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Chinese Physics B

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

Ning Tang, Tian-Tian Xu, Hao‐Sheng Zeng

Published Mar 1, 2013• 14 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/22/3/030304

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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.
The decoherence of two initially entangled qubits in non-Markovian reservoirs has been investigated exactly without Born-Markovian approximation or rotating-wave approximation (RWA). The nonperturbative quantum master equation is derived and its exact solution is obtained. The decoherence behavior of two qubits, initially entangled in Bell states, has been investigated in the weak coupling regime and strong coupling regime. The results show that the counter-rotating wave terms have obvious influence on the decoherence behavior in the non-Markovian regime.
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 (Fleming and Cummings 2011 Phys. Rev. E 83 031117), cannot be accurately calculated using any second-order perturbative master equation, such as RWA, Born–Markov, Redfield, etc. The 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 the previous work. We also use our solution, without invoking the RWA, to characterize the necessary conditions for Dicke subradiance at finite temperature. We find that the sub-radiant 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 driven two-level system is a useful model to describe many quantum objects, particularly in quantum information processing. However, the exact master equation for such a system has barely been explored. Making use of the Feynman-Vernon influence functional theory, we derive an exact non-Markovian master equation for the driven two-level system and show the lost feature in the perturbative treatment for this system. The perturbative treatment leads to the time-convolutionless (TCL) and Nakajima-Zwanzig (NZ) master equations. So to this end we derive the TCL and NZ master equations for the system and compare the dynamics given by the three master equations. We find the validity condition for the TCL and NZ master equations. Based on the exact non-Markovian master equation, we analyze the regime of validity for the secular approximation in the time-convolutionless master equation and discuss the leading corrections of the nonsecular terms to the quantum dynamics. Significant effects are found in the dynamics of the driven system.
We study the effect of the rotating-wave approximation (RWA) and the secular approximation (SA) on the non-Markovian behavior in the spin-boson model at zero temperature. We find that both the RWA and SA lead to a dramatic reduction in the observed non-Markovianity. In general, non-Markovian dynamics is observed for the whole relaxation time of the system, whereas the RWA and SA lead to such dynamics only on the short time scale of the environmental correlation time. Thus, the RWA and SA are not necessarily justified in the studies of non-Markovianity, although they can estimate the state of the system precisely. Furthermore, we derive an accurate analytical expression for the non-Markovianity measure without the RWA or SA. This expression yields important insight into the physics of the problem.