Current fluctuations in open quantum systems: Bridging the gap between quantum continuous measurements and full counting statistics

TL;DR

This work unifies stochastic master equation and full counting statistics approaches to current fluctuations in continuously measured open quantum systems. By introducing jump and diffusion unravelings, tilted Liouvillians, and cumulant-generating formalisms, the authors provide a coherent toolkit for computing currents, fluctuations, waiting times, and higher-order statistics across diverse quantum platforms. The tutorial combines four pedagogical examples with practical solution methods and connections to topical areas such as quantum sensing, fluctuation theorems, TURs/KURs, and QPC-based readouts, highlighting both deep theoretical links and actionable numerical techniques. The resulting framework enables cross-disciplinary insights, enables efficient numerical analyses, and offers a structured route to extract physically meaningful fluctuation phenomena from open-quantum-system dynamics.

Abstract

Continuously measured quantum systems are characterized by an output current, in the form of a stochastic and correlated time series which conveys crucial information about the underlying quantum system. The many tools used to describe current fluctuations are scattered across different communities: quantum opticians often use stochastic master equations, while a prevalent approach in condensed matter physics is provided by full counting statistics. These, however, are simply different sides of the same coin. Our goal with this tutorial is to provide a unified toolbox for describing current fluctuations. This not only provides novel insights, by bringing together different fields in physics, but also yields various analytical and numerical tools for computing quantities of interest. We illustrate our results with various pedagogical examples, and connect them with topical fields of research, such as waiting-time statistics, quantum metrology, thermodynamic uncertainty relations, quantum point contacts and Maxwell's demons.

Figures (24)

• Figure 1: Continuously measured quantum systems and their output currents. Left: paradigmatic example of photo-detector collecting the photons that leak out of an optical cavity. Middle: the measurement gives rise to a stochastic output current; i.e., a correlated time-series which contains information about the underlying quantum system. These currents usually come in two types: quantum jumps, which involve discrete clicks like in photo detection; and quantum diffusion, where the current is a continuous noisy signal. In this tutorial we provide the tools and the intuition to analyze the statistical properties of these currents, such as their multi-time correlations. We also show how this is connected to the framework of Full Counting Statistics (right), which analyzes the properties of the integrated current (called the net charge).
• Figure 2: Structure of the tutorial, divided into core and optional material.
• Figure 3: Overview of the main results in this tutorial, with the corresponding sections.
• Figure 4: Depictions of the 4 basic examples discussed throughout the tutorial. See Table \ref{['tab:examples']} for more details.
• Figure 5: Steady-state Wigner function $W(x,p)=\frac{1}{\pi} \int_{-\infty}^\infty dy~\langle x-y| \rho_{\rm ss}|x+y\rangle e^{2i py}$ (where $|x\rangle$ are the eigenstates of position) for Example D (Sec. \ref{['ssec:exampleD']}). The horizontal and vertical axes refer to the field quadratures (position and momentum), not shown for clarity. Parameters: (a) $(G, U, \Delta) = (0.3i, 0, 0)$, (b) $(1, 1/3, 0)$, and (c) $(1, 1/3, -2)$, with $\kappa = 1$. In all cases, the Wigner function is non-negative. The color scales are omitted for visibility. These three examples exhibit the rich landscape of possible states that can be studied in Example D.