Observation of mesoscopic crystalline structures in a two-dimensional Rydberg gas

Peter Schauß1,∗, Marc Cheneau1, Manuel Endres1, Takeshi Fukuhara1, Sebastian Hild1,Ahmed Omran1, Thomas Pohl2, Christian Gross1, Stefan Kuhr1,3, and Immanuel Bloch1,4

1Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany2Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany

3University of Strathclyde, Department of Physics, SUPA, Glasgow G4 0NG, UK and4Ludwig-Maximilians-Universität, Fakultät für Physik, 80799 München, Germany

(Dated: 29 August 2012)

The ability to control and tune interactions in ultracold atomic gases has paved the way towardsthe realization of new phases of matter. Whereas experiments have so far achieved a high degreeof control over short-ranged interactions, the realization of long-range interactions would open upa whole new realm of many-body physics and has become a central focus of research. Rydbergatoms are very well-suited to achieve this goal, as the van der Waals forces between them are manyorders of magnitude larger than for ground state atoms [1]. Consequently, the mere laser excitationof ultracold gases can cause strongly correlated many-body states to emerge directly when atomsare transferred to Rydberg states. A key example are quantum crystals, composed of coherentsuperpositions of different spatially ordered configurations of collective excitations [2–5]. Here wereport on the direct measurement of strong correlations in a laser excited two-dimensional atomicMott insulator [6] using high-resolution, in-situ Rydberg atom imaging. The observations reveal theemergence of spatially ordered excitation patterns in the high-density components of the preparedmany-body state. They have random orientation, but well defined geometry, forming mesoscopiccrystals of collective excitations delocalised throughout the gas. Our experiment demonstrates thepotential of Rydberg gases to realise exotic phases of matter, thereby laying the basis for quantumsimulations of long-range interacting quantum magnets.

The strongly enhanced interaction between Rydbergatoms makes them unique building blocks for a vari-ety of applications ranging from quantum optics andquantum information processing [1, 7, 8] to engineer-ing of exotic quantum many-body phases [9–11]. Forthe latter purpose, two main ideas have been exploredtheoretically. On the one hand, the weak admixing ofa Rydberg state to the atomic ground state using off-resonant laser coupling was suggested as a way to benefitfrom the long-range interactions without persistent pop-ulation in the Rydberg state [9, 10, 12]. On the otherhand, direct laser excitation leads to the formation ofa gas of Rydberg excitations, also called Rydberg gas.This strongly correlated system [13] can exhibit highlynon-classical states characterized by the coherent super-position of ordered structures in the spatial distributionof the Rydberg excitations [2–5, 14–16]. Here the excit-ation dynamics proceeds on a timescale of a few micro-seconds, on which the atoms can be considered frozen inspace, representing strongly interacting effective spins.At the heart of the formation of such correlated stateslies the dipole blockade effect [1, 7, 8] that prevents sim-ultaneous Rydberg excitation of two close-by atoms [17–21]. Recent experiments using two trapped atoms haveshown how this blockade effect can be used to imple-ment fast two-qubit quantum gates [22, 23]. In largerultracold atomic ensembles, the coherence of the collect-ive excitation was demonstrated [24–26] and evidence for

∗ Electronic address: peter.schauss@mpq.mpg.de

strong correlations could be found by observing univer-sal scaling laws for the number of excited Rydberg atoms[27, 28]. However, direct measurements of spatial order-ing have remained an outstanding challenge. Import-ant steps in this direction were recently explored using afield-ion-microscope [20], allowing for the measurement ofthe blockade radius in a three-dimensional Rydberg gas.Recent theoretical works, on the other hand, have pro-posed detection schemes with potential resolution belowthe blockade radius, based on conditional Raman transfer[29] or electromagnetically induced transparency [30].

Here, we demonstrate an alternative approach thatpermits direct imaging of spatial excitation patterns, andprecise measurements of correlation functions. This al-lows to probe the underlying spatially ordered constitu-ents of the excited many-body state, revealing crystal-line excitation patterns of its high-density components.Two key advancements form the basis of our observations.First, a two-dimensional atomic Mott insulator providesa dense and well-ordered initial system that maximisescoherence times during the excitation dynamics. Second,we developed an all-optical technique to image individualRydberg atoms in-situ with high spatial and temporalresolution.

