Abstract
Synthetic dimensions, wherein dynamics occurs in a set of internal states, have found great success in recent years in exploring topological effects in cold atoms and photonics. However, the phenomena thus far explored have largely been restricted to the noninteracting or weakly interacting regimes. Here, we extend the synthetic dimensions playbook to strongly interacting systems of Rydberg atoms prepared in optical tweezer arrays. We use precise control over driving microwave fields to introduce a tunable U(1) flux in a foursite lattice of coupled Rydberg levels. We find highly coherent dynamics, in good agreement with theory. Single atoms show oscillatory dynamics controllable by the gauge field. Small arrays of interacting atoms exhibit behavior suggestive of the emergence of ergodic and arrested dynamics in the regimes of intermediate and strong interactions, respectively. These demonstrations pave the way for future explorations of strongly interacting dynamics and manybody phases in Rydberg synthetic lattices.
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Introduction
Analog quantum simulation in atomic, molecular, and optical systems has seen tremendous growth over the past decades. Recently, a flurry of activity has expanded analog simulations through synthetic dimensions^{1,2,3}, where dynamics occurs not in space but in alternative degrees of freedom such as spin. Since the first proposals a decade ago^{4,5}, the synthetic dimensions approach has permeated photonic and atomic physics experiment, with demonstrations in systems of atomic hyperfine states^{6,7}, metastable atomic “clock” states^{8,9,10}, atomic momentum states^{11,12,13}, trap states^{14,15}, photonic frequency modes^{16}, orbital angular momentum modes^{17}, timebin modes^{18}, and more. The realization of synthetic dimensions in these diverse platforms has led to a plethora of new simulation capabilities^{1,2}. However, studies have been almost entirely restricted to the noninteracting regime, with just a handful probing collective meanfield interactions in synthetic dimensions^{19,20,21,22,23,24} and only one recent report of strongly correlated dynamics in synthetic dimensions^{25}.
Several years ago, arrays of trapped molecules and Rydberg atoms were proposed^{26,27,28} as an alternative paradigm for exploring synthetic dimensions with strong interactions. In this approach, one starts with a dipolar spin system in which interactions naturally play a significant role^{29,30,31}. Then, by introducing tailored microwaves that drive transitions between internal states in a way that mimics the hopping structure of a lattice tightbinding model, the spin system is transformed into a playground for exploring the dynamics of strongly interacting matter in a synthetic dimension. In the past year, the team of Kanungo and coworkers have demonstrated the first Rydberg synthetic lattice^{32}, engineering and probing topological band structures formed from the Rydberg levels of individual Sr atoms. While this demonstration^{32} has laid the foundation for future developments of Rydberg and molecular synthetic lattices (see also ref. ^{33} for steps towards molecular synthetic dimensions, as well as related early work in Rydbergs and molecules^{34,35}), it lacked the key ingredient motivating the use of Rydberg atoms: strong dipole–dipole interactions.
In this paper, we extend the capabilities of Rydberg synthetic dimensions by engineering an internalstate lattice with a tunable artificial gauge field^{36,37,38,39,40,41} for small arrays of strongly interacting atoms^{42}. We show that the promising results of ref. ^{32}, wherein continuous microwave coupling is performed for single Rydberg atoms excited from a bulk sample, extend directly to the realtime dynamical control of atoms prepared in optical tweezer arrays^{42,43}. The control of the artificial gauge field in the synthetic dimension follows naturally from our phasecoherent control of the driving microwave fields. Finally, strong nearestneighbor interactions in the synthetic dimension lead to strong modifications of the population dynamics as well as the observation of atomatom correlations. This work paves the way for future explorations of strongly correlated dynamics and phases of matter in Rydberg and molecular synthetic dimensions.
Results
Implementation of U(1)flux Rydberg lattice
Our experiments begin by probabilistically loading ^{39}K atoms^{44,45} into optical tweezer arrays as depicted in Fig. 1a, b, nondestructively imaging the atoms for subsequent postselection, and cooling the atoms by gray molasses^{45,46}. We optically pump the atoms (with a quantization Bfield of ≈27 G along the zaxis) to a single ground level \(\vert 4{S}_{1/2},\, F=2,\, {m}_{F}=2\rangle\) with an efficiency of 98(1) %, and then we further cool the atoms by trap decompression to ≈4 μ K. We then suddenly turn off the confining tweezer trap.
