Quantum Dissipation – From Path Integral to Hierarchical Equations of Motion and to Continued Fraction Formalisms

This article reviews the path integral formulation and its time derivative that leads to a set of hierarchical equations of motion (EOM) for treating nonperturbative and non-Markovian quantum dissipation at an arbitrary finite temperature. Quantum dissipation refers to the reduced dynamics of a system embedded in a medium (or bath). The latter consists of macroscopic degrees of freedom whose effects on the system should be treated in a statistical manner. Consequently, the system of primary interest undergoes energy relaxation and dephasing processes, evolving eventually to the thermal equilibrium state. The development of quantum dissipation theory has involved diversified fields of research, such as nuclear magnetic resonance, quantum optics, molecular spectroscopy, condensed phase physics, and chemical physics [1–20]. The key quantity in quantum dissipation theory is the reduced density operator, ρ(t) ≡ trBρT(t), i.e., the partial trace of the total density operator over the bath space. The reduced dynamics of ρ(t) can be formulated exactly. One standard formalism is the path integral expression, in which the effect of harmonic bath linearly coupled to the reduced system is described by the Feynman-Vernon influence functional [15–18]. This formalism is, however, numerically expensive and feasible only to few-level systems with moderate parameters for the nonMarkovian bath couplings. Alternatively, differential EOM that has the numerical advantage over the integral formulation, has been derived from path integral for both harmonic [18–20] and anharmonic systems [21–25]. Our previous work has discussed the general construction on the basis of calculus on path integral, the truncation schemes to the resulting hierarchical EOM, and the application to the kinetics and thermodynamics of electron transfer in Debye solvents [25–27]. To familiarize ourselves with the path integral formalism, let us start with the nondissipative (Hilbert-space) propagator U(t, t0) for a wave function,