Full counting statistics and first-passage times in quantum Markovian processes: Ensemble relations, metastability, and fluctuation theorems

TL;DR

This work develops a comprehensive framework that unifies full counting statistics and first-passage time distributions for open quantum systems governed by Lindblad dynamics. It derives finite-time ensemble relations linking the counting-field generator $\mathsf L(z)$ and the first-passage propagator $\mathsf F_1(s)$, and extends these correspondences to two-way currents with $\mathsf J_+$ and $\mathsf J_-$. The authors identify conditions under which such correspondences hold or fail, notably in metastable regimes where cumulant relations break down, and they formulate a fluctuation theorem for rare first-passage events connected to spectral properties of $\mathsf L(z)$. Through three concrete applications—the two-state emitter, the SSH model, and a driven qubit—the paper demonstrates practical computation of the generators and propagators and illuminates how time-integrated and time-resolved fluctuations encode dynamical regimes. The results offer a versatile, experimentally relevant toolkit for quantum transport and thermodynamics, with implications for quantum optics, superconducting circuits, and nanoscale engines.

Abstract

We develop a comprehensive framework for characterizing fluctuations in quantum transport and nonequilibrium thermodynamics using two complementary approaches: full counting statistics and first-passage times. Focusing on open quantum systems governed by Markovian Lindblad dynamics, we derive general ensemble relations that connect the two approaches at all times, and we clarify how the steady states reached at long times relate to those reached at large jump counts. In regimes of metastability, long-lived intermediate states cause violations of experimentally testable cumulant relations, as we discuss. We also formulate a fluctuation theorem governing the probability of rare fluctuations in the first-passage time distributions based on results from full counting statistics. Our results apply to general integer-valued trajectory observables that do not necessarily increase monotonically in time. Three illustrative applications, a two-state emitter, a driven qubit, and a variant of the Su-Schrieffer-Heeger model, highlight the physical implications of our results and provide guidelines for practical calculations. Our framework provides a complete picture of first-passage time statistics in Markovian quantum systems, encompassing multiple earlier results, and it has direct implications for current experiments in quantum optics, superconducting circuits, and nanoscale heat engines.

Figures (9)

• Figure 1: Full counting statistics and first-passage times. As an example, we consider particles that tunnel between an open quantum system and two external leads. The tunneling events to and from the drain are detected and change the number of transferred particles $n$ by plus or minus one.The resulting stochastic trajectories show the number of transferred particles as a function of time. The first-passage time (FPT) is the first time that a given target value of $n$ is reached.The full counting statistics (FCS), by contrast, is the distribution of the particle number $n$ at a fixed time $t$, here $t=15$ (a.u.).The first-passage time distribution for the target value $n=5$.The highlighted example trajectory in panel (b) falls into the highlighted bins in the two histograms.
• Figure 2: Applications of full counting statistics and first-passage times. Photon emitter, which switches stochastically between two states at the rate $\Gamma$. Photons are emitted at the rates $\gamma_1$ or $\gamma_2$, depending on the state of the system. This setup is discussed in Sec. \ref{['subsec:exampleTWE']}.Su-Schrieffer-Heeger (SSH) model, where each unit cell $i$ contains two sites, $\lvert iA\rangle$ and $\lvert iB\rangle$. The tunneling amplitude between sites within a cell is denoted by $v$, while $w$ connects neighboring cells. We consider two unit cells and impose periodic boundary conditions. The orange arrows represent a collective dissipation channel, which induces quantum jumps from $\lvert iB\rangle$ to $\lvert iA\rangle$. We discuss the SSH model in Sec. \ref{['sec:exampleSSH']}.Driven qubit, where transitions between two states are driven coherently with the Rabi frequency $\Omega$ and incoherently at the rates $\gamma_\mathrm{d}$ and $\gamma_\mathrm{u}$. We consider this setup in Sec. \ref{['subsec:example3']}.
• Figure 3: Two-state emitter. (a,b) Spectral plots of the generators $\mathsf L(z)$ and $\mathsf F_1(s)$. The points $(z{=}1,s{=}0)$, where derivatives are taken to find the cumulants at long times, are indicated by dotted lines. The largest eigenvalues around these points (highlighted in purple) are the scaled generating functions $c(z)$ and $\bar{c}(s)$. The inverse function relationship derived in Ref. BudiniJStatMech2014 can be seen in the mirror symmetry between them. Our generalized, finite-time ensemble correspondence shows that not only $c(z)$ and $\bar{c}(s)$, but the entire spectral plots (a) and (b) are mirror images of each other. The mirror symmetry is indicated by diagonal dotted lines. (c) Number of emitted photons as a function of time for 150 randomly generated trajectories. The result shows that in this example involving only one-way currents, the counted number can only increase. In all panels, we use arbitrary units with $\gamma_1 = 1/2$, $\gamma_2 = 2$, and $\Gamma = 1$.
• Figure 4: Large-deviation functions for the two-state emitter. Large-deviation function for the full counting statistics.Large-deviation function for the first-passage time.The results in this figure illustrate the relationship \ref{['eq:ldf_identity']} between the large-deviation functions GingrichPhysRevLett2017. By transforming the large-deviation function in panel (a) according to Eq. \ref{['eq:ldf_identity']}, we obtain the dashed lines in panel (b), which match the large-deviation function for the first-passage time. The large-deviation functions are shown for two values of $\Gamma$, featuring a unique steady state ($\Gamma = 1$) and multiple steady states ($\Gamma = 0$), highlighting that this relationship continues to hold even without a unique steady state. The remaining parameters are the same as in Fig. \ref{['fig:ex1']}.
• Figure 5: Two-state emitter. We show the average current, $\langle n\rangle_t / t$, as a function of the observation time $t$ (blue solid lines) together with the average first-passage time per jump, $\langle t\rangle_n / n$, as a function of the discrete target count $n$ (orange points). These results show that the emitter becomes metastable as the switching rate $\Gamma$ becomes smaller. For small switching rates, we find a crossover between small $t$ and $n$, where the emitter effectively has multiple steady states and the results can be described approximately by Eqs. \ref{['eq:ex1:regime2']} and \ref{['eq:ex1:regime2_2']} (dotted gray lines), and large $t$ and $n$, where the emitter reaches the true steady states and the results match Eqs. \ref{['eq:ex1:regime1']} and \ref{['eq:ex1:regime1_2']}. We have used the same parameters as in Fig. \ref{['fig:ex1']} with the initial state $\rho_0 = (1/4, 3/4)$.