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AIP conference proceedings

Symmetry adapted states and the quantum phase transition in the Dicke model

Octavio Castaños, Eduardo Nahmad-Achar, Ramón López–Peña, Jorge G. Hirsch, Kurt Bernardo Wolf, Luis Benet, Juan Mauricio Torres, Peter Oto Hess

Published Jan 1, 2010• 6 citations
Publisher

Abstract

We study the phase transition properties of the ground state of the Dicke model by means of a variational procedure together with the catastrophe formalism.The proposed variational state is the tensorial product of the SU(2) and Weyl coherent states, and the expectation value of the Dicke Hamiltonian with respect this state is calculated.The stability properties of the energy surface exhibit a phase transition from the normal to the super-radiant behavior of the two-level atoms, when ωa ωF = ±4 γ 2 .At the same time, the analytic form of the ground state in terms of the minimum critical points of the system is obtained.The Dicke Hamiltonian is invariant under transformations of the point group C 2 = {I, exp (iπ Λ)}, with Λ the excitation number operator.To restore this parity symmetry, we separate the variational proposed states into orthogonal even and odd components, which give rise to analytic expressions for the ground and first excited states.The fidelity of these projected variational states is verified by studying their overlap with the exact quantum results.

DOI
10.1063/1.3537864

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We study the phase transition properties of the ground state of the Dicke model by means of a variational procedure together with the catastrophe formalism.The proposed variational state is the tensorial product of the SU(2) and Weyl coherent states, and the expectation value of the Dicke Hamiltonian with respect this state is calculated.The stability properties of the energy surface exhibit a phase transition from the normal to the super-radiant behavior of the two-level atoms, when ωa ωF = ±4 γ 2 .At the same time, the analytic form of the ground state in terms of the minimum critical points of the system is obtained.The Dicke Hamiltonian is invariant under transformations of the point group C 2 = {I, exp (iπ Λ)}, with Λ the excitation number operator.To restore this parity symmetry, we separate the variational proposed states into orthogonal even and odd components, which give rise to analytic expressions for the ground and first excited states.The fidelity of these projected variational states is verified by studying their overlap with the exact quantum results.
The transition to the super-radiant regime in the Dicke model of two-level atoms interacting with a one-mode radiation field has been argued to be impossible for many years. Recently, however, experimental results show that this transition is possible in an open system [1, 2]. Here we exhibit the power of the use of coherent states with restored symmetry to describe the superradiant regime of the Dicke Hamiltonian. These symmetry-adapted states allow for an analytic description of both the ground and first excited states. From these, analytical expressions for all expectation values of interest are obtained, valid for any number of atoms. All these results are exact in the thermodynamic limit.
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 energy spectrum of Dicke Hamiltonians with and without the rotating wave approximation for an arbitrary atom number is obtained analytically by means of the variational method, in which the effective pseudo-spin Hamiltonian resulting from the expectation value in the boson-field coherent state is diagonalized by the spin-coherent-state transformation. In addition to the ground-state energy, an excited macroscopic quantum-state is found corresponding to the south- and north-pole gauges of the spin-coherent states, respectively. Our results of ground-state energies in exact agreement with various approaches show that these models exhibit a zero-temperature quantum phase transition of the second order for any number of atoms, which was commonly considered as a phenomenon of the thermodynamic limit with the atom number tending to infinity. The critical behavior of the geometric phase is analyzed.
The critical value of the atom-field coupling strength for a finite number of atoms is determined by means of both semiclassical and exact solutions. In the semiclassical approach we use a variational procedure with coherent and symmetry-adapted states, while for the exact quantum solution the concept of fidelity is employed. These procedures allow for the determination of the phase transitions in the model and coincide in the thermodynamic limit. For the three cases mentioned above, universal parametric curves are obtained for the expectation values of both the first quadrature of the electromagnetic field and the atomic relative population as implicit functions of the atom-field coupling parameter, valid for the ground and first-excited states.
The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: NNN two-level atoms inside a perfectly reflecting cavity, which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits that behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase from zero. By employing symmetry-adapted coherent states, it is shown that these properties scale with the number of atoms, their reported divergences at the critical point represent the limit when this number goes to infinity, and, in this limit, they remain divergent in the superradiant phase. Analytical expressions are presented for all observables of interest, for any number of atoms. Comparisons with exact numerical solutions strongly support the results.
The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: N two level atoms inside a perfectly reflecting cavity which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits which behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and of atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase from zero. Employing symmetry-adapted coherent states it is shown that these properties scale with the number of atoms, that their reported divergences at the critical point represent the limit when this number goes to infinity, and that in this limit they remain divergent in the superradiant phase. Analytical expressions are presented for all observables of interest, for any number of atoms. Comparisons with exact numerical solutions strongly support the results.