The physical system considered here is a two-dimensional gas of alkali atoms trapped in a rotationallyinvariant harmonic confinement potential and pinned ina square optical lattice. The gas was prepared deep in theMott-insulating phase, ensuring uniform filling with oneatom per site within a disk of radius R '

√Nata2lat/π,

where Nat is the total number of atoms and alat the

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Figure 1. Schematics of the many-body excitation. a,Energy spectrum in the absence of optical driving. Stateswith more than one excitation form a broad energy band(gray shading) above the degenerate manifold comprising theground state and all singly excited states. For each excita-tion number Ne > 1, the states with lowest energy correspondto spatially ordered configurations, which maximise the dis-tance between the Rydberg excitations. The minimal interac-tion energy (black arrows) is determined by the finite systemsize and increases with Ne. Possible spatial configurations ofthe excitations (blue dots) in the initial Mott-insulating state(black dots) are shown schematically as circular insets nextto their respective interaction energy. The blockade radius isdepicted by the blue shaded disc around the excitation. b,Simplified level scheme of 87Rb showing the transitions usedfor the Rydberg excitation and detection.

lattice spacing. The atoms were initially in their elec-tronic ground state, |g〉, and then resonantly coupled toa Rydberg state, |e〉. In the interaction picture, the in-ternal dynamics of the atoms is governed by the many-body Hamiltonian:

H =~Ω2

∑i

(σ(i)eg + σ(i)

ge

)+∑i 6=j

Vij2σ(i)ee σ

(j)ee . (1)

Here, the vectors i = (ix, iy) label the lattice sites inthe plane. The first term in this Hamiltonian describesthe coherent coupling of the ground and excited stateswith Rabi frequency Ω, where σ(i)

ge = |ei〉〈gi| and σ(i)eg =

|gi〉〈ei| are the local transition operators. The secondterm is the van der Waals interaction potential betweentwo atoms in the Rydberg state. In our case it is repulsiveand takes the asymptotic form: Vij = −C6/r

6ij , with

the van der Waals coefficient C6 < 0 and rij = alat|i −j| the distance between the two atoms at sites i andj. The projection operator σ(i)

ee = |ei〉〈ei| measures thepopulation of the Rydberg state at site i. This model isvalid as long as the mechanical motion of the atoms andall decoherence effects can be neglected (SupplementaryInformation).

The dynamics of this strongly correlated system can

be understood intuitively from its energy spectrum inthe absence of optical driving. It is instructive to groupthe large number of many-body states, 2Nat , according tothe number of Rydberg excitations, Ne, contained in eachstate (Fig. 1a). All singly excited states (Ne = 1) withdifferent positions of the Rydberg atom have identicalenergies and form a Nat-fold degenerate manifold. Formultiply excited states (Ne > 1), this degeneracy is lif-ted by the strong van der Waals interaction, giving rise toa broad energy band (Fig. 1a). Starting from the groundstate, the creation of the first excitation is resonant, whilethe sequential coupling to many-body states with largernumber of excitations is rapidly detuned by the interac-tions. In fact, the rapid variation of the van der Waals po-tential with distance prevents the excitation of all thosestates where Rydberg atoms are separated by less thanthe blockade radius, Rb, defined by ~Ω = −C6/R

6b. The

existence of this exclusion radius is expected to have astriking consequence: while the total many-body stateexhibits finite-range correlations on a scale of Rb [13], itshigh-density components with a Rydberg density closeto 1/R2

b should display a crystalline structure, meaningthat the position of the Rydberg atoms is correlated overa distance comparable to the system size.

The excitation dynamics of all configurations shouldoccur in an entirely coherent fashion, resulting in highlynon-classical many-body states. First, the approximaterotational symmetry of our system leads to symmetricsuperpositions of all microscopic configurations with dif-ferent orientation but identical relative positions of theRydberg atoms. Second, since the coupling addressesall states within an energy range ∼ ~Ω, it produces acoherent superposition of many-body states with differ-ent number of excitations and slightly different separa-tion between the Rydberg atoms (Fig. 1a). This collect-ive nature of the excited many-body states dramaticallychanges the timescale on which their dynamics occurs.The coupling strength to the state with a single excita-tion is enhanced by a factor

√Nat 1 [8] and the coup-

ling to states with Ne > 1 is similarly enhanced, withNat replaced by the number of energetically accessibleconfigurations in each Ne-manifold [3].

Our experiments began with the preparation of a two-dimensional degenerate gas of 150 to 390 87Rb atomsconfined to a single antinode of a vertical (z-axis) opticallattice [31]. The gas was brought deep into the Mott-insulating phase by adiabatically turning on a square op-tical lattice with period alat = 532 nm in the xy-plane.Within the system radius, R = 3.5 µm to 5 µm, the prob-ability of a lattice site to be occupied by a single atom wastypically 80%. The atoms were then initialised in the hy-perfine ground state |g〉 ≡ |5S1/2, F = 2,mF = −2〉 andcoupled to the Rydberg state |e〉 ≡ |43S1/2,mJ = −1/2〉(Fig. 1b). The coupling was achieved through a two-photon process via the intermediate state |5P3/2, F =3,mF = −3〉 using lasers of wavelengths 780 nm and480 nm and σ− and σ+ polarisation, respectively (Fig. 1band Methods). The resulting two-photon Rabi frequency

