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arXiv:2110.11554 · Physical Review A

Effect of the atomic dipole-dipole interaction on the phase diagrams of field-matter interactions: Variational procedure

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

Published Mar 21, 2022• 1 citation
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Abstract

We establish, within the second quantization method, the general dipole-dipole Hamiltonian interaction of a system of $n$-level atoms. The variational energy surface of the $n$-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.

DOI
10.1103/physreva.105.033712

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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 \ensuremathℓ\ensuremath{\ell}\ensuremathℓ-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.
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.
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.
We compute the energies of the interaction between two atoms placed in the middle of a perfectly reflecting planar cavity, in the weak-coupling nondegenerate regime. Both inhibition and enhancement of the interactions can be obtained by varying the size of the cavity. We derive exact expressions for the dyadic Green's function of the cavity field which mediates the interactions and apply time-dependent quantum perturbation theory in the adiabatic approximation. We provide explicit expressions for the van der Waals potentials of two polarizable atomic dipoles and the electrostatic potential of two induced dipoles. We compute the van der Waals potentials in three different scenarios: two atoms in their ground states, two atoms excited, and two dissimilar atoms with one of them excited. In addition, we calculate the phase-shift rate of the two-atom wave function in each case. The effect of the two-dimensional confinement of the electromagnetic field on the dipole-dipole interactions is analyzed. This effect depends on the atomic polarization. For dipole moments oriented parallel to the cavity plates, both the electrostatic and the van der Waals interactions are exponentially suppressed for values of the cavity width much less than the interatomic distance, whereas for values of the width close to the interatomic distance, the strength of both interactions is higher than their values in the absence of cavity. For dipole moments perpendicular to the plates, the strength of the van der Waals interaction decreases for values of the cavity width close to the interatomic distance, while it increases for values of the width much less than the interatomic distance with respect to its strength in the absence of cavity. We illustrate these effects by computing the dipole-dipole interactions between two alkali atoms in circular Rydberg states.
A general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for N a = 1 and N a = 4 particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with 2 modes of electromagnetic field. For N a = 1, the reduced density matrix of the matter allows us to determine the phase regions in a 2-simplex (associated to a general three dimensional density matrix), on the different 3-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established.
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 .