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arXiv:1303.2716 · Physical Review A

Quantum phase transitions of three-level atoms interacting with a one-mode electromagnetic field

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

Published Feb 7, 2013• 27 citations
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Abstract

We apply the energy surface method to study a system of ${N}_{a}$ three-level atoms interacting with a one-mode radiation field in the $\ensuremath{\Xi}$, $\ensuremath{\Lambda}$, and $\mathsf{V}$ configurations. We obtain an estimation of the ground-state energy, the expectation value of the total number of excitations, and the phase diagram of the model in the interaction parameter space and compare the results with the exact solutions. The phase diagram separates the behavior of, in general, a dominant single state without excitations ($M=0$) from a collective state with excitations ($M\ensuremath{\ne}0$). We have first- and second-order phase transitions, except for the $\mathsf{V}$ configuration, which only presents second-order phase transitions.

DOI
10.1103/physreva.87.023805

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We apply the energy surface method to study a system of Na{N}_{a}Na​ three-level atoms interacting with a one-mode radiation field in the \ensuremathΞ\ensuremath{\Xi}\ensuremathΞ, \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ, and V\mathsf{V}V configurations. We obtain an estimation of the ground-state energy, the expectation value of the total number of excitations, and the phase diagram of the model in the interaction parameter space and compare the results with the exact solutions. The phase diagram separates the behavior of, in general, a dominant single state without excitations (M=0M=0M=0) from a collective state with excitations (M\ensuremath≠0M\ensuremath{\ne}0M\ensuremath=0). We have first- and second-order phase transitions, except for the V\mathsf{V}V configuration, which only presents second-order phase transitions.
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.
A system of three-level atoms interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations is studied. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. A semi-classical approximation to the exact quantum solution is built by projecting coherent field and matter states to a specific value of the total number of excitations. The states in parameter-space and the expectation values of relevant observables are obtained explicitly, and it is shown that those for the projected states offer an excellent approximation to the exact quantum solution. The exact form of the ground state near the triple point in the Ξ configuration is given.
We consider N a three-level atoms (or systems) interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. The semi-classical and exact quantum calculations for both the expectation values of the total number of excitations and photon number have excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the and n distribution functions. Therefore, their corresponding fluctuations are not well described by the semi-classical approximation. We show that this can be corrected by projecting the variational state to a definite value of .
We study the structure of the phase diagram for systems consisting of two- and three-level particles dipolarly interacting with a one-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 three-level systems is studied.
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.