Tables of integrals

Open quantum systems (OQSs) cannot always be described with the Markov approximation, which requires a large separation of system and environment time scales. An overview is given of some of the most important techniques available to tackle the dynamics of an OQS beyond the Markov approximation. Some of these techniques, such as master equations, Heisenberg equations, and stochastic methods, are based on solving the reduced OQS dynamics, while others, such as path integral Monte Carlo or chain mapping approaches, are based on solving the dynamics of the full system. The physical interpretation and derivation of the various approaches are emphasized, how they are connected is explored, and how different methods may be suitable for solving different problems is examined.

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Dynamics of non-Markovian open quantum systems

Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mossbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscop...

Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

We propose an approximate method for calculating Kubo-transformed real-time correlation functions involving position-dependent operators, based on path integral (Parrinello-Rahman) molecular dynamics. The method gives the exact quantum mechanical correlation function at time zero, exactly satisfies the quantum mechanical detailed balance condition, and for correlation functions of the form C(Ax)(t) and C(xB)(t) it gives the exact result for a harmonic potential. It also works reasonably well at short times for more general potentials and correlation functions, as we illustrate with some example calculations. The method provides a consistent improvement over purely classical molecular dynamics that is most apparent in the low-temperature regime.

Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics

The semiclassical (SC) initial value representation (IVR) provides a potentially practical way for adding quantum mechanical effects to classical molecular dynamics (MD) simulations of the dynamics of complex molecular systems (i.e., those with many degrees of freedom). It does this by replacing the nonlinear boundary value problem of semiclassical theory by an average over the initial conditions of classical trajectories. This paper reviews the background and rebirth of interest in such approaches and surveys a variety of their recent applications. Special focus is on the ability to treat the dynamics of complex systems, and in this regard, the forward−backward (FB) version of the approach is especially promising. Several examples of the FB-IVR applied to model problems of many degrees of freedom show it to be capable of describing quantum effects quite well and also how these effects are quenched when some of the degrees of freedom are averaged over (“decoherence”).

The Semiclassical Initial Value Representation: A Potentially Practical Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations

Using a forward−backward representation of all degrees of freedom in a system, we present a rigorous formulation of semiclassical correlation functions or expectation values where the contribution of the prefactor is compensated for by the semiclassical phase This procedure eliminates the need for computing the semiclassical prefactor whose determination amounts to evaluating the full stability matrix, resulting in a scheme that scales linearly with the number of degrees of freedom Numerical calculations show that, while some interference is lost, the scheme is capable of capturing most features important to the vibrational dynamics and spectroscopy of multidimensional systems

Forward−Backward Semiclassical Dynamics without Prefactors

A forward−backward semiclassical method is presented for calculating correlation functions of polyatomic systems. Unlike conventional semiclassical theories, this formulation does not require evaluation of the prefactor that contains a determinant with elements defined by the stability matrix. It is shown rigorously that the contribution of the semiclassical prefactor in the present formulation can be absorbed in the semiclassical phase and initial density if the momentum jump at the end of the forward propagation is chosen to be one-half of that dictated by the classical equations of motion. As a consequence, the number of equations of motion to be solved in its implementation is linearly proportional to the number of degrees of freedom. The method is applied to the dynamics of water clusters which involve strongly anharmonic interactions.

Forward−Backward Semiclassical Dynamics with Linear Scaling

The dynamics of a two-level system which is coupled to an environmental sea of infinitely many spin-1/2 particles is investigated by use of the resolvent operator approach. Only at zero temperature does this spin-spin-bath model exhibit identical behavior as the more familiar spin-boson model. It is found that increasing temperature favors coherent dynamics. At high temperatures, the spin-spin-bath model for an Ohmic spectral density sustains a coherent dynamics if the dissipation coefficient $\ensuremath{\alpha}$ is sufficiently small, i.e., $\ensuremath{\alpha}l1/2$; while the decoherence exhibits pure exponential decay if $\ensuremath{\alpha}g1/2$.

https://www.physik.uni-augsburg.de/theo1/hanggi/Papers/226.pdf

Decoherent Dynamics of a Two-Level System Coupled to a Sea of Spins

The exact quantum expression for the thermal rate of reaction is the trace of a product of two operators It may therefore be written exactly as a phase space integral over the Wigner phase space representations of the two operators The two are a projection operator onto the product’s space, which is difficult to compute, and the symmetrized thermal flux operator, which can be computed using Monte Carlo methods A quantum transition state theory was presented recently, in which the exact projection operator was replaced by its parabolic barrier limit Alternatively, the exact projection operator may be replaced by its classical limit Both approximations give thermodynamic estimates for the quantum rates In this paper, we derive a perturbation theory expansion for the projection operator about the parabolic barrier limit and the classical limit The correction terms are then used to evaluate the leading order corrections to the rate estimates based on the parabolic barrier or classical limits of the projection operator The expansion is applied to a symmetric and an asymmetric Eckart barrier The first two terms in the expansion give excellent results for temperatures above the crossover between quantum tunneling and thermal activation For deep tunneling and asymmetric systems, the use of variational transition state theory, the classical limit, and perturbation theory leads to significant improvement in the estimate of the tunneling rate Multidimensional extensions are presented and discussed

Quantum transition state theory: Perturbation expansion

We present an iterative path integral algorithm for computing multitime correlation functions of a quantum system coupled to a dissipative bath of harmonic oscillators. By splitting the Boltzmann operator into two parts and reordering the propagators in the expression for canonical correlation functions, we are able to transform the evolution time contour into a symmetric one so that a forward propagation and a backward one are specified. Because the memory induced by the bath through the Feynman–Vernon influence functional decays rapidly in the complex time plane, long-time correlations are negligible. Taking advantage of this fact, we show that the correlation function can be obtained via an iterative procedure. The method is used to calculate three-time correlation functions of a dissipative two-level system.

Jiushu Shao

Papers

Forward−Backward Semiclassical Dynamics without Prefactors

Forward−Backward Semiclassical Dynamics with Linear Scaling

Decoherent Dynamics of a Two-Level System Coupled to a Sea of Spins

Quantum transition state theory: Perturbation expansion

Iterative path integral formulation of equilibrium correlation functions for quantum dissipative systems