Quantum fluctuation-dissipation theorem: a time-domain formulation

A time-domain formulation of the equilibrium quantum fluctuation–dissipation theorem (FDT) in the whole range of temperatures is presented. In the classical limit, the FDT establishes a proportionality relation between the dissipative part of the linear response function and the derivative of the corresponding equilibrium correlation function. At zero temperature, the FDT takes the form of Hilbert transform relations between the dissipative part of the response function and the corresponding symmetrized equilibrium correlation function, and thus allows to establish a connection with the analytic signal theory. The harmonic oscillator is treated as a basic example. The time-domain formulation of the relation between dissipation and fluctuations is especially valuable in out-of-equilibrium dynamics. We show on the specific example of the free particle quantum Brownian motion that it is the case in the discussion of the aging phenomena.