Breakdown of the few-level approximation in collective systems

The validity of the few-level approximation in dipole-dipole interacting collective systems is discussed. As an example system, we study the archetype case of two dipole-dipole interacting atoms, each modeled by two complete sets of angular momentum multiplets. We establish the breakdown of the few-level approximation by first proving the intuitive result that the dipole-dipole induced energy shifts between collective two-atom states depend on the length of the vector connecting the atoms, but not on its orientation, if complete and degenerate multiplets are considered. A careful analysis of our findings reveals that the simplification of the atomic level scheme by artificially omitting Zeeman sublevels in a few-level approximation generally leads to incorrect predictions. We find that this breakdown can be traced back to the dipole-dipole coupling of transitions with orthogonal dipole moments. Our interpretation enables us to identify special geometries in which partial few-level approximations to two- or three-level systems are valid.

Figure 1

(a) The system of interest is comprised of two identical atoms that are located at ${\mathit{r}}_1$ and ${\mathit{r}}_2$, respectively. $\mathit{R}={\mathit{r}}_2-{\mathit{r}}_1$ is the relative position of atom 2 with respect to atom 1. (b) Level structure of atom $\mu ∊\{1,2\}$ which we employ to illustrate our results. The ground state is a $S_0$ singlet state, and the three excited levels are Zeeman sublevels of a $P_1$ triplet. $\delta$ is the frequency splitting of the upper levels.

Figure 2

(Color online) Setup considered in Sec. 3, which provides an illustration of the physical mechanisms responsible for the breakdown of the few-level approximation. In this example, an external laser field is used for the sake of illustration. Our main results starting from Sec. 4 do not rely on external driving fields.

Figure 3

(Color online) Population in the subspace $S$ obtained by applying the few-level approximation to the setup in Fig. 2. The common parameters are $\theta =\pi ∕2$, $\varphi =\pi ∕2$, and $\delta =0$. Further, in (I), ${\Omega }_L=2\gamma$, $R=0.3{\lambda }_0$, and $\Delta =0.58\gamma$, where $\Delta ={\omega }_L-{\omega }_0$. Curve (II) shows the case ${\Omega }_L=5.4\gamma$, $R=0.1{\lambda }_0$, and $\Delta =5.2\gamma$.

Figure 4

(Color online) Population of the subspace $V$, which contains all population which was lost from subspace $S$ in Fig. 3, such that the population in $S+V$ remains unity for all times. The parameters are as in Fig. 3.