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arXiv:0912.3273

Quantum phase transitions and quantum chaos in generalized Dicke and Jahn-Teller polaron model and finite-size effects

Eva Majerníková, Serge Shpyrko

Published Dec 16, 2009• 0 citations
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Abstract

The Dicke model extended to two bosons of different frequencies or equivalent generalized Jahn-Teller lattice model are shown to exhibit a spontaneous quantum phase transition between the polaron-modified "quasi-normal" and squeezed "radiation" phase with the transition point dependent on the frequencies. In a finite lattice a mixed domain of coexistence of the quasi-normal and modified radiation phase is created within the quasi-normal phase domain. There occurs a field-directed oscillation-assisted tunneling (hopping). The field is driven by simultaneous squeezing and polaron-dressing of the collective boson level mode due to the additional boson mode. In a finite lattice in the radiation domain there occurs a sequence of local tunnelings (oscillations) between two minima of a local potential weakly coupled to two assisting oscillations. The "radiation" phase reveals itself as an almost ideal instanton--anti-instanton gas phase. The correlations among the energy levels mediated by the additional mode in the mixed domain considerably reduce the level repulsions. As a consequence, the Wigner level spacing probability distribution of the two-boson Dicke model is non-universally reduced from the Wigner to the semi-Poisson and asymptotically to the Poisson distribution of level spacings. The correlations cause a suppression of the coherence of the radiation phase as finite-size effect. Possible applications of the present theory are suggested.

DOI
10.48550/arxiv.0912.3273

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The Dicke model extended to two bosons of different frequencies or equivalent generalized Jahn-Teller lattice model are shown to exhibit a spontaneous quantum phase transition between the polaron-modified "quasi-normal" and squeezed "radiation" phase with the transition point dependent on the frequencies. In a finite lattice a mixed domain of coexistence of the quasi-normal and modified radiation phase is created within the quasi-normal phase domain. There occurs a field-directed oscillation-assisted tunneling (hopping). The field is driven by simultaneous squeezing and polaron-dressing of the collective boson level mode due to the additional boson mode. In a finite lattice in the radiation domain there occurs a sequence of local tunnelings (oscillations) between two minima of a local potential weakly coupled to two assisting oscillations. The "radiation" phase reveals itself as an almost ideal instanton--anti-instanton gas phase. The correlations among the energy levels mediated by the additional mode in the mixed domain considerably reduce the level repulsions. As a consequence, the Wigner level spacing probability distribution of the two-boson Dicke model is non-universally reduced from the Wigner to the semi-Poisson and asymptotically to the Poisson distribution of level spacings. The correlations cause a suppression of the coherence of the radiation phase as finite-size effect. Possible applications of the present theory are suggested.
The generalized multispin Jahn-Teller model on a finite lattice or formally equivalent Dicke model extended to two long-wavelength coherent bosons of different frequencies is shown to exhibit a crossover between the polaron-modified "quasi-normal" and the squeezed "radiation" domains. We investigate effects of two kinds of interfering fluctuations on the phase crossover and on statistical characteristics of boson complex spectra: (i) Fluctuations in the electron subsystem- finite-size quantum fluctuations- are responsible for the dephasing of the coherence in the radiation domain and for the moderate occupation of the excited states in the normal domain. In the quasiclassical limit, radiation phase implies existence of a coherent acoustic super-radiant phase. (ii) Level-spacing fluctuations in excited boson level subsystem with strong level repulsions. Related probability distributions are shown to be non-universally spread between the limiting universal Wigner-Dyson and Poisson distributions. We proved that the difference in boson frequencies is responsible for reaching the most stochastic limit of the Wigner-Dyson distribution. Instanton lattice as a sequence of tunneling events in the most chaotic radiation domain exhibits maximal number of level-avoidings (repulsions). The non-universality of the distributions is caused by boson correlations which compete the level repulsions.
We introduce a group-theoretical extension of the Dicke model, which describes an ensemble of two-level atoms interacting with a finite radiation field. The latter is described by a spin model whose main feature is that it possesses a maximum number of excitations. The approach adopted here leads to a nonlinear extension of the Dicke model that takes into account both the intensity dependent coupling between the atoms and the radiation field and an additional nonlinear Kerr-like or Pösch–Teller-like oscillator term, depending on the degree of nonlinearity. We use the energy surface minimization method to demonstrate that the extended Dicke model exhibits a quantum phase transition, and we analyze its dependence upon the maximum number of excitations of the model. Our analysis is carried out via three methods: through mean-field analysis (i.e., by using the tensor product of coherent states), by using parity-preserving symmetry-adapted states (using the critical values obtained in the mean-field analysis and numerically minimizing the energy surface), and by means of the exact quantum solution (i.e., by numerically diagonalizing the Hamiltonian). Possible connections with the qp-deformed algebras are also discussed.
The Dicke model describes NNN two-level atoms interacting with a single-mode bosonic field and exhibits a second-order phase transition from the normal to the superradiant phase. The energy levels are not degenerate in the normal phase but have degeneracy in the superradiant phase, where quantum tunneling occurs. By means of the Born-Oppenheimer approximation and the instanton method in quantum field theory, the tunneling splitting, inversely proportional to the tunneling rate for the adiabatic Dicke model, in the superradiant phase can be evaluated explicitly. It is shown that the tunneling splitting vanishes as $\mathrm{exp}(\ensuremath{-}N)$ for large NNN, whereas for small NNN it disappears as N∕exp(N)\sqrt{N}∕\mathrm{exp}(N)N​∕exp(N). The dependence of the tunneling splitting on the relevant parameters, especially on the atom-field coupling strength, is also discussed.
We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.
We investigate the quantum-chaotic properties of the Dicke Hamiltonian; a quantum-optical model that 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 --> infinity limit, which we describe exactly in an effective Hamiltonian approach. We then numerically investigate the system at finite N, and by analyzing 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 wave function indicate that this is connected with a delocalization of the system and the emergence of macroscopic coherence. We also derive a semiclassical Dicke model that exhibits analogues of all the important features of the quantum model, such as the phase transition and the concurrent onset of chaos.
The dipole-coupled two-level atoms (qubits) in a single-mode resonant cavity are studied by extended bosonic coherent states. The numerical solution is presented. The first-order quantum phase transitions may occur in finite systems, in contrast to the original Dicke model. This system also exhibits a second-order quantum phase transition from the normal to the superradiant phase, as in the original Dicke model. Finite-size scaling for several observables---such as average fidelity susceptibility, order parameter, and concurrence---is performed for different interatomic interactions. The scaling exponents obtained suggest that interatomic interactions do not change the universality class.