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

Gabriel T. Landi, Michael J. Kewming+2 more|arXiv (Cornell University)|Mar 7, 2023

Advanced Thermodynamics and Statistical Mechanics266 references4 citations

TL;DR

This paper unifies quantum continuous measurement and full counting statistics (FCS) by deriving a common framework for analyzing current fluctuations in open quantum systems. It introduces a systematic toolbox based on stochastic master equations and cumulant generating functions, demonstrating how quantum jumps and diffusive currents can be treated on equal footing, with key results including a unified treatment of noise spectra, correlation functions, and fluctuation theorems across diverse quantum systems such as qubits, quantum dots, and parametric oscillators.

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.

Motivation & Objective

• To bridge the conceptual and formal gap between quantum continuous measurement theory and full counting statistics (FCS), which are typically treated as separate frameworks in quantum optics and condensed matter physics.
• To develop a unified analytical and numerical toolbox for computing current fluctuations, including correlation functions, power spectra, and cumulants, applicable to both quantum jump and diffusive measurement scenarios.
• To establish connections between current fluctuation theory and topical research areas such as thermodynamic uncertainty relations, quantum metrology, quantum point contacts, and Maxwell’s demon protocols.
• To provide a systematic method for computing the stochastic current, its fluctuations, and higher-order statistics using Lindblad master equations, Drazin inverses, and saddle-point approximations.
• To demonstrate the equivalence of different measurement paradigms—such as homodyne, heterodyne, and photo-detection—within a single formalism based on quantum trajectories and positive operator-valued measures (POVMs).

Proposed method

• Derives the output current as a stochastic time series from the stochastic master equation, using a general Lindblad generator with time-dependent jump operators and continuous measurement backaction.
• Introduces the use of the Drazin inverse to compute the two-point correlation function and power spectrum of the current, particularly in the weak dissipation limit.
• Applies the cumulant generating function (CGF) formalism to compute higher-order current statistics, including the scaled cumulant generating function (SCGF) for both quantum jump and diffusive processes.
• Uses vectorized notation and matrix representations of Liouvillians to enable efficient numerical computation of current fluctuations and correlation functions.
• Employs the saddle-point approximation to compute full probability distributions of current counts, particularly for systems with Poissonian or Gaussian statistics.
• Establishes the equivalence between quantum jump and diffusive descriptions in the limit of weak coupling, using a limiting procedure from discrete to continuous measurement outcomes.

Experimental results

Research questions

• RQ1 How can quantum jump and diffusive current fluctuations be described within a single, unified formalism?
• RQ2 What is the relationship between the power spectrum of the output current and the underlying system’s Liouvillian dynamics in open quantum systems?
• RQ3 How do fluctuation theorems and the fluctuation-dissipation theorem emerge from the stochastic dynamics of continuous measurements?
• RQ4 What is the role of the Drazin inverse in computing the noise and correlation functions of the output current in non-Markovian or non-ergodic systems?
• RQ5 How can full counting statistics be used to extract information about quantum correlations, entanglement, and thermodynamic bounds in driven quantum systems?

Key findings

• The power spectrum of the output current for a coherently driven qubit coupled to a thermal reservoir exhibits a Mollow triplet structure, directly linking the spectral features to the underlying Rabi oscillations and vacuum Rabi splitting.
• For a parametrically driven Kerr oscillator, the second-order coherence function $ g^{(2)} $ reveals non-classical statistics, with $ g^{(2)}(0) < 1 $, indicating sub-Poissonian photon statistics.
• The Drazin inverse provides an exact expression for the current correlation function in quantum jump processes, even in the presence of non-ergodic dynamics, enabling precise noise calculations.
• The scaled cumulant generating function (SCGF) for a quantum dot coupled to two fermionic leads satisfies a fluctuation theorem, with $ C(m{ heta}) = C(-m{ heta} + im{ u}) $, confirming the validity of the fluctuation-dissipation relation in non-equilibrium steady states.
• The noise power spectrum for weakly dissipative systems can be computed analytically using a perturbative expansion in the Drazin inverse, yielding a leading-order correction to the Markovian result.
• The saddle-point approximation accurately recovers the full probability distribution of current counts in a cavity, with results matching those from direct photo-detection and homodyne measurement schemes when properly accounting for measurement backaction.