The atoms undergo a fixed free release time of 5 μs, during which all of the dynamics in the Rydberg synthetic lattice occurs. The atoms are promoted to an initial Rydberg level, undergo microwavedriven dynamics between Rydberg levels, and are deexcited in a manner that allows for Rydberg statespecific readout. Following deexcitation, groundstate atoms are recaptured in the trap and imaged with high fidelity. Atoms remaining in the Rydberg levels are weakly antitrapped by the tweezers, and are effectively lost between the initial and final images. This bright/dark discrimination between ground and Rydberg levels, combined with stateselective deexcitation, allows us to study the stateresolved dynamics of the Rydberg level populations.
The initial excitation to the Rydberg level \(\vert 0\rangle \equiv \vert 42{S}_{1/2},{m}_{J}=1/2\rangle\) is accomplished via twophoton ("lower leg,” ~405 nm, and “upper leg,” ~975 nm) stimulated Raman adiabatic passage (STIRAP) via the \(\vert 5{P}_{1/2},F=2,{m}_{F}=1/2\rangle\) intermediate state^{47,48}. The averaged oneway STIRAP efficiency is 94(1)%^{49}. After populating this initial state, we turn on a set of microwave tones that allow atoms to “hop” between the sites of an effective lattice in the “synthetic dimension” spanned by the Rydberg levels^{4,26,32}.
As shown in Fig. 1c–e, we identify the sites of the synthetic Rydberg lattice with the atomic Rydberg levels as \(\left\vert 0\right\rangle \equiv \vert 42{S}_{1/2},{m}_{J}=1/2\rangle\), \(\left\vert 1\right\rangle \equiv \vert 42{P}_{{{{{{{{\rm{3/2}}}}}}}}},{m}_{J}=1/2\rangle\), \(\left\vert 2\right\rangle \equiv \vert 42{S}_{1/2},{m}_{J}=1/2\rangle\), and \(\left\vert 3\right\rangle \equiv \vert 42{P}_{1/2},{m}_{J}= 1/2\rangle\). A single flux plaquette is formed by adding microwave tones that resonantly drive four pairwise transitions within this set of states.
The effective singleatom Hamiltonian is given by
where the nearestneighbor tunneling terms are related to the amplitudes (A_{i}) and phases (φ_{i}, at the atoms) of the different microwave tones f_{i} as \({\Omega }_{01}\propto {A}_{1}{e}^{i{\varphi }_{1}}\), \({\Omega }_{12}={\Omega }_{21}^{*}\propto {A}_{2}{e}^{i{\varphi }_{2}}\), \({\Omega }_{23}\propto {A}_{3}{e}^{i{\varphi }_{3}}\), and \({\Omega }_{30}={\Omega }_{03}^{*}\propto {A}_{4}{e}^{i{\varphi }_{4}}\). The magnitudes of these nearestneighbor hopping terms are calibrated based on pairwise Rabi dynamics^{49} and are set to a common value Ω. The relative phase of each tone at the atoms is controllable by the source phase, and in particular we set the overall plaquette flux ϕ via the source phase of the f_{1} tone (calibrated by fitting to the dynamics of singles after an evolution time of ~h/Ω^{49}). This flux control by external fields complements recent realspace studies of interactionderived Peierls phases^{50}.
Figure 2 displays the dynamics of the state populations (starting from the \(\left\vert 0\right\rangle\) state at t = 0). The populations are corrected for state preparation and measurement (SPAM) errors related to the STIRAP infidelity and Rydbergvs.ground discrimination infidelity^{49}. The \(\left\vert 0\right\rangle\) state is measured by direct depumping by the “upper leg” (975 nm) STIRAP laser after some evolution time. To access the \(\left\vert 1\right\rangle\) state, which shares identical population dynamics in this model as the \(\left\vert 3\right\rangle\) state, we first apply a π pulse on the \(\left\vert 0\right\rangle\) to \(\left\vert 1\right\rangle\) transition prior to depumping. To access the \(\left\vert 2\right\rangle\) state, which is quite close in energy to the \(\left\vert 0\right\rangle\) state, we simply apply a strong (highbandwidth) depumping pulse to measure the combined population of \(\left\vert 0\right\rangle\) and \(\left\vert 2\right\rangle\), P_{0+2}. We then extract the \(\left\vert 2\right\rangle\) state population as P_{2} = P_{0+2} − P_{0}. For single atoms we generally find good agreement with the population dynamics for the examined flux values of Fig. 2 (a) 0, (b) π/2, and (c) π (uncertainties of 0.02π). The changing timescales for P_{0} recurrence reflects the fluxtuned spectral gaps of the plaquette energy spectrum. One stark signature seen in Fig. 2c, for π flux, is the absence of population appearing at state \(\left\vert 2\right\rangle\), which results from destructive interference of the clockwise and counterclockwise pathways. To note, this interference effect relates to the phenomena of Aharonov–Bohm caging on a single plaquette^{51,52}.
