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arXiv:1306.6435 · Physical Review E

Excited-state quantum phase transitions in Dicke superradiance models

Tobias Brandes

Published Sep 23, 2013• 137 citations
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Abstract

We derive analytical results for various quantities related to the excited-state quantum phase transitions in a class of Dicke superradiance models in the semiclassical limit. Based on a calculation of a partition sum restricted to Dicke states, we discuss the singular behavior of the derivative of the density of states and find observables such as the mean (atomic) inversion and the boson (photon) number and its fluctuations at arbitrary energies. Criticality depends on energy and a parameter that quantifies the relative weight of rotating versus counterrotating terms, and we find a close analogy to the logarithmic and jump-type nonanalyticities known from the Lipkin-Meshkov-Glick model.

DOI
10.1103/physreve.88.032133

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•May 14, 2014•0 citations
We derive analytical results for various quantities related to the excited-state quantum phase transitions in a class of Dicke superradiance models in the semiclassical limit. Based on a calculation of a partition sum restricted to Dicke states, we discuss the singular behavior of the derivative of the density of states and find observables such as the mean (atomic) inversion and the boson (photon) number and its fluctuations at arbitrary energies. Criticality depends on energy and a parameter that quantifies the relative weight of rotating versus counterrotating terms, and we find a close analogy to the logarithmic and jump-type nonanalyticities known from the Lipkin-Meshkov-Glick model.
We show that semiclassical states adapted to the symmetry of the Hamiltonian are an excellent approximation to the exact quantum solution of the ground and first excited states of the Dicke model. Their overlap with the exact quantum states is very close to 1 except in a close vicinity of the quantum phase transition. Furthermore, they have analytic forms in terms of the model parameters and allow us to calculate analytically the expectation values of field and matter observables. Some of these differ considerably from results obtained via the standard coherent states and by means of Holstein-Primakoff series expansion of the Dicke Hamiltonian. Comparison with exact solutions obtained numerically supports our results. In particular, it is shown that the expectation values of the number of photons and of the number of excited atoms have no singularities at the phase transition. We comment on why other authors have previously found otherwise.
The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M.-J. Hwang et al., Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher-energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which show a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states lead to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.
The Dicke model generalized to the case where the dependence of the coupling constant on individual atoms is taken into account is investigated. It is shown that the generalized Dicke model exhibits a quantum phase transition from the normal phase to the superradiant phase in the thermodynamic limit, as well as the standard Dicke model. The mean photon number, the atomic inversion, the lowest excitation energy, and the entanglement entropy between the field and the atoms are evaluated analytically or numerically, and the critical behavior is examined. As a result, it turns out that the critical behavior in the generalized model is essentially the same as that in the standard model. This implies universality in the Dicke model.
We systematically analyze the various phase transitions of the anisotropic Dicke model that is endowed with both rotating and counterrotating light-matter couplings. In addition to the ground-state quantum phase transition (QPT) from the normal to the superradiant phase, the anisotropic Dicke model also exhibits other transitions, namely, the excited state quantum phase transition (ESQPT), ergodic to nonergodic transition (ENET), and the temperature-dependent phase transition. We show that these phase transitions are profitably studied not only with the standard consecutive level spacing ratio, but also with the aid of various eigenvector quantities such as von Neumann entanglement entropy, the participation ratio, multifractal dimension, and mutual information. For ENET, both the statics and dynamics of the participation ratio offer a consistent and useful picture. An exciting finding from our work is that the ESQPT and the ENET are closely related to each other. We show this with the aid of two characteristic energies in the spectrum corresponding to jumps in von Neumann entropy.
We study the nonintegrable Dicke model and its integrable approximation, the Tavis-Cummings model, as functions of both the coupling constant and the excitation energy. Excited-state quantum phase transitions (ESQPT) are found analyzing the density of states in the semiclassical limit and comparing it with numerical results for the quantum case in large Hilbert spaces, taking advantage of efficient methods recently developed. Two different ESQPTs are identified in both models, which are signaled as singularities in the semiclassical density of states; one static ESQPT occurs for any coupling, whereas a dynamic ESQPT is observed only in the superradiant phase. The role of the unstable fixed points of the Hamiltonian semiclassical flux in the occurrence of the ESQPTs is discussed and determined. Numerical evidence is provided that shows that the semiclassical results describe very well the tendency of the quantum energy spectrum for any coupling in both models. Therefore, the semiclassical density of states can be used to study the statistical properties of the fluctuation in the spectra, a study that is presented in a companion paper.
We investigate signatures of the excited-state quantum phase transition in the periodic dynamics of the Lipkin-Meshkov-Glick model. With this aim, we use the semiclassical trajectories to calculate quantum mechanical quantities such as the density of states and expectation values of observables in eigenstates of the system. Motivated by this approach, we suggest an experimental protocol, based on the time evolution of the finite-size system, to find singularities in observables, which arise due to the excited-state quantum phase transition. We also discuss the influence of the participation ratio on the deviations of the quantum mechanical finite-size time evolution from the classical one.