We study the most general continuous transformation on the generators of bilinear master equations of a quantum oscillator. We find that transformation operators that preserve the hermiticity of density operators and conserve the probability of reduced dynamics should be adjoint-symmetric, and they are not limited to the pure product of unitary operators in the bra and ket space but could be a mixture of them. We need to include the more general transformation operators to explore the full symmetry of generic reduced dynamics. We discuss how the operators are related to those considered in previous works, and illustrate how they leave the reduced dynamics form invariant, or map one into the other. The positive semidefinite requirement on the density operator can be imposed to give a valid range of transformation parameters.

The quantum Markovian master equation of the reduced dynamics of a harmonic oscillator coupled to a thermal reservoir is shown to possess thermal symmetry. This symmetry is revealed by a Bogoliubov transformation that can be represented by a hyperbolic rotation acting on the Liouville space of the reduced dynamics. The Liouville space is obtained as an extension of the Hilbert space through the introduction of tilde variables used in the thermofield dynamics formalism. The angle of rotation depends on the temperature of the reservoir, as well as the value of Planck's constant. This symmetry relates the thermal states of the system at any two temperatures. This includes absolute zero, at which purely quantum effects are revealed. The Caldeira-Leggett equation and the classical Fokker-Planck equation also possess thermal symmetry. We compare the thermal symmetry obtained from the Bogoliubov transformation in related fields and discuss the effects of the symmetry on the shape of a Gaussian wave packet.

The solution to a quantum Markovian master equation of a harmonic oscillator weakly coupled to a thermal reservoir is investigated as a non-Hermitian eigenvalue problem in space coordinates. In terms of a pair of quantum action-angle variables, the equation becomes separable and a complete set of biorthogonal eigenfunctions can be constructed. Properties of quantum states, such as the change in the quantum coherence length, damping in the motion, and disappearance of the spatial interference pattern, can then be described as the decay of the nonequilibrium modes in the eigenbasis expansion. It is found that the process of gaining quantum coherence from the environment takes a longer time than the opposite process of losing quantum coherence to the environment. An estimate of the time scales of these processes is obtained.

We derive an exact Markovian kinetic equation for an oscillator linearly coupled to a heat bath, describing quantum Brownian motion. Our work is based on the subdynamics formulation developed by Prigogine and collaborators. The space of distribution functions is decomposed into independent subspaces that remain invariant under Liouville dynamics. For integrable systems in Poincaré's sense the invariant subspaces follow the dynamics of uncoupled, renormalized particles. In contrast, for nonintegrable systems, the invariant subspaces follow a dynamics with broken time symmetry, involving generalized functions. This result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces. We comment on the relation between our Markovian kinetic equation and the Hu-Paz-Zhang equation.

We obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the Gaussian form. The well-behavedness and positive semidefiniteness of the stationary states could be characterized by a three-dimensional Minkowski vector. By requiring the stationary states to satisfy a factorized condition, we obtain a generic class of master equations that includes the well-known ones and their generalizations, some of which are completely positive. A further subset of the master equations with the Gibbs states as stationary states is also obtained. For master equations with not completely positive generators, an analysis on the stationary states for a given initial state isuggests conditions on the coefficients of the master equations that generate positive evolution.When we obtain the master equations of a system in contact with an environment, there are often assumptions made to facilitate the derivation of the effective influence of the environment on the system [1,2]. For instance, we assume that the initial density matrix of the system and environment is factorized, the coupling between the system and its environment is weak, the rotating wave approximation is valid, the memory effects are negligible, or the generators of the master equation should be completely positive, and etc. Appropriate derivations should produce master equations with well-behaved solutions.However, anomaly could arise if we carelessly apply the master equations to situations that are not consistent with the assumptions made in their derivations, such as studying low temperature behavior of a system with the Caldeira-Leggett equation [3,4,5], obtaining an Markovian equation by ignoring the memory effects or using the rotating wave approximation [6,7,8,9], or starting with initial conditions inconsistent with the factorized assumptions [10,11].An issue that has received wide attention is the completely positive [12,13] nature of the generators of the master equations. Initial states that are positive could evolve outside their positive domain during some interval of the time evolution when the master equations do not have completely positive generators. However, there is no a priori reason why master equations must be completely positive [10]. Not completely positive generators with appropriately chosen coefficients could maintain the positivity of the states for a given initial state for all time [14]. The systems are then properly behaved as far as positivity is concerned.It is in this spirit that we start with a generic master equation that satisfies the essential hermitian and trace preserving requirements of a reduced dynamics [15,16,17]. We obtain its explicit solutions in Gaussian form and analyzed the stationary states. We further identify a factorized condition on the stationary states that is strong enough to produce a generic class of master equations. This class includes the well-known master equations as special cases, i.e., the Kossakowski-Lindblad (KL) equation for quantum optical ...