We propose a method to compute expectation values of observables in the stationary state of a (Markovian) bosonic open quantum system. Using a hierarchy of semi-definite relaxations, we obtain finer and finer upper and lower bounds to any expectation value of interest. The bounds are rigorous, robust to stationary state degeneracies, and numerically improve as the occupation number increases on the examples we considered. This makes it adapted to the simulation of stationary states of bosonic qubits and in particular dissipatively stabilized cat qubits.
We point out that the quantum dynamical map of an open quantum system can be generated by an effective Liouville operator. The effective Liouville shows the dynamical breaking of time reversibility. This breaking of reversibility is expressed by the effective Liouville's discrete spectrum having negative imaginary parts. This generated dynamics of open quantum systems is capable of memory effects described by a frequency dependence of the spectrum. When memory effects can be neglected or smoothed out, the effective Liouville generates the well known semi-group dynamics of open systems in the Markov approximation. The spectral analysis of the effective Liouville -with or without memory- allows to represent the quantum dynamical map and expectation values of physical quantities by metric expressions using complex eigenvalues and bi-orthonormal eigenmodes. The long time dynamics is dominated by the zero-mode, which forms the stationary state of the open system. The relaxation time scale
τ is set by the inverse of the smallest negative imaginary part of the non-vanishing effective eigenvalues of the effective Liouville. In generic cases of system-environment-coupling this time scale
τ is the relevant scale for both, the relaxation of diagonal elements (populations) and the decay of off-diagonal elements (coherences) of an initial density matrix in the stationary state's diagonal representation. It also sets the time scale after which the entropy tends to its stationary value. The effective Liouville and its spectral content thus give a framework to study the dynamic breaking of time reversibility, memory effects, decoherence, relaxation to a stationary state and the role of entropy in open quantum systems.
Left alone, open quantum systems relax toward a Kubo--Martin--Schwinger (KMS) equilibrium state, yet the inner workings of this process remain opaque. It remains unclear why the intricate fourth-order time-convolutionless (TCL4) population generator reproduces the comparatively simple second-order stationary state corrections. Even more surprisingly, stationary-state corrections of such precision can be compressed into a simple virtual coherence pathway: an open quantum systems analogue of virtual transitions, in which populations communicate through coherences that are never occupied.
Here we show that this simplification arises through a sequence of stationary-state-preserving transformations that progressively eliminate asymptotically redundant components while preserving the stationary state, ultimately yielding the virtual coherence pathway. This resolves the longstanding stationary-state problem of the Redfield equation and reveals that much of the apparent complexity of the TCL4 generator is asymptotically redundant.