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Relation among various expressions of the complex admittance for quantumsystems in contact with heat reservoir is studied. Exact expressions of thecomplex admittance are derived in various types of formulations of equations ofmotion under contact with heat reservoir. Namely, the complex admittance isstudied in the relaxation method and the external-field method. In the formermethod, the admittance is calculated using the Kubo formula for quantum systemsin contact with heat reservoir in no external driving fields, while in thelatter method the admittance is directly calculated from equations of motionwith external driving terms. In each method, two types of equation of motionsare considered, i.e., the time-convolution (TC) equation andtime-convolutionless (TCL) equation. That is, the full of the four cases arestudied. It is turned out that the expression of the complex admittanceobtained by using the relaxation method with the TC equation exactly coincideswith that obtained by using the external-field method with the TC equation,while other two methods give different forms. It is also explicitlydemonstrated that all the expressions of the complex admittance coincide witheach other in the lowest Born approximation for the systemreservoirinteraction. The formulae necessary for the higher order expansions in powersof the system-reservoir interaction are derived, and also the expressions ofthe admittance in the n-th order approximation are given. To characterize theTC and TCL methods, we study the expressions of the admittances of two exactlysolvable models. Each exact form of admittance is compared with the results ofthe two methods in the lowest Born approximation. It is found that depending onthe model, either of TC and TCL would be the better method.

Comparison among Various Expressions of Complex Admittance for Quantum System in Contact with Heat Reservoir