Computing Large Deviations of First-Passage-Time Statistics in Open Quantum Systems: Two Methods

TL;DR

This work tackles the challenge of computing large deviations in first-passage-time statistics for open quantum systems. It develops two complementary approaches: a pole-based analytic method that identifies the region of convergence in the $(\nu,z)$ plane by solving $\det[\nu-\mathcal{L}_z]=0$, and a wave function cloning simulation that scales to large Hilbert spaces. The authors derive analytical SCGFs for field-driven two-level and three-level systems and validate both methods on a model of two interacting two-level atoms, showing consistency with inverse relations between counting statistics and FPT statistics and confirming fluctuation theorems. The results provide practical tools for evaluating rare-event statistics in quantum dynamics, including regimes where semi-Markov descriptions fail, thereby enabling accurate large-deviation analyses in many-body open quantum systems.

Abstract

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 is then obtained as the logarithm of the boundary values within this region. The theoretical basis lies in the facts that the dynamics of the open quantum systems can be unraveled into a piecewise deterministic process and there exists a tilted Liouville master equation in Hilbert space. The second method uses a simulation-based approach 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 alongside cloning simulations for a field-driven system comprising two interacting two-level atoms.

Figures (2)

• Figure 1: Schematic representations of a two-level system (a), three-level system (b), and two interacting two-level atoms (c). The lines marked with double arrows indicate driving, while the lines denoted by a single arrow represent the jumps occurring within the quantum systems.
• Figure 2: For the two interacting two-level atoms, the open circles in the real $\nu$-$\lambda$ plane represent the real solutions of the equation of poles (\ref{['equationofpolestiltedquantummasterequation']}), where $\nu$ is free parameter and $z(\nu)$ are the real solutions or roots. We obtain these data at discrete $\nu$-values and do not connect them with curves. Nevertheless, we can still easily identify the continuous boundaries of the regions of convergence (ROCs). Both boundaries lie on the rightmost side of the plane. They pass through the point $(0,0)$ (denoted by the large crosses), and extend towards the right-hand side. Thus, $\lambda(\nu)$ on these boundaries precisely corresponds the SCGFs $\Psi_{\pm}(\nu)$ or $\Psi(\nu)$ in the FPT statistics. The solid curves represent the SCGFs in the counting statistics, $\Phi(\lambda)$. These are obtained by finding the roots of the same Eq. (\ref{['equationofpolestiltedquantummasterequation']}) with respect to the variable $\nu$ and then applying Eq. (\ref{['SCGFformulacountingstatistics']}). In this case, the free parameter is $z=\exp(\lambda)$. The asterisks symbols denote the SCGFs data obtained from simulations built on the wave function cloning algorithm. The fixed small time-step for the quantum trajectory simulation is $dt=0.01$, and $10000$ initial copies of the quantum system are used for the population dynamics. (a) The current-like counting variable. (b) The simple counting variables. Their respective weights are described in the main text. The values of the parameters are $\Delta=0.1$, $\Omega=1$, $V=5$, $r_-=2.0$, $r_+=0.6$, respectively.