Experimental realization of a 3D random hopping model

Scientific advance is often driven by identifying conceptually simple models underlying complex phenomena. This process commonly ignores imperfections which, however, might give rise to non-trivial collective behavior. For example, already a small amount of disorder can dramatically change the transport properties of a system compared to the underlying simple model. While systems with disordered potentials were already studied in detail, experimental investigations on systems with disordered hopping are still in its infancy. To this end, we experimentally study a dipole–dipole-interacting three-dimensional Rydberg system and map it onto a simple XY model with random couplings by spectroscopic evidence. We discuss the localization–delocalization crossover emerging in the model and present experimental signatures of it. Our results demonstrate that Rydberg systems are a useful platform to study random hopping models with the ability to access the microscopic degrees of freedom. This will allow to study transport processes and localization phenomena in random hopping models with a high level of control.

because the product term n i =j δ ik δ jk vanishes. i.e. Ω P V dd ij , we directly excite the coherent many-body 30 state formed by the n seeds and one additional atom ex-31 cited by the probe pulse. The off-diagonal nature of the 32 dipole-dipole interaction intrinsically provides coherence 33 on the two particle level. In order to destroy this coher-34 ence any decoherence process (laser noise, atomic motion, 35 internal decay) must be of similar magnitude.  (Fig. 2a), where the probe pulse is delayed 51 by τ = 300 µs, we obtain the signal created by the seeds 52 alone (gray area) and the spectroscopic signal of the |↑ -53 excitations without the presence of the seeds (green area).

54
For the much smaller pump-probe delay τ = 1 µs, which 55 we use for the interacting spectra, the signals generated Normalized ion signal Supplementary Figure 1: Influence of the on-site energy. Spectra for the excitation of the |51P 3/2 -state after the creation of different numbers of initial seeds in the |51S 1/2 -state with two-photon Rabi frequencies Ω S a 2π ×18 kHz, b 2π ×37 kHz, c 2π × 74 kHz, d 2π × 111 kHz. The shaded areas denote the standard error of the mean. The same spectra are already shown in Fig. 2c-f. In addition to the fitted numerical model with (C 6 = 0, orange), a comparison for a model without (C 6 = 0, red) on-site energy is shown. The non-interacting spectra (green) are given as reference.
by both excitation pulses overlap (Fig. 2b). We therefore 57 use the knowledge of the isolated seed contribution from 58 the reference measurement (gray area) and, by subtrac-59 tion, isolate the contribution from the |↑ excitation (blue 60 area).

61
The integrated number of events in the blue area com-  numerically more stable way, we make use of the relation between the integral signal and its amplitude A and 84 lifetime τ Rb + . The lifetime is now obtained by di- Supplementary Figure 3: Localization-delocalization crossover. We plot the probability to find a localized state p E,n a for a pure random XY model and b with additional C 6 interaction between the seeds as realized in the experiment. In contrast to Fig. 3 in the main text, here the probe atom is excited into another fine structure state, the |51P 1/2 -state. The regime of predominantly delocalized states spreads for increasing number of seeds n, shifting the localization-delocalization crossover to larger energies. The estimated energies where the corresponding spectra show a transition towards algebraic |∆ P | −2 scaling are denoted by dashed lines. Supplementary Figure 4: Influence of the Rydberg state on localization. Comparison of the equiprobability lines L 0.5 for the pure random XY model (blue) and with additional C 6 interaction (orange) for a probe atom in a the |51P 3/2 -state and b the |51P 1/2 -state. The energy range where the states are predominantly delocalized is substantially narrower in the model with C 6 interaction than in the pure random XY model. Additionally, the relative difference between both models is more pronounced for the |51P 1/2 -state. Note that the energy scales are different.

Supplementary
In the dense atomic samples investigated here the Ryd-  Figure 6: Comparison between spectra χ and probability of localization P L . a shows the simulated spectra for n = 4 seeds with (blue) or without (orange) C 6 interaction. The dashed lines indicate the detuning ∆ CO where the transition towards |∆ P | −2 scaling (dotted gray line) sets in. For the calculation of these detunings see Methods. b shows the probability P L to find a localized state. The detuning ∆ CO calculated from the spectra matches the crossover to localized states. The insets c and d show the Probability P L for different seed numbers and are identical to Fig. 3, with the position of the cuts shown in b marked. The dashed lines show ∆ CO from the simulated spectra as shown in a.