Quantum Chaos Triggered by Precursors of a Quantum Phase Transition: The Dicke Model

Abstract

We consider the Dicke Hamiltonian, a simple quantum-optical model which exhibits a zerotemperature quantum phase transition. We present numerical results demonstrating that at this transition the system changes from being quasi-integrable to quantum chaotic. By deriving an exact solution in the thermodynamic limit we relate this phenomenon to a localisation-delocalisation transition in which a macroscopic superposition is generated. We also describe the classical analogues of this behaviour.At zero temperature, systems of N interacting particles can exhibit a quantum phase transition (QPT) as a function of a coupling parameter λ in the limit that N → ∞. How do the precursors of such a transition influence quantum chaotic (and non-chaotic) behaviour of the same system for finite N ?One of the most direct indicators of the emergence of quantum chaos is the change in energy level spacing statistics from Poissonian to being described by the Gaussian ensembles of Random Matrix Theory. Although this change-over has been observed in many systems [1,2,3,4], only in a comparatively few, isolated cases has the onset of quantum chaos been correlated with the presence of a QPT. Important examples include spin glass shards, which have recently been used in modeling the onset of chaos in quantum computers [5], the Lipkin model [6], the interacting boson model [7], and the three-dimensional Anderson model [8,9], where the change in level statistics occurs at the metalinsulator (localisation-delocalisation) transition found in disordered electronic systems.In this Letter we consider the Dicke Hamiltonian (DH) [10], a quantum-optical model describing the interaction of N two-level atoms with a number of bosonic modes. We demonstrate that a crossover between Poisson and Wigner-Dyson statistics in this model for finite N is intimately connected to a mean-field type superradiance QPT.The simplicity and generality of the Dicke Hamiltonian have afforded it appeal both for the investigation of quantum chaos, and as a model for phase transitions at a critical coupling λ c induced by the interaction with light. The level statistics for finite N have revealed the existence of quantum chaos in certain isolated regimes of the model [11,12]. On the other hand, the QPT aspect for N → ∞ has been discussed in the context of superradiance [13,14], and recently for exciton condensation [15]. Here, we derive an exact solution for all eigenstates, eigenvalues and critical exponents in the thermodynamic limit, and show that above the critical point λ = λ c the ground-state wavefunction bifurcates into a macroscopic superposition for any N < ∞. Our numerical results indicate that a localisation-delocalisation transition for N → ∞ underlies the cross-over between Poissonian and Wigner level-spacing distributions for finite N . Furthermore, we use an exact Holstein-Primakoff transformation to derive the classical limit of the model for arbitrary N and find a transition at λ = λ c from regular to chaotic trajectories. The latter are delocalised around ...

Quantum criticality in a generalized Dicke model

“…In order to describe excitations in the parameter region above the critical point, we now incorporate the fact that both the field and the atomic collective excitations acquire macroscopic occupations, namely, the above approximation to neglect the number of excitations over N is no longer valid [2]. To this aspect, the introduced collective operators B † and C † in Eq.…”

“…where α, β and γ are generally complex parameters in the order of O( √ N ) [2] to be determined later. This is equivalent to assume that all modes behave as the nonzero, macroscopic mean fields above λ (n) c .…”

“…When the coupling parameter λ varies from that less than the critical value λ c to that larger than λ c , the system goes from the normal phase to the super-radiant one in the presence of the symmetrybreaking. As the precursors of the QPT, the onset of chaos [2] and the entanglement properties [10] were studied in detail. However, it is noticed that these studies focused only on atomic ensembles with small dimensions compared with the optical wavelength, in which the dipole approximation can be used [2,3,10].…”

Ultrastrong-coupling quantum-phase-transition phenomena in a few-qubit circuit QED system

“…It was later found that in the thermodynamical limit, i.e., the atom number N → ∞, and in the strong-coupling regime, the model exhibits a superradiant quantum phase transition (QPT) [10][11][12][13][14][15][16]. In a large N limit and a single-qubit coupling strength beyond the critical point, the effective coupling between atoms and the radiation mode becomes comparable to the bare frequencies of the atom and radiation mode, which leads to the occurrence of a superradiant phase transition.…”

Signatures of quantum fluctuations in the Dicke model by means of Rényi uncertainty

“…To illustrate it we selected the Dicke model that proved to be very useful in studying quantum optical [34][35][36][37][38], chaotic [38,39] or entanglement [40] properties. It has been realized with a superfluid gas in an optical cavity [41] and the spontaneous symmetry breaking has been observed recently [42].…”