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 analog. 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
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),
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 environments 13 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
\hat{A}=\frac{1}{2\sqrt{\Omega \omega}}(\zeta_{+}\hat{a}+\zeta_{-}\hat{a}^{\dag}),\qquad\hat{A}^{\dag}=\frac{1}{2\sqrt{\Omega \omega}}(\zeta_{+}^{*}\hat{a}^{\dag}+\zeta_{-}^{*}\hat{a}),
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 $\hat{A},~\hat{A}^{\dag}$
reads as
On the other hand, the atom-field interaction in its quantum approach has unlocked a wide area of research in quantum optics studies. Jaynes and Cummings suggested a model by which one can study the interaction between a two-level atom with a single-mode cavity field 3. Despite the simplicity of this model, is noteworthy its ability to create some non-classical phenomena observed in the laboratory. Due to these facts, the model has been extended widely. In this regard, we may refer to the generalization of the intensity-dependent atom-field coupling 27–29,
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.
After characterization of the considered system,
we will get on the probability amplitudes of the entangled atom-field system.
In addition to that, the time behavior of some physical properties such as atomic population inversion, the degree of entanglement, quasiprobability distribution function and fidelity are investigated, numerically and study the effect of damping parameter on this properties.\\
The paper is prepared as follows. In
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 summary and conclusion is provided
in Section 4