Quantum many-body systems are characterized by their correlations.While equal-time correlators and unequal-time commutators between operators are standard observables, the direct access to unequal-time anticommutators poses a formidable experimental challenge.Here, we propose a general technique for measuring unequal-time anticommutators using the linear response of a system to a non-Hermitian perturbation.We illustrate the protocol at the example of a Bose-Hubbard model, where the approach to thermal equilibrium in a closed quantum system can be tracked by measuring both sides of the fluctuation-dissipation relation.We relate the scheme to the quantum Zeno effect and weak measurements, and illustrate possible implementations at the example of a cold-atom system.Our proposal provides a way of characterizing dynamical correlations in quantum many-body systems with potential applications in understanding strongly correlated matter as well as for novel quantum technologies.
Quantum many-body systems are characterized by their correlations.While equal-time correlators and unequal-time commutators between operators are standard observables, the direct access to unequal-time anticommutators poses a formidable experimental challenge.Here, we propose a general technique for measuring unequal-time anticommutators using the linear response of a system to a non-Hermitian perturbation.We illustrate the protocol at the example of a Bose-Hubbard model, where the approach to thermal equilibrium in a closed quantum system can be tracked by measuring both sides of the fluctuation-dissipation relation.We relate the scheme to the quantum Zeno effect and weak measurements, and illustrate possible implementations at the example of a cold-atom system.Our proposal provides a way of characterizing dynamical correlations in quantum many-body systems with potential applications in understanding strongly correlated matter as well as for novel quantum technologies.
Linear response theory is the theoretical foundation of modern experiments. In particular, it plays a vital role in measuring quantum matters. Its main idea is to take the external probe signal of the physical system as a perturbation and use the correlation function in the unperturbed equilibrium state to depict the response to the observable in system. In recent half century, the linear response theory for the closed quantum system has achieved great success in experiments on quantum matters. In recent years, with the tremendous progress of the precise manipulation of the light-matter interaction, the ultracold atom experiments can precisely control dissipative quantum many-body systems. With the discovery of many novel phenomena, dissipative quantum systems and non-Hermitian physics have attracted extensive attention in theory and experiment. We developed a linear response theory, named non-Hermitian linear response theory, to deal with open quantum systems. This theory takes the non-Hermitian term and quantum noise, which are induced by dissipation, as an external perturbative input, to detect the properties of the quantum system, and relates the time evolution of the observable with the correlation function in the unperturbed state of the system. The non-Hermitian linear response theory provides a new method for distinguishing the exotic quantum phase from the normal phase. The theoretical predictions are highly consistent with the recent experimental results of cold atom systems. This paper will review the non-Hermitian linear response theory and discuss its applications in quantum many-body and time-reversal symmetric quantum systems.
Experimental access to many-body quantum systems is often limited by measurement backaction, and key dynamical properties are typically obtained by perturbing a system and measuring its response. Here we replace this active paradigm with a minimally invasive protocol based on a pair of weak quantum measurements that leverages measurement backaction as a strength. By correlating time-separated measurements with the first detecting fluctuations -- of any sort -- and the second tracking their time evolution, our method directly measures dynamical correlation functions without external perturbation. We demonstrate this technique in an atomic Bose-Einstein condensate using phase-contrast imaging to obtain the two-time density-density correlation function known as the Van Hove function and, through its Fourier transform, the dynamical structure factor. Due to the role of spatial correlations in scattering, these quantities underpin neutron and X-ray scattering and atomic Bragg spectroscopy. This approach is broadly applicable, providing access to correlation functions between any pair of observables amenable to weak measurement, thereby going beyond the capabilities of conventional strong measurements. We further isolate the role of quantum backaction through Aharonov's post-selection-based quantum weak values.
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.
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.
Linear response theory and Green's functions provide a universal framework for understanding dynamical correlations in strongly correlated open quantum systems. While the theoretical foundation for non-Hermitian linear response has been recently established to describe dissipation and fluctuation-dissipation relations (FDR), generalizing these predictions onto practical quantum computers remains a formidable algorithmic challenge due to the intrinsically non-unitary nature of the dynamics. In this work, we present a systematic algorithmic framework that seamlessly transforms non-unitary multi-time correlation functions into a unitary form viable for digital quantum hardware. By mapping the vectorization of the Lindblad master equation into an expanded continuous-variable Liouville space, we employ the Schrödingerization technique to deterministically evaluate the non-Hermitian response. Furthermore, through hardware-aware simulations utilizing a 133-qubit device noise model, we demonstrate that our unitary framework robustly preserves the fundamental spectral information -- specifically the phase and oscillatory frequency -- against realistic depolarizing channels. By bypassing explicit non-unitary mid-circuit measurements, this approach intrinsically supports the integration of standard quantum error mitigation protocols, providing a scalable algorithmic blueprint for probing open-system universality on near-term hardware.
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.