  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 systemreservoir
interaction. The formulae necessary for the higher order expansions in powers
of the system-reservoir interaction are derived, and also the expressions of
the admittance in the n-th order approximation are given. 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.
