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Physica Scripta

A general system of n levels interacting with ℓ electromagnetic modes

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

Published Mar 27, 2017• 9 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/aa6363

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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.
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 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.
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.
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.
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.