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Register a new userApproximated integrability of the Dicke model
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Added by
Sergio Lerma-Hernandez
Publication date
2016
fields
Physics
and research's language is
English
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A very approximate second integral of motion of the Dicke model is identified within a broad region above the ground state, and for a wide range of values of the external parameters. This second integral, obtained from a Born Oppenheimer approximation, classifies the whole regular part of the spectrum in bands labelled by its corresponding eigenvalues. Results obtained from this approximation are compared with exact numerical diagonalization for finite systems in the superradiant phase, obtaining a remarkable accord. The region of validity of our approach in the parameter space, which includes the resonant case, is unveiled. The energy range of validity goes from the ground state up to a certain upper energy where chaos sets in, and extends far beyond the range of applicability of a simple harmonic approximation around the minimal energy configuration. The upper energy validity limit increases for larger values of the coupling constant and the ratio between the level splitting and the frequency of the field. These results show that the Dicke model behaves like a two-degree of freedom integrable model for a wide range of energies and values of the external parameters.
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We report the experimental implementation of the Dicke model in the semiclassical approximation, which describes a large number of two-level atoms interacting with a single-mode electromagnetic field in a perfectly reflecting cavity. This is managed by making use of two non-linearly coupled active, synthetic LC circuits, implemented by means of analog electrical components. The simplicity and versatility of our platform allows us not only to experimentally explore the coexistence of regular and chaotic trajectories in the Dicke model but also to directly observe the so-called ground-state and excited-state ``quantum phase transitions. In this analysis, the trajectories in phase space, Lyapunov exponents and the recently introduced Out-of-Time-Order-Correlator (OTOC) are used to identify the different operating regimes of our electronic device. Exhaustive numerical simulations are performed to show the quantitative and qualitative agreement between theory and experiment.
The Dicke-Ising model describes cavity quantum electrodynamics setups in which dipoles couple not only with the photonic cavity field but also to each other through dipole-dipole interaction. In this work we diagonalise such a model in terms of bosonic polaritonic operators for arbitrarily large values of the light-matter coupling and for values of the dipole-dipole interaction until the onset of the ferromagnetic Ising phase transition. In order to accomplish this task we exploit higher order terms of the Holstein-Primakoff transformation, developing a general method which allows to solve the normal phase of the Ising model in term of bosonic excitations for large values of the dipole-dipole interaction. Our results shed light on the interplay between the dipole-dipole and the light-matter coupling strengths, and their effect on the virtual excitations which populate the ground-state when the interactions become comparable with the bare transition frequency.
Using Hills determinant method we show that the set of Judds solutions is only a subset of all the eigenvalues with the form $E_n=nomega-g^2/omega$ in the spectrum of the Rabi model. Therefore Braaks solution of the quantum Rabi model is not complete.
We study effective light-matter interactions in a circuit QED system consisting of a single $LC$ resonator, which is coupled symmetrically to multiple superconducting qubits. Starting from a minimal circuit model, we demonstrate that in addition to the usual collective qubit-photon coupling the resulting Hamiltonian contains direct qubit-qubit interactions, which have a drastic effect on the ground and excited state properties of such circuits in the ultrastrong coupling regime. In contrast to a superradiant phase transition expected from the standard Dicke model, we find an opposite mechanism, which at very strong interactions completely decouples the photon mode and projects the qubits into a highly entangled ground state. These findings resolve previous controversies over the existence of superradiant phases in circuit QED, but they more generally show that the physics of two- or multi-atom cavity QED settings can differ significantly from what is commonly assumed.
We study the ergodic -- non-ergodic transition in a generalized Dicke model with independent co- and counter rotating light-matter coupling terms. By studying level statistics, the average ratio of consecutive level spacings, and the quantum butterfly effect (out-of-time correlation) as a dynamical probe, we show that the ergodic -- non-ergodic transition in the Dicke model is a consequence of the proximity to the integrable limit of the model when one of the couplings is set to zero. This can be interpreted as a hint for the existence of a quantum analogue of the classical Kolmogorov-Arnold-Moser theorem. Besides, we show that there is no intrinsic relation between the ergodic -- non-ergodic transition and the precursors of the normal -- superradiant quantum phase transition.
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