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arXiv:2407.01940 · Physical review. E

Semi-Markov processes in open quantum systems. III. Large deviations of first-passage-time statistics

Fei Liu, Shihao Xia, Shanhe Su

Published Dec 23, 2024• 5 citations
arXivHTMLPDF

Abstract

In a specific class of open quantum systems with finite and fixed numbers of collapsed quantum states, the semi-Markov process method is used to calculate the large deviations of the first passage time statistics. The core formula is an equation of poles, which is also applied in determining the scaled generating functions (SCGFs) of the counting statistics. For simple counting variables, the SCGFs of the first passage time statistics are derived by finding the largest modulus of the roots of this equation with respect to the z-transform parameter and then calculating its logarithm. The procedure is analogous to that of solving for the SCGFs of the counting statistics. However, for current-like variables, the method generally fails unless the equation of pole is simplified to a quadratic form. The fundamental reason for this lies in the nonuniqueness between the roots and the region of convergence for the joint transform. We illustrate these results via a resonantly driven two-level quantum system, where for several counting variables the solutions to the SCGFs of the first passage time are analytically obtained. Furthermore, we apply these functions to investigate quantum violations of the classical kinetic and thermodynamic uncertainty relations.

DOI
10.1103/physreve.110.064145

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In a specific class of open quantum systems with finite and fixed numbers of collapsed quantum states, the semi-Markov process method is used to calculate the large deviations of the first passage time statistics. The core formula is an equation of poles, which is also applied in determining the scaled generating functions (SCGFs) of the counting statistics. For simple counting variables, the SCGFs of the first passage time statistics are derived by finding the largest modulus of the roots of this equation with respect to the z-transform parameter and then calculating its logarithm. The procedure is analogous to that of solving for the SCGFs of the counting statistics. However, for current-like variables, the method generally fails unless the equation of pole is simplified to a quadratic form. The fundamental reason for this lies in the nonuniqueness between the roots and the region of convergence for the joint transform. We illustrate these results via a resonantly driven two-level quantum system, where for several counting variables the solutions to the SCGFs of the first passage time are analytically obtained. Furthermore, we apply these functions to investigate quantum violations of the classical kinetic and thermodynamic uncertainty relations.
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.
We use a semi-Markov process method to calculate large deviations of counting statistics for three open quantum systems, including a resonant two-level system and resonant three-level systems in the Λ and V configurations. In the first two systems, radical solutions to the scaled cumulant generating functions are obtained. Although this is impossible in the third system, since a general sixth-degree polynomial equation is present, we still obtain asymptotically large deviations of the complex system. Our results show that, in these open quantum systems, the large deviation rate functions at zero current are equal to two times the largest nonzero real parts of the eigenvalues of operator -iH[over ̂], where H[over ̂] is a non-Hermitian Hamiltonian, while at a large current these functions possess a unified formula.
A semi-Markov process method for obtaining general counting statistics for open quantum systems is extended to the scenario of resetting. The simultaneous presence of random resets and wave function collapses means that the quantum jump trajectories are no longer semi-Markov. However, focusing on trajectories and using simple probability formulas, general counting statistics can still be constructed from reset-free statistics. An exact tilted matrix equation is also obtained. The inputs of these methods are the survival distributions and waiting-time density distributions instead of quantum operators. In addition, a continuous-time cloning algorithm is introduced to simulate the large-deviation properties of open quantum systems. Several quantum optics systems are used to demonstrate these results.
Using the age-structure formalism, we definitely establish connections between semi-Markov processes and the dynamics of open quantum systems that satisfy the Markov quantum master equations. A generalized Feynman-Kac formula of the semi-Markov processes is also proposed. In addition to inheriting all statistical properties possessed by the piecewise deterministic processes of wave functions, the semi-Markov processes show their unique advantages in quantum counting statistics. Compared with the conventional method of the tilted quantum master equation, they can be applied to more general counting quantities. In particular, the terms involved in the method have precise probability meanings. We use a driven two-level quantum system to exemplify these results.
First-passage phenomena play a fundamental role in classical stochastic processes. We here exactly solve a quantum first-passage time problem for quantum diffusion driven by measurement noise, a generalization of classical Brownian motion. Such continuous monitoring may trap the measured quantum system in a decoherence-free subspace, a fraction of the available state space that is isolated from the surroundings, and thus plays an important role in quantum information science. We analytically determine the first-passage time distribution, whose form neither depends on the system Hamiltonian nor on the measurement operator, and is therefore universal. These results provide a general framework to investigate the first-passage statistics of diffusive quantum trajectories.
We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.