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

Abstract

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.

22pages, no figure; title changed, minor corrections, references added; minor corrections

Cited by in corpus (10)

• Generalized Gibbs state with modified Redfield solution: Exact agreement up to second order
• Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between $\mathcal{P}\mathcal{T}$ and anti-$\mathcal{P}\mathcal{T}$ symmetries
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations
• Point-gap topology with complete bulk-boundary correspondence in dissipative quantum systems
• Intrinsic mechanisms for drive-dependent Purcell decay in superconducting quantum circuits
• Eigenvalues of the Liouvillians of Quantum Master Equation for a Harmonic Oscillator
• Analytic approach to dynamics of the resonant and off-resonant Jaynes-Cummings systems with cavity losses
• Liouvillian exceptional points in continuous variable system
• Lindblad dynamics of open multi-mode bosonic systems: Algebra of bilinear superoperators, exceptional points and speed of evolution
• Evolution of Gaussian mixed states under the Markovian master equation for a driven quantum oscillator