Relaxation properties of two-level systems in condensed phases

Generalized Gibbs state with modified Redfield solution: Exact agreement up to second order

“…The discrepancy arises due to the second order diagonal elements which require contributions from the higher order relaxation tensor for their correct evaluation. In view of these findings it is not possible to capture correctly the effects of finite coupling up to second order by use of the Redfield formalism; -this feature also corrects some inadequately stated claims contained in the previous literature 42,43 .…”

“…It is interesting to know if this distribution can be obtained from a full nonMarkovian dynamical theory of a system weakly coupled to a heat bath. Although this seems reasonable there is no agreed consensus on this issue from the viewpoint that the literature deals with a variety of perturbative quantum (2-nd order) master equations 42,43,57,58 . Since the Redfield formalism is rigorously valid only in the λ → 0 limit it is expected that in this very limit the canonical form ρ eq ∝ e −βHS emerges.…”

“…With this Appendix we outline the basic reasoning underlying canonical perturbation theory 40,42,43 (CPT). This will assist us in determining the correct equilibrium reduced density matrix up to second order in coupling strength for a harmonic bath which is coupled bi-linearly to a general system.…”

On the weak-coupling limit and complete positivity

“…It is sometimes argued [12][13][14] that complete positivity is the consequence of two auxiliary technical simplifications that are essential in the standard [1][2][3] derivation of quantum dynamical semigroups from the closed dynamics of the system S plus its environment. It is in fact assumed a) that the initial state of S be uncorrelated to that of the environment, and b) that a Markov approximation is possible on rescaled times τ = λ 2 t, where λ << 1 is the strength of the system-environment interaction [2][3][4][5][6][7][8][9] (the weak-coupling limit).…”

“…It is in fact assumed a) that the initial state of S be uncorrelated to that of the environment, and b) that a Markov approximation is possible on rescaled times τ = λ 2 t, where λ << 1 is the strength of the system-environment interaction [2][3][4][5][6][7][8][9] (the weak-coupling limit). However, requests a) ad b) are not always physically plausible [12][13][14][15][16][17][18][19][20][21]; in particular, it might be necessary to examine the subsystem dynamics on times of the order of λ 4 t, hence beyond the weak-coupling limit [15,[17][18][19][20] and, in such instances, dynamics not of completely positive type may appear.…”