Facilitation Dynamics and Localization Phenomena in Rydberg Lattice Gases with Position Disorder

We explore the dynamics of Rydberg excitations in an optical tweezer array under antiblockade (or facilitation) conditions. Because of the finite temperature the atomic positions are randomly spread, an effect that leads to quenched correlated disorder in the interatomic interaction strengths. This drastically affects the facilitation dynamics as we demonstrate experimentally on the elementary example of two atoms. To shed light on the role of disorder in a many-body setting we show that here the dynamics is governed by an Anderson-Fock model, i.e., an Anderson model formulated on a lattice with sites corresponding to many-body Fock states. We first consider a one-dimensional atom chain in a limit that is described by a one-dimensional Anderson-Fock model with disorder on every other site, featuring both localized and delocalized states. We then illustrate the effect of disorder experimentally in a situation in which the system maps on a two-dimensional Anderson-Fock model on a trimmed square lattice. We observe a clear suppression of excitation propagation, which we ascribe to the localization of the many-body wave functions in Hilbert space.

Figure 1

Two-atom setting. (a) The harmonic traps are disposed in a line along $i=3$ with average separation $r_0$ and widths ${\sigma }_i$. (b),(c) Level structure for the two-atom case for the resonant ($\mathrm{\Delta }=0$) and facilitated ($\mathrm{\Delta }=-V_{\mathrm{NN}}$) conditions, respectively. The experimental data for the time evolution of the excitation probabilities $P_{\uparrow \downarrow }$, $P_{\downarrow \uparrow }$ are shown as full circles in panels (d),(e). The data are averaged over at least 100 realizations of the disorder. The solid lines show numerical solutions of the dynamics obtained averaging over 30 realizations of the disorder. The experimental data here and in the following were obtained using ${}^{87}\mathrm{Rb}$ atoms held at a temperature $T=50\mu \mathrm{K}$ in the traps with frequencies ${\omega }_1=2\pi \times 11$, ${\omega }_{2,3}=2\pi \times 91.5\mathrm{kHz}$ resulting in the position uncertainties ${\sigma }_1=1\mu \mathrm{m}$ and ${\sigma }_{2,3}=120\mathrm{nm}$. The internal levels are $|\downarrow ⟩=|5S_{1/2},F=2,M=2⟩$ and $|\uparrow ⟩=|100D_{3/2},F=3,M=3⟩$ with $r_0=14.2\mu \mathrm{m}$, $\mathrm{\Omega }=2\pi \times 1.25\mathrm{MHz}$, $C_6=-2\pi \times 7.3\times 10^7\mathrm{MHz}{\mu \mathrm{m}}^6$. Consequently, $V_{\mathrm{NN}}=C_6/r_0^6=-2\pi \times 8.9$ and $|\delta V|\sim 2\pi \times 0.64\mathrm{MHz}$ (all energies are in units of $\hslash$).

Figure 2

Fock space structure for three atoms prior and after applying the facilitation condition. States with similar energy occupy the same row. (a) Internal structure of the Fock space. Linked states are connected by one spin flip. (b) Fock states spanning the reduced Hilbert space under facilitation conditions are shown in red (see text for details).

Figure 3

Lyapunov exponent for the one-dimensional Anderson-Fock model. All data shown in this figure are obtained with the same parameters given in Fig. 1. In the main figure we report the Lyapunov exponent as a function of the energy $E$ (measured in units of $\mathrm{\Omega }/2$). The inset shows a comparison between the shapes of the wave functions obtained from a numerical reconstruction (left panel) and from the corresponding prediction associated to the Lyapunov exponent (right panel) for a chain of $L=20$ sites and a specific realization of the disorder. In the right panel the envelopes $\propto \exp [-4\gamma (E)|k-k_{\max }(E)|]$ are centered at the position $k_{\max }(E)$ at which the corresponding set of excitation probabilities in the left panel reaches its maximum. The factor 4 in the exponent stems from considering probabilities instead of amplitudes and the fact that the length of the atomic chain is about half the one in Fock space.

Figure 4

Eight-atom experiment and two-dimensional Anderson-Fock model. (a) Experimental data for the dynamics of the site-resolved excitation probability averaged over more than 100 realizations. Here, $|\uparrow ⟩=|56D_{3/2},F=3,M=3⟩$, $r_0=4.1\mu \mathrm{m}$; $\mathrm{\Omega }=2\pi \times 2.1$, $\mathrm{\Delta }=-V_{\mathrm{NN}}=-2\pi \times 8.4$, and $|\delta V|\sim 2\pi \times 2.1\mathrm{MHz}$ (in units of $\hslash$). The data are compared with numerical data from exact diagonalization of the Hamiltonian (b),(c), and the 2D Anderson-Fock model (d),(e), with and without disorder. Disorder averages are made over 100 realizations. In the absence and for low disorder ($|\delta V|\lesssim 2\pi \times 0.4\mathrm{MHz}$), excitations still propagate ballistically. (f) Lattice structure of the effective model for $L=4$ atoms and $N_{\mathrm{cl}}=1$. (g) Inverse participation ratio $I$ as a function of the energy $E$ (measured in units of $\mathrm{\Omega }/2$) for a chain of $L=20$ atoms. The amplitude of the wave function (projecting $|E⟩$ on the Fock basis $|c⟩$) is reported for four representative states on a lattice whose structure follows the one shown in panel (f). From left to right they display: a state localized in both Fock space and real space, the special state $|{\psi }_0⟩$ and a similar state found for small $E>0$ (see text for details).