Analyzing Density Operator in Thermal State for Complicated Time-Dependent Optical Systems

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. 1. Introduction 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 [1–11]. As the knowledge of quantum mechanics deepen, the understanding of quantum characteristics of optical systems that have time-dependent parameters has become important as the counterpart of classical ones. Several methods for solving quantum solutions of TDOSs are known so far. They are invariant operator method [12], propagator method [13], and unitary or canonical transformation method [14]. Invariant operator method is the most common among them. In traditional quantum mechanics, the state of a quantized system is described in terms of the state vector . However, if we use the density operator, the state can be described more generally. For instance, it is possible to treat the interaction between light and matter in terms of the density operator. The density operator of a system is obtained by making use of wave functions satisfying Schr？dinger equation and can be used to derive various expectation values of physical quantities in thermal state. 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 [15]. The density operator representation is a complete description of the information for a given system because it gives the knowledge that enables us to evaluate the outcome of any statistical measurement. If we consider that any quantum state regardless of being pure or mixed can be described by a single density matrix, the density operator