}{\pi}$ and $\frac{\gamma}{G}$, in the resonance case with
$\theta=\frac{\pi}{3}$, $\varphi=0$ and $\alpha=3$, is displayed in
Fig.\ref{w3}a.
From Fig.\ref{w3}a, if $\frac{\gamma}{G}=0$, one can observe that
the behavior of curve of $S_{qu}$ is not periodical.
In addition, we see that the time evolution curve of qubit entropy
has local extreme at the revival time
$\Big(t_{R}=\frac{2\pi|\alpha|}{G}=3\sqrt{8I_{m}
\omega/E_{J}\omega_{z}{f^{'}}^{2}}\Big)$, indicating that the qubit
state does not return most closely to its initial state at this
time.
Also, one can observe that the value of minima for $S_{qu}$ in
Fig.\ref{w3}a is increased when the time increases.
Finally, the $S_{qu}$ oscillate around its maxima as time becomes
very large.
Besides that, in the presence of the dispersive reservoir
$\frac{\gamma}{G}\neq0$, we cannot use the $S_{qu}$ as a measure for
entanglement but we use it as a measure to the amount of the
coherence loss of the qubit state.
At a weak dispersive reservoir, one can find the amplitudes of the
oscillations of entropy diminish rapidly, and the local extreme of
the purity at half-revival time for the mentioned measure are still
apparent, where at this time the qubit and vibration mode are
approximately disentangled.
If $\frac{\gamma}{G}=0.1$, the amplitudes of the local extreme of
the purity oscillations are more rapidly deteriorated.
Due to damping, the amplitude of the purity of qubit is suppressed.
Therefore, the qubit coherence loss may occur due to dephasing
interactions of qubit with the vibration mode.
However, the purity loses its coherence more than its gain due to
dispersive reservoir, and finally completely loses coherence and
fall into a statistically mixed state.
\vspace{0.2cm}\\  \indent
As well as, we investigate in Fig.\ref{w3}b and Fig.\ref{w3}c,
\ref{w3}d), the evolution of purity in off-resonance case with
$\varphi=0$ and $\alpha=3$ for a different values of $\theta$ and
$\frac{\delta}{G}$.
In the absence of $\frac{\gamma}{G}$, see Fig.\ref{w3}b,
Fig.\ref{w3}c and Fig.\ref{w3}d, one can note that the detuning
parameter affects the fluctuations in the temporal evolution for
$S_{qu}$, and results in deterioration of the oscillations.
One can also note from these figures, the local extreme for the
mentioned measure are pronounced and still apparent in all curves.
However, we see that the amplitudes of oscillations of purity are
decreasing due to the increase of the value for $\frac{\delta}{G}$
in these different figures, that is the detuning parameter improves
the influence of dispersive reservoir on this phenomenon.
In addition, when $\frac{\gamma}{G}\neq0$, one can find from
Fig.\ref{w3}b, Fig.\ref{w3}c and Fig.\ref{w3}d at a weak value of
$\frac{\gamma}{G}$, that the local extreme for the mentioned measure
is about to disappear.
Also, one can observe at the high values of $\frac{\gamma}{G}$, that
the amplitudes of oscillations for all curves are deteriorated
completely, and the purity is increased monotonically and no longer
equal zero due to the influence of the dispersive reservoir.
\\ \indent
However, from Fig.\ref{w3}c for $\theta=\frac{\pi}{4}$ (phenomenon
of coherent trapping) and $\frac{\delta}{G}=10$, and from
Fig.\ref{w3}d for $\theta=\frac{\pi}{8}$ and $\frac{\delta}{G}=25$.
We note that the behavior for two curves of the purity in these
cases of detuning is periodical, and note that the oscillations
periods for two curves are apparent after a large period of time on
the case in which $\frac{\delta}{G}=4$.
Then, we conclude that the behavior of curves of purity is also
periodical for the values more than $\frac{\delta}{G}=25$ for the
different values of $\theta$.
Finally, when $\frac{\gamma}{G}=1$, we get
$\lambda^{\pm}_{qu}(Gt\rightarrow\infty)=
\frac{1}{2}\pm\frac{1}{2}e^{-|\alpha|^{2}}\cos^{2}{\theta}$, then
$S_{qu}(Gt\rightarrow\infty)\sim\ln2=0.693$, which means that the
asymptotic qubit state is a statistically mixed state, that is the
mixedness of the phase qubit state is complete.
\subsubsection{Dynamical properties of  entanglement phenomenon}
\hspace{0.5cm}
Because of the existence of dispersive reservoir parameter in
Eq.\ref{a4}, the purity cannot be a measure for the entanglement.
Therefore, we use the negativity $N(\rho)$ to measure the amount of
the entanglement that can be expressed as \cite{b20}
\begin{eqnarray}
N(\rho) &=& \max\Bigg(0, -\sum_{s}\lambda_{s}\Bigg),
\end{eqnarray}
where $\lambda_{s}$ is a negative eigenvalue of the partially
transposed density matrix of the vibrational mode-phase qubit.
Here, $\lambda_{s}$ is computed by using numerical calculations.
For an entangled mixed state, $N(\rho)$ is positive.
