Iterative evaluation of the path integral for a system coupled to an anharmonic bath

Abstract

An iterative algorithm is presented for evaluating the path integral expression for the reduced density matrix of a quantum system interacting with an anharmonic dissipative bath whose influence functional is obtained via numerical methods. The method allows calculation of the reduced density matrix over very long time periods. © 1999 American Institute of Physics. ͓S0021-9606͑99͒03238-9͔In spite of persistent efforts, the problem of calculating the quantum time evolution of a wave-function or density matrix in a multidimensional Hamiltonian remains unsolved. Recent work has revolved around methods based on mean field, quantum-classical, or semiclassical ideas. The most rigorous of these approaches, semiclassical evolution with the Van Vleck propagator, 1,2 is often highly accurate, 3-8 yet extremely demanding numerically because it requires multidimensional integration of oscillatory functions as well as evaluation of a prefactor that scales nonlinearly with the number of particles. Current efforts to make it practical exploit filtering techniques, 9-11 the self-cancellation achieved via combined forward-backward propagation, 12-16 or formulations which avoid calculation of the prefactor. 11,[17][18][19] While these approaches appear promising, they are bound to fail at long propagation times or if tunneling effects are prominent. 20 Treatment at a higher level becomes necessary in such situations.In a series of papers by our group, we have argued that the path integral-influence functional formulation of quantum dynamics 21,22 offers significant advantages when dealing with large-dimensional problems. One begins by identifying the observable ''system'' ͓the degree͑s͒ of freedom s being probed in the calculation͔ and the remaining ''bath'' degrees of freedom x which interact with the system and thus affect its dynamics but whose precise state is not followed. Thus, the Hamiltonian is split into two terms,Expressing the full propagator as a path integral, and collecting all bath variables into an influence functional, one arrives at a formal path integral representation where only paths of the low-dimensional system are summed over. For example, the reduced density matrix of the system takes the formHere, s ϩ , s Ϫ are forward and backward paths of the observable system respectively, S 0 is the corresponding action, and the influence functional is given by the expressionwhere U b is the time evolution operator of the bath along a chosen system path. Note that the time parametrization of these paths makes the bath Hamiltonian time-dependent. There are numerous advantages of this representation, as well as severe obstacles. The explicit form of the influence functional-an intrinsically quantum mechanical quantity not obtainable by classical molecular dynamics methods-is not available except in very restrictive situations, the most notable of which is the case of a harmonic bath. 22 Yet the simple structure of the influence functional, where the only operators appearing are the time propagators and the initial dens...

A semiclassical generalized quantum master equation for an arbitrary system-bath coupling

“…69,168,169 Similarly to the memory kernel, the IF provides a formally exact and compact parameterization of the influence of the bath on the system dynamics. 69,168,169 Similarly to the memory kernel, the IF provides a formally exact and compact parameterization of the influence of the bath on the system dynamics.…”

Dissipation in media with memory: A master equation in the statistical resonance approximation

“…4 Environmental effects are also of importance for laser control scenarios for chemical reactions, 5 for quantum computing devices, 6 and for the operation of molecular wires. [8][9][10][11][12][13][14][15][16][17][18][19][20][21] Two well known exact theories of subsystem dynamics are the Feynman-Vernon influence functional theory [22][23][24] and the Nakajima-Zwanzig master equation. Efforts have thus been focused on developing direct methods for calculating the reduced probability density matrix of the subsystem.…”

Finite quantum environments as thermostats: an analysis based on the Hilbert space average method

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