Relaxation at finite temperatures
Herbert E. Spohn
Relaxation at finite temperaturesThe weak coupling theory of chapter 17 is the workhorse of quantum optics and serves very well in practice, also at nonzero temperatures.From the viewpoint of the theory one might wonder about the structure at fixed small, but nonzero, coupling strength, which needs to go beyond the analysis of the weak coupling theory.Much effort has been invested to achieve this goal, basically by trying to identify corrections within time-dependent perturbation theory.Unfortunately, since the long-time behavior must be extracted, the details quickly become unwieldy and one has to rely on ad hoc approximations.Over recent years a novel approach has been pursued which investigates the pole structure of the analytic continuation of the resolvent of HA. across the real axis through complex dilations; compare with section 17.3.The techniques are demanding but simplify substantially at finite temperatures when the Hamiltonian is replaced by the Liouvillean and, since its spectrum is the full real line, complex dilations are replaced by complex translations which can be handled more easily.From the pole structure a fairly complete picture of the long-time dynamics can be extracted with the potential of computing systematically higher corrections to the weak coupling theory.The finite temperature relaxation is a digression into the realm of timedependent statistical mechanics with small deviations from thermal equilibrium.While this is of independent interest and has important applications in quantum optics and condensed matter, our goal is merely to illustrate the power of complex translations and make the connection to the weak coupling theory.For completeness we recall once again the set-up.In the dipole approximation and N -level approximation the Hamiltonian is (18.1)see ( 17 .I 0).Hat acts on eN.Diverging from our previous convention, the energies c: j are labeled as cJ ::S c:2 ... ::S c: N allowing for
Cambridge University Press eBooks•Jul 13, 2023•0 citations