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 in general will 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. In either form, the RWA gives an inaccurate quantum state which makes it inappropriate for calculating entanglement dynamics or similar detailed properties of the state dynamics. We also emphasize the fact that even under the RWA the master equation for a combination of systems and external forces is not a simple combination of the master equations of the systems and forces. Such a combination may be tempting, because the RWA guarantees that a master equation so constructed will have a valid mathematical form; however, it will not accurately reflect the dynamics of the physical system. To obtain the correct master equation for the composite system a proper consideration of the non-Markovian dynamics is required.
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 in general will 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. In either form, the RWA gives an inaccurate quantum state which makes it inappropriate for calculating entanglement dynamics or similar detailed properties of the state dynamics. We also emphasize the fact that even under the RWA the master equation for a combination of systems and external forces is not a simple combination of the master equations of the systems and forces. Such a combination may be tempting, because the RWA guarantees that a master equation so constructed will have a valid mathematical form; however, it will not accurately reflect the dynamics of the physical system. To obtain the correct master equation for the composite system a proper consideration of the non-Markovian dynamics is required.
We study the stochastic dynamics of a two-level quantum system interacting with a stochastic magnetic field, and a single frequency electromagnetic field, with and without making the rotating wave approximation (RWA). The transformation to the rotating frame does not commute with the stochastic Hamiltonian if the stochastic field has nonvanishing components in the transverse direction. Hence, making the RWA modifies the stochastic terms in the Hamiltonian. Modification of the decay terms is also required in a master equation approach (i.e., the Liouville--von Neumann density matrix equation) for describing the dynamics. For isotropic Gaussian white noise, the RWA dynamics remains Markovian, although the Lindblad terms in the master equation for the density matrix become time-dependent when the non-commutation of the RWA transformation and the noise Hamiltonian is properly accounted for. We also treat Ornstein--Uhlenbeck noise, and find, in contra-distinction to the white noise case, a significant difference in the dynamics calculated with the RWA when the non-commutation of the RWA transformation and the noise Hamiltonian is taken into account. These findings are applicable to the modeling of any open quantum system coupled to an electromagentic field.
Away from equilibrium, the properties of open quantum systems depend on the details of their environment. A microscopic derivation of a master equation (ME) is therefore crucial. Of particular interest are Lindblad-type equations, not only because they provide the most general class of Markovian MEs, but also since they are the starting point for efficient quantum trajectory simulations. Lindblad-type MEs are commonly derived from the Born-Markov-Redfield equation via a rotating-wave approximation (RWA). However the RWA is valid only for ultraweak system-bath coupling and often fails to accurately describe nonequilibrium processes. Here we derive an alternative Lindbladian approximation to the Redfield equation, which does not rely on ultraweak system-bath coupling. Applying it to an extended Hubbard model coupled to Ohmic baths, we show that, especially away from equilibrium, it provides a good approximation in large parameter regimes where the RWA fails.
Quantum master equations are an invaluable and highly successful tool for the approximate, yet accurate, description of undriven open quantum systems weakly coupled to their environment. In contrast, results on the description of the equally important case of driven open quantum systems using quantum master equations remain scarce. Existing approaches typically impose restrictions on the driving protocol, including, for instance, Floquet master equations for periodic driving and adiabatic master equations for slow driving. However, no generally accepted formulation for arbitrary driving has yet been established. A promising candidate from which all existing approaches can be derived by approximations based on protocol constraints is the driven Redfield master equation. Strikingly, it has received very little attention so far. A main reason is that, in analogy to its undriven counterpart, the driven Redfield master equation notoriously violates complete positivity. In the undriven case, this problem can be successfully addressed using the rotating wave approximation (RWA). For approaches derived from the driven Redfield master equation for specific protocols, such as Floquet master equations and adiabatic master equations, a generalized form of the RWA can still be applied. However, a full generalization directly applicable to the driven Redfield master equation under arbitrary driving has not yet been developed. To address this issue, we propose a straightforward and natural generalization of the RWA based on the version of the RWA used for adiabatic master equations. We then apply the driven Redfield master equation to the paradigmatic, exactly solvable model of a driven quantum harmonic oscillator coupled to a bosonic bath and benchmark it against the exact solution. In particular, we compare the original driven Redfield master equation without an RWA to its generalized RWA version. We find that the driven Redfield master equation without an RWA exhibits excellent agreement with the exact solution across all driving speeds, while the violation of complete positivity shows no notable effect. By contrast, the RWA-based version of the driven Redfield master equation fails to reproduce the dynamics accurately beyond slow driving at short times. Additionally, we analyze known adiabatic master equations, both without and with an RWA. Remarkably, the non-RWA versions agree with the exact dynamics far beyond the strictly adiabatic regime, while the RWA-based versions already show noticeable deviations even for slow driving, contrasting previous observations in the literature. Taken together, these results indicate that the driven Redfield master equation and adiabatic master equations without RWA have been significantly underestimated in their practical usefulness, whereas RWA-based approaches in driven settings may have been overestimated. This strongly suggests that future work should focus on alternative approximation methods applied to the driven Redfield master equation that preserve its high numerical accuracy while restoring complete positivity.
With increasing performance of actual qubit devices, even subtle effects in the interaction between qubits and environmental degrees of freedom become progressively relevant and experimentally visible. This applies particularly to the timescale separations that are at the basis of the most commonly used numerical simulation platform for qubit operations, namely, the conventional Lindblad master equation (LE): the Markov approximation and the rotating wave approximation (RWA). In this contribution, we shed light on the questions (i) to which extent it is possible to monitor violations of either of these timescale separations experimentally and (ii) which of them is the most severe to provide highly accurate predictions within (approximate) numerical schemes in relevant parameter ranges. For this purpose, we compare three simulation methods for the reduced density matrix with progressively growing accuracy. In particular, predictions for relaxation and decoherence of a qubit system in the presence of reservoirs with Ohmic and sub-Ohmic spectral densities are explored and, with the aid of proper protocols based on Ramsey experiments, the role of non-Markovianity and RWA are revealed. We discuss potential implications for future experiments and the design of approximate yet accurate numerical approaches.
We derive an easily computable quantum speed limit (QSL) time bound for open systems whose initial states can be chosen as either pure or mixed states. Moreover, this QSL time is applicable to either Markovian or non-Markovian dynamics. By using of a hierarchy equation method, we numerically study the QSL time bound in a qubit system interacting with a single broadened cavity mode without rotating-wave, Born and Markovian approximation. By comparing with rotating-wave approximation (RWA) results, we show that the counter-rotating terms are helpful to increase evolution speed. The problem of non-Markovianity is also considered. We find that for non-RWA cases, increasing system-bath coupling can not always enhance the non-Markovianity, which is qualitatively different from the results with RWA. When considering the relation between QSL and non-Markovianity, we find that for small broadening widths of the cavity mode, non-Markovianity can increase the evolution speed in either RWA or non-RWA cases, while, for larger broadening widths, it is not true for non-RWA cases.