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arXiv:0907.1512 · Physical Review E

Comparison among various expressions of complex admittance for quantum systems in contact with a heat reservoir

Mizuhiko Saeki, Chikako Uchiyama, Takashi Mori, Seiji Miyashita

Published Mar 30, 2010• 24 citations
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Abstract

Relation among various expressions of the complex admittance for quantum systems in contact with heat reservoir is studied. Exact expressions of the complex admittance are derived in various types of formulations of equations of motion under contact with heat reservoir. Namely, the complex admittance is studied in the relaxation method and the external-field method. In the former method, the admittance is calculated using the Kubo formula for quantum systems in contact with heat reservoir in no external driving fields, while in the latter method the admittance is directly calculated from equations of motion with external driving terms. In each method, two types of equation of motions are considered, i.e., the time-convolution (TC) equation and time-convolutionless (TCL) equation. That is, the full of the four cases are studied. It is turned out that the expression of the complex admittance obtained by using the relaxation method with the TC equation exactly coincides with that obtained by using the external-field method with the TC equation, while other two methods give different forms. It is also explicitly demonstrated that all the expressions of the complex admittance coincide with each other in the lowest Born approximation for the system-reservoir interaction. The formulas necessary for the higher-order expansions in powers of the system-reservoir interaction are derived, and also the expressions of the admittance in the nth order approximation are given. By transforming inverse-temperature integrals into time integrals, the admittances are also given in the formulas with time-integrals alone. To characterize the TC and TCL methods, we study the expressions of the admittances of two exactly solvable models. Each exact form of admittance is compared with the results of the two methods in the lowest Born approximation. It is found that depending on the model, either of TC and TCL would be the better method.

DOI
10.1103/physreve.81.031131

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Relation among various expressions of the complex admittance for quantum systems in contact with heat reservoir is studied. Exact expressions of the complex admittance are derived in various types of formulations of equations of motion under contact with heat reservoir. Namely, the complex admittance is studied in the relaxation method and the external-field method. In the former method, the admittance is calculated using the Kubo formula for quantum systems in contact with heat reservoir in no external driving fields, while in the latter method the admittance is directly calculated from equations of motion with external driving terms. In each method, two types of equation of motions are considered, i.e., the time-convolution (TC) equation and time-convolutionless (TCL) equation. That is, the full of the four cases are studied. It is turned out that the expression of the complex admittance obtained by using the relaxation method with the TC equation exactly coincides with that obtained by using the external-field method with the TC equation, while other two methods give different forms. It is also explicitly demonstrated that all the expressions of the complex admittance coincide with each other in the lowest Born approximation for the system-reservoir interaction. The formulas necessary for the higher-order expansions in powers of the system-reservoir interaction are derived, and also the expressions of the admittance in the nth order approximation are given. By transforming inverse-temperature integrals into time integrals, the admittances are also given in the formulas with time-integrals alone. To characterize the TC and TCL methods, we study the expressions of the admittances of two exactly solvable models. Each exact form of admittance is compared with the results of the two methods in the lowest Born approximation. It is found that depending on the model, either of TC and TCL would be the better method.
The relation between the method of TCL equation with external driving terms and the relaxation method is discussed analytically in the quantal bath model. It is shown that the interference term in the former method gives the part of second order in powers of the system-bath interaction in the Kubo formula, and that the former method is simpler than the latter method. It is also shown that the relaxation method neglects the low temperature effects which are given by the first part of interference term, and that the relaxation method in the narrowing limit neglects the effects of the interference term completely. The general form of admittance for the case of decoupled initial condition is given by the method of TCL equation with external driving terms, and can be equal to that for the case of thermal equilibrium initial condition. The above facts are verified by applying to a quantal oscillator. Validity of the second-order perturbation calculations is discussed.
The TCLE method (a method in which the admittance of a system is directly calculated from time-convolutionless equations with external driving terms) formulated in terms of thermo-field dynamics, is applied to a boson system interacting with its surroundings, referred to as the “heat reservoir”, and with an external driving field. For a boson system interacting with its heat reservoir, the fluctuation-dissipation theorem, a dispersion relation, and sum rules are studied through application of the TCLE method. The effects of the interference between the external driving field and the heat reservoir, which give the effects of the initial correlation and memory, as well as the effects of the boson-boson interaction, are numerically investigated for a non-degenerate boson system interacting with its heat reservoir, up to first order in powers of the boson-boson interaction.
The relation between the method of TCL equation with external driving terms and the relaxation method is discussed analytically in the quantal bath model. It is shown that the interference term in the former method gives the part of second order in powers of the system-bath interaction in the Kubo formula, and that the former method is simpler than the latter method. It is also shown that the relaxation method neglects the low temperature effects which are given by the first part of interference term, and that the relaxation method in the narrowing limit neglects the effects of the interference term completely. The general form of admittance for the case of decoupled initial condition is given by the method of TCL equation with external driving terms, and can be equal to that for the case of thermal equilibrium initial condition. The above facts are verified by applying to a quantal oscillator. Validity of the second-order perturbation calculations is discussed.
In the context of non-equilibrium statistical mechanics, the TCLE method (a method in which the admittance of a system is directly calculated from time-convolutionless equations with external driving terms) for the linear response, is formulated in terms of thermo-field dynamics and in this way, thermo-field dynamics is generalized to the case of a quantal system interacting with its surroundings and with an external driving field. It is applied to interacting boson systems, and an admittance formula for such systems is derived using the TCLE method.
A general form of admittance is derived using the relaxation method under thermal equilibrium initial conditions from a second-order TCL equation with no external driving terms. Its admittance is shown to include the two interference terms in the TCLE method (method of TCL equations with external driving terms). Effects of the first and second interference terms in the TCLE method are numerically discussed for a quantal spin (magnitude 1/2) interacting with a spin-like bath and with a boson-like bath, respectively. It is shown that the first interference term which represents effects of the initial correlation and which is neglected in the relaxation method under decoupled initial conditions of the spin and bath, produces the effects which enhance the peaks of line shapes slightly and which abase the line shapes at the small frequency region, and becomes larger as the characteristic frequency of the heat bath becomes smaller.
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.