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arXiv:0911.4600 · Physica Scripta

Non-Markovian master equation for a damped driven two-state system

P Haikka

Published Sep 1, 2010• 8 citations
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Abstract

We present a detailed microscopic derivation for a non-Markovian master equation for a driven two-state system interacting with a general structured reservoir. The master equation is derived using the time-convolutionless projection operator technique in the limit of weak coupling between the two-state quantum system and its environment. We briefly discuss the Markov approximation, the secular approximation and their validity.

DOI
10.1088/0031-8949/2010/t140/014047

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We present a detailed microscopic derivation for a non-Markovian master equation for a driven two-state system interacting with a general structured reservoir. The master equation is derived using the time-convolutionless projection operator technique in the limit of weak coupling between the two-state quantum system and its environment. We briefly discuss the Markov approximation, the secular approximation and their validity.
We study a driven two-state system interacting with a generic structured environment. We outline the derivation of the time-local microscopic non-Markovian master equation, in the limit of weak coupling between the system and the reservoir, and we derive its analytic solution for general reservoir spectra in the regime of validity of the secular approximation. We also consider the non-Markovian master equation without the secular approximation and study the effect of nonsecular terms on the system dynamics for two classes of reservoir spectra: the Ohmic and the Lorentzian reservoir. Finally, we derive the analytic conditions for complete positivity of the dynamical map, with and without the secular approximation. Interestingly, the complete positivity conditions have a transparent physical interpretation in terms of the characteristic time scales of phase diffusion and relaxation processes.
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 present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima—Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.
We study the dynamics of open quantum systems driven by an external time-periodic force. Combining Floquet theory and the time-convolutionless projection operator technique we derive a time-local quantum master equation which exactly takes into account the periodic driving, while treating the system-environment interaction within second order in the coupling strength without performing the Markov approximation. The resulting equation of motion for the reduced density matrix is called Floquet time-convolutionless master equation. Employing the example of the driven spin-boson system, we demonstrate that this master equation is capable of describing strong non-Markovian effects, while yielding the Floquet-Lindblad master equation in the Markovian limit. A characteristic feature of memory effects in such driven dissipative systems is the emergence of sharp peaks of the trace-distance based non-Markovianity measure as a function of the driving amplitude, which can be traced back to quasienergy crossings leading to almost decoherence-protected subspaces through a quasienergy-induced dissipative decoupling mechanism.
The reduced dynamics of two interacting qubits coupled to two independent bosonic baths is investigated. The one-excitation dynamics is derived and compared with that based on the resolution of appropriate non-Markovian master equations. The Nakajima-Zwanzig and the time-convolutionless projection operator techniques are exploited to provide a description of the non-Markovian features of the dynamics of the two-qubit system. The validity of such approximate methods and their range of validity in correspondence to different choices of the parameters describing the system are brought to light.
The master equation for a two-level atom driven by a strong classical field and damped into a tailored reservoir with nonflat density of modes is derived under the Born-Markov approximation. To derive the master equation, the dressing transformation on the atomic operators is performed first, and, next, the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The effects of a strong field and/or structured reservoir are seen as nonstandard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs. The master equation leads to the generalized optical Bloch equations that can easily be solved for the steady state and, together with the quantum regression theorem, allow for analytical expressions for the fluorescence, as well as absorption spectra.