Polychromatic phase diagram for $n$-level atoms interacting with $\ell$ modes of an electromagnetic field

A system of $N_a$ atoms of $n$ levels interacting dipolarly with $\ell$ modes of an electromagnetic field is considered. The energy surface of the system is constructed from the direct product of the coherent states of $\mathrm{U}\left(n\right)$ in the totally symmetric representation for the matter times the $\ell$ coherent states of the electromagnetic field. A variational analysis shows that the collective region is divided into $\ell$ 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 $\mathrm{\Xi }$ configuration in the presence of two modes, the variational calculation is compared with the exact quantum solution showing that both are in agreement.

Figure 1

Configurations of a three-level atom interacting with two modes of an electromagnetic field. For the $\mathrm{\Xi }$ configuration ${\mu }_{13}=0$, for the $\mathrm{\Lambda }$ configuration ${\mu }_{12}=0$, and for the $V$ configuration ${\mu }_{23}=0$. The mode of the electromagnetic field ${\mathrm{\Omega }}_{jk}$ for each transition is indicated.

Figure 2

Energies and phase diagrams for atoms in the $\mathrm{\Xi }, V$, and $\mathrm{\Lambda }$ configurations. The separatrices (white lines) and the order of the transitions are shown. The normal regions are labeled by $N$ (black). The collective regime is divided into the regions: (a) $S_{12}$ and $S_{23}$ corresponding to the active modes ${\mathrm{\Omega }}_{12}$ and ${\mathrm{\Omega }}_{23}$, respectively, shown here for ${\mathrm{\Omega }}_{12}=1,{\mathrm{\Omega }}_{23}=0.5,{\omega }_2=1$, and ${\omega }_3=1.3$; (b) $S_{12}$ and $S_{13}$ corresponding to the active modes ${\mathrm{\Omega }}_{12}$ and ${\mathrm{\Omega }}_{13}$, respectively, with ${\mathrm{\Omega }}_{13}=1,{\mathrm{\Omega }}_{12}=0.8,{\omega }_2=0.8$, and ${\omega }_3=1$; and (c) $S_{13}$ and $S_{23}$ corresponding to the active modes ${\mathrm{\Omega }}_{13}$ and ${\mathrm{\Omega }}_{23}$, respectively, with ${\mathrm{\Omega }}_{13}=1,{\mathrm{\Omega }}_{23}=0.8,{\omega }_2=0.2$, and ${\omega }_3=1$.

Figure 3

Behavior of the expectation values of the photon and atomic level populations, as functions of the interaction parameters. See text for details.

Figure 4

Under the condition $\langle {\mathit{A}}_{44}\rangle =0$ for the ground state, a system of four-level atoms in the $\mathrm{\Xi }$ configuration is reduced to three-level atoms in the $\mathrm{\Xi }$ configuration.

Figure 5

Schematic procedure to obtain, by a simple consideration ($\langle {\mathit{A}}_{kk}\rangle =0$), a three-level system from a four-level system. The $\lambda$ configuration is reduced to the $\mathrm{\Xi }$ or $\mathrm{\Lambda }$ configuration and the $♦$ configuration is reduced to the $\mathrm{\Xi }$ or $V$ configuration. In both cases the simple condition on the ground state is given by $\langle {\mathit{A}}_{22}\rangle =0$ and $\langle {\mathit{A}}_{44}\rangle =0$, respectively.

Figure 6

Phase diagram as a function of the dipolar intensities for atoms of four levels in the $\mathrm{\Xi }$ configuration. Parameters are ${\mathrm{\Omega }}_{12}=1,{\mathrm{\Omega }}_{23}=0.7,{\mathrm{\Omega }}_{34}=0.3,{\omega }_1=0,{\omega }_2=1,{\omega }_3=1.7$, and ${\omega }_4=2$.

Figure 7

The quantum ground energy surface $E_{\min }$ (color graded scale) in comparison with the corresponding variational ground energy (mesh surface) as a function of the control parameters for a number of particles (a) $N_a=1$ and (b) $N_a=2$.

Figure 8

The rate of the difference to the sum of the expectation value of the number of photons, $\delta \nu$, of the exact ground state for (a) $N_a=1$ and (b) $N_a=2$.

Figure 9

Expectation value of the number of photons $\langle {\mathbf{\nu }}_{12}\rangle$ of the exact ground state for (a) $N_a=1$ and (b) $N_a=2$. The classical separatrix is indicated (white line).