Abstract The eigenstate thermalization hypothesis (ETH) is a mechanism for the thermalization of quantum chaotic systems. Most researches on quantum chaotic systems involve the two-body interaction between spins, fermions, and bosons, but for the JCH model, the emergence of quantum-chaotic properties comes from the competition between the hopping strength of the photon and the photon-atom coupling strength. We use exact diagonalization to prove that the JCH system has the property of the quantum chaotic system and verify the validity of ETH ansatz both for diagonal and off-diagonal matrix elements of the photon observables with reasonable experimental parameters.
Abstract The eigenstate thermalization hypothesis (ETH) is a mechanism for the thermalization of quantum chaotic systems. Most researches on quantum chaotic systems involve the two-body interaction between spins, fermions, and bosons, but for the JCH model, the emergence of quantum-chaotic properties comes from the competition between the hopping strength of the photon and the photon-atom coupling strength. We use exact diagonalization to prove that the JCH system has the property of the quantum chaotic system and verify the validity of ETH ansatz both for diagonal and off-diagonal matrix elements of the photon observables with reasonable experimental parameters.
The eigenstate thermalization hypothesis (ETH) is a successful theory that provides sufficient criteria for ergodicity in quantum many-body systems. Most studies were carried out for Hamiltonians relevant for ultracold quantum gases and single-component systems of spins, fermions, or bosons. The paradigmatic example for thermalization in solid-state physics are phonons serving as a bath for electrons. This situation is often viewed from an open-quantum-system perspective. Here, we ask whether a minimal microscopic model for electron-phonon coupling is quantum chaotic and whether it obeys ETH, if viewed as a closed quantum system. Using exact diagonalization, we address this question in the framework of the Holstein polaron model. Even though the model describes only a single itinerant electron, whose coupling to dispersionless phonons is the only integrability-breaking term, we find that the spectral statistics and the structure of Hamiltonian eigenstates exhibit essential properties of the corresponding random-matrix ensemble. Moreover, we verify the ETH ansatz both for diagonal and off-diagonal matrix elements of typical phonon and electron observables, and show that the ratio of their variances equals the value predicted from random-matrix theory.
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called "efficient basis" over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.
In the one-dimensional low-density Jaynes-Cummings Hubbard (JCH) model, we find that when the hopping strength is much smaller than the coupling strength, the average restricted energy gap ratio exhibits an abnormal statistical behavior that is neither a Poisson nor a Gaussian orthogonal ensemble. But the average half-chain entanglement entropy exhibits ergodicity, and the eigenstate thermalization hypothesis (ETH) is valid for the observable. These results are quite different from those of the standard JCH model. In addition, when the hopping and the coupling strengths are of the same order, quantum chaos still appears in the low-density JCH model, which is in contrast to the integrability of the one-dimensional hard-core bosons. Finally, the dipole-dipole interaction breaks the particle-hole symmetry and leads the abnormal statistical properties to be closer to those of the integrable system at the weak hopping strength limit, but the quantum chaos properties cannot be affected when the hopping strength is of the same order as the coupling strength. Our results demonstrate the counterintuitive behavior in the low-density JCH model and explain the physics behind them from the perspective of the energy spectrum.
The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deutsch, who showed how a small interaction, which is modeled by a random matrix, leads to ETH (J.M. Deutsch, Phys. Rev. A 43, 1991). This thesis aims to find justification for the assumptions made by Deutsch from a microscopic point of view. The idea is to analyze whether a generic quantum system can be brought to a form that satisfies Deutsch's ansatz. The method is to use flow equations to define unitary transformations to map the initial Hamiltonian to an effective one, which can be compared to a random matrix. The test system is a one-dimensional hardcore boson model, which is set up using exact diagonalization and numerous symmetries. We analyze the statistical properties of the numerically obtained Hamiltonian matrices and test for normality and compare them to a random transformation. In a second step we verify that the continuous unitary transformations do not destroy the structure of typical few-body observables by studying the scaling of the inverse participation ratio. The obtained results support our hypothesis that it is possible to extend Deutsch's argument to generic microscopic many-body Hamiltonians.
In one-dimensional low-density Jaynes-Cummings Hubbard (JCH) models [Phys. Rev. E 106, 064107 (2022)2470-004510.1103/PhysRevE.106.064107], we proved that the eigenstate thermalization hypothesis (ETH) is valid when the tunneling strength and coupling strength are of the same order. Surprisingly, at the weak tunneling limit, we observed that the entanglement entropy and scaling law of kinetic energy operators also exhibit obvious quantum chaotic properties, this is an unexpected result. To substantiate these findings, we further discuss their nonequilibrium dynamics in this paper. Our analysis reveals that when the model is a weak tunneling limit after the quench and the initial state is an equilibrium state of chaos, the system reaches an equilibrium state. This observation supports the conclusion that the low-density JCH model at the weak tunneling limit is nonintegrable, corroborating our previous results [Phys. Rev. E 106, 064107 (2022)2470-004510.1103/PhysRevE.106.064107]. Additionally, by discussing the validity of the fluctuation-dissipation theorem (FDT) and the evolution behavior of entanglement entropy and fidelity, we numerically demonstrate the differences between the one-dimensional low-density JCH model and general nonintegrable systems. Specifically, in the low-density JCH model, when the Hamiltonian after the quench is integrable, the validity of FDT depends on the thermal behavior of the initial Hamiltonian, and a metastable state is observed during the evolution of entanglement entropy. Our research presents an an intriguing and unique nonintegrable model, enriching the current understanding of nonintegrable systems.
Disorder in one-dimensional (1D) many-body systems facilitates abundant phases such as many-body localization (MBL) and thermalization. However, it remains unclear regarding their existence and behavior within hybrid quantum systems. Here, based on a simple bosonic-spin hybrid model, known as the Jaynes-Cummings-Hubbard (JCH) array, we investigate the effect of disorder compared to the phenomena in the clean system with the variation of atom-photon coupling strength. By using the level-spacing ratio, entanglement entropy, and the properties of observable diagonal and off-diagonal matrix elements, we find that strong disorder results in the appearance of a MBL phase in the JCH model that strongly violates the eigenstate thermalization hypothesis (ETH), while a conditional prethermal behavior can exist in the weak disorder regime. The conditional prethermal dynamics is based on the choice of initial product states. This work systematically reveals abundant many-body phases in the 1D JCH model and clarifies the discrepancies in the thermalization properties of systems with and without disorder.