## The widening attention in the investigation of dissipative systems has its origin in the early days of experimental works in quantum mechanics [1

The widening attention in the investigation of dissipative systems has its origin in the early days of experimental works in quantum mechanics 1, 2. Since the combination of the quantum mechanics and damping concept is a difficult problem, there have been many endeavors to produce different realistic models 3, 4. For example, the electromagnetic field is modeled with a damped oscillator 9 which is arisen from its classical analogue. in one of the most fundamental approaches to the damped systems in quantum mechanics, In this way and after pioneer work of Bateman 1,the Hamiltonian which was presented by Caldirola and Kanai has provided a vast area of research. The Caldirola–Kanai (CK) Hamiltonian illustrates a damped harmonic oscillator with a time-dependent mass 10, 11
\begin{eqnarray}\label{eq:I1}
H_{CK}=\frac{\hat{p}^{2}}{2m_{0}}\exp(-2\gamma t)+\frac{1}{2}m_{0}\omega^{2}\hat{q}^{2}\exp(2\gamma t),
\end{eqnarray}
with, the damping parameter $\gamma$, frequency $\omega$,
and the initial mass which is given by $m_{0}$ \\
In fact. The Caldirola–Kanai Hamiltonian is used in numerous quantum systems to study the various physical properties, for instance, plasma environments13 and mesoscopic RLC circuits 12. In this regard, by some canonical transformations, it is possible to construct generalized creation and annihilation operators which are defined in terms of the standard $\hat{a}^{\dag}$ and $\hat{a}$as
\begin{eqnarray}\label{eq:I2}
\end{eqnarray}
with $\Omega=\omega\sqrt{1-\eta^{2}}$, $\eta=\frac{\gamma}{\omega}$ and $\zeta_{\pm}=\Omega+i\gamma\pm\omega$. It can be easily verified that the relation $\hat{A},\hat{A}^{\dag}=1$ is satisfied. Then the transformed Hamiltonian (\ref{eq:I1}) in terms of $\hat{A},~\hat{A}^{\dag}$
multi-mode field instead of a single-mode field 26 and multi-level atoms 23–25 instead of two-level atoms etc. In fact, all of these investigations 30–33 have been taken in an ideal form, where the damping influence is ignored. Especially, a $\Lambda$-type three-level atom has been considered in many studies . Based on the above. In this article, we will study the damping effect on the interaction between a $\Lambda$-type three-level atom and a single-mode field, where the Hamiltonian of the field is performed based on the Caldirola– Kanai damping Hamiltonian.
Section 2, we obtain the state vector of the whole system. In Section 3, some interesting physical properties such as DEM, atomic population inversion, fidelity and, quasiprobability distribution function $Q$ are collected for the system, numerically. Finally, the summery and conclusion is provided