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Journal of the Physical Society of Japan

A Generalized Master Equation for a Driven Quantal Oscillator

Mizuhiko Saeki

Published Jun 15, 1986• 1 citation
Publisher

Abstract

The time-convolutionless master equation for driven open system derived in a previous paper, is applied to a simple system of a driven quantal oscillator. This system is assumed to be interacting weakly with its surroundings (heat bath). The operator form of equation is transcribed into the c -number one by the phase-space method. The equations of motion for the expectation values of the boson operator and boson number operator are derived from the c -number equations and are solved for the case of a non-adiabatic oscillator-bath interaction. The results are compared with the ones in the conventional Markoffian approximation for the time-convolution equation in the narrowing limit.

DOI
10.1143/jpsj.55.1861

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Time-convolutionless forms of the quantal master equations for driven open system are derived from the Liouville equation for the total system under an arbitrary initial condition for the two cases of weak external driving fields and of arbitrary ones. They have forms convenient for the perturbational expansions, and are respectively compared with the time-convolution forms of equations in the lowest Born approximation for the interaction H SB of the system with its surroundings. It is shown that the time-convolutionless forms of equations prevail over the time-convolution ones, and that the conventional Markoffian approximation for the latter in the narrowing limit leads to the incorrect conclusion. The time-convolutionless form of master equation is expanded up to n -th order in powers of the external driving fields in the lowest Born approximation for H SB .
The time-convolutionless equation of motion for the linear response derived in a previous paper is applied to a simple system of a quantal spin interacting weakly with its surroundings (heat bath). The operator form of equation is derived and is transcribed into the c-number one by the generalized phase-space method for spin operators. The equations of motion for the longitudinal and transverse expectation values of the spin are derived and are solved for the case of S=1/2 (S: the magnitude of spin). The longitudinal and transverse susceptibilities are derived from the stationary solutions, and are compared with the ones derived by the method of spin relaxation theory and Kubo theory for the linear response.
We construct a quantum Markovian master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the master equation to be extended beyond the adiabatic limit. The nonadiabatic master equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic master equation, the NAME leads to coupled equations of motion for the population and coherence. We employ the NAME to solve the example of an open driven time-dependent harmonic oscillator. For the harmonic oscillator the NAME predicts the emergence of coherence associated with the dissipation term. As a result of the nonadiabatic driving the thermalization rate is suppressed. The solution is compared with both numerical calculations and the adiabatic master equation.
A time-convolutionless expression for the quantal equation of motion for the linear response of a system in contact with a heat bath to an external, time-varying driving field is derived from the quantal Liouville equation of motion for the total system. This is done by renormalizing the memory effect by the method of Shibata, Takahashi and Hashitsume. It has a form convenient for the perturbational expansion. It is shown that this equation coincides with the time-convolution equation of Argyres and Kelley in their Markoffian approximation in the narrowing limit and in the lowest Born approximation for the system-bath interaction. This equation is expanded up to fourth order in powers of the system-bath interaction.
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.