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Physical Review A

Relationship between the adiabatic-following solutions of Bloch equations and the solutions of non-Markovian Pauli-type master equations

G. S. Agarwal

Published Oct 1, 1978• 8 citations
Publisher

Abstract

A general derivation of non-Markovian Pauli-type master equations is given for systems interacting with pump and relaxation mechanisms. It is shown that the adiabatic-following solutions of Bloch equations, due to Grischkowski and co-workers, are obtained as a special case from Pauli equations of non-Markovian type. Finally, it is proved that for a two-level system the non-Markovian master equation obtained in the Born approximation is also exact.

DOI
10.1103/physreva.18.1618

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A general derivation of non-Markovian Pauli-type master equations is given for systems interacting with pump and relaxation mechanisms. It is shown that the adiabatic-following solutions of Bloch equations, due to Grischkowski and co-workers, are obtained as a special case from Pauli equations of non-Markovian type. Finally, it is proved that for a two-level system the non-Markovian master equation obtained in the Born approximation is also exact.
We provide the exact non-Markovian master equation for a two-level system interacting with a thermal bosonic bath, and we write the solution of such a master equation in terms of the Bloch vector. We show that previous approximated results are particular limits of our exact master equation. We generalize these results to more complex systems involving an arbitrary number of two-level systems coupled to different thermal baths, providing the exact master equations also for these systems. As an example of this general case we derive the master equation for the Jaynes-Cummings model.
For adiabatically and periodically manipulated dissipative quantum systems we derive, using Floquet theory, a simple Markovian master equation. Contrary to some previous works we explicitly take into account the time dependence of the Hamiltonian and, therefore, obtain a master equation with a time-dependent dissipative part. We illustrate our theory with two examples and compare our results with the previously proposed master equations. In particular, we consider the problem of Cooper pair pumping and demonstrate the inadequacy of the secular (rotating wave) approximation when calculating the pumped charge. The secular approximation producing a master equation of the Lindblad type approximates well the quantum state (density matrix) of the system, while to determine the pumped charge a non-Lindblad master equation beyond the rotating wave approximation is necessary.
Non-Markovian master equations describe general open quantum systems when no approximation is made. We provide the exact closed master equation for the class of Gaussian, completely positive, trace preserving, non-Markovian dynamics. This very general result allows us to investigate a vast variety of physical systems. We show that the master equation for non-Markovian quantum Brownian motion is a particular case of our general result. Furthermore, we derive the master equation unraveled by a non-Markovian, dissipative stochastic Schrödinger equation, paving the way for the analysis of dissipative non-Markovian collapse models.
For adiabatically and periodically manipulated dissipative quantum systems, we derive, using Floquet theory, a simple Markovian master equation. Contrary to some previous works, we explicitly take into account the time dependence of the Hamiltonian and, therefore, obtain a master equation with a time-dependent dissipative part. We illustrate our theory with two examples and compare our results with the previously proposed master equations. In particular, we consider the problem of Cooper-pair pumping and demonstrate the inadequacy of the secular (rotating-wave) approximation when calculating the pumped charge. The secular approximation producing a master equation of the Lindblad-type approximates well the quantum state (density matrix) of the system, while to determine the pumped charge, a non-Lindblad master equation beyond the rotating-wave approximation is necessary.
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We present a succinct and intuitive derivation of a formally exact master equation for general open quantum systems, without the use of an "inverse" map which was invoked in previous works on formally exact master equations. This formalism is applicable to non-Markovian regimes. We derive a second-order equation of motion for the illustrative spin-boson model at arbitrary temperatures, observing non-exponential decoherence and relaxation. Limiting our generic derivation to zero temperature, we also reproduce the result for the special case of a vacuum bath in Phys. Rev. A 81, 042103 (2010).