3

was Ω/(2π) = 170(20) kHz, yielding a blockade radiusof Rb = 4.9(1) µm. Following the initial preparation, wesuddenly switched on the excitation lasers and let thesystem evolve for a variable duration t. After the excit-ation pulse, we detected the Rydberg excitations by firstremoving all atoms in the ground state with a resonantlaser pulse, then deexciting the Rydberg atoms to theground state via stimulated emission towards the inter-mediate state (Fig. 1b and Methods) and finally record-ing their position using high-resolution fluorescence ima-ging [31]. The accuracy of the measurement was limitedby the probability of 75(10)% to detect a Rydberg atomand by a background signal due to on average 0.2(1) non-removed ground state atoms per picture (Supplement-ary Information). The spatial resolution of our detectiontechnique is limited to about one lattice site by the resid-ual motion of the atoms in the Rydberg state before deex-citation (Supplementary Information). Repeating the ex-periment many times allowed for sampling the differentspatial configurations of Rydberg atoms constituting themany-body state and to measure their respective statist-ical weight.

In Fig. 2a we show typical images of microscopic config-urations with Ne = 2-5. In order to analyse the structureof the many-body state, we group the individual imagesaccording to their number of excitations and determinethe spatial distributions of the excitations, ρe(i) = 〈σ(i)

ee 〉,where 〈 · 〉 denotes the average from repeated measure-ments. These distributions display a typical ring-shapedprofile (Fig. 5), which results from the blockade effect andfrom the rotational symmetry of the system. Crystallinestructures become visible once each microscopic config-uration has been centred and aligned to a fixed referenceaxis (Fig. 2b and Methods).

For our smallest sample (R ≈ 3.5 µm), we observestrong correlations between Ne = 2 excitations thatare localized at a distance ∼ 6 µm, due to the inter-action blockade. In the same dataset, configurationswith Ne = 3 show an arrangement on an equilateral tri-angle, revealing both strong radial and azimuthal or-dering. These correlations persist for larger numbersof Rydberg excitations, which we can prepare in lar-ger samples (R ≈ 5 µm). They form quadratic andpentagonal configurations for Ne = 4 and Ne = 5, re-spectively. However, since their interaction energy is lar-ger, these states are populated only with low probability,leading to a reduced signal-to-noise ratio. Our experi-mental data is in good agreement with numerical sim-ulations of the many-body dynamics according to theHamiltonian of Eq. (1), for the same atom numbers,temperature and laser parameters as in the experiment(Fig. 2c and Supplementary Information). These simula-tions are based on a truncation of the underlying Hilbertspace, exploiting the dipole blockade, and neglect anydissipative effects ([3] and Supplementary Information).The spatial distributions of excitations provided by thesimulation reproduce all the features observed in the ex-periment. The only apparent discrepancy is the overall

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Figure 2. Mesoscopic crystalline components of themany-body states. Spatial distribution of excitations forthe observed microscopic configurations sorted according totheir number of excitations Ne = 2-5 (top to bottom row). a,Examples of false-colour fluorescence images in which deex-cited Rydberg atoms are directly visible as dark-blue spots. b,Histograms of the spatial distribution of Rydberg atoms ob-tained after centring and aligning the individual microscopicconfigurations to a reference axis (Methods). The initial atomdistribution had a diameter of 7.2(8) µm and 10.8(8) µm forNe = 2-3 and Ne = 4-5, respectively. c, Theoretical predic-tion from numerical simulations of the excitation dynamicsgoverned by the many-body Hamiltonian of Eq. (1) for thesame conditions as in the experiment (Supplementary Inform-ation).

slightly larger size of the measured structures, which canbe attributed to the spatial resolution of our detectionmethod, as discussed below.

For a more quantitative analysis of spatial correlations,we also measured the pair correlation function (Fig. 3a)

g(2)(r) =

∑i6=j δr,rij 〈σ

(i)ee σ

(j)ee 〉∑

i 6=j δr,rij 〈σ(i)ee 〉〈σ(j)

ee 〉, (2)

which characterizes the occurrence of two excitations be-ing separated by a distance r. Here δr,rij is the Kroneckersymbol that restricts the sum to sites (i, j) for whichrij = r. In contrast to the spatial distributions presentedabove, the average is now taken over all values of Ne. Thepair correlation function g(2)(r) shows a strong suppres-sion at distances smaller than r = 4.8(2) µm, which coin-cides with the expected blockade radius Rb = 4.9(1) µm.Moreover, we find a clear peak at r = 5.6(2) µm and evid-

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g(2) (r

)

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φ)

~g(2

) (∆φ

)~

g(2) (∆

φ)

