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Journal of Physics Conference Series

Universal critical behaviour of 3-level atoms interacting dipolarly with radiation

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

Published Aug 1, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/1071/1/012006

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A system of Na{N}_{a}Na​ atoms of nnn 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 U(n)\mathrm{U}(n)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.
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.
We introduce a combination of coherent states as variational test functions for the atomic and radiation sectors to describe a system of N a three-level atoms interacting with a one-mode quantised electromagnetic field, with and without the rotating wave approximation, which preserves the symmetry presented by the Hamiltonian. These provide us with the possibility of finding analytical solutions for the ground and first excited states. We study the properties of these solutions for the V -configuration in the double resonance condition, and calculate the expectation values of the number of photons, the atomic populations, the total number of excitations, and their corresponding fluctuations. We also calculate the photon number distribution and the linear entropy of the reduced density matrix to estimate the entanglement between matter and radiation. For the first time, we exhibit analytical expressions for all of these quantities, as well as an analytical description for the phase diagram in parameter space, which distinguishes the normal and collective regions, and which gives us all the quantum phase transitions of the ground state from one region to the other as we vary the interaction parameters (the matter-field coupling constants) of the model, in functional form.
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 study the structure of the phase diagram for systems consisting of 2- and 3- level particles dipolarly interacting with a 1-mode electromagnetic field, inside a cavity, paying particular attention to the case of a finite number of particles, and showing that the divergences that appear in other treatments are a consequence of the mathematical approximations employed and can be avoided by studying the system in an exact manner quantum-mechanically or via a catastrophe formalism with variational trial states that satisfy the symmetries of the appropriate Hamiltonians. These variational states give an excellent approximation not only to the exact quantum phase space, but also to the energy spectrum and the expectation values of the atomic and field operators. Furthermore, they allow for analytic expressions in many of the cases studied. We find the loci of the transitions in phase space from one phase to the other, and the order of the quantum phase transitions are determined explicitly for each of the configurations, with and without detuning. We also derive the critical exponents for the various systems, and the phase structure at the triple point present in the Ξ-configuration of 3-level systems is studied.