Experimental realization of a 3D random hopping model

The study addresses how disorder in hopping amplitudes, rather than in on-site energies, affects transport and localization in quantum systems. While Anderson localization and many-body localization are well studied for on-site disorder, random hopping (off-diagonal disorder) with long-range dipole-dipole couplings is less explored experimentally. The authors use a three-dimensional ultracold Rydberg gas to realize an effective XY spin-1/2 model with random couplings arising from position disorder and dipole-dipole interactions. They aim to spectroscopically validate the mapping to the random XY model and identify an energy-dependent localization–delocalization crossover predicted for such systems, thereby providing an experimentally accessible platform for random hopping physics relevant to energy transport in various physical, chemical, and biological contexts.

The paper situates the work within the broader context of disorder-driven transport phenomena, citing Anderson localization, many-body localization, and glassy dynamics. Classic Anderson models focus on on-site disorder and nearest-neighbor hopping, while extensions include interactions and long-range terms. Random hopping models (off-diagonal disorder) exhibit rich effects such as many-body relaxation, glassy dynamics, localization, and superfluid stiffness, and are relevant to systems with dipole-dipole couplings (e.g., Rydberg gases, waveguides, NV centers, molecular transport). Prior theoretical studies of dipolar systems in 3D showed that long-range hopping can delocalize states in the presence of on-site disorder, but adding hopping disorder can restore localization and yield an energy-resolved crossover between delocalized and pair-localized states. Previous evidence was mainly through level statistics or eigenstate properties; an experimentally accessible criterion had been lacking. Rydberg systems are highlighted as natural platforms due to strong, tunable dipole interactions and intrinsic position disorder.

Model and mapping: The effective spin-1/2 is encoded in two Rydberg states, |S> = |↓> and |P> = |↑>, coupled by resonant dipole-dipole interaction with anisotropic 1/R^3 dependence. The system maps to an XY Hamiltonian with random couplings Jij ∝ (1 − 3 cos^2 θij)/Rij^3 set by random interparticle separations and orientations; weak van der Waals interactions contribute a small Ising-like term. In the weak probing limit, the C6 contribution acts as a random longitudinal field, reducing the dynamics to a random-hopping XY model for spectroscopy.

Experimental setup: A 3D frozen ultracold gas of 87Rb is prepared as a Bose–Einstein condensate (∼90×10^3 atoms, peak density 3×10^14 cm^-3) in a crossed optical dipole trap. A pump–probe scheme is used: a 1 µs two-photon resonant pump pulse creates a controlled number of |51S1/2, mJ=1/2> (|↓>) seed excitations with effective two-photon Rabi frequency Ωs/2π in the range 18–111 kHz, set by laser power under Rydberg blockade conditions. After a delay τ, a 1 µs weak probe pulse drives a single-photon transition from |5S1/2> to |51P3/2, mJ=1/2> (|↑>) with Ωp ≈ 2π×4.5 kHz and detuning Δp. Two delays are used: τ=1 µs (interacting, seeds present) and τ=300 µs (non-interacting reference, seeds decayed). Rydberg populations are monitored via spontaneous ionization to Rb+ with continuous extraction and time-resolved detection. Spectra are obtained by integrating time windows isolating the probe response and scanning Δp.

Spectroscopic analysis: Increasing Ωs varies the mean number of seeds. Interacting spectra exhibit on-resonance suppression (C3 blockade) and enhanced wings (anti-blockade). Asymmetry towards negative detunings is observed and attributed to many-body correlations in hopping matrix elements rather than molecular formation.

Numerical simulations: Monte Carlo simulations of the random XY model are performed in the single-excitation subspace (weak probing). For each realization, n seed positions (subject to blockade radius rB) and one additional |P> atom are sampled from the BEC’s Thomas–Fermi density. The Hamiltonian (dipole-dipole exchange plus weak C6 term) is diagonalized to obtain eigenstates |ξ> and eigenenergies Eξ. For each parameter set (n, rB), 10^5 random realizations are simulated. The coupling of the probe to eigenstates is approximated by the |n+1> component, yielding normalized spectra χn(Δp). Realistic seed-number fluctuations are incorporated via a Poisson distribution with mean μ, and the total spectrum is χ(Δp)=Σn p(n) χn(Δp); the n=0 contribution uses the measured non-interacting spectrum to include molecular features. Fit parameters (μ, rB, amplitude) are obtained by least-squares fits to experimental spectra.

Localization diagnostics: Eigenstate coherence C(ξ)=Σi Σj |<i|ξ><j|ξ>|^2 is computed; localized states satisfy C<2 and, to ensure real-space localization for small systems, a spatial extent below the blockade radius. Conditional probability PL(Δp, n) of localized states versus energy and seed number is extracted. An experimentally accessible crossover indicator is defined via the scaling of the spectral density: the crossover detuning Δco is the energy where the spectrum approaches an algebraic |Δp|^-2 tail, identified by a scaling exponent threshold from smoothed spectra.

