Non-Markovian Dynamics and Entanglement of Two-level Atoms in a Common Field

We derive the stochastic equations and consider the non-Markovian dynamics of a system of multiple two-level atoms in a common quantum field. We make only the dipole approximation for the atoms and assume weak atom-field interactions. From these assumptions we use a combination of non-secular open- and closed-system perturbation theory, and we abstain from any additional approximation schemes. These more accurate solutions are necessary to explore several regimes: in particular, near-resonance dynamics and low-temperature behavior. In detuned atomic systems, small variations in the system energy levels engender timescales which, in general, cannot be safely ignored, as would be the case in the rotating-wave approximation (RWA). More problematic are the second-order solutions, which, as has been recently pointed out, cannot be accurately calculated using any second-order perturbative master equation, whether RWA, Born-Markov, Redfield, etc.. This latter problem, which applies to all perturbative open-system master equations, has a profound effect upon calculation of entanglement at low temperatures. We find that even at zero temperature all initial states will undergo finite-time disentanglement (sometimes termed sudden death), in contrast to previous work. We also use our solution, without invoking RWA, to characterize the necessary conditions for Dickie subradiance at finite temperature. We find that the subradiant states fall into two categories at finite temperature: one that is temperature independent and one that acquires temperature dependence. With the RWA there is no temperature dependence in any case.

The Rotating-Wave Approximation: Consistency and Applicability from an Open Quantum System Analysis

We provide an in-depth and thorough treatment of the validity of the rotating-wave approximation (RWA) in an open quantum system. We find that when it is introduced after tracing out the environment, all timescales of the open system are correctly reproduced, but the details of the quantum state may not be. The RWA made before the trace is more problematic: it results in incorrect values for environmentally-induced shifts to system frequencies, and the resulting theory has no Markovian limit. We point out that great care must be taken when coupling two open systems together under the RWA. Though the RWA can yield a master equation of Lindblad form similar to what one might get in the Markovian limit with white noise, the master equation for the two coupled systems is not a simple combination of the master equation for each system, as is possible in the Markovian limit. Such a naive combination yields inaccurate dynamics. To obtain the correct master equation for the composite system a proper consideration of the non-Markovian dynamics is required.