Biorthonormal eigenbasis of a Markovian master equation for the quantum Brownian motion

Abstract

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.

Lindblad Dynamics and Disentanglement in Multi-Mode Bosonic Systems

“…Though the general physics behind Lindblad-type master equations has been extensively discussed (see, e.g., References [15][16][17][18]), the mathematical techniques developed for the analysis of bosonic systems are mostly applicable to the single-mode Lindblad equation [19][20][21][22][23][24][25] and cannot be employed to treat its multi-mode generalizations (the most general form of the multi-mode Lindblad equation is described, e.g., in [26]).…”

Spectral resolution of the Liouvillian of the Lindblad master equation for a harmonic oscillator

“…There have been trials to solve this equation (see Refs. [2][3][4]16 ). Briegel and Englert have obtained the eigenvalues and the biorthogonal eigenbasis in terms of creation and annihilation operators by using normal ordering 2 , and Tay and Petrosky also have done in terms of the Wigner function 16 .…”

Dynamics of Characteristic and One-Point Correlation Functions of Multi-Mode Bosonic Systems: Exactly Solvable Model

“…Though the Lindblad-type master equations has been extensively discussed [10][11][12][13], the bulk of the mathematical methods developed for analysis of bosonic systems are applicable to the special case of single-mode Lindblad Equations [14][15][16][17][18][19]. These techniques cannot be employed to treat its multi-mode generalizations.…”