[Paper Review] Coherent Fluctuations in Noisy Mesoscopic Systems, the Open Quantum SSEP and Free Probability

Ludwig Hruza, Denis Bernard|arXiv (Cornell University)|Apr 25, 2022

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper establishes a connection between coherent fluctuations in noisy mesoscopic quantum systems and free probability theory, demonstrating that the time evolution of connected coherence fluctuations in the open quantum symmetric simple exclusion process (Q-SSEP) is governed by free cumulants. The key contribution is a novel framework where free probability provides a natural mathematical language for describing non-equilibrium quantum coherence dynamics, leading to a simple steady-state solution for coherence correlations.

ABSTRACT

Quantum coherences characterise the ability of particles to quantum mechanically interfere within some given distances. In the context of noisy many-body quantum systems these coherences can fluctuate. A simple toy model to study such fluctuations in an out-of-equilibrium setting is the open quantum symmetric simple exclusion process (Q-SSEP) which describes spinless fermions in one dimension hopping to neighbouring sites with random amplitudes coupled between two reservoirs. Here we show that the dynamics of fluctuations of coherences in Q-SSEP have a natural interpretation as free cumulants, a concept from free probability theory. Based on this insight we provide heuristic arguments why we expect free probability theory to be an appropriate framework to describe coherent fluctuations in generic mesoscopic systems where the noise emerges from a coarse-grained description. In the case of Q-SSEP we show how the link to free probability theory can be used to derive the time evolution of connected fluctuations of coherences as well as a simple steady state solution.

Motivation & Objective

• To develop a theoretical framework for describing coherent fluctuations in out-of-equilibrium quantum mesoscopic systems, particularly those involving quantum interference and entanglement.
• To identify a universal mathematical structure—free cumulants—that naturally describes the dynamics of coherence fluctuations in such systems.
• To demonstrate that the open quantum SSEP serves as a tractable model where free probability theory provides exact solutions for connected coherence correlations.
• To extend macroscopic fluctuation theory to quantum systems by incorporating quantum coherences as dynamical variables governed by free cumulant dynamics.

Proposed method

• The authors model the open quantum SSEP as a one-dimensional system of spinless fermions with stochastic hopping amplitudes, coupled to reservoirs, to study non-equilibrium coherence dynamics.
• They express the time evolution of coherence correlation functions using loop expectation values over non-crossing partitions, a central construct in free probability theory.
• The dynamics of connected coherence fluctuations are derived via an expansion into free cumulants, with time evolution governed by a master equation involving derivatives and delta functions on partition blocks.
• The steady-state solution for coherence correlations is shown to be the minimum function of spatial coordinates, derived using Heaviside and delta functions, and verified to satisfy the steady-state equation.
• A bijection is established between non-crossing partitions and factorized partitions arising from boundary coupling, linking the structure of free cumulants to physical dynamics.
• The framework is validated by showing that the time evolution of connected 2-point functions and higher-order correlations align with free probability identities.

Experimental results

Research questions

• RQ1 Can free probability theory provide a natural mathematical framework for describing coherent fluctuations in non-equilibrium quantum mesoscopic systems?
• RQ2 How do the dynamics of connected coherence fluctuations in the open quantum SSEP relate to free cumulants and non-crossing partitions?
• RQ3 What is the analytical form of the steady-state solution for coherence correlations in the open quantum SSEP, and how does it emerge from free probability?
• RQ4 To what extent can the macroscopic fluctuation theory be generalized to include quantum coherence effects via free cumulant dynamics?
• RQ5 Is there a structural correspondence between the factorization of correlation functions in the Q-SSEP and the combinatorics of non-crossing partitions in free probability?

Key findings

• The time evolution of connected coherence fluctuations in the open quantum SSEP is governed by a master equation that maps directly to free cumulant dynamics on non-crossing partitions.
• The steady-state solution for the n-point coherence correlation function is given by the minimum of the spatial coordinates, φ∞(x₁,…,xn) = min(x₁,…,xn), which satisfies the steady-state equation derived from the dynamics.
• The dynamics of coherence fluctuations are shown to be equivalent to the time evolution of free cumulants, with the structure of the equations reflecting the non-crossing nature of the partitions involved.
• A bijection between non-crossing partitions and factorized correlation structures arising from boundary coupling confirms the consistency of the free probability framework with the physical model.
• The connected 2-point function admits an analytic solution that matches the predictions of the free probability approach, validating the method at the lowest non-trivial order.
• The use of Heaviside and delta functions allows exact derivation of the steady-state solution, confirming that the minimum function is the unique solution to the steady-state equation.