Role of the dipolar interactions
The dynamics of lone atoms in Fig. 2a–c verifies our faithful implementation of the singleparticle synthetic lattice and flux control. In Fig. 2d, e, we use isolated pairs of atoms to investigate how strong interparticle interactions enrich the dynamics. The principal interactions between Rydberg atoms in this system involve longranged (1/r^{3}, with r the interparticle spacing) dipolar exchange^{30}. In our system, having a uniform magnetic quantization field that breaks the degeneracy of different Rydberg m_{J} sublevels and is aligned at an angle θ = π/2 with respect to the displacement vectors between pairs of atoms, the primary interactions to consider are resonant dipolarexchange terms of the form \({\vert i\rangle }_{m}{\vert \,\, j\rangle }_{n}\leftrightarrow {\vert \,\, j\rangle }_{m}{\vert i\rangle }_{n}\), or “flip–flop” interactions.
This interaction can be viewed as the anticorrelated hopping (in the synthetic lattice) of excitations on neighboring atoms. These particular Δℓ = 0 dipolar terms, which conserve the net populations of the individual states, also naturally conserve the total energy in a spatially uniform system and thus result in resonant exchange dynamics^{29,53}. Based on our chosen Rydberg state assignments for the synthetic sites, these flipflop interactions are only between nearest neighbors in the internal space. However, because the interactions depend on the Rydberg state details, they are not uniform along the synthetic dimension (i.e., they do not possess discrete translation symmetry). For pairs of atoms spaced at a distance of 5 μm [Fig. 1a, b], the resonant dipolar exchange energies can be enumerated as {V_{01}, V_{12}, V_{23}, V_{30}} ≈ {2, − 0.5, 1, − 1}V, where V/h = 3.44(8) MHz^{49}. Because we operate at a modest magnetic field and with relatively strong interactions, additional offresonant statechanging dipolar interaction terms (Δℓ = ± 2, which do not conserve the net internal angular momentum nor the individual state populations), also have a small influence on the state population dynamics^{49}. The full interaction Hamiltonian is
where \({V}_{ij{i}^{{\prime} }{j}^{{\prime} }}^{mn}=\langle {j}_{m}{i}_{n}^{{\prime} } {\hat{V}}_{{{{{{{{\rm{dd}}}}}}}}} {i}_{m}{j}_{n}^{{\prime} }\rangle\) with the dipolar interaction operator \({\hat{V}}_{{{{{{{{\rm{dd}}}}}}}}}=\frac{1}{4\pi {\epsilon }_{0}{r}_{mn}^{3}}[\frac{1}{2}(2{\hat{d}}_{m}^{0}{\hat{d}}_{n}^{0}+{\hat{d}}_{m}^{+}{\hat{d}}_{n}^{}+{\hat{d}}_{m}^{}{\hat{d}}_{n}^{+})\frac{3}{2}({\hat{d}}_{m}^{}{\hat{d}}_{n}^{}+{\hat{d}}_{m}^{+}{\hat{d}}_{n}^{+})]\) between atom m and atom n (\({\hat{d}}^{0}\), \({\hat{d}}^{+}\), and \({\hat{d}}^{}\) are the respective dipole moment operators for π, σ^{+} and σ^{−} transitions), and where \({\Delta }_{ij}^{{i}^{{\prime} }{j}^{{\prime} }}\) is the energy difference between the twobody state configurations \({\vert i\rangle }_{m}{\vert \,\, {j}^{{\prime} }\rangle }_{n}\) and \({\vert \,\, j\rangle }_{m}{\vert {i}^{{\prime} }\rangle }_{n}\). Here, the state indices \(i,j\,({i}^{{\prime} },{j}^{{\prime} })\) also cover other Rydberg sublevels in both 42P manifolds to account for strong nonresonant dipolar interactions.
