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Physical Review A

Quantum master equation for dephasing of a two-level system with an initial correlation

Masashi Ban

Published Dec 10, 2009• 28 citations
Publisher

Abstract

Exact quantum master equation is derived for dephasing of a two-level system, which is an homogeneous time-convolutionless equation even though there is an initial correlation between a two-level system and a thermal reservoir. The result is compared with the quantum master equation derived by means of the projection operator method. Furthermore, the effects of the initial correlation on the dephasing process are examined.

DOI
10.1103/physreva.80.064103

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Exact quantum master equation is derived for dephasing of a two-level system, which is an homogeneous time-convolutionless equation even though there is an initial correlation between a two-level system and a thermal reservoir. The result is compared with the quantum master equation derived by means of the projection operator method. Furthermore, the effects of the initial correlation on the dephasing process are examined.
Time-evolution of the dispersive Jaynes–Cummings model interacting with a bosonic reservoir is considered in the presence of an initial correlation between the cavity field and reservoir. When there is at most one excitation in the whole system, the exact time-evolution can be obtained. The effects of the initial correlation on the decoherence of the two-level system and cavity field are investigated. Furthermore, the validity of the time-convolutionless quantum master equation in the second-order approximation with respect to the system-reservoir interaction is examined by means of the exact solution.
A double-time correlation function of arbitrary two quantum operations is studied for a nonstationary open quantum system which is in contact with a thermal reservoir. It includes a usual correlation function, a linear response function, and a weak value of an observable. Time evolution of the correlation function can be derived by means of the time-convolution and time-convolutionless projection operator techniques. For this purpose, a quasidensity operator accompanied by a fictitious field is introduced, which makes it possible to derive explicit formulas for calculating a double-time correlation function in the second-order approximation with respect to a system-reservoir interaction. The derived formula explicitly shows that the quantum regression theorem for calculating the double-time correlation function cannot be used if a thermal reservoir has a finite correlation time. Furthermore, the formula is applied for a pure dephasing process and a linear dissipative process. The quantum regression theorem and the the Leggett-Garg inequality are investigated for an open two-level system. The results are compared with those obtained by exact calculation to examine whether the formula is a good approximation.
A phase relaxation process of interacting two qubits is studied by means of the quantum master equation, where each qubit is influenced by an independent thermal reservoir. The damping operator of the quantum master equation consists of two parts; one includes the effects of the interaction between the qubits and the other does not. It is shown that the interaction significantly affects the phase relaxation process. The decoherence of entanglement is investigated for the two qubits initially prepared in the XXX state.
We show that the exact master equation incorporating initial correlations for open quantum systems, within the Nakajima-Zwanzig operator-projection method, is a homegenous master equation for the reduced density matrix. We also derive explicitly the exact master equation for a large class of bosonic and fermionic open quantum systems incorporating initial correlations, the resulting master equation is homogenous. We find that the effects of the initial correlations can be fully embedded into the fluctuation dynamics through the exact homogenous master equation. Also a generalized nonequilibrium fluctuation-dissipation theorem incorporating the initial correlations is obtained.
A linear response theory for open quantum systems is formulated by means of the time-local and time-nonlocal quantum master equations, where a relevant quantum system interacts with a thermal reservoir as well as with an external classical field. A linear response function that characterizes how a relaxation process deviates from its intrinsic process by a weak external field is obtained by extracting the linear terms with respect to the external field from the quantum master equation. It consists of four parts. One represents the linear response of a quantum system when system-reservoir correlation at an initial time and correlation between reservoir states at different times are neglected. The others are correction terms due to these effects. The linear response function is compared with the Kubo formula in the usual linear response theory. To investigate the properties of the linear response of an open quantum system, an exactly solvable model for a stochastic dephasing of a two-level system is examined. Furthermore, the method for deriving the linear response function is applied for calculating two-time correlation functions of open quantum systems. It is shown that the quantum regression theorem is not valid for open quantum systems unless their reduced time evolution is Markovian.
Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level open quantum system with finite number NNN of quantum eigenstates interacting with arbitrary external deterministic fields and dissipative environment simultaneously. Unlike conventional master equations derived by the projection operator method for the quantum systems of this kind so far, these equations are homogeneous in the reduced statistical operator while, at the same time, taking into account initial correlations between the multi-level system and the environment. Consistent representation of the multi-level system evolution in terms of the SU(N)SU(N)SU(N) operator algebra facilitates efficient numerical analysis of these equations derived for various physically realistic quantum models.