Effect of the Atomic Dipole-Dipole Interaction on the Phase Diagrams of Field-Matter Interactions

Abstract

Quantum information measures are used to study the quantum phase diagrams of the two-level Dicke model including the atomic dipole-dipole interaction, for a finite number of particles, with and without the rotating-wave approximation, which yields the conservation of the total number of excitations in the first case and its parity in the general case. We show that the quantum phase transitions can be observed in the fluctuation of the atomic populations and that of the number of photons, and also that the conditional probability distribution of the population of the excited state with zero photons carries the information of the quantum phase transitions when the matter-field interaction is weak.

Figures (11)

• Figure 1: (colour online) Schematic representation of the atom-atom dipolar interaction in the case when dipoles are oriented in opposite direction (first two arrows from left), and when they are oriented in the same direction (last four arrows from left). See text.
• Figure 2: Variational minimum energy surface Eq. (\ref{['eq.minEv']}) as a function of the dimensionless control parameters, for $\omega_1=0$ and $\omega_a=1$. The variational separatrix is plotted as a solid black line, while white lines correspond to the limiting cases $\chi=0$ (the Dicke model without the dipole-dipole interaction) and $x_{12}=0$ (the Hamiltonian without matter-field interaction).
• Figure 3: First five energy values ${\cal E}_m^{(0)}$ per particle, which determine the ground state $E_g$ [Eq. (\ref{['eq.eminRWA']})] as a function of the dimensionless dipolar strength $x_{12}$, for a constant value of the dipole-dipole interaction $\zeta=0.5$. (a) is the case for $N_a=2$ atoms, and (b) for $N_a=5$ atoms. The parameters of the system are in this case $\omega_1=0$ and $\omega_2=\Omega=1$. Crossings of the curves for different $m$'s determine quantum phase transitions.
• Figure 4: Quantum phase diagram in terms of the control parameters $x_{12}$ and $\zeta$ when the RWA approximation is considered for (a) $N_a=2$ atoms and (b) $N_a=5$ atoms. The total number of excitations in each region is indicated in colors, the solid black line corresponds to the variational separatrix. The parameters are the same as those in Fig. \ref{['f.energy']}.
• Figure 5: Quantum phase diagram defined by a fidelity criterion, in terms of the control parameters $x_{12}$ and $\zeta$ for $N_a=2$ atoms, fixing the dipolar strength parameter at (a) $\eta=-1$, (b) $\eta=0$ and (c) $\eta=1$. The solid white line marks a discontinuous (first order) transition. The solid black line corresponds to the variational separatrix. Other parameters are the same as those in Fig. \ref{['f.energy']}.