We study the time dynamics of a single boson coupled to a bath of two-level systems (spins 1/2) with different excitation energies, described by an inhomogeneous Dicke model. Analyzing the time-dependent Schr\"odinger equation exactly, we find that at resonance the boson decays in time to an oscillatory state with a finite amplitude characterized by a single Rabi frequency if the inhomogeneity is below a certain threshold. In the limit of small inhomogeneity, the decay is suppressed and exhibits a complex (mainly Gaussian-like) behavior, whereas the decay is complete and of exponential form in the opposite limit. For intermediate inhomogeneity, the boson decay is partial and governed by a combination of exponential and power laws.
We consider an exactly solvable inhomogeneous Dicke model which describes an interaction between a disordered ensemble of two-level systems with single mode boson field. The existing method for evaluation of Richardson–Gaudin equations in the thermodynamical limit is extended to the case of Bethe equations in Dicke model. Using this extension, we present expressions both for the ground state and lowest excited states energies as well as leading-order finite-size corrections to these quantities for an arbitrary distribution of individual spin energies. We then evaluate these quantities for an equally-spaced distribution (constant density of states). In particular, we study evolution of the spectral gap and other related quantities. We also reveal regions on the phase diagram, where finite-size corrections are of particular importance.
We show that in the few-excitation regime, the classical and quantum time evolution of the inhomogeneous Dicke model for N two-level systems coupled to a single boson mode agree for N⪢1. In the presence of a single excitation only, the leading term in an 1/N expansion of the classical equations of motion reproduces the result of the Schr\"odinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for N sufficiently large. By solving the Schr\"odinger equation exactly for two excitations and a particular inhomogeneity, we obtain 1/N corrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by ${\ensuremath{\tau}}_{E}=\sqrt{N/⟨{g}^{2}⟩}$, where ⟨g2⟩ is an effective coupling strength between the two-level systems and the boson.
Previous experimental realizations of Dicke model in atomic or ionic systems are based on global observables assuming uniform spin-boson coupling, while inevitable experimental nonuniformity on the one hand requires site-resolved measurement of spin states, and on the other hand provides potential quantum advantage on the simulation of multi-spin distributions. Here we report the quantum simulation of a nonuniform Dicke-like model in a two-dimensional (2D) crystal of up to 200 ions. We explicitly demonstrate the sensitivity of few-spin observables and multi-spin distributions to the spatial inhomogeneity of the model, and examine the thermalization dynamics of the nonuniform model by measuring the subsystem entropies of selected ion groups. Our work enables the study of Dicke-like models beyond the symmetric subspace, paving the way toward understanding the role of disorder in its thermalization and quantum chaos behavior.