~

ba

Figure 3. Correlation functions of Rydberg excitations. a, Pair correlation function. The blockade effect results in astrong suppression of the probability to find two excitations separated by a distance less than the blockade radiusRb = 4.9(1)µm.Moreover, we observe a peak at r ' 5.6µm and a weak oscillation at larger distances. The initial atom distribution had adiameter of 10.8(8) µm. The experimental data (blue circles) are compared to the theoretical prediction both taking into accountthe independently characterized imperfections of our detection method (green line) and disregarding these imperfections (grayline). The dashed line marks the value of g(2) in the absence of correlations. The error bars represent the standard error of themean (s.e.m.) of g(2)(r). b, Azimuthal correlation function. The crystalline structure of the high-density components is bestvisible in the angular correlations around the centre of mass of the distribution of excitations, characterized by the correlationfunction g(2)(∆φ) defined in Eq. (3). By construction, this function is symmetric around 180. Correlations are observed at theangles expected for the respective crystals shown in the insets. The peaks close to 180 are more pronounced since the centreof mass of a configuration is likely to lie close to the intersections of the diagonals, due to the blockade effect. Error bars, s.e.m.

ence for weak oscillations extending to the boundaries ofour system. This indicates that the overall many-bodystate only exhibits finite-range correlations. Our theor-etical calculation of g(2)(r) (gray line in Fig. 3a) exhibitssimilar features, but shows more pronounced oscillationsand vanishes perfectly within the blockade radius. Thesediscrepancies can be attributed to several imperfectionsof the detection technique. The sharp peak at short dis-tances r . 1 µm results from hopping of single atoms toadjacent sites during fluorescence imaging with a smallprobability of approximately 1%, which is falsely detec-ted as two neighbouring excitations. The non-zero valueof g(2)(r) for distances r . 3 µm arises from the imperfectremoval of the ground state atoms. Finally, the shift andslight broadening of the peak in the correlation functionis attributed to the residual motion of the Rydberg atomsbefore imaging (Supplementary Information). When ac-counting for these independently characterized effects inthe theoretical calculations (green line in Fig. 3a), werecover excellent agreement with the measurements.

Since our system size is comparable to the blockade ra-dius, the excitations in states with Ne > 1 are localisedalong the circumference of the system. We characterizethe resulting angular order by introducing an azimuthalcorrelation function that reflects the probability to findtwo excitations with a relative angle ∆φ measured withrespect to the centre of mass of the distribution of excit-ations:

g(2)(∆φ) =

∫dφ2π

〈n(φ)n(φ+∆φ)〉〈n(φ)〉〈n(φ+∆φ)〉

. (3)

Here n(φ) =∑

i δφ,φi σ(i)ee is the azimuthal distribution of

excitations, with (ri, φi) the polar coordinates of the sitei. As can be seen in Fig. 3b, the spatially ordered struc-ture is clearly visible as correlations at relative angles∆φ = ν × 360/Ne, with ν = 1, 2, . . . , Ne, even for thelargest excitation numbers.

We finally analyse the many-body excitation dynam-ics of the system. In Fig. 4a we show the time evol-ution of the average number of Rydberg excitations,Ne =

∑i〈σ

(i)ee 〉, which quickly saturates to a small value

Ne ' 1.5, much smaller than the total number of atomsin the system, Nat = 150(30). The saturation is reachedin ∼ 500 ns, a factor of ten faster than the Rabi period2π/Ω, due to the collective enhancement of the op-tical coupling strength. The probability to observe NeRydberg excitations shows a similar saturation profile foreach excitation number Ne (Fig. 4b-d), but on a times-cale that increases with Ne, from about 200 ns for Ne = 1to about 600 ns for Ne = 3. This can be attributed to thevariation of the collective enhancement factor associatedwith the number of energetically accessible microscopicconfigurations for a given Ne. The theoretical excitationdynamics corresponding to the Hamiltonian (1) showsremarkable agreement with the experimental data whenincluding the finite detection efficiency. This providesevidence that the dynamics observed in the experimentis coherent, as expected on these timescales, which aremuch shorter than the lifetime of the Rydberg state of25(5) µs in the lattice and the timescale of other decoher-ence effects (Supplementary Information). The absenceof high-contrast Rabi oscillations in the time evolution of

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Pulse duration, t (µs)

0

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tions

Pulse duration, t (µs)0 0.5 1 1.5

Pulse duration, t (µs)0 0.5 1 1.5

Pulse duration, t (µs)

a b c dNe = 1 Ne = 2 Ne = 3

Figure 4. Time evolution of the number of Rydberg excitations. a, Average number of detected Rydberg atoms as afunction of the excitation pulse duration. Error bars, s.e.m. b-d, Time evolution of the probability to observe Ne = 1 (b),Ne = 2 (c) and Ne = 3 (d) Rydberg excitations. The experimental data (blue circles) are compared to the theoretical prediction(green line), which is based on initial ground state atom distributions observed in the experiment and neglects all decoherenceeffects. It takes into account the finite detection efficiency as a free parameter (75%). Error bars, s.e.m.

the average number of Rydberg excitations is caused bythe strong dephasing between many-body states with dif-ferent interaction energies arising from the different spa-tial distribution of excitations. However, remnant sig-natures of Rabi oscillations can still be observed. Inparticular, the population of the singly excited statesshows a peak around t = 200(50)ns (Fig. 4b), whichmatches the π-pulse time of the enhanced Rabi frequencyπ/(√NatΩ

)= 240(40) ns. Further evidence for the co-

herence of the dynamics can be found in the spatially re-solved analysis of the excitation dynamics (Supplement-ary Information).

