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Level 2.5 large deviations for piecewise deterministic processes

Fei Liu

Published Jan 28, 2021• 0 citations
OpenAlex

Abstract

Level 2.5 large deviations of piecewise deterministic processes (PDPs) are obtained from the perspective of trajectory. As a simple application, the tilted Liouville master equation for PDPs in Hilbert space is proven to be the unraveling of the tilted quantum master equation for stochastic heat production.

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Abstract From the perspective of Markovian piecewise deterministic processes (PDPs), we investigate the derivation of a kinetic uncertainty relation (KUR), which was originally proposed in Markovian open quantum systems. First, stationary distributions of classical PDPs are explicitly constructed. Then, a tilting method is used to derive a rate functional of large deviations. Finally, based on an improved approximation scheme, we recover the KUR. These classical results are directly extended to the open quantum systems. We use a driven two-level quantum system to exemplify the quantum results.
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We simulate a piecewise deterministic Markovian jump process that represents an unraveling of a nonlinear quantum master equation describing the evolution of a quantum subsystem in contact with a heat bath. This process relies on running ensemble averages and state vectors normalized only on average. The latter causes the normalization of a small subset of trajectories to grow exponentially, resulting in numerical instability. We propose an efficient solution to this problem and illustrate the general ideas for a harmonic oscillator and a two-level system, each of them being weakly coupled to a heat reservoir. Our findings lay the foundation for a powerful simulation technique for dissipative quantum systems surrounded by general classical nonequilibrium environments.
Recently a generalized master equation was derived that extends the Lindblad theory to highly non-Markovian quantum processes [H.-P. Breuer, Phys. Rev. A 75, 022103 (2007)]. We perform a stochastic unraveling of this master equation by considering nnn random state vectors that satisfy the corresponding stochastic differential equation for a piecewise deterministic process. As an application we consider a two-state system randomly coupled to an environment consisting of two energy bands with finite number of levels. Our numerical results are compared to results obtained from the time-convolutionless projection operator method using correlated projection superoperators and the exact solution of the Schr\"odinger equation for this system.
We propose two methods for computing the large deviations of the first-passage-time statistics in general open quantum systems. The first method determines the region of convergence of the joint Laplace transform and the z transform of the first-passage time distribution by solving an equation of poles with respect to the z-transform parameter. The scaled cumulant generating function, which is the logarithm of the boundary values within this region, is subsequently obtained. The theoretical basis is that the dynamics of open quantum systems can be unraveled into a piecewise deterministic process and that a tilted Liouville master equation exists in Hilbert space. The second method uses a simulation-based approach that is built on the wave-function cloning algorithm. To validate both methods, we derive analytical expressions for the scaled cumulant generating functions in field-driven two-level and three-level systems. In addition, we present numerical results and cloning simulations for a field-driven system comprising two interacting two-level atoms.