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

Abstract

Biorthonormal eigenbasis of a Markovian master equation for the quantum Brownian motionHarmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system A Lindblad master equation for a harmonic oscillator, which describes the dynamics of an open system, is formally solved. The solution yields the spectral resolution of the Liouvillian, that is, all eigenvalues and eigenprojections are obtained. This spectral resolution is discussed in depth in the context of the biorthogonal system and the rigged Hilbert space, and the contribution of each eigenprojection to expectation values of physical quantities is revealed. We also construct the ladder operators of the Liouvillian, which clarify the structure of the spectral resolution.

Solving the quantum master equation of coupled harmonic oscillators with Lie-algebra methods

“…The reason is that, in general, the dual space does not equal the Liouville space itself which is especially true for non-Hermitian open systems. This problem can be treated rigorously in the framework of rigged Hilbert spaces [15], and it can be shown [16] that the Liouville space constructed from the Hilbert space spanned by bosonic Fock states does possess a complete basis.…”

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]).…”

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

“…where α 0 = α(0). It is not difficult to obtain the solution along the characteristic curves (18) given by…”

“…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.…”