In conclusion, we have characterised the stronglycorrelated excitation dynamics of a resonantly drivenRydberg gas using optical detection with unpreceden-ted spatial resolution, and observed mesoscopic Rydbergcrystals in the high-density components of the producedmany-body states. One future challenge lies in the de-terministic preparation of ground-state Rydberg crystalswith a well-defined number of excitations via adiabatic

sweeps of the laser parameters [3, 4, 14]. Together withthe demonstrated imaging technique, this would enableprecise studies of quantum phase transitions in long-range interacting quantum systems on the microscopiclevel [2–4, 15]. Combining the dipole blockade effect withthe single-atom addressing demonstrated already in ourexperimental setup, one could also engineer mesoscopicquantum gates [32], which can serve as an experimental“toolbox” for digital quantum simulations of a broad classof spin models, including such fundamental systems asKitaev’s toric code [33].

ACKNOWLEDGEMENTS

We thank R. Löw for discussions. We acknowledgefunding by MPG, DFG, EU (NAMEQUAM, AQUTE,Marie Curie Fellowship to M.C.) and JSPS (PostdoctoralFellowship for Research Abroad to T.F.).

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METHODS

Rydberg excitation and detection scheme Thetwo excitation laser beams were counterpropagating

along the z-axis, with an intermediate-state detuningδ/(2π) = 742(2)MHz (Fig. 1b). During the sequence, amagnetic offset field of B ' 30G along the z-axis definedthe quantisation axis. The excitation pulse was per-formed by switching the laser at 780 nm while the laserat 480 nm was on. The temporal resolution of our meas-urement was thus set by the rise time of the 780 nmlight, which was ' 40 ns. Immediately after the excit-ation pulse, we used near-resonant circularly-polarisedlaser beams to drive the transitions |5S1/2, F = 1〉 →|5P3/2, F = 2〉 and |5S1/2, F = 2〉 → |5P3/2, F = 3〉 andremove all ground state atoms, with a fidelity of 99.9%in 10 µs. Subsequently, the Rydberg atoms were stimu-lated down to the ground state by resonantly driving the|43S1/2,mJ = −1/2〉 → |5P3/2, F = 3,mF = −3〉 trans-ition for 2 µs.

Computation of the histograms The histogramsshown in Fig. 2b are based on the digitised atom dis-tribution reconstructed from the raw images [31]. Theyreflect the Rydberg atom distribution in a region of in-terest covering a disc of radius Rmax = 1.5 × R. Eachindividual image was aligned in the following way. First,we set the origin of the coordinate system to the centreof mass of the atom distribution. Then, for each atom wedetermined the angle between its position vector and areference axis, and rotated the images about the origin bythe mean value of these angles (repeating this operationwould leave the configuration unchanged). The histo-grams contain data taken at different evolution times upto 4 µs, as we found no significant temporal dependenceof the excitation patterns. The theoretical calculationsused the same parameters as in the experiment (includingtemperature and atom number distribution of the initialstate) and followed the same procedure to determine theRydberg atom densities. Both the experimental and the-oretical histograms were normalised such that the valueat each bin represents the probability to observe a mi-croscopic configuration with a Rydberg atom located atthis position.

SUPPLEMENTARY INFORMATION

I. SPATIAL DISTRIBUTION OF THEEXCITATIONS WITHOUT ROTATIONAL

ALIGNMENT

Here we show the spatial distribution of excitationsbased on the same data as in Fig. 2b and 2c of the maintext but without the rotational alignment procedure de-scribed in the Methods section.

