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Physica Scripta

Stochastic simulation of long-time nonadiabatic dynamics

Daniel A. Uken, Alessandro Sergi, Francesco Petruccione

Published Feb 1, 2011• 11 citations
Publisher

Abstract

Using a generalized transition probability, we show 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, we provide substantial numerical evidence that the statistical error can be reduced for a wide range of parameter values.

DOI
10.1088/0031-8949/2011/t143/014024

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Using a generalized transition probability, we show 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, we provide substantial numerical evidence that the statistical error can be reduced for a wide range of parameter values.
Simulation of quantum dynamics for many-body systems is an open area of research. For interacting many-body quantum systems, the computer memory necessary to perform calculations has an astronomical value, so that approximated models are needed to reduce the required computational resources. A useful approximation that can often be made is that of quantum-classical dynamics, where the majority of the degrees are treated classically, while a few of them must be treated quantum mechanically. When energy is exchanged very quickly between the quantum subsystem and classical environment, the dynamics is nonadiabatic. Most theories for nonadiabatic dynamics are unsatisfactory, as they fail to properly describe the quantum backreaction of the subsystem on the environment. However, an approach based on the quantum-classical Liouville equation solves this problem. Even so, nonadiabatic dynamics is difficult to implement on a computer, and longer simulation times are often inaccessible due to statistical error. There is thus a need for improved algorithms for nonadiabatic dynamics. In this thesis, two algorithms that utilise the quantum-classical Liouville equation will be qualitatively and quantitatively compared. In addition, stochastic sampling schemes for nonadiabatic transitions will be studied, and a new sampling scheme is introduced [D. A. Uken et al., Phys. Rev. E. 88, 033301 (2013)] which proves to have a dramatic advantage over existing techniques, allowing far longer simulation times to be calculated reliably.
The simulation of the dynamics of quantum systems is very difficult, due to the fact that, in general, it cannot be calculated exactly for interacting many-body systems. Brute force simulations of quantum dynamics are simply not feasible, and approximations need to be made. In many instances a quantum system can be approximated as a quantum-classical system, where only a subsystem of interest is treated quantum mechanically, and the rest is considered as a classical bath. When energy is free to be exchanged between the subsystem and its environment, the dynamics that occur is said to be nonadiabatic. This type of dynamics is challenging to calculate on a computer, as it can lead to large statistical errors at long times. Hence, there is a need for improved algorithms for nonadiabatic dynamics. In this thesis, a recently introduced nonadiabatic sampling scheme [A. Sergi and F. Petruccione, Phys. Rev. E 81, 032101 (2010)] is used to calculate the long-time dynamics of a model system comprising a quantum spin coupled to a bath of harmonic oscillators. Also, various technical aspects of the algorithm are investigated.
We study a number of filtering schemes for the reduction of the statistical error in nonadiabatic calculations by means of the quantum-classical Liouville equation. In particular, we focus on a scheme based on setting a threshold value on the sampling weights, so that when the threshold is overcome the value of the weight is reset, and on another approach which prunes the ensemble of the allowed nonadiabatic transitions according to a generalized sampling probability. Both methods have advantages and drawbacks, however, their combination drastically improves the performance of an algorithm known as the sequential short-time step propagation [MacKernan et al., J. Phys: Condens. Matter 14, 9069 (2002)], which is derived from a simple first order expansion of the quantum-classical propagator. Such an algorithm together with the combined filtering procedures produce results that compare very well with those obtained by means of numerically accurate path integral quantum calculations for the spin-boson model, even for intermediate and strong coupling regimes.
Phase-space representations based on coherent states (PPP,QQQ, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high-dimensional quantum systems. However, many problems using these techniques remain intractable over long integration times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase-space representations. This process requires an effective sampling of a nonclassical probability distribution. We describe and demonstrate a method of achieving this sampling using stochastic weights.