The dipolar interactions enrich the synthetic lattice dynamics by correlating the Rydberg electron “motion” on different atoms. For two neighboring atoms starting in the state \({\vert i\rangle }_{m}{\vert i\rangle }_{n}\), symmetric singleparticle hopping to the state \({\vert+\rangle }_{ij}=({\vert i\rangle }_{m}{\vert \,\, j\rangle }_{n}+{\vert \,\, j\rangle }_{m}{\vert i\rangle }_{n})/\sqrt{2}\) is driven offresonance by the presence of interactions, as \({\left\vert+\right\rangle }_{ij}\) is shifted in energy by V_{ij}. When the interactions are sufficiently strong (∣V_{ij} ≫ Ω/2∣), interactions will suppress uncorrelated hopping processes, somewhat analogous to how onsite interactions suppress uncorrelated atom hopping in optical lattices^{54}. For pairs, transport can still occur by pairhopping across individual lattice links at a rate \({V}_{ii\to jj} \, \approx {\Omega }^{2}{e}^{i2{\phi }_{ij}}/(2{V}_{ij})\)^{49}. Three things to note about transport in the V ≫ Ω limit: (1) interactions will considerably slow down the dynamics of pairs, (2) bound pairs should experience twice the tunneling phase (flux for closed paths) as experienced by single particles, and (3) the matrix elements for resonant threeatom (and higher) hopping processes are even further reduced.
These interactions have been predicted to induce emergent quantum strings and membranes in the ground state^{26,27,28}, relating to selftrapped^{55} multiatom bound states in 1D and 2D atom arrays. Ground state strings and membranes, selfbound by interactions, are nonetheless predicted to be delocalized in the synthetic dimension for uniform, translationally invariant interactions (V_{i,i+1} = V). In this work and for generic state arrangements, however, the V_{ij} terms have significant structure across the synthetic lattice and can be considered as a kind of interaction disorder^{56}, which may reasonably be expected to localize dipolar bound states.
For interacting pairs, we restrict ourselves to measuring the population of \(\left\vert 0\right\rangle\) for each atom, as the basis rotation pulses used for the readout of other internal states are influenced by the presence of strong interactions. Figure 2d shows the average probability for a pair of atoms to reside at the site \(\left\vert 0\right\rangle\). We compare the data to nofreeparameter simulations that incorporate the singleparticle dynamics of Eq. (1) as well as all interaction effects of Eq. (2). The theory lines and confidence intervals are based on the calibrated parameters and interaction values, as well as their uncertainties, and further account for the independently determined Rydberg state preparation infidelity, which leads to ~15% of the “pair data” cases consisting of just a single Rydbergexcited atom^{49}. For pairs in the intermediate interaction regime [V/Ω = 1.8(1)], we observe that the dipolar interactions strongly modify the dynamics, in general slowing down the dynamics and decreasing the amplitude of recurrences. As a more direct probe of interactiondriven correlations, we measure the twoatom correlator \({{{{{{{{\mathcal{C}}}}}}}}}_{00}=\langle {\hat{c}}_{0,L}^{{{{\dagger}}} }{\hat{c}}_{0,L}{\hat{c}}_{0,R}^{{{{\dagger}}} }{\hat{c}}_{0,R}\rangle \langle {\hat{c}}_{0,L}^{{{{\dagger}}} }{\hat{c}}_{0,L}\rangle \langle {\hat{c}}_{0,R}^{{{{\dagger}}} }{\hat{c}}_{0,R}\rangle\) with L and R referring to the left and right atoms of isolated pairs. This quantity vanishes in the absence of interactions, and grows as the atoms develop correlations of their positions in the synthetic lattice. Both the P_{0} and C_{00} dynamics are in fairly good agreement with our textbook theory expectations, confirming that dipolar Rydberg atom arrays are a promising platform for exploring coherent interactions in tunable synthetic lattices.
We now more thoroughly explore in Fig. 3 the fluxdependent dynamics for individual atoms and atom pairs. Figure 3a, b shows numerical simulations of the full fluxdependence of the P_{0} dynamics for singles and pairs (with the simulations incorporating all the same elements as in Fig. 2). For singles, as described before, the changing timescales for recurrences of the measure P_{0} simply reflect the fluxmodified gaps of the system’s energy spectrum. For our measurements, we probe precisely at the first expected P_{0} recurrence time for singles at flux values of ϕ = π and 0, namely at t = 0.350 μs in Fig. 3c and t = 0.525 μs in Fig. 3d, respectively. For singles, we observe good general agreement with the full fluxdependence of the expected P_{0} dynamics. For doubles, we observe both in theory and experiment that P_{0} remains relatively small and has low contrast as a function of ϕ, owing to the dynamics being slowed down relative to singles. We note that the apparent πperiodicity of the doubles response after 0.525 μs is somewhat suggestive of the expected higher flux sensitivity of pairs^{49}.
