Exact solution for the inhomogeneous Dicke model in the canonical ensemble: Thermodynamical limit and finite-size corrections

Abstract

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. PACS numbers: 02.30Ik, 42.50.Ct, 03.65.Fd I. INTRODUCTIONRecent progress in engineering of artificial quantum systems for information technologies renewed an interest to Dicke (Tavis-Cummings) model [1] and its exact solution, already well known for a long time [2][3][4][5][6][7]. Dicke model describes an interaction between a collection of two-level systems and single mode radiation field, while physical realizations range from superconducting qubits coupled to microwave resonators to polaritons in quantum wells, see e.g. Refs. [8,9] and references therein; furthermore, it can be also applied to Fermi-Bose condensates near the Feshbach resonance [10].The characteristic feature of macroscopic artificial quantum systems such as superconducting qubits is a disorder in excitation frequencies and inhomogeneous broadening of the density of states. This feature is due to fundamental mechanisms: for example, an excitation energy of flux qubits depends exponentially on Josephson energies [11], which makes it extremely sensitive to characteristics of nanometer-scale Josephson junctions. Inhomogeneous broadening appears even in the case of microscopic two-level systems, such as NV-centers, where it is induced by spatial fluctuations of background magnetic moments [12]. In the case of NV-centers, the density of states is characterized by the q-Gaussian distribution [13]. Moreover, there are prospect to utilize the broadening for the construction of a multimodal quantum memory [14]. It is also possible to engineer a density of states profile by using, e.g., a so-called spectral arXiv:1612.01774v2 [cond-mat.stat-mech]

Fluctuations and photon statistics in a quantum metamaterial near a superradiant transition

“…Zero temperature description * Electronic address: shapiro.dima@gmail.com for a limit of large excitations number was obtained in Ref. [19] by means of Bethe-ansatz technique.…”

Radiation trapping effect versus superradiance in quantum simulation of light-matter interaction

“…Notice that counterrotating terms however are essential in some special situation, for instance, under the parametric driving which can give rise to the dynamical Lamb effect [32][33][34]. Also note that in absence of dissipation, the system we study can be addressed using an exact solution through Bethe-ansatz technique [25,35].…”

Universal fluctuations and squeezing in generalized Dicke model near the superradiant phase transition

“…Hence, there exist sectors with different excitation numbers N ex or with different excitation densities ρ = N ex /N , provided the thermodynamical limit N → ∞ is considered. The leading in 1/N contribution to the ground state energy density at given ρ is [33]…”

“…We stress that all other contributions to the energy are negligible in the limit N → ∞, i.e., non-extensive, but they can be evaluated using the approach of Ref. [33]. At fixed ρ, parameters ξ and λ also determine energies of excited dressed states (RWA Hamiltonian eigenstates) given by (ε − λ) 2 + ξ 2 .…”

“…Zero temperature description for RWA in a limit of large excitations number was obtained in Ref. [33] using the Bethe ansatz technique. Recently, the analysis of quantum chaos in the symmetric Dicke model via the method of out-of-timeordered correlators attracted a great interest [34][35][36].…”