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arXiv:1206.0797 · Physical Review A

Universal critical behavior in the Dicke model

Octavio Castaños, Eduardo Nahmad-Achar, Ramón López–Peña, Jorge G. Hirsch

Published Aug 8, 2012• 25 citations
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Abstract

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.

DOI
10.1103/physreva.86.023814

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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 critical value of the atom-field coupling strength in the Dicke model, for a finite number of atoms interacting with photons in a cavity, is determined by means of semiclassical (via coherent states and symmetry-adapted states) and exact solutions. These procedures allow in turn for the determination of the phase transitions in the model, and coincide in the thermodynamic limit. For the three cases mentioned above, a universal parametric curve is obtained for the expectation values of the first quadrature of the electromagnetic field, as a function of the atomic relative population, valid for all values of the atom-field coupling parameter, and valid for the ground-and first-excited states.
We show how various mathematical formalisms, specifically the catastrophe formalism and group theory, aid in the study of relevant systems in quantum optics. We describe the phase transition of the Dicke model for a finite number N of atoms, via three different methods, which led to universal parametric curves for the expectation value of the first quadrature of the electromagnetic field and the expectation value of the number operator, as functions of the atomic relative population. These are valid for all values of the matter–field coupling parameter, and valid for both the ground and first-excited states. Using these mathematical tools, the critical value of the atom–field coupling parameter is found as a function of the number of atoms, from which its critical exponent is derived.
Two-level atoms interacting with a one-mode cavity field at zero temperature have order parameters which reflect the presence of a quantum phase transition at a critical value of the atom–cavity coupling strength. Two popular examples are the number of photons inside the cavity and the number of excited atoms. Coherent states provide a mean field description, which becomes exact in the thermodynamic limit. Employing symmetry-adapted (SA) SU (2) coherent states the quantum crossover, precursor of the critical behavior, can be described for a finite number of atoms. A variation after projection treatment, involving a numerical minimization of the SA energy surface, associates the quantum crossover with a discontinuity in the order parameters, which originates from competition between two local minima in the SA energy surface. Although this discontinuity is not present in finite systems, it provides a good description of 1/ N effects in the observables.
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 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 calculate numerically the fidelity and its susceptibility for the ground state of the Dicke model. A minimum in the fidelity identifies the critical value of the interaction where a quantum phase crossover, the precursor of a phase transition for finite number of atoms , takes place. The evolution of these observables is studied as a function of , and their critical exponents evaluated. Using the critical exponents the universal curve for the specific susceptibility is recovered. An estimate to the precision to which the ground state wave function is numerically calculated is given, and found to have its lowest value, for a fixed truncation, in a vicinity of the critical coupling.