Rydberg Composites

We introduce the Rydberg composite, a new class of Rydberg matter where a single Rydberg atom is interfaced with a dense environment of neutral ground state atoms. The properties of the composite depend on both the Rydberg excitation, which provides the gross energetic and spatial scales, and the distribution of ground state atoms within the volume of the Rydberg wave function, which sculpt the electronic states. The latter range from the “trilobites,” for small numbers of scatterers, to delocalized and chaotic eigenstates, for disordered scatterer arrays, culminating in the dense scatterer limit in symmetry-dominated wave functions which promise good control in future experiments. We discuss one-, two-, and three-dimensional arrangements of scatterers using different theoretical methods, enabling us to obtain scaling behavior for the regular spectrum and measures of chaos and delocalization in the disordered regime. We also show that analogous quantum dot composites can elucidate in particular the dense scatterer limit. Thus, we obtain a systematic description of the composite states. The two-dimensional monolayer composite possesses the richest spectrum with an intricate band structure in the limit of homogeneous scatterers, experimentally accessible with pancake-shaped condensates.

Popular Summary

In a Rydberg atom, one or more electrons are excited to a very high state. In this state, the wave function of the excited electron occupies a relatively large volume (easily $1\times {10}^9$ times larger than the ground state) and has highly degenerate energies. Here, we show that this degeneracy can be lifted in a controlled manner by filling the particle’s wave function with other neutral atoms, or “scatterers,” creating a system we call a Rydberg composite.

By smoothly tuning the number of scatterers in the composite, the energy degeneracy can be lifted in two different ways. When just a few scatterers are present, each additional scatterer causes one of the energy levels to deviate from the degenerate value until all energy levels are no longer degenerate. At the opposite extreme of many scatterers, the density of scatterers becomes so dense that the Rydberg wave function no longer “sees” individual scatterers but rather a homogenous fog of them. The geometry of this fog is incompatible with any Rydberg wave function and therefore breaks the degeneracy. If the geometry is planar, for example, then the Rydberg spectrum changes to one with well-structured energy bands, which we predict to be observable in a dense pancake-shaped Bose-Einstein condensate.

Analogous composites may be formed with suitable highly excited quantum dot states. Generally, by varying the scatterer density, excitation composites introduce an approach to systematically create and describe properties of systems that occupy territory between atomic and condensed matter.

Figure 1

Schematics of the three scenarios we consider: (a) a linear lattice (1D), (b) a monolayer (2D), and (c) a cubic lattice (3D) with the Rydberg ion in the center, respectively. In each panel the black spheres represent scatterers sitting on lattice sites and the red lines give the lattice spacing $d$. The missing scatterers in (b) represent a situation with incomplete filling. The volume of scatterers situated within the Rydberg wave function is represented by the blue circle in (b) and by the sphere in (c). The exemplary densities shown give different representations of the composite’s electronic wave function in these three scenarios. In (a) the strongly perturbed wave function is shown with three surfaces of constant density, revealing the exotic nature of these wave functions. The full 3D contour is cut away in front to reveal the interior structure. Figure 3 provides details of the plot parameters. Panel (b) shows a contour of the angular dependence of a typical circular state, which plays a crucial role in the 2D composite’s properties (see Sec. 4). Panel (c) shows an illustration of the Rydberg atom, illustrating that the spatially varying probability cloud spans many lattice sites of the 3D lattice.

Figure 2

Density of states $dN/d\tilde{E}$ for a $\nu =30$ composite in 1D (a), 2D (b), and 3D (c) as a function of scaled lattice spacing $d/\nu$.

Figure 3

Electronic densities of a 1D lattice of scatterers for $\nu =30$ and different lattice spacings $d$ in atomic units. The yellow, green, and blue surfaces correspond to contours at wave function densities spanning factors of 10. The full three-dimensional surfaces are cut away in front to reveal the interior structure. The left-hand column gives the density for the most deeply perturbed state, while the right-hand column gives the density just above the degenerate band.

Figure 4

The electron density $|\mathrm{\Psi }(\mathbf{r}){|}^{1/2}$ of selected eigenstates of a $\nu =30$ 2D composite. Both panels display from bottom to top the first, third, and fifth eigenstates. Top: Probability densities for a dense lattice ($d=20$) increasing in fill factor from left to right in steps of 0.1 excluding the first panel where only five scatterers are present. Bottom: Probability densities for a full lattice with $d=\{1000,500,300,200,100,90,80,70,60,54,20\}$, respectively.