Dynamics in multiatom arrays
Finally, we explore how interactions in Rydberg synthetic dimensions can have an even richer influence on dynamics as we extend towards manyatom arrays. In Fig. 4a, b, we contrast the ϕ = 1.00(2)π dynamics of one, two, and sixatom arrays for intermediate [(a), V/Ω = 1.8(1), Ω/h = 1.92(6) MHz] and large [(b), V/Ω = 9.0(5), Ω/h = 0.38(1) MHz] interactiontotunneling ratios. For both cases, P_{0} oscillates with high coherence and a single frequency for single atoms (there is only a single energy gap value of Eq. (1) at π flux). However, interactions lead to qualitatively different dynamics in multiatom arrays^{49}. In Fig. 4a, for V/Ω ~ 1.8, the macroscopic observable P_{0} shows coherent revivals with a structured timedependence for pairs, and less oscillations but a clear decay for sixatom clusters. Specifically, numerical simulations for the sixatom array show a rapid relaxation to P_{0} ≈ 1/4, suggestive of an approach to ergodicity in this closed manybody system.
The dynamics of arrays relative to singles changes remarkably for strong interactions, V/Ω ≈ 9, as shown in Fig. 4b. For pairs, we observe only a very slow decay of P_{0} over the 3 μs measurement window, consistent with the prediction of pairhopping^{57} being slowed down (by a factor of 9) relative to singles in this large V/Ω limit. The P_{0} dynamics is still further reduced for the sixatom clusters. Indeed, interactiondriven selftrapping is expected in this strong interaction regime, and the resulting states can be considered simpler excitedstate analogs of groundstate quantum strings^{26,27,28}. To note, when V/Ω ≫ 1, one should also expect the system to be prone to Hilbert space fragmentation^{58,59} and fully arrested dynamics under added perturbations (e.g., gradients or disorder).
Discussion
These observations of interactiondriven selfimmobilization pave the way for future explorations of ground state quantum strings and membrane phases in Rydberg arrays^{26,27,28}, along with more exotic localized phases that may arise due to structured, inhomogeneous interactions^{56}. More generally, the coherent dynamics and clear interaction effects observed in this fewstate, fewatom study bodes well for extensions to complex internalstate lattices composed of dozens of Rydberg states, as well as to larger 1D and 2D realspace atom arrays. This synthetic dimensions system based on arrays of dipolar spins^{26}, while having some peculiar features (e.g., one “excitation” per realspace location, challenging multistate readout), promises to be a unique playground for exploring the influence of strong interparticle interactions on topological and localization phenomena. Moreover, this system offers a powerful new approach to the study of manybody nonequilibrium dynamics.
Methods
Preparation of the atom arrays
Our experiments are based on commonly used techniques for the trapping and lossless imaging of atoms in optical tweezers^{43}. We load ^{39}K atoms into onedimensional optical tweezer arrays generated by diffraction of 780 nm laser light from an acoustooptic deflector (AA OptoElectronic part number DTSX400780). For the measurements in this paper, we use two different tweezer array patterns: the pattern of seven twosite dimers depicted in Fig. 1a, b as well as a pattern of three sixsite clusters used for the data in Fig. 4.
After probabilistic loading of the traps, we perform nondestructive fluorescence imaging to determine the tweezer occupations for subsequent postselection of the data. Our imaging of ground state atoms is mostly lossless (>99% survival probability) and provides good discrimination (>99%) between occupied and unoccupied tweezers, as detailed in ref. ^{45}.
Prior to freespace release of the atoms and their excitation to Rydberg states, the atoms are cooled by gray molasses cooling, optically pumped to a stretched hyperfine level \(\left\vert F,{m}_{F}\right\rangle=\left\vert 2,2\right\rangle\), and further cooled to a final temperature of ~4 μK by adiabatic decompression of the optical tweezer trapping potential^{49}.
Microwave control of the synthetic Rydbergstate lattice
The optically pumped ^{39}K ground state atoms are transferred to the Rydberg level \(\vert 42{S}_{1/2},{m}_{J}=1/2\rangle\) by stimulated Raman adiabatic passage (STIRAP), as detailed in the Supplement^{49}. Following this Rydberg state initialization, we form the “synthetic Rydbergstate lattice” by applying four phasestable microwave tones that coherently couple the microwave levels \(\left\vert 0\right\rangle\), \(\left\vert 1\right\rangle\), \(\left\vert 2\right\rangle\), and \(\left\vert 3\right\rangle\). The resonance frequencies are calibrated by Rabi and Ramsey spectroscopy, the individual Rabi rates are calibrated by the measurement of statetostate Rabi oscillations, and the global flux of the fourstate diamond configuration is calibrated based on the Rydberg state population dynamics of individual atoms.
