Spectral backbone of excitation transport in ultracold Rydberg gases

The spectral structure underlying excitonic energy transfer in ultracold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics, in particular, the density of states and the nearest-neighbor level-spacing statistics, exhibits a transition from approximately a 1-stable Lévy to a Gaussian orthogonal ensemble. Deviations from random matrix statistics are shown to stem from correlations between interatomic interaction strengths that lead to an asymmetry of the spectral density and profoundly affect localization properties. We discuss approximations to the self-consistent Matsubara-Toyozawa locator expansion that incorporate these effects.

Figure 1

Density of states $f_{\Lambda }$ for the ultracold Rydberg gases described by the Hamiltonian $H$ in Eq. (4), with the blockade radius $r_{\mathrm{b}}$ set to zero. The main diagram shows plots for $N=10$ (dotted dashed line; average over $15660000$ disorder realizations), $N={10}^2$ (dashed line; $171000$ realizations), $N={10}^3$ (dotted line; $101550$ realizations), and $N={10}^4$ (solid line; $22790$ realizations) Rydberg atoms. The common qualitative features are unimodality (the existence of a single maximum corresponding to the most probable eigenvalue) and a slight skewness to positive energies. The most striking differences are a trend to a smaller central peak and a more pronounced skewness for larger $N$. The log-log plots in the insets illustrate that, asymptotically, $f_{\Lambda }\left(\lambda \right)$ falls off like ${\left|\lambda \right|}^{-2}$ [red (gray)] on both sides, irrespective of the system size $N$.

Figure 2

Level-spacing density $f_S$ for Rydberg gases modeled by Eq. (4) with $N={10}^4$ atoms and no dipole blockade $r_{\mathrm{b}}=0$. Eigenvalues $\Lambda$ with $\left|\Lambda \right|\ge 100$ (dashed line) follow Poissonian spacing statistics [red (gray) dotted line)], whereas spacings of eigenvalues from the spectrum's center ($\left|\Lambda \right|\le 0.2$, dotted line) are distributed according to Rayleigh (Wigner-Dyson) statistics [red (gray) dashed line]. The overall statistics (that is, including all eigenvalues) is mixed and not shown because it is virtually identical to the spacing statistics in the remaining intervals $(-100,-0.2)$ and $\left(0.2,100\right)$ (solid line). The inset shows a double-logarithmic plot of the same quantities for comparison with algebraic level repulsion for small spacings.

Figure 3

Transition between Poissonian and Wigner-Dyson statistics in the level-spacing statistics of unblockaded (i.e., $r_{\mathrm{b}}=0$) Rydberg gases with $N={10}^4$ atoms. Shown are the root-mean-square deviations $\Delta [f_S,f_S^{\mathrm{P}}]$, $\Delta [f_S,f_S^{\mathrm{WD}}]$ to both references, Poisson [red (gray) solid line, Eq. (11)] and Wigner [red (gray) dashed line, Eq. (12)], as a function of energy $\lambda$. The dots are numerical data, the curves are polynomial interpolations. The energies of intersection are ${\Lambda }_{\mathrm{tr}}^{-}=-4.51$ and ${\Lambda }_{\mathrm{tr}}^{+}=2.14$.

Figure 4

Density of states $f_{\Lambda }$ for the Rydberg Hamiltonian (4) for $N={10}^4$ atoms and $r_{\mathrm{b}}=0$ [red (gray) solid line] against the result for the same model but with all correlations set to zero (dashed line) and the density of states obtained for the corresponding 1-stable Lévy ensemble (black solid line) (see Sec. 4 ). All densities have the same tail asymptotics, but only the Rydberg model displays the characteristic asymmetry. This shows that the asymmetry originates from correlations between the Rydberg Hamiltonian's matrix elements.

Figure 5

Spectral density $f_{\Lambda }$ for $r_{\mathrm{b}}=0$, $0.25$, $0.5$, and $0.75$. Shown are the respective results of numerically exact diagonalization [red (gray) solid line] of an ensemble of realizations of Eq. (4) for $N={10}^4$ atoms, the low-concentration approximation to the Matsubara-Toyozawa locator expansion with $l\le 1$ (dotted line) as well as $l\le 2$ (black solid line), and the approximation to the high-concentration locator expansion (dashed line). The diagram also depicts the Wigner semicircle law [Eq. (15)] and the boundary of its support [Eq. (16)] as a function of $r_{\mathrm{b}}$ (both light gray).