If $N(\rho)=0$, the states are separable.
Moreover, the negativity is used to estimate the amount of
entanglement of the final state if we start with a pure or mixed
state.
\\ \indent
The influence of dispersive reservoir of qubit on the negativity as
a function of $\frac{Gt}{\pi}$ and a parameter $\frac{\theta}{\pi}$
in the resonance case when  $\varphi=\frac{\pi}{4}$ and $\alpha=3$
is shown in Fig.\ref{w4}a and Fig.\ref{w4}b.
From Fig.\ref{w4}a, in the absence of $\frac{\gamma}{G}$, we note
that the local extremes of $N(\rho)$ approximately are regularly
distributed at small times $ Gt\simeq10\pi$ with the different
values of $\theta$.
After that, the values of minimum for $N(\rho)$ are increased when
the time increases.
Also, the time evolution curve of $N(\rho)$ has a local extreme at
the revival time $t_{R}$.
However, we note that, the value of $N(\rho)$ is oscillated around
its maxima for the different values of $\theta$ and $Gt$.
If we take into account the dispersive reservoir parameter, the
amplitudes of the fast oscillations of the negativity measure
diminish, and $N(\rho)$ arrives to its minima (maxima) value at
half-revival time.
The negativity has a very strong sensitivity to $\frac{\gamma}{G}$
parameter.
Therefore, when we take $\frac{\gamma}{G}=0.1$, see Fig.\ref{w4}b,
the $N(\rho)$ quite vanishes at particular times for the different
values of $\theta$.
In this case, the dispersive reservoir at these particular times,
the entanglement between the vibrational mode and qubit is
completely destroyed.
If $N(\rho)=0$, the final state shown in Eq.\ref{a4} disentangles
completely and abruptly in just a finite time and evolves to the
vacuum state, and its coherence is lost completely.
\begin{figure}
\hspace{-1cm}\includegraphics[height=9cm,width=10cm]{n1.pdf}\includegraphics[height=9cm,width=10cm]{n2.pdf}\\
\hspace{-1cm}\includegraphics[height=9cm,width=10cm]{n3.pdf}\includegraphics[height=9cm,width=10cm]{n4.pdf}
\caption{The time evolution of the entanglement $N(\rho)$ in the
resonance case for all $\theta$ and off-resonance case for
$\theta=\frac{\pi}{7}$, with $\varphi=\frac{\pi}{4}$ and $\alpha=3$,
for the different values of dispersive reservoir:
$\frac{\gamma}{G}=0$ in (a, c) and $\frac{\gamma}{G}=0.1$ in (b,
d).}\label{w4}
\end{figure}
\\ \indent
To visualize the influence of detuning parameter and
$\frac{\gamma}{G}$ parameter on the temporal evolution for
negativity, the behavior of the negativity in off-resonance case
with $\theta=\frac{\pi}{7}$, $\alpha=3$ and $\varphi=\frac{\pi}{4}$
is displayed in Fig.\ref{w4}c and Fig.\ref{w4}d.
One can find that, the detuning parameter $\frac{\delta}{G}\neq0$
affects the amplitudes of fluctuations in the evolution of $N(\rho)$
either by increasing or by decreasing on the case in which
$\frac{\delta}{G}=0$ for the different values of $\theta$.
Therefore, if $\theta=\frac{\pi}{7}$, see Fig.\ref{w4}c, the
amplitudes of oscillations of $N(\rho)$ are less than their
counterparts of the case $\frac{\delta}{G}=0$.
Because of the detuning parameter, some of oscillations are
deteriorated, the local extremes of the negativity are pronounced
and still apparent in this surface.
In addition, with $\frac{\gamma}{G}\neq0$, the oscillations of the
local extreme of the $N(\rho)$ are disappeared completely.
When the dispersive reservoir is increased, say $\frac{\gamma}{G} =
0.1$, see Fig.\ref{w4}d, the amplitude of most oscillations are
suppressed and $N(\rho)$ quite vanishes at particular times.
This means that, after a particular time, the dispersive reservoir
destroys the entanglement of the global total system state.
Therefore, one can determine a particular region in which there is
no entanglement between the qubit and the torsional resonator due to
the complete destruction of coherence in the presence of
$\frac{\gamma}{G}$.
\section{Conclusion}
In this article, the system of SC phase qubit coupled to a torsional
resonator is governed by a master equation.
An analytical solution for the master equation is obtained with the
general initial state.
Several physical phenomena have been studied, such as qubit
inversion, purity and negativity in the resonance and off-resonance
cases for many different initial states.
It is found that, the qubit inversion and purity, for the resonance
and off-resonance cases, are quite sensitive to any change
$\frac{\gamma}{G}$ parameter.
By using the negativity the amount of the entanglement is studied.
The dispersive reservoir destroys the entanglement of the global
vibrational mode-phase qubit state.
In addition, we found that in off-resonance case, the detuning
parameter improve the influence of the damping on qubit inversion,
purity and negativity.
Therefore, one can determine a particular region in which there is
no entanglement between the qubit and the torsional resonator due to
the presence of $\frac{\gamma}{G}$.