Figure 5

Spectra for a $\nu =30$ 2D Rydberg composite as a function of lattice spacing. The line color shows $P_{\mathrm{PR}}$ for each state as defined by Eq. (11). The black line marks the border between homogeneous and inhomogeneous regimes [see Eq. (34)].

Figure 6

The 2D Rydberg composite’s spectrum displayed as a dispersion relation, emphasizing the formation of energy bands. The black curves show $\tilde{E}(m/\nu )$ for $\nu =100$. The dashed red lines are the asymptotic linearization of Eq. (17).

Figure 7

Dependence of the band structure of a 2D Rydberg composite on the projection of $Y_{lm}(\theta ,\phi )$ into the plane. Shown are a few spherical harmonics, starting in the bottom left-hand corner with the maximally circular state. Horizontal black (vertical black) arrows represent a decrease in $m$ ($l$). States in the same column therefore are from the same block diagonal. Red diagonal arrows represent increases in $k=\nu -l$, starting from $k=1$ in the bottom row. The inset shows the projection $\mathcal{P}$ as a function of $l$ for $\nu =30$ for several $b$ bands. Notice how much larger the drop in energy is between bands compared to the gaps between $k$ states within a band.

Figure 8

Density of states computed by convolving the spectrum using a Gaussian distribution for three $\nu$ values: 40 (yellow), 70 (green), and 100 (blue). (a) The energies are scaled with ${\nu }^{5.5}$, the “band” scaling. Individual band contributions for $\nu =100$, obtained from the dispersion curves in Fig. 6, are shown in gray scale. (b) The energies are scaled with ${\nu }^6$, the “overlap” scaling. (c) The energies and densities of states are scaled via the “universal” scaling connecting the band and overlap regions. The interpolating tanh function connecting the two regimes is overlayed. The FWHM of the convolving Gaussian distribution is approximately 0.0235; for details, see Appendix pp4.

Figure 9

The $\mu =100$ eigenspectrum for a 2D SQD composite in the $d\to 0$ limit. The oscillator frequency is set to $\omega =1$.

Figure 10

Density of states for a $\mu =30$ SQD composite as a function of lattice spacing $d$ in atomic units. The oscillator frequency is set to $\omega =1$ and the color is the same as in Fig. 2.

Figure 11

(a) Average AGR of 2000 realizations of a lattice of scatterers as a function of $F$ for $d=5$, 10, 30, and 70 in sequence from dark to lighter line color, $\nu =30$. (b) AGR for a filled lattice as a function of $(d_2-d)/\nu$ with $d_2$ from $d_D$ in Table 1 for $D=2$. The arrow marks $d_c(l,m)$ from Eq. (34) with $m=l=\nu -1$, while the horizontal line shows the RMT prediction for AGR incorporating the appropriate symmetries of the scatterers on the lattice; see text. Note that in both plots increasing values of the abscissa mean higher density of scatterers.

Figure 12

Numerically obtained AGR for a Rydberg composite with $\nu =30$ (blue) and $\nu =100$ (gray) for a filled lattice ($F=1$) as a function of $d/\nu$.

Figure 13

The DOS for a $\nu =200$ Rydberg composite formed with scatterers in a quasi-2D BEC in pancake geometry. The scatterers are randomly positioned with a uniform distribution in the $xy$ plane and a Gaussian profile in the $z$ direction. The cloud’s peak density and width (standard deviation) in the $z$ direction are, respectively, $\rho =4\times {10}^{15}{\mathrm{cm}}^{-3}$ and ${\delta }_z=0.15\mu \mathrm{m}$ in (a) and $\rho =8\times {10}^{15}{\mathrm{cm}}^{-3}$ and ${\delta }_z=0.3\mu \mathrm{m}$ in (b). We have convolved the line spectra with a Gaussian distribution of 1 MHz width and use the zero-energy electron-rubidium scattering length $a_s=-16.1a_0$. The DOS is normalized by multiplying by a factor of $1/\nu$ as in Fig. 2.

Figure 14

Absorption spectra in arbitrary units from absorbing two circularly polarized photons (see text) for the pancake Rydberg composite of Fig. 13 (black line) and the corresponding DOS from Fig. 13 shown as blue shaded area. All parameters are as in Fig. 13.