Figure 6

Probability density function $f_{H_{ij}}$ [see Eq. (B7)] for the interactions between $N={10}^4$ atoms in the Hamiltonian (4). The black solid curve in the main figure is the graph of Eq. (B7) in the limit $r_{\mathrm{b}}\to 0$. The one-stable Lévy distribution of Eq. (17) is plotted in black as a dashed line. Both distributions feature the same heavy-tailed asymptotics. In the main figure, they lie on top of each other, but in the inset, which magnifies the small-$\left|h\right|$ region, the graph of $f_{H_{ij}}$ appears lopsided. This is a direct consequence of the anisotropic dipolar interaction (3). The red (gray) solid line is a plot of Eq. (B7) for $r_{\mathrm{b}}=0.75$. It illustrates the radically different asymptotics for the cases $r_{\mathrm{b}}=0$ and $r_{\mathrm{b}}>0$. In contrast, the inset shows that, for small $\left|h\right|<5\times {10}^4$, $f_{H_{ij}}$ is virtually independent of $r_{\mathrm{b}}$, the curve for $r_{\mathrm{b}}=0.75$ [red (gray) solid line] covers that for $r_{\mathrm{b}}=0$ (black solid line) completely. For $r_{\mathrm{b}}=0.75$, the variance of the Hamiltonian matrix elements is finite, $\overline{\left(H_{ij}\right){}^2}\simeq 2.86\times {10}^{-4}$. For comparison, a normal distribution with the same variance is plotted as red (gray) dashed line.

Figure 7

Examples of paths starting and ending at the home atom (white circle), and thereby contributing to the spectral density. Thin arrows represent single directed transitions from one atom (black or white circle) to another one. A thick double-headed arrow represents arbitrarily many round trip transitions between the two respective atoms. (a) The low-concentration limit ($l=1$) takes into account sequences of arbitrarily many transitions (double-headed thick arrow) between pairs of atoms arranged in a treelike structure. (b) Low-concentration limit ($l=2$): similar as (a), but involving transitions between three atoms in addition to pairs of atoms. (c) The high-concentration limit includes round trips on arbitrarily many atoms each of which is visited only a single time within this round trip. To each atom, however, other round trips may be attached. (d) Example of a path ($0\to 1\to 2\to 3\to 1\to 2\to 0$) not taken into account in the classes of paths represented by (a), (b), or (c). (e) Irreducible loop diagram similar to (d).

Figure 8

Comparison between the numerically obtained density of states $f_{\Lambda }$ for the Hamiltonian (4) with parameters $N={10}^4$, $r_{\mathrm{b}}=0$ [red (gray) solid line] and the spectral densities derived in Ref. [46] from the self-consistent Matsubara-Toyozawa locator expansion for low concentrations in first [dashed line, Eq. (21)] and second order [black solid line, Eq. (25)] approximation. The dotted line is a plot of the density of states of the 1-stable Lévy ensemble.

Figure 9

Breakdown of the two-level approximation for small distances. Shown are the coherent population dynamics of a pair of rubidium-85 Rydberg atoms at different interatomic distances $R$, (a) $18.5\mu \mathrm{m}$, (b) $8.60\mu \mathrm{m}$, and (c) $3.99\mu \mathrm{m}$, but for equal orientation $\widehat{\mathbf{R}}={(-1,0,\sqrt{2})}^t/\sqrt{3}$. In each case, the system is initialized in the state $|P,S\rangle =|46P_{3/2,3/2},46S_{1/2,1/2}\rangle$. Plotted are the populations $p_{PS}$ and $p_{SP}$ of the states $|P,S\rangle$ and $|S,P\rangle$, respectively, obtained in two-level approximation [black dashed line and red (gray) dotted line] and when treated with the full Hamiltonian [black dotted line and red (gray) dashed line]. For the sake of error estimation, we also include $1-p_{PS}-p_{SP}$ (black solid line). The results are discussed in the text.