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arXiv:1508.07064 · Physical Review A

Polychromatic phase diagram for n -level atoms interacting with ℓ modes of an electromagnetic field

Sergio Cordero, Eduardo Nahmad-Achar, Ramón López–Peña, Octavio Castaños

Published Nov 19, 2015• 28 citations
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Abstract

A system of ${N}_{a}$ atoms of $n$ levels interacting dipolarly with $\ensuremath{\ell}$ modes of an electromagnetic field is considered. The energy surface of the system is constructed from the direct product of the coherent states of $\mathrm{U}(n)$ in the totally symmetric representation for the matter times the $\ensuremath{\ell}$ coherent states of the electromagnetic field. A variational analysis shows that the collective region is divided into $\ensuremath{\ell}$ zones, inside each of which only one mode of the electromagnetic field contributes to the ground state. In consequence, the polychromatic phase diagram for the ground state naturally divides itself into monochromatic regions. For the case of three-level atoms in the $\mathrm{\ensuremath{\Xi}}$ configuration in the presence of two modes, the variational calculation is compared with the exact quantum solution showing that both are in agreement.

DOI
10.1103/physreva.92.053843

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A system of 3-level atoms interacting with 2-modes of electromagnetic radiation is studied by means of a variational procedure. This allows us to establish the corresponding quantum phase diagram and demonstrate that the super-radiant region is divided in monochromatic sub-regions where only one coupling strength μ ij and one radiation field mode rule the behaviour of the system. In this contribution we prove that the variational coherent states, the symmetry-adapted states, and the exact quantum solution exhibit a universal parametric curve associated essentially with the two-level population and the number of photons exciting these levels.
A study of the $\ensuremath{\lambda}$ and NNN atomic configurations under dipolar interaction with two modes of electromagnetic radiation is presented. The corresponding quantum phase diagrams are obtained by means of a variational procedure. Both configurations exhibit normal and collective (super-radiant) regimes. While the latter in the $\ensuremath{\lambda}$ configuration divides itself into two subregions, corresponding to each of the modes, that in the NNN configuration may be divided into two or three subregions depending on whether the field modes divide the atomic system into two separate subsystems or not. Our variational procedure compares well with the exact quantum solution. The properties of the relevant field and matter observables are obtained.
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We establish, within the second quantization method, the general dipole-dipole Hamiltonian interaction of a system of nnn-level atoms. The variational energy surface of the nnn-level atoms interacting with $\ensuremath{\ell}$-mode fields and under the Van Der Waals forces is calculated with respect to the tensorial product of matter and electromagnetic field coherent states. This is used to determine the quantum phase diagram associated with the ground state of the system and quantify the effect of the dipole-dipole Hamiltonian interaction. By considering real induced electric dipole moments, we find the quantum phase transitions for 2- and 3-level atomic systems interacting with one and two modes of the electromagnetic field, respectively. The corresponding order of the transitions is established by means of Ehrenfest classification; for undetermined cases, we propose two procedures: one based on the difference of the expectation value of the Casimir operators of the 2-level subsystems, and another based on maximizing the Bures distance between neighbor variational solutions.
A semiclassical analysis is presented to determine the quantum phase transition from single to collective regimes in three-level atoms in the presence of a radiation field. The energy surfaces of the Ξ, V, and Λ configurations are constructed by taking the expectation value with respect to U(3) coherent states that carry the totally symmetric representation, determined by the total number of atoms. The corresponding stability and equilibrium properties are calculated by means of the catastrophe theory, discovering the bifurcation and Maxwell sets. We establish that the atoms with Λ and Ξ configurations have the presence of double points, i.e., there are two independent quantum states with the same energy that can be obtained depending on the values of the dipolar strengths of the interaction between the atoms and the radiation field. Additionally, the Ξ configuration exhibits a fixed triple point.