• Spectroscopy of the |51P3/2> state in the presence of |51S1/2> seeds shows strong, pump-power-dependent modifications: on-resonance suppression (blockade) and enhanced far-detuned wings (anti-blockade), with a notable asymmetry towards negative detunings attributable to many-body hopping correlations rather than molecular formation.
• Monte Carlo simulations of the random XY model in the single-excitation subspace quantitatively reproduce the measured spectra across pump powers, including asymmetry, blockade suppression, and enhanced wings.
• Fitted average numbers of coherently coupled seeds increase with pump Rabi frequency, with extracted values approximately 0.6, 2.4, 5.0, and 5.7 for Ωs/2π = 18, 37, 74, and 111 kHz, respectively.
• A consistent blockade radius rB ≈ 3.4 µm is obtained, in agreement with expectations.
• The weak C6 interaction only slightly alters spectral shapes but significantly affects eigenstate structure by acting as an effective random longitudinal field in the weak probing limit, shifting the localization–delocalization crossover and narrowing the delocalized regime.
• Simulations predict an energy-resolved crossover: low-energy states are predominantly delocalized, while high-energy states are pair-localized; increasing the number of seeds shifts the crossover to higher energies.
• An experimentally accessible crossover indicator is observed: at large detunings the spectral density follows an algebraic |Δp|^-2 tail, signaling predominance of localized pair states. The detuning where this scaling sets in increases with seed number, consistent with simulations.
• Lifetimes of the Rb+ ion signal (τRb+) decrease with increasing detuning, consistent with a growing fraction of short-lived, pair-localized states at small interparticle distances. For higher seed numbers, the lifetime decreases more slowly versus detuning, indicating a larger energy range of delocalized states.
• Repeating measurements with |51P1/2, m=1/2> yields similar agreement, indicating minor sensitivity to fine-structure details and validating the effective two-level spin description.
• Overall, the data provide experimental signatures of a localization–delocalization crossover in a 3D random hopping (random XY) model realized with Rydberg atoms.

The experiments validate that a 3D frozen Rydberg gas with dipole-dipole exchange realizes a random-hopping XY model and exhibits an energy-dependent transition from delocalized to pair-localized eigenstates. The excellent agreement between measured and simulated spectra, including asymmetric broadening and anti-blockade features, supports the effective model. The observation of |Δp|^-2 spectral tails at large detuning provides a practical, experimentally accessible signature of pair localization that correlates with the inferred localization probabilities from eigenstate measures. The detuning-dependent reduction in Rb+ lifetimes further corroborates the increased weight of localized, short-range pairs at high energies. The weak van der Waals term effectively acts as a random field in the weak probing regime, shifting the crossover and compressing the delocalized window, consistent with Anderson-type localization effects. Together, these results address the central question of how hopping disorder and long-range couplings govern transport and localization in 3D, demonstrating the feasibility of controlled studies of random hopping and localization phenomena in Rydberg systems.

The work provides an experimental realization of a three-dimensional random hopping (random XY) model using a dipole-dipole-coupled Rydberg gas, with quantitative spectroscopic agreement to an effective spin model. It identifies experimentally accessible signatures of an energy-resolved localization–delocalization crossover via algebraic spectral tails and detuning-dependent lifetimes. The platform enables microscopic access to transport and localization physics in disordered hopping systems. Future directions include tuning the relative strength of hopping and on-site disorder via choice of Rydberg states, employing 3D tweezer arrays to engineer controlled disorder and probe spatial structure of localized states, adding local excitation and readout to explore open-system dynamics, and moving to the strong probing regime to study many-body correlated transport with comparable populations of both spin states.

• Direct measurement of eigenstate coherence and full spatial structure is not available; localization is inferred from spectral scaling and model-based eigenstate metrics.
• The classical rate and weak-probing single-excitation approximations may miss microscopic details of the excitation process; the observed crossover appears at smaller detunings than in simulations, suggesting model limitations.
• At high pump powers, deviations in inferred seed numbers likely arise from fast redistribution (e.g., l-changing collisions) and saturation effects due to blockade, leading to seeds in low-density wings that do not contribute coherently.
• The exact number of |P> excitations during probing is difficult to determine precisely; simulations assume the single-excitation limit.
• Molecular contributions are only modeled in the n=0 reference via measured spectra; multilevel atomic structure is simplified to an effective two-level model.
• Coherent dynamics are inferred under conditions where decoherence sources are estimated to be small but not directly measured across all parameters.