7

II. SPATIALLY RESOLVED ANALYSIS OF THEEXCITATION DYNAMICS

The coherence of the dynamics can be revealed in amore obvious way by studying the time evolution of thespatial distribution of the Rydberg excitations. For thispurpose, we considered the subset of microscopic config-urations with only one excitation. Because the block-ade radius is only slightly smaller than the system dia-meter, only those configurations in which the excitationis located close to the edge of the system are significantlycoupled to configurations with two excitations. This res-ults in different time constants for the dynamics at dif-ferent distance r from the centre. We have investigatedthis effect theoretically by calculating the time evolutionof the relative probability for the excitation to be locatedclose to the centre of the system (green line in Fig. 6). Incontrast to Fig. 4 of the main text, we now observe Rabi-like oscillations with notable amplitude over long times-cales. We performed the corresponding measurement inthe experiment for two pulse durations (blue circles) andfind reasonable agreement. This provides further evid-

17.6

µm

0

2

4

6

Even

t pro

babi

lity

(10-3

)

0

1

2

Even

t pro

babi

lity

(10-3

)

0

1

Even

t pro

babi

lity

(10-3

)

0

0.2

0.4

0.6

0.8

Even

t pro

babi

lity

(10-3

)

a b

Ne=2

Ne=3

Ne=4

Ne=5

Figure 5. Spatial distribution of the distribution ofexcitations before rotation. a, Histograms constructedfrom the experimental data. The Rydberg atoms are excitedwith higher probability close to the edge than in the centreof the cloud. The resulting ring-shaped excitation region isclearly visible for Ne = 2 and 3. The contrast decreases forNe = 4 and 5 due to the lower number of occurences in theexperiment. b, Theoretical predictions for the excitation frominitial clouds of same temperature and atom number as in theexperiment (see Fig. 2 for details).

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

Cen

tral f

ract

ion

of R

ydbe

rg e

xcita

tions

Pulse duration, t (µs)

0

0.02

0.04

0.06

0 1 2 3Pulse duration, t (µs)

Cen

tral f

ract

ion

Figure 6. Excitation dynamics at the centre of thesystem. Relative number of excitations in the central ninesites as a function of the excitation pulse duration for micro-scopic configurations with a single excitation Ne = 1. Thetheoretical calculation (green line, inset) reveals the coher-ent evolution, which is hardly visible in the time evolution ofthe total excitation number. Two experimental points (bluecircles) were obtained from an additional dataset containingabout 800 images per pulse duration. It was characterized bythe temperature of the initial state T = 9(2) nK, the atomnumber Nat = 210(30) and the radius R = 4.2(5) µm. Theseexperimental parameters were included in the numerical sim-ulation. The error bars denote one standard deviation of themean (s.e.m.).

ence for the coherence of the many-body dynamics inthe experiment.

III. ADDITIONAL INFORMATION ON THEDATASETS

The experimental data results from three differentdatasets (A, B and C). Each dataset was characterizedby a temperature, T , atom number, Nat, and diameter,2R, which we extracted from a fit to the ground stateatom distribution in the initial state [34] (Table I). Thedatasets A and B were used for Fig. 2 and 3, while thedataset C was used for Fig. 4. The distribution of thenumber of excitations in the datasets A and B is detailedin Table II, where we also indicated which subset of im-ages was used for which figures. The dataset C consistedof 54 images per pulse duration and the relative distri-bution of excitations is directly visible in Fig. 4.

8

dataset A dataset B dataset C

T (nK) 8(4) 13(2) 9(4)Nat 150(30) 390(30) 150(30)R (µm) 3.6(4) 5.4(4) 3.6(4)

Table I. Temperature T , atom number Nat and radius R forthe datasets A, B and C. Errors, s.d.

dataset A dataset B

number ofexcitations

number ofimages figures number of

images figures

0 177 – 321 –1 235 – 375 3a2 191 2a, 3b 390 3a3 65 2a, 3b 308 3a4 7 – 177 2a, 3a, 3b5 1 – 64 2a, 3a, 3b6 0 – 14 –7 0 – 5 –

Table II. Distribution of the number of excitations in the data-sets A and B. For each subset of images we have also indicatedin which figure of the main text it has been used.

IV. NUMERICAL CALCULATIONS

In order to determine the dynamics governed by theHamiltonian in Eq. (1), we expand the many-body wavefunction, |ψ〉, of the Nat-atom system in terms of Fock-states

|ψ〉 = c(0)|0〉+∑i1

c(1)i1|i1〉+

∑i1,i2

c(2)i1,i2|i1, i2〉+ . . .

+∑

i1,...,iNat

c(Nat)i1,...,iNat

|i1, . . . , iNat〉 ,

where |i1, . . . , iNe〉 corresponds to a state with Ne

Rydberg excitations located at lattice sites i1 to iNe ,and c

(Ne)i1,...,iNe

denotes the respective time dependentamplitude. The basis states are eigenfunctions of theHamiltonian (1) in the absence of laser driving, with en-ergy eigenvalues E(Ne)

i1,...,iNe=∑Neα<β Viαiβ . For a system

of Nat atoms, this basis set expansion yields a set of2Nat coupled differential equations (Fig. 7). Due to theexponential growth of the number of many-body stateswith Nat, a direct numerical propagation is practicallyimpossible for the large number of atoms in our experi-ments, Nat ∼ 100. In order to make the calculations feas-ible, we exploit the blockade effect and discard all many-body states containing Rydberg atom pairs separated byless than a critical distance Rc. For the present simu-lations, we obtain well-converged results for Rc ' Rb/2,where Rb is the blockade radius. The resulting geometricconstraint not only reduces the number of relevant many-body states within a given Ne-manifold, but, due to thefinite system size, also restricts the total number of excit-

ations Ne necessary to obtain converged results. For theparameters considered in this work, a maximum numberof Rydberg excitations of Ne

(max) = 6 was found suffi-cient. This procedure allows to significantly mitigate theotherwise strong exponential scaling of the underlyingHilbert space dimension, and yields a power-law depend-ence ∼ NN(max)

eat of the number of relevant basis states on

the total number of atoms. This makes the computationsfeasible, albeit still demanding, for such large systems asin our experiment.

