[Paper Review] Stochastic Simulation of Nonadiabatic Dynamics at Long Time

Daniel A. Uken, Alessandro Sergi+1 more|arXiv (Cornell University)|Mar 16, 2010

Quantum chaos and dynamical systems1 references3 citations

TL;DR

This paper introduces a generalized stochastic sampling scheme for nonadiabatic dynamics in the Wigner-Heisenberg representation, significantly reducing statistical error growth over long times. By filtering trajectories that cause excessive energy fluctuations through a tailored transition probability weight, the method extends accessible integration times by up to an order of magnitude compared to primitive sampling, as demonstrated in spin-boson model simulations with reduced error bars at long times.

ABSTRACT

Using a generalized energy-conserving transition probability, it is shown how nonadiabatic calculations, within the Wigner-Heisenberg representation of quantum mechanics, can be reliably extended to far longer times than those allowed by a primitive sampling scheme. Tackling the spin-boson model as a paradigmatic example, substantial numerical evidence is provided that effective integration of the dynamics can be achieved for a wide range of temperatures and friction.

Motivation & Objective

• To address the long-standing problem of statistical error growth in stochastic simulations of nonadiabatic quantum dynamics over extended time intervals.
• To improve numerical stability in nonadiabatic dynamics simulations by refining the stochastic sampling of quantum transitions.
• To enable reliable simulation of nonadiabatic processes beyond the limits of primitive sampling schemes, particularly in systems with strong system-bath coupling.
• To provide a framework for filtering unphysical trajectories that induce spurious energy fluctuations during stochastic propagation.
• To demonstrate the effectiveness of the generalized sampling approach across a range of friction and coupling parameters in the spin-boson model.

Proposed method

• The method employs a generalized transition probability weight in the stochastic sampling of nonadiabatic transitions, derived from a linearized Wigner-Heisenberg propagator.
• The weight function is designed to suppress trajectories that lead to large energy fluctuations, effectively filtering out unphysical stochastic jumps.
• The approach is based on a piecewise adiabatic trajectory representation, where transitions between subsystem energy levels are stochastically sampled using the modified probability distribution.
• The sampling scheme is implemented within the Momentum-Jump (MJ) approximation, allowing for consistent energy conservation and back-reaction effects in the hybrid quantum-classical dynamics.
• The generalized sampling is applied to the spin-boson model, with time evolution computed via ensemble averaging over trajectories.
• The method uses a threshold parameter $ c_{ m E} $ to control the filtering of high-energy fluctuation events, ensuring numerical stability.

Experimental results

Research questions

• RQ1 Can a generalized sampling scheme reduce statistical error growth in long-time nonadiabatic dynamics simulations?
• RQ2 How does the filtering of high-energy fluctuation trajectories affect the long-time stability of stochastic nonadiabatic dynamics?
• RQ3 To what extent can the integration time be extended using the proposed sampling method compared to primitive sampling?
• RQ4 Does the generalized sampling preserve energy conservation and physical consistency in nonadiabatic dynamics?
• RQ5 Can the method be applied effectively across a broad range of friction and coupling parameters in the spin-boson model?

Key findings

• The generalized sampling scheme reduces statistical error growth significantly, with error bars remaining small up to $ t = 20 $, while primitive sampling shows unacceptably large errors by $ t = 10 $.
• For $ eta = 3 $, $ eta = 0.25 $, and $ eta = 1.0 $, the generalized sampling maintains low error levels beyond $ t = 16 $, whereas primitive sampling fails after $ t = 10 $.
• The fraction of trajectories with zero transitions ($ n=0 $) increases over time under generalized sampling, indicating suppression of unphysical jumps.
• The number of trajectories with one or two transitions remains stable and physically plausible, while primitive sampling shows uncontrolled fluctuations in transition statistics.
• The method enables integration times up to an order of magnitude longer than previous schemes, depending on numerical parameters.
• The results provide substantial numerical evidence that the generalized sampling reduces statistical error across a wide range of parameter values, including high friction and strong coupling regimes.