QUANTUM MECHANICS OF DISSIPATIVE SYSTEMS

YiJing Yan, RuiXue Xu
2005 Annual review of physical chemistry (Print)  
Quantum dissipation involves both energy relaxation and decoherence, leading toward quantum thermal equilibrium. There are several theoretical prescriptions of quantum dissipation but none of them is simple enough to be treated exactly in real applications. As a result, formulations in different prescriptions are practically used with different approximation schemes. This review examines both theoretical and application aspects on various perturbative formulations, especially those that are
more » ... those that are exact up to second-order but nonequivalent in high-order system-bath coupling contributions. Discrimination is made in favor of an unconventional formulation that in a sense combines the merits of both the conventional time-local and memory-kernel prescriptions, where the latter is least favorite in terms of the applicability range of parameters for system-bath coupling, non-Markovian, and temperature. Also highlighted is the importance of correlated driving and disspation effects, not only on the dynamics under strong external field driving, but also in the calculation of field-free correlation and response functions. QUANTUM MECHANICS OF DISSIPATIVE SYSTEMS 189 mechanics. Correlation/response functions have been widely used in the study of physical problems such as spectroscopy (24-34), transport (38-43), and reaction rate (68, 95-97). Correlation and Response Functions Versus Linear Response Theory Consider the measurement on a dynamical variable A with a classical probe field (t) that couples with the system as −B (t). For simplicity, both operators A and B are assumed to be Hermite. The total composite material system was initially at the thermal equilibrium, ρ eq M = e −β H M /Tre −β H M , before external field disturbance. The field-induced deviation in Ā(t) from its equilibrium expectation value is δ , where ρ T (t) is the total composite system density operator in the presence of external field. To the first order of the external disturbance, we have with the material response function given by Here, [. . ., . . .] denotes a commutator, O(t) ≡ e i H M t Oe −i H M t , and . . . M ≡ Tr(. . . ρ eq M ). Physically, the response function χ AB (t) is needed only for t ≥ 0 due to causality (cf. Equation 2.1). Its extension to t < 0 is formally made with Equation 2 .2 as Obviously, the response function χ AB (t) for Hermitian operators is real. We now turn to the correlation function, denoted as Either χ AB (t − τ ) of Equation 2 .2 or CAB (t − τ ) of Equation 2 .4 depends only on the duration t − τ . This is a property of the stationary statistics as [H M , ρ eq M ] = 0. We have also that A(t) M = A M , which does not depend on time, and 2.5. Moreover, the correlation function satisfies the following symmetry and detailedbalance relations: Note that χ AB (t) = −2Im CAB (t). The common phenomenon of statistical independence as t → ∞ implies that CAB (t → ∞) = A M B M in a general dissipative 192 YAN XU We have also the following expressions, 2.17. The first expression is obtained when the Kramer-Kronig relation is at zero frequency, together with χ(+) AB (0) = χAB (0) as inferred from Equation 2 .15 that χ(−) AB (0) = 0. Note that as χ AB (t) is real for Hermite operators, χAB (0) must be real (cf. Equation 2 .12). The second expression in Equation 2 .17 is obtained by taking the time derivative of the inverse Fourier transform of Equation 2.13b, followed by setting t = 0. Later in this review, the bath correlation/response functions for a set of bath operators {F a (t) ≡ e ih B t F a e −ih B t } will be exploited to describe the energy relaxation and decoherence processes in the reduced system of primary interest. The bath correlation functions will be denoted similarly as 2.18.
doi:10.1146/annurev.physchem.55.091602.094425 pmid:15796700 fatcat:hwsulizfabgufmoeq6eqotsqty