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Universal Fluctuation-Dissipation Theorem for Non-Hermitian Quantum Systems

Revista, Zen, PHYSICS, 10

Published Nov 28, 2025• 0 citations
Publisher

Abstract

The fluctuation-dissipation theorem (FDT) is a cornerstone of statistical mechanics, linking the linear response of a system to external perturbations with the equilibrium fluctuations of its constituent degrees of freedom. Traditionally formulated for Hermitian quantum systems in thermal equilibrium, its applicability to non-Hermitian quantum systems, which are increasingly relevant in areas such as open quantum systems, PT-symmetric quantum mechanics, and topological phases, remains a complex and largely unresolved challenge. This paper presents a theoretical framework for a universal fluctuation-dissipation theorem applicable to a broad class of non-Hermitian quantum systems. We demonstrate how the presence of non-Hermiticity, characterized by complex energy eigenvalues and non-orthogonal eigenstates, fundamentally modifies the relationship between correlation functions and response functions. Our derivation utilizes a generalized time-evolution operator and incorporates concepts from non-Hermitian spectral theory, revealing a modified FDT that accounts for energy gain or loss inherent in these systems. The resulting theorem provides insights into the interplay between quantum noise and dissipation in environments far from conventional thermal equilibrium, offering a crucial tool for understanding and engineering non-Hermitian quantum devices. We discuss the implications for non-equilibrium steady states and the breakdown of standard notions of temperature and entropy, proposing a path towards experimental verification in platforms like photonic lattices and cold atoms with controlled losses.

DOI
10.5281/zenodo.17750607

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The fluctuation-dissipation theorem (FDT) is a cornerstone of statistical mechanics, linking the linear response of a system to external perturbations with the equilibrium fluctuations of its constituent degrees of freedom. Traditionally formulated for Hermitian quantum systems in thermal equilibrium, its applicability to non-Hermitian quantum systems, which are increasingly relevant in areas such as open quantum systems, PT-symmetric quantum mechanics, and topological phases, remains a complex and largely unresolved challenge. This paper presents a theoretical framework for a universal fluctuation-dissipation theorem applicable to a broad class of non-Hermitian quantum systems. We demonstrate how the presence of non-Hermiticity, characterized by complex energy eigenvalues and non-orthogonal eigenstates, fundamentally modifies the relationship between correlation functions and response functions. Our derivation utilizes a generalized time-evolution operator and incorporates concepts from non-Hermitian spectral theory, revealing a modified FDT that accounts for energy gain or loss inherent in these systems. The resulting theorem provides insights into the interplay between quantum noise and dissipation in environments far from conventional thermal equilibrium, offering a crucial tool for understanding and engineering non-Hermitian quantum devices. We discuss the implications for non-equilibrium steady states and the breakdown of standard notions of temperature and entropy, proposing a path towards experimental verification in platforms like photonic lattices and cold atoms with controlled losses.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. Third, it clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state.
Fluctuation dissipation theorems (FDTs) connect the linear response of a physical system to a perturbation to the steady-state correlation functions. Until now, most of these theorems have been derived for finite-dimensional systems. However, many relevant physical processes are described by systems of infinite dimension in the Gaussian regime. In this work, we find a linear response theory for quantum Gaussian systems subject to time dependent Gaussian channels. In particular, we establish a FDT for the covariance matrix that connects its linear response at any time to the steady state two-time correlations. The theorem covers non-equilibrium scenarios as it does not require the steady state to be at thermal equilibrium. We further show how our results simplify the study of Gaussian systems subject to a time dependent Lindbladian master equation. Finally, we illustrate the usage of our new scheme through some examples. Due to broad generality of the Gaussian formalism, we expect our results to find an application in many physical platforms, such as opto-mechanical systems in the presence of external noise or driven quantum heat devices.
Abstract Fluctuation dissipation theorems (FDTs) connect the linear response of a physical system to a perturbation to the steady-state correlation functions. Until now, most of these theorems have been derived for finite-dimensional systems. However, many relevant physical processes are described by systems of infinite dimension in the Gaussian regime. In this work, we find a linear response theory for quantum Gaussian systems subject to time dependent Gaussian channels. In particular, we establish a FDT for the covariance matrix that connects its linear response at any time to the steady state two-time correlations. The theorem covers non-equilibrium scenarios as it does not require the steady state to be at thermal equilibrium. We further show how our results simplify the study of Gaussian systems subject to a time dependent Lindbladian master equation. Finally, we illustrate the usage of our new scheme through some examples. Due to broad generality of the Gaussian formalism, we expect our results to find an application in many physical platforms, such as opto-mechanical systems in the presence of external noise or driven quantum heat devices.
We present a fluctuation-dissipation theorem (FDT) for nonequilibrium quantum systems with detailed-balance breaking, which deviates from the conventional form of the FDT for equilibrium systems preserving detailed balance. Using the phase space formulation of quantum mechanics and the potential-flux landscape framework, we find that the response function of nonequilibrium quantum systems to external perturbations contains a nontrivial contribution from the quantum curl flux quantifying detailed-balance breaking in the steady state, in addition to the correlation function of observables in the steady state representing the contribution of spontaneous fluctuations which is also present in the equilibrium FDT. We illustrate our general formalism with a harmonic oscillator coupled to two heat baths, and show that the nonequilibrium FDT reduces to the conventional expression at the equilibrium condition when the flux contribution vanishes.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.
Traditional thermodynamics, built upon the foundation of Hermitian quantum mechanics, primarily describes closed or weakly open systems where energy is conserved. However, the ubiquitous presence of dissipation and gain in realistic physical and biological systems necessitates a more generalized framework. Non-Hermitian quantum mechanics, characterized by Hamiltonians with complex eigenvalues, naturally accounts for such open systems and their interaction with the environment. This paper explores the burgeoning field of non-Hermitian thermodynamics, which aims to extend classical and quantum thermodynamic principles to systems exhibiting energy exchange with their surroundings. We delve into how non-Hermiticity provides a natural language for describing processes involving dissipation, gain, and decoherence, thereby offering a unified perspective on these phenomena. Key concepts such as PT-symmetry, exceptional points, and generalized definitions of work, heat, and entropy in non-Hermitian settings are reviewed. The paper discusses the implications for fundamental thermodynamic laws, the performance of non-Hermitian heat engines, and the role of coherence in energy transport. By embracing the mathematical elegance and physical insights of non-Hermitian physics, we propose a comprehensive framework that unifies the seemingly disparate concepts of dissipation and coherence within a consistent thermodynamic theory, paving the way for advanced applications in quantum technologies, materials science, and biological energy transfer.