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arXiv:1409.5789 · Physica Scripta

The Husimi distribution, the Wehrl entropy and the superradiant phase in spin–boson interactions

R del Real, Manuel Calixto, E Romera

Published Mar 1, 2013• 15 citations
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Abstract

We study the Husimi distribution of the ground state in the Dicke model of field–matter interactions to visualize the quantum phase transition, from normal to superradiant, in phase space. We follow an exact numerical and variational analysis, without making use of the usual Holstein–Primakoff approximation. We find that the Wehrl entropy of the Husimi distribution provides an indication of the sharp changes in symmetry through the critical point. Additionally, we note that the zeros of the Husimi distribution characterize the Dicke model quantum phase transition.

DOI
10.1088/0031-8949/2013/t153/014016

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We study the Husimi distribution of the ground state in the Dicke model of field–matter interactions to visualize the quantum phase transition, from normal to superradiant, in phase space. We follow an exact numerical and variational analysis, without making use of the usual Holstein–Primakoff approximation. We find that the Wehrl entropy of the Husimi distribution provides an indication of the sharp changes in symmetry through the critical point. Additionally, we note that the zeros of the Husimi distribution characterize the Dicke model quantum phase transition.
The Husimi distribution is proposed for a phase-space analysis of quantum phase transitions in the Dicke model of spin-boson interactions. We show that the inverse participation ratio and Wehrl entropy of the Husimi distribution give sharp signatures of the quantum phase transition. The analysis is done using two frameworks: a numerical treatment and an analytical variational approximation. Additionally, we propose a characterization of the Dicke model quantum phase transition by means of the zeros of the Husimi distribution in the variational approach.
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.
The Husimi distribution is proposed for a phase-space analysis of quantum phase transitions in the two-dimensional U(3) vibron model for NNN-size molecules. We show that the inverse participation ratio and Wehrl's entropy of the Husimi distribution give sharp signatures of the quantum (shape) phase transition from linear to bent. Numerical results are complemented with a variational approach using parity-symmetry-adapted U(3) coherent states, which reach the minimum Wehrl entropy N(3+2N)(N+1)(N+2)\frac{N(3+2N)}{(N+1)(N+2)}(N+1)(N+2)N(3+2N)​, in the rigidly linear phase, according to a generalized Wehrl-Lieb conjecture. We also propose a characterization of the vibron-model quantum phase transition by means of the zeros of the Husimi distribution.
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 occurrence of a second-order quantum phase transition in the Dicke model is a well-established feature. On the contrary, a comprehensive understanding of the corresponding open system, particularly in the proximity of the critical point, remains elusive. When approaching the critical point, the system inevitably enters first the system-bath ultrastrong coupling regime and finally the deepstrong coupling regime, causing the failure of usual approximations adopted to describe open quantum systems. We study the interaction of the Dicke model with bosonic bath fields in the absence of additional approximations, which usually relies on the weakness of the system-bath coupling. We find that the critical point is not affected by the interaction with the environment. Moreover, the interaction with the environment is not able to affect the system ground-state condensates in the superradiant phase, whereas the bath fields are i n f e c t e d by the system and acquire macroscopic occupations. The obtained reflection spectra display lineshapes which become increasingly asymmetric, both in the normal and superradiant phases, when approaching the critical point.
In this paper, we reveal a second-order phase transition for the Dicke model with the dipole–dipole interaction between the atoms. By means of Holstein–Primakoff transformation, the energy spectra for both the normal and superradiant phases are explicitly obtained and therefore the scaling behaviour near the critical transition point can be given clearly. It is also shown that the dipole–dipole interactions between the atoms deeply affect the critical transition point, the ground-state and excitation energies and the scaled atomic inversion in the ground state.