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The Journal of Chemical Physics

Real time path integral methods for a system coupled to an anharmonic bath

Gregory Ilk, Nancy Makri

Published Oct 15, 1994• 50 citations
Publisher

Abstract

We propose an efficient path integral scheme for calculating the quantum dynamics of an arbitrary one-dimensional system coupled nonlinearly to many anharmonic noninteracting ‘‘bath’’ degrees of freedom. The starting point is an improved discretization of the path integral in terms of numerically constructed propagators [Chem. Phys. Lett. 193, 435 (1992)]. The resulting influence functional is comprised of one-dimensional correlation functions with step-structured time-dependent potentials and therefore is similar in structure to that employed in the spin-boson calculations of Coalson [J. Chem. Phys. 86, 995 (1987)]. In the present case, though, the influence functional is nonlinear and is computed using numerical iterative wave function propagation methods. Numerical tests on a system coupled to ten anharmonic oscillators demonstrate the efficiency of the proposed scheme, which requires numerical effort that scales only linearly with the number of anharmonic bath degrees of freedom.

DOI
10.1063/1.468364

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We propose an efficient path integral scheme for calculating the quantum dynamics of an arbitrary one-dimensional system coupled nonlinearly to many anharmonic noninteracting ‘‘bath’’ degrees of freedom. The starting point is an improved discretization of the path integral in terms of numerically constructed propagators [Chem. Phys. Lett. 193, 435 (1992)]. The resulting influence functional is comprised of one-dimensional correlation functions with step-structured time-dependent potentials and therefore is similar in structure to that employed in the spin-boson calculations of Coalson [J. Chem. Phys. 86, 995 (1987)]. In the present case, though, the influence functional is nonlinear and is computed using numerical iterative wave function propagation methods. Numerical tests on a system coupled to ten anharmonic oscillators demonstrate the efficiency of the proposed scheme, which requires numerical effort that scales only linearly with the number of anharmonic bath degrees of freedom.
In a recent letter [Chem. Phys. Lett. 291, 101 (1998)] we presented a semiclassical methodology for calculating influence functionals arising from many-body anharmonic environments in the path integral formulation of quantum dynamics. Taking advantage of the trace operation associated with the unobservable medium, we express the influence functional in terms of a single propagator along a combined forward–backward system path. This propagator is evaluated according to time-dependent semiclassical theory in a coherent state initial value representation. Because the action associated with propagation in combined forward and backward time is governed by the net force experienced by the environment due to its interaction with the system, the resulting propagator is generally a smooth function of coordinates and thus amenable to Monte Carlo sampling; yet, the interference between forward and reverse propagators is fully accounted for. In the present paper we present a more elaborate version of the semiclassical influence functional formalism, along with algorithms for evaluating the coherent state transform of the Boltzmann operator that enters the influence functional. This factor is evaluated by performing an imaginary time path integral, and various approximations of the resulting expression as well as sampling schemes are discussed. The feasibility of the approach is demonstrated via numerous test calculations involving a two-level system coupled to (a) a dissipative harmonic bath and (b) a ten-dimensional bath of coupled anharmonic oscillators.
Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.
The reduced density matrix of a system coupled to a harmonic dissipative bath can be obtained accurately using an iterative path integral methodology that involves a propagator tensor, whose rank is roughly equal to the number of time steps that span the bath-induced memory. The present work shows that even in situations where the bath is characterized by high-frequency modes (which necessitate a small time step) and slow oscillators (which give rise to very long memory), efficient propagation that takes into account the memory on all time scales is feasible by renormalizing the propagator matrix to include successively larger-length influence functional correlations. At its crudest level, the renormalized propagator is an ordinary matrix, thus the computational demands of the method are very modest, similar to those for propagating the bare system. Yet, test applications on dissipative two-level systems show that the method produces accurate results in demanding situations. By including memory on successively increasing time scales, the method is capable of capturing nonexponential kinetics induced by sluggish solvents.
We review an iterative path integral method that allows the computation of exact, long-time dynamics of small systems interacting with a dissipative bath. The method takes advantage of the finite memory of large environments at finite temperature to allow an iterative evaluation of the dynamics, thereby replacing an exponential scaling in simulation time with a linear scaling. This method is applied to calculate the dynamics of two model systems. The first consists of two-qubits interacting with a common bath. In this case we observe a variety of entanglement effects. If the qubits are initially separable, through an indirect coupling, the bath can create steady state entanglement between the qubits. This effect is due to the existence of a decoherence-free subspace as a result of the form of the qubit Hamiltonian and system-bath coupling. Entanglement created by the bath is shown to decrease with increasing temperature and system-bath coupling strength. However large system-bath coupling causes a faster increase in the entanglement. Initially entangled qubits lose their entanglement as a result of interactions with the bath, an effect that is heightened at higher temperatures. Direct coupling between the qubits is shown to slow the decay of entanglement and preserve some entanglement at long times; however at high temperatures this steady state entanglement becomes negligible. The second system we consider is vibrational degree of freedom coupled to a bath of harmonic oscillators or two-level systems. We compute four-time correlation functions which are used to calculate response functions relevant to third order infrared or seventh order Raman experiments for harmonic, Morse, and quadratic-quartic potentials. Our calculations reveal the role of potential features (anharmonicity and eigenvalue spectrum), both on short and long time scales, on the response function. Further, thermal excitation causes dramatic changes in the appearance of the response function, introducing symmetry with respect to the main diagonal. Finally, coupling to harmonic dissipative baths leads to decay of the response function (primarily along the t3 direction) and a broadening of the peaks in its Fourier transform. At high temperatures two-level-system baths are less efficient in destroying coherence than harmonic baths of similar parameters.
Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.