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Density operator of oscillatory optical systems with time-dependent parameters is analyzed. In this case, a system is described by a time-dependent Hamiltonian. Invariant operator theory is introduced in order to describe time-varying behavior of the system. Due to the time dependence of parameters, the frequency of oscillation, so-called a modified frequency of the system, is somewhat different from the natural frequency. In general, density operator of a time-dependent optical system is represented in terms of the modified frequency. We showed how to determine density operator of complicated time-dependent optical systems in thermal state. Usually, density operator description of quantum states is more general than the one described in terms of the state vector.

Quantum behavior of time-dependent optical systems (TDOSs) is an interesting topic of study, that has attracted great concern in the literature of quantum physics for the last several decades [

In traditional quantum mechanics, the state of a quantized system is described in terms of the state vector

The density operator will be described in thermal state using standard formalism relevant to invariant operator method. We study how to construct density operator in a general way by paying attention to the fact that density operator is represented in terms of the modified frequency of the system [

This paper is organized as follows. Preliminary quantum treatment of TDOSs is represented in Section

Sometimes, parameters of an optical system may vary with time via its interaction with surroundings. In this case, the system is described by a time-dependent Hamiltonian. Let us consider a time-dependent Hamiltonian of the form

To see the quantum features of TDOSs, it is useful to find a constant of motion (invariant operator). From

Quantum theory for describing the statistical state of optical systems requires the formalism of the density operator which is the quantum analogue of a phase space density given in classical statistical mechanics. From the Liouville-von Neumann equation,

In fact

Recall that, in the previous section, the density operator of the TDOS is represented in terms of

As analogy to (

In terms of

Now we will show how to determine

Hence, if we consider that

In case of

Let us consider another case that the Hamiltonian has the time function of the form

Finally, we consider a more complicated case that the Hamiltonian has the time function of the form

From the choice of different time dependence of parameters, we can obtain different time evolutions of density operators with the different initial states. Density operator representation of a quantum optical system is equivalent to the wave function representation. However, there are many actual advantages of the use of density operators for quantum description of a certain TDOS, in particular, for the description of linear and nonlinear spectroscopy, thermal characteristics, relaxation in the condensed phase, and representation of photon states in time-varying media.

Density operator of a quantum TDOS is investigated in thermal state using invariant operator method. It is shown from Liouville-von Neumann equation for density operator that density operator is expressed in more general form, when the system contains time-dependent parameters, as given in (

We applied our theory to several examples of actual systems by choosing appropriate time functions. All calculations have been fulfilled within the theory of a quantized optical system with time-dependent parameters. As the system becomes complicate, the expression of the density operator is more or less intricate. On the other hand, for the case of optical wave described by the damped harmonic oscillator, the density operator reduces to the well known one [

Here, we show under the limit of standard damped harmonic oscillator that

We represent a somewhat different way for determining

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant no. NRF-2013R1A1A2062907).