V. OPTICAL DETECTION SCHEME

We developed a fully-optical detection technique forRydberg atoms, which represents an alternative to theusual detection schemes based on the ionisation ofRydberg atoms [18, 20, 35, 36]. Here, we provide addi-tional details to those given in the main text, especiallyregarding the detection efficiency and spatial resolution.

A. Stimulated deexcitation of the Rydberg atoms

We stimulated the Rydberg atoms down to the groundstate by resonantly driving the |43S1/2,mJ = −1/2〉 →|5P3/2, F = 3,mF = −3〉 transition. The Rabi fre-quency associated with this resonant single-photon trans-ition was typically several MHz. In combination withthe short lifetime of the 5P3/2 state (27 ns), this allowsfor a very efficient and fast (< 2 µs) pumping to theground state. The laser light resonant with the trans-ition between the Rydberg and the intermediate stateswas produced by a resonant electro-optical modulator(EOM) in the path of the excitation laser at 480 nm,which created a sideband at the desired intermediate-state detuning δ/(2π) = 742MHz. The other sidebandand the carrier have negligible influence on the atoms inthe deexcitation phase since they are off-resonant. TheEOM also allows for the required fast switching of thedeexcitation light within < 1 µs.

B. Spatial resolution

Two effects can in principle limit the spatial resolutionof our detection technique. The first one is the residualhopping of the atoms during the fluorescence imagingphase. We found that such an event can occur in ourexperiment with a probability of approximately 1% perparticle. When this happens, the moving atom will yielda fluorescence signal on two adjacent sites, which canbe falsely attributed to two distinct atoms by the recon-struction algorithm. This detection artifact results in acorrelation signal at short distances r < 1 µm (Fig. 3a).However, due to their rarity these events have negligibleinfluence on the spatial resolution. The second effect is

9

V(a)

V(a)

2V(a)

Ne = 0

Ne = 1

Ne = 2

Ne = 3

Figure 7. Schematics of the numerical calculations. The underlying many-body level structure is shown for the exampleof a one-dimensional chain of five atoms. The atomic states are symbolised by effective spins, with spin-down (blue arrows)and spin-up (red arrows) corresponding to the atomic ground and Rydberg state, respectively. In the displayed example,we consider strong interactions V (a) ~Ω between adjacent Rydberg excitations, while next-nearest neighbour interactionsV (2a) = V (a)/64 < ~Ω are assumed to be smaller than the laser coupling strength. Consequently, the dynamics of nearresonant basis states (orange boxes) is explicitly calculated in the simulations, while strongly shifted states (grey boxes) donot participate in the excitation dynamics and are discarded (see text for further details). The near-resonant laser couplingbetween relevant many-body states is indicated by the green arrows. Due to the strong geometrical constraint imposed by theinteraction blockade combined with the finite system size, many-body states containing more than Ne = 3 Rydberg excitationsdo not need to be considered.

the possible motion of the Rydberg atoms in the opticallattice potential before the imaging phase. For Rydbergatoms, the lattice potential has similar amplitude butopposite sign compared to ground state atoms [37]. Anexcited atom therefore finds itself at a maximum of theperiodic potential and can move in the xy-plane with anaverage velocity of ∼ 30 nm/µs, for a typical depth of theoptical lattice potential of Vlat = 40Er. Both effects leadto a possible motion of the Rydberg atoms by about onelattice site during the 10 µs of the removal pulse. Therecoil velocity acquired in the two-photon excitation pro-cess is insignificant in comparison as it is oriented alongz-direction and only of smaller magnitude ' 4 nm/µs.

C. Detection efficiency

The detection efficiency for Rydberg atoms in oursetup is limited by the lifetime of the Rydberg state andby the anti-confining character of the optical lattice po-tential for the Rydberg atoms. If a Rydberg atom decaysto the ground state during the removal pulse, it will beremoved as well and not detected. The residual motionof the Rydberg atoms in the lattice potential also leadsto a reduction of the detection efficiency when the atomsmove away from the focal plane of the imaging system,which has a depth of focus of order 1 µm. Both effects canbe reduced by minimising the time the atoms spend inthe Rydberg state after the excitation pulse, by reducing

the duration of the removal pulse for the ground stateatoms. A removal pulse duration of 10 µs turned out tooffer the best compromise between a good detection effi-ciency of Rydberg atoms and a low survival probabilityof ground state atoms.

