Comparison of Matsubara dynamics with exact quantum dynamics for an oscillator coupled to a dissipative bath

TL;DR

This study extends Matsubara dynamics simulations to approximately 200 modes for a dissipative oscillator, demonstrating near-perfect agreement with exact quantum results for certain non-linear correlations, thus validating its quantum-classical correspondence.

Contribution

It introduces a novel approach to simulate Matsubara dynamics with many modes by reformulating equations as a generalized Langevin equation and approximating the noise as real.

Findings

Matsubara dynamics accurately reproduces quantum correlation functions for a dissipative Morse oscillator.

The method remains stable and efficient up to a certain number of modes depending on bath coupling and anharmonicity.

Approximate real-noise Matsubara dynamics shows strong agreement with exact quantum results for non-linear operators.

Abstract

Matsubara dynamics is the classical dynamics which results when imaginary-time path-integrals are smoothed; it conserves the quantum Boltzmann distribution and appears in drastically approximated form in path-integral dynamics methods such as (thermostatted) ring-polymer molecular dynamics (T)RPMD and centroid molecular dynamics (CMD). However, it has never been compared directly with exact quantum dynamics for non-linear operators, because the difficulty of treating the phase has limited the number of Matsubara modes to fewer than 10. Here, we treat up to $\sim$200 Matsubara modes in simulations of a Morse oscillator coupled to a dissipative bath of harmonic oscillators. This is done by expressing the Matsubara equations of motion in the form of a generalised Langevin equation, approximating the noise to be real, and analytically continuing the momenta to convert the Matsubara phase into ring-polymer springs. The resulting equations of motion are stable up to a maximum value of modes which increases with bath coupling strength and decreases with system anharmonicity. The dynamics of the tail of highly oscillatory Matsubara modes is found to be harmonic, and can thus be computed efficiently. For a moderately anharmonic oscillator with a strong but subcritical coupling to the bath, the Matsubara simulations yield non-linear $\large\langle{\hat q^2\hat q^2(t)}\large\rangle$ time-correlation functions in almost perfect agreement with the exact quantum results. Reasonable agreement is also obtained for weaker coupling strengths, where errors arise because of the real-noise approximation. These results give strong evidence that Matsubara dynamics correctly explains how classical dynamics arises in quantum systems which are in thermal equilibrium.