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Physical Review A

Reduced bases for a three-level atom interacting with a two-mode radiation field

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

Published Mar 6, 2019• 6 citations
Publisher

Abstract

Reduced bases are obtained for a single three-level atom interacting dipolarly with a two-mode electromagnetic field in a cavity. A truncation scheme for the infinite dimensional Hilbert space of the system is proposed, which we take it as the exact ground-state solution. This is used to determine the quantum phase diagram of the atomic $\mathrm{\ensuremath{\Lambda}}$ configuration and to judge the goodness of the reduced bases. This provides us with a mathematical technique that can be used to solve systems where the number of atoms and excitations grow, yielding a Hilbert space with enormous dimensions, more effectively than with the currently available methods. Additionally, the sudden changes suffered by the ground-state solution can be observed through the calculation of the purity and the occupation probabilities.

DOI
10.1103/physreva.99.033811

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Reduced bases are obtained for a single three-level atom interacting dipolarly with a two-mode electromagnetic field in a cavity. A truncation scheme for the infinite dimensional Hilbert space of the system is proposed, which we take it as the exact ground-state solution. This is used to determine the quantum phase diagram of the atomic \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ configuration and to judge the goodness of the reduced bases. This provides us with a mathematical technique that can be used to solve systems where the number of atoms and excitations grow, yielding a Hilbert space with enormous dimensions, more effectively than with the currently available methods. Additionally, the sudden changes suffered by the ground-state solution can be observed through the calculation of the purity and the occupation probabilities.
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 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 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 $\ensuremath{\Xi}$, $\ensuremath{\Lambda}$, 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 ($M\ensuremath{\ne}0$). We have first- and second-order phase transitions, except for the V\mathsf{V}V configuration, which only presents second-order phase transitions.
Abstract We present a study of the structure of phase diagrams for matter-radiation systems, based on the use of coherent states and the catastrophe formalism, that compares very well with the exact quantum solutions as well as providing analytical expressions. Emphasis is made on 2- and 3-level systems, but in general n -level systems in the presence of ℓ electromagnetic modes are described. Due to the infinite-dimensional nature of the Hilbert space, and using the results of the analyses and the behaviour of the solutions, we construct a sequence of ever-approximating reduced bases, which make possible the study of larger systems both, in the number of atoms and in the number of excitations. These studies are of importance in fundamental quantum optics, quantum information, and quantum cryptography scenarios.
Abstract The phase diagram corresponding to the ground state of a completely general system of atoms interacting dipolarly with a radiation field, in a cavity, is established. The system is most general in that we consider any finite number N a of atoms, any number n of atomic levels, any number ℓ of field modes, and we do not restrict a radiation mode to interact with a sole pair of atomic levels. The study is carried out through a variational procedure. The test function is constructed from the tensorial product of coherent states associated to the matter and field components. In the parameter space of the dipolar field–matter couplings, we show that the ground state of the system suffers quantum phase transitions described by a phase diagram which divides itself into monochromatic regions. Linear combinations of coherent states are considered to restore the symmetries of the Hamiltonian and, in the thermodynamic limit, we show that the corresponding symmetry-adapted energy surface approaches that of the separable coherent 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.