Path-integral treatment of the real-time dynamics of few-mode spin-boson models

The real-time dynamics of electronic two-level systems coupled to a finite set of harmonic vibrational modes is treated by a path-integral approach. The sum over paths in electronic state space, which becomes intractable for more than ≈ 30 elementary time slices, is replaced by a summation over suitable subsets (classes) of paths and the recursive computation of propagator averages over classes. In the present work the class averages have been evaluated in the so-called class-average factorization (CAF) approximation, which represents the lowest order of a systematic expansion in terms of mode correlations. This method allows the approximate evaluation of the path integral for several hundreds of time slices, which represents a gain of at least one order of magnitude compared to the direct numerical spin enumeration. The accuracy of the CAF approximation has been tested by calculations of the autocorrelation function and the time-dependent electronic population probability for spin-boson models with three and four vibrational degrees of freedom, for which exact solutions can be obtained by standard wave-packet propagation techniques. To demonstrate the potential of the PI approach for the treatment of the real-time dynamics of larger systems, results are reported for a 31-mode model.