An exactly solvable model for the multi-level system interacting with several reservoirs at zero temperatures is presented. Population decay rates and decoherence rates predicted by exact solution and several approximate master equations, which are widespread in physical literature, are compared. The space of parameters is classified with respect to different inequalities between the exact and approximate rates.

Non-Markovian Evolution of Multi-level System Interacting with Several Reservoirs. Exact and Approximate

We study the perturbative corrections to the { Gorini-Kossakowski-Sudarshan-Lindblad equation which arises in the weak coupling limit}. The spin-boson model in the rotating wave approximation at zero temperature is considered. We show that the perturbative part of the density matrix satisfies the time-independent Gorini-Kossakowski-Sudarshan-Lindblad equation for arbitrary order of the perturbation theory (if all the moments of the reservoir correlation function are finite). But to reproduce the right asymptotic precision at long times, one should use { an initial condition different} from the one for exact dynamics. Moreover, we show that the initial condition for this master equation even fails to be a density matrix under certain resonance conditions.

Non-perturbative effects in corrections to quantum master equation arising in Bogolubov-van Hove limit.

The model of a multilevel system interacting with several reservoirs is considered. The exact reduced evolution of a system’s density matrix can be obtained for this model without using the Markov approximation. Namely, this evolution is completely defined by the finite set of linear differential equations. The results obtained earlier for one Lorentz peak in the spectral density are generalized to the case of an arbitrary number of such peaks. The contribution of Ohmic spectral density is also considered.

Exact Non-Markovian Evolution with Several Reservoirs

The Jaynes–Cummings model with or without rotating-wave approximation plays a major role to study the interaction between atom and light. We investigate the Jaynes–Cummings model beyond the rotating-wave approximation. Treating the counter-rotating terms as periodic drivings, we solve the model in the extended Floquet space. It is found that the full energy spectrum folded in the quasi-energy bands can be described by an effective Hamiltonian derived in the high-frequency regime. In contrast to the Z 2 symmetry of the original model, the effective Hamiltonian bears an enlarged U(1) symmetry with a unique photon-dependent atom-light detuning and coupling strength. We further analyze the energy spectrum, eigenstate fidelity and mean photon number of the resultant polaritons, which are shown to be in accordance with the numerical simulations in the extended Floquet space up to an ultra-strong coupling regime and are not altered significantly for a finite atom-light detuning. Our results suggest that the effective model provides a good starting point to investigate the rich physics brought by counter-rotating terms in the frame of Floquet theory.

Effective Hamiltonian of the Jaynes–Cummings model beyond rotating-wave approximation*

We study the entanglement (measured by negativity) evolution and the non-Markovianity for the dynamical process of a spin-S system embedded in dephasing environments. The exact analytical solution is presented, which shows that the decoherence function governs the evolutions of coherence, entanglement, and the non-Markovianity of the corresponding dynamical processes. For Ohmic and sub-Ohmic reservoirs, the negativity decreases monotonically in time and the corresponding dynamics is Markovian. While for super-Ohmic reservoirs with non-monotonic decoherence function, the negativity appears as the phenomenon of revival and the corresponding dynamics is non-Markovian. The relation between non-Markovianity and the system dimension is studied.

Entanglement and non-Markovianity of a spin-S system in a dephasing environment

In the limit of weak coupling between a system and its reservoir, we derive the time-convolutionless (TCL) non-Markovian master equation for a two-level system interacting with a zero-temperature structured environment with no rotating wave approximation (NRWA). By comparing the dynamics with RWA, we demonstrate the impact of RWA on the system dynamics, as well as the effects of non-Markovianity on the preservation of atomic coherence, squeezing, and entanglement.

https://iopscience.iop.org/article/10.1088/1674-1056/22/3/030304/pdf

Xu Tian-Tian

Papers

Comparison between non-Markovian dynamics with and without rotating wave approximation