We estimated the detection efficiency in three differ-ent ways. First, we measured the lifetime of the atomsin the Rydberg state in the lattice by varying the dur-ation of the removal pulse. We fitted a 1/e-decay timeτ = 25(5) µs, which corresponds to a detection efficiencyof 65(5)%. A second estimation is provided by the timeevolution of the number of Rydberg excitation displayedin Fig. 4. Here the theoretical prediction matches thedata best when assuming a detection efficiency of ∼ 75%,which is compatible with the previous estimate. Finally,the statistical weight of the different number of excita-tions can also be related to the detection efficiency. Thislast estimation points to a higher detection efficiency of∼ 80%. Combining all these values with equal weight,we finally obtain a detection efficiency of 75% with anuncertainty of about 10%.

VI. VALIDITY OF THE THE MODEL

The validity of the Hamiltonian in Eq. (1) for our ex-perimental system relies on two main assumptions, whichare discussed in this section: the positions of the atoms isfrozen during the dynamics and all decoherence sources

10

can be neglected.

A. Movement of the atoms during the dynamics

The ground state atoms were confined in a three-dimensional optical lattice of depth Vxy = 40(3)Er inthe xy-plane and Vz = 75(5)Er along the z-axis, whereEr = (2π~)2/(8ma2lat) denotes the recoil energy of the lat-tice, and m the atomic mass of 87Rb. For the minimumlattice depth used in the experiment of Vlat = 40Er, thetime associated to the inverse of the tunnelling matrixelement was ~/J ' 700ms, and therefore negligible com-pared to the timescale of the internal dynamics. TheRydberg atoms move in the lattice potential with a typ-ical velocity of 30 nm/µs (see discussion in Section VB),which can also be neglected.

B. Light scattering from the intermediate state

One source of decoherence in our experimental sys-tem is light scattering from the intermediate state usedin the two-photon excitation process. The laser beamoff-resonantly driving the ground-to-intermediate-statetransition had a detuning of 742MHz and an intensity of∼ 450mW/cm2, yielding a scattering rate of 9× 104 s−1.This corresponds to a coherence time of 11 µs, which isa factor of ten longer than the typical timescale of themany-body dynamics.

C. Laser linewidth

The finite spectral width of the optical radiation driv-ing the transition to the Rydberg state acts as a deco-herence source and was reduced by carefully stabilisingthe frequency of the excitation lasers. We could achievea two-photon linewidth of ≈ 70 kHz, leading to a coher-ence time of 14 µs. Technical details on the laser setupare provided in Section VII.

D. Two-photon Rabi frequency

We determined the two-photon Rabi frequency by driv-ing Rabi oscillation in a very dilute system, where theaverage distance between two atoms was larger than theblockade radius of the 43S state, for which the van derWaals coefficient is C6 = −1.6× 10−60 Jm6 [38]. Themeasurement was very time-consuming in our experi-mental setup since at such densities only a few atoms arelocated within the waist of the excitation lasers, resultingin a very low signal (about one Rydberg excitation perimage) with relatively large fluctuations. We could ob-serve one period of the Rabi oscillation, as expected fromthe combined coherence time of light scattering and laserlinewidth of 6 µs, and extracted from the data a Rabifrequency of Ω/(2π) = 170(20) kHz.

The waists of the two laser beams of wavelength 780 nmand 480 nm [39] driving the two-photon transition were57(2) µm and 17(5) µm, respectively. The largest systemswe studied had a radius of 5.4 µm, leading to a variationin the coupling strength to the Rydberg state by < 20%over the whole system.

VII. LASER SETUP

The light at a wavelength of 780 nm was producedby a diode laser whose frequency was stabilised using amodulation transfer spectroscopy in a rubidium vapourcell. The light at 480 nm was generated by frequency-doubling light at 960 nm, which was emitted from a di-ode laser and amplified by a tampered amplifier. Thissecond laser was stabilised by a phase-lock to a masterlaser, which allows for tuning its frequency while main-taining the narrow laser linewidth. The master laser at960 nm was locked to a temperature stabilised ULE cav-ity in a vacuum chamber. The short-term linewidth ofboth excitation lasers was measured using an independ-ent resonator (EagleEye, Sirah Laser- und PlasmatechnikGmbH, Germany), which can resolve linewidths down to∼ 20 kHz. We obtained a linewidth of 20 kHz for the laserat 480 nm and 50 kHz for the laser at 780 nm. We meas-ured the long-term stability of the two-photon excitationto be 50 kHz over several hours (FWHM of the centre ofthe line) using EIT-spectroscopy in a rubidium vapourcell [40].