Chaos and the quantum phase transition in the Dicke model

Abstract

We investigate the quantum chaotic properties of the Dicke Hamiltonian; a quantum-optical model which describes a single-mode bosonic field interacting with an ensemble of N two-level atoms. This model exhibits a zero-temperature quantum phase transition in the N → ∞ limit, which we describe exactly in an effective Hamiltonian approach. We then numerically investigate the system at finite N and, by analysing the level statistics, we demonstrate that the system undergoes a transition from quasi-integrability to quantum chaotic, and that this transition is caused by the precursors of the quantum phase-transition. Our considerations of the wavefunction indicate that this is connected with a delocalisation of the system and the emergence of macroscopic coherence. We also derive a semi-classical Dicke model, which exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos.

Instabilities of the Dicke model with field interactions

“…The average BP γ 1 /N is equal to π at λ < 1 and increases abruptly at discontinuous "critical" points when λ > 1. Note that the plateau is formed clearly for N = 1, 2, 4 , the width of the plateau becomes narrower and narrower with the increasing N , which are quite different from the phenomenon of the quantum phase transition in the full DM [7,8,9]. A clear picture of the instability in the ground state of the RWA DM is given in terms of the BP with N = 64 atoms shown in the inset of Fig.…”

“…It has been attracted considerable attentions recently, mainly due to the fact that the Dicke model is closely related to many recent interesting fields in quantum optics and condensed matter physics, such as the superradiant behavior by an ensemble of quantum dots [2] and Bose-Einstein condensates [3], coupled arrays of optical cavities used to simulate and study the behavior of strongly correlated systems [4], and superconducting charge qubits [5,6]. It is known from the previous studies [7,8,9] that the full DM undergoes the second-order quantum phase transition [10].…”

Heisenberg uncertainty principle as a probe of entanglement entropy: Application to superradiant quantum phase transitions

“…, the DH can be proved [25] to be equivalent to H N →∞ =∆ + e † + e + +∆ − e † − e − + E G . E G is the fundamental energy, and the normal eigenfrequencies (gaps) are such that 2∆ 2 ± = (ω 0 /µ) 2 + ω 2 ± ((ω 0 /µ) 2 − ω 2 ) 2 + 16λ 2 ωω 0 µ, with µ = 1 if λ < λ cr and µ = ω0ω 4λ 2 if λ > λ cr .…”

“…The total angular momentum operators J z and J ± read J z = N l=1 σ l z and J ± = N l=1 σ l ± , where σ l z and σ l ± are the usual Pauli matrices for the l th pseudo-spin, so that the angular commutation relations are [J z , J ± ] = ±2J ± and [J + , J − ] = 2J z . In the thermodynamic limit (N → ∞), this Hamiltonian undergoes a superradiant QPT for λ = λ cr = √ ωω 0 /2 [25]. When λ < λ cr , the system is in a Normal Phase (NP), with a squeezed and non-degenerate vacuum, while the Superradiant Phase (SP) occurring for λ > λ cr is embodied by the appearance of a double degeneracy with atomic and photonic macroscopic coherences.…”

“…1 (in red). For λ → λ cr ,∆ 1 ∼ |λ − λ cr | 1/2 [25] and the HP diverges like |λ − λ cr | −1/4 . Thus, in the thermodynamic limit, where the DH in Eq.…”