The study of dissipative systems and their quantization is of a great theoretical and practical value in view of many different situations in which dissipative phenomena with a quantum origin manifest themselves [1] . Within these dissipative systems, the damped harmonic oscillator (DHO) is the simplest quantum systems displaying the dissipation of energy. It is of great physical importance and has found many applications especially in quantum optics [2] [3] . For example, it plays a central role in the quantum theory of lasers and masers [4] . Moreover, damped harmonic oscillators are used to investigate the quantum decoherence (QD) phenomenon whose role became relevant in many interesting physical problems such as quantum computation and quantum information processing [5] , material science [6] - [8] , heavy ion collisions [9] , quantum gravity and cosmology [10] - [22] , and condensed matter physics [23] - [25] . In many cases, physicists are interested in understanding the causes of QD in order to prevent decoherence from damaging quantum states and protect the information stored in these states. Thus, decoherence is responsible for washing out the quantum interference effects which are desirable to be seen as signals in some experiments. However, QD has negative effects in many areas such as quantum computation and quantum control of atomic and molecular processes. The physics of information and computation is a domain where decoherence is an obvious major obstacle in the implementation of information-processing hardware. It takes advantage on the superposition principle [26] . QD is a condition that has to be satisfied in order that a system could be considered as classical. This condition requires that the system should be in one of relatively permanent states (called by Zurek “preferred states”) and the interference between different states should be negligible [27] [28] . The loss of coherence can be achieved by introducing an interaction between the system and environment [23] [29] [30] .

Nowadays, a great deal of research is dedicated to understanding decoherence in harmonic oscillator [31] [32] . Isar et al. [33] determine the degree of quantum decoherence of a harmonic oscillator interacting with a thermal bath using Lindblad theory [34] [35] . Other authors [36] use a semi-classical approach to examine decoherence in a harmonic oscillator coupled to a thermal harmonic bath. Darius et al. [37] exploit the Feynman path integral to study the memory in a non-locally damped oscillator. Moreover, Ozgur et al. [32] determine the time dependence of Leipnik’s entropy in the damped harmonic oscillator via path integral techniques. Another strategy to describe dissipative quantum systems is based on the idea of Bateman [38] .

In this paper, we investigate the coherence of the damped harmonic oscillator using the Caldirola-Kanai model [39] but based on the idea of Bateman [38] . This model is known as a popular model used to describe dissipative systems coupled to a harmonic bath. It has many applications like reproducing classical effects or giving a good Hamiltonian necessary to exhibit the phenomenon of decoherence [38] . This paper is organized as follows. In Section 2, we present the mathematical tools based on the path integral formalism. We also discuss the case of the damped harmonic oscillator and build the associated propagator. In Section 3, we derive the general expressions of the thermodynamic parameters of the system, and analyze the effects of the damping constant on the distribution probability and the Shannon entropy for a single Gaussian wave function. The influence of the system’s frequency has also been measured on these parameters. Hence, we consider a double Gaussian wave function and make numerical investigations appreciate the impact of the damping constant, the distance separating the two wave functions and the system’s frequency on the thermodynamic characteristics of the system. Discussion and concluding remarks are given in the last section.

We start by presenting the model which consists of a particle of mass, labeled by the position variable and the momentum. Then follows the description of the used mathematical tools which is the path integral formalism introduced by Feynman [40] . These tools suggest that the transformation function called propagator is

analogue to in which stands for the action, solution of Hamilton-Jacobi equation. On the other

way, the transition amplitude of the particle (of mass) from the position at time to the position at time, known as the propagator, represents the solution of the Schrodinger equation. Nowadays, several problems of physics are solved via these techniques [32] [41] .

Next, we consider the Bateman Hamiltonian [38] defined as:

where and are the mirror variables corresponding to the coordinate and the momentum. The quantities and are respectively the damped coefficient and the system frequency. The associated lagrangian is given by

Using Euler-Lagrange equation, we derive the following two motion equations [42] :

Bateman’s dual Hamiltonian describes classical mechanics correctly, but this model faces some difficulties. It violates Heisenberg’s principle for. Therefore, to solve quantum mechanical problem, Caldirola-Kanai [39] build a theory based on the idea of Bateman dissipative system by considering the standard Hamiltonian of harmonic oscillator with time dependent mass given by [39] . Hence, the Hamiltonian and the Lagrangian of the system become respectively:

Here, is the system frequency. From the Lagrangian theory and exploiting quantities (4), the equation of motion takes the form:

The classical solution of (5) is given by

where in and are complex quantities defined as:, with. The integration constants and are evaluated when the particle moves from the position at the time to the position at time. The determination of the propagator is convenient for founding quantum mechanical solution for this Hamiltonian. Therefore, the classical action is defined as:

whose computation for the current study case leads to:

in which we set:

From the classical action, the expression of the corresponding propagator is defined below.

Substituting (7) into (8), we obtain the following expression for the quantum propagator of damped harmonic oscillator:

This result is identical to the one establish in [41] using the propagator method developed by Um et al. [43] . It also appears from (9) that the propagator depends on the damped coefficient that links the system with the environment in which it evolves.

Hereafter, we intend to use the propagator (9) and derive some characteristic parameters (such as the distribution probability and the Shannon entropy) of the system subjected respectively to single and double Gaussian wave functions. These investigations aim to measure the impact of the environment on the behavior of the system when the latter progresses.

In this paper, we have examined the dynamics of the modified damped harmonic oscillator constructed from the Caldirola-Kanaï model and based on the idea of Bateman. For this purpose, the Feynman path integral method has been used to investigate the time dependent probability and the entanglement entropy exploiting the single and double Gaussian initial states. In these two initial states, we have shown that the Shannon entropy decreases with the system frequency and grows with the others parameters (such as time or damped factor). For both cases, we have established that the distribution probability possesses the same behavior. In the specific case of the double Gaussian approximation, we have obtained that the distribution probability and the Shannon’s entropy have been improved by the distance between two consecutive peaks of the wave.

In the absence of the damped factor, we have recovered the results that link with the system in its pure state. Here, the information is exchanged periodically between the corresponding harmonic oscillator and the environment. For non-zero values of the damped factor, it has appeared that this transfer of information loses its periodicity traducing the loss of information in the system (decoherence phenomena). These results could be of great interest for engineering purposes since it becomes necessary to control the effects of the environment on the evolution of the system in order to reduce its decoherence. This phenomenon of controlling decoherence in an evolving system is essential in the construction of quantum computers that need the use of systems taken in their

different superposition states. Our study has also shown that such aim could be achieved by acting on the damped coefficient, the frequency of oscillation and/or the type of state as control parameter.