Coherently delocalized states in dipole interacting Rydberg ensembles: The role of internal degeneracies

We investigate the effect of degenerate atomic states on the exciton delocalization of dipole-dipole interacting Rydberg assemblies. Using a frozen gas and regular one-, two-, and three-dimensional lattice arrangements as examples, we see that degeneracies can enhance the delocalization compared to the situation when there is no degeneracy. This enhancement is particularly large in the case of the three-dimensional (3D) random gas, but is absent for 1D arrangements. Using the Zeeman splitting provided by a magnetic field, we controllably lift the degeneracy to study in detail the transition between degenerate and nondegenerate regimes. These observations, although specific to the experimentally clean Rydberg gas, have generic implications for various dipole-interacting systems.

Figure 1

(a) Energy structure of a single Rydberg atom in the relevant subspace without (left) and with (right) an applied magnetic field. The $l=0$, $m=0$ levels are set to be at zero energy while the $l=1$, $m=0$ state sits at energy $\epsilon$. (b) Definition of the angles that enter the interaction matrix elements. The angle ${\theta }_{ij}$ is defined with respect to the quantization axis, while the angle ${\varphi }_{ij}$ is given with respect to an arbitrarily chosen $x$-axis. When a magnetic field is present, we choose the quantization axis $\vec{z}\left|\right|\vec{B}$. (c) We structure our study around four generic arrangements: a three-dimensional random gas, and one-, two-, and three-dimensional regular lattice arrangements.

Figure 2

The frozen Rydberg gas. All plots are for $N=1000$ Rydberg atoms averaged over $\sim {10}^3$ realizations. (a) Probability densities for the PTM measure, displayed for several magnetic field strengths. The black curve gives the $B\to \infty$ reference value, obtained by setting the coupling between $m$-levels to zero. (b) Probability densities to find a certain PTM value for an eigenstate with a certain energy, for the same magnetic fields as in panel (a). (c) Comparison between the $B=0$ and 50 case. Although we use $m$ as a label to distinguish the three distributions, we note that they are calculated in the same way for each magnetic field value, and these labels are only approximate, since the Zeeman splitting allows these distributions to be characterized by $m$ to a very good approximation. Below the $m=0$ distribution, we show the distribution in the asymptotic limit $B\to \infty$ (flipped to better see the close agreement with the $m=0$ distribution for $B=50$) . The energies are given in units of $E_{\mathrm{ref}}={\mu }_{\text{sp}}^2/3a_{\mathrm{ref}}^3$, where $a_{\mathrm{ref}}={(3/4\pi N)}^{1/3}L$ is the Wigner-Seitz radius. The zero of energy is at the energy $\epsilon$ of the noninteracting atoms, which is introduced in Eq. (3).

Figure 3

Same as Fig. 2, but for small magnetic fields. The vertical bars indicate the positions of the Zeeman energies, $\pm mB$.

Figure 4

Distribution of PTM for (a) a 1D chain, (b) a 2D, and (c) a 3D lattice. In all three cases we have roughly the same number of atoms as in the frozen gas case [990 atoms in the 1D case ($N_x=990$), 992 atoms in the 2D case ($N_x=32$, $N_y=31$), and 990 atoms in the 3D case ($N_x=11$, $N_y=10$, $N_z=9$)], and the quantization axis is set parallel to the magnetic field, pointing along the $z$-direction. The blue bars are the case without magnetic field ($B=0$), and the magenta (mirrored) bars are the infinite B-field limit. The bottom row shows the upper plots on the same $x$-axis.

Figure 5

Transition from lattice to gas. Comparison of a 3D lattice with different filling fractions (legend provided in the upper panel) to the frozen Rydberg gas (black) for different strengths of the magnetic field (provided in the panels). For all cases we fix the number of atoms at $N=990$. For the lattice we apply $5\%$ uniform fluctuations around the perfect lattice positions.