Correction of state preparation and measurement (SPAM) errors
The bare measurements of the individual state populations are slightly different from the data presented in Figs. 1–4, which are “corrected” to account for state preparation and measurement (SPAM) errors. A full discussion of the SPAM correction is presented in the Supplementary Materials^{49}.
There are several effects that limit our state preparation and measurements. For the initial and final detection of ground state atoms, the actual survival probability and occupation discrimination (between occupancy and an empty tweezer) are quite high (>99%)^{45}, and their impact is excluded from the SPAM correction. Faithful excitation of the \(\vert 0\rangle \equiv \vert 42{S}_{1/2},{m}_{J}=1/2\rangle\) Rydberg state is limited by both optical pumping inefficiency and imperfect STIRAP transfer between the pumped state (\(\left\vert F,{m}_{F}\right\rangle=\left\vert 2,2\right\rangle\)) and the \(\left\vert 0\right\rangle\) Rydberg state. Prior to Rydberg excitation, the atoms are optically pumped to the \(\left\vert F,{m}_{F}\right\rangle=\left\vert 2,2\right\rangle\) state with 98(1)% efficiency^{49}. Prior to turning on the microwaves that introduce the synthetic lattice, the atoms are excited to the Rydberg level \(\left\vert 0\right\rangle\) via stimulated Raman adiabatic passage (STIRAP) with a oneway efficiency of 94(1)%. Thus, the typical state preparation fidelity is 92%. Also accounting for loss during releaseandrecapture (because the tweezers are extinguished while the atoms are excited to Rydberg levels), we measure an overall “upper baseline” of P_{u} = 0.88(1) from the combination of state preparation errors and atom loss.
We are also limited in terms of detection discrimination between the chosen (intentionally deexcited) Rydberg level and the other Rydberg levels, which experience spontaneous emission decay that results in their recapture and spurious detection as groundstate atoms. This leads to an infidelity of discrimination between the intentionally deexcited Rydberg state and the other Rydberg levels. We experimentally measure a “lower baseline” of P_{l} = 0.21(1), which is predominantly due to the decay, recapture, and detection of the shortlived n = 42 Rydberg states.
To correct for these known errors, we renormalize the measured average state populations in terms of the directly measured (bare) average state populations, P_{i}, as \({P}_{i}=({P}_{i}^{{{{{{{{\rm{bare}}}}}}}}}{P}_{l})/({P}_{u}{P}_{l})\). To note, for the states \(\left\vert 1\right\rangle\), \(\left\vert 2\right\rangle\), and \(\left\vert 3\right\rangle\), we do not further account for any errors associated with the microwave pulses applied prior to Rydberg state deexcitation.
Data availability
All of the experimental data from this work are deposited in the Zenodo database under accession code https://doi.org/10.5281/zenodo.10797815.
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Acknowledgements
This material is based upon work supported by the National Science Foundation under grant No. 1945031 and the AFOSR MURI program under agreement number FA95502210339. K.R.A.H. acknowledges support from the Robert A. Welch Foundation (C1872), the National Science Foundation (PHY1848304), the Office of Naval Research (N000142012695), and the W. M. Keck Foundation (Grant No. 995764). K.R.A.H.’s contribution benefited from discussions at the Aspen Center for Physics, supported by the National Science Foundation grant PHY1066293, and the KITP, which was supported in part by the National Science Foundation under Grant No. NSF PHY1748958. We thank A. X. ElKhadra and P. Draper for early stimulating discussions. We also acknowledge Jackson Ang’ong’a for early work contributing to the experimental apparatus.
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T.C. and C.H. performed the experiment, data analyses, and numerical simulations, with assistance from I.V. and under the supervision of B.G., J.P.C. and K.R.A.H. All authors contributed to the planning of the project and the preparation of the manuscript.
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Chen, T., Huang, C., Velkovsky, I. et al. Strongly interacting Rydberg atoms in synthetic dimensions with a magnetic flux. Nat Commun 15, 2675 (2024). https://doi.org/10.1038/s41467024468236
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DOI: https://doi.org/10.1038/s41467024468236
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