High-energy Bragg scattering measurements of a dipolar supersolid

D. Petter,1 A. Patscheider,1 G. Natale,1 M. J. Mark,1, 2 M. A. Baranov,2 R. v. Bijnen,2, 3

S. M. Roccuzzo,4, 5 A. Recati,4, 5 B. Blakie,6, 7 D. Baillie,6, 7 L. Chomaz,1 and F. Ferlaino1, 2

1Institut fur Experimentalphysik, Universitat Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria2Institut fur Quantenoptik und Quanteninformation,

Osterreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria3Center for Quantum Physics, University of Innsbruck, Austria

4INO-CNR BEC Center and Dipartimento di Fisica,Universita degli Studi di Trento, 38123 Povo, Italy

5Trento Institute for Fundamental Physics and Applications, INFN, 38123, Trento, Italy6The Dodd-Walls Centre for Photonic and Quantum Technologies,

University of Otago, Dunedin 9054, New Zealand7Department of Physics, University of Otago, Dunedin 9016, New Zealand

(Dated: August 16, 2021)

We present an experimental and theoretical study of the high-energy excitation spectra of adipolar supersolid. Using Bragg spectroscopy, we study the scattering response of the system toa high-energy probe, enabling measurements of the dynamic structure factor. We experimentallyobserve a continuous reduction of the response when tuning the contact interaction from an ordinaryBose-Einstein condensate to a supersolid state. Yet the observed reduction is faster than the onetheoretically predicted by the Bogoliubov-de-Gennes theory. Based on an intuitive semi-analyticmodel and real-time simulations, we primarily attribute such a discrepancy to the out-of-equilibriumphase dynamics, which although not affecting the system global coherence, reduces its response.

The field of quantum gases has moved towards thestudy and realization of novel quantum states of matter,often showing exotic properties [1]. A recent example,still challenging scientist’s intuition, is the long-soughtsupersolid phase, recently observed in atom-light-coupledsystems [2, 3] and dipolar quantum gases [4–6]. A super-solid state spontaneously develops a density modulationin space, breaking the translation symmetry, and a globalphase coherence, breaking the gauge symmetry [7–10]. Ina dipolar gas, the supersolid phase (SSP) lives in a nar-row interaction-parameter range, sandwiched between anordinary Bose-Einstein condensate (BEC) and an inco-herent array of droplets (ID), showing extreme density-modulation [5, 6, 11–16].

In a dipolar supersolid, fundamental properties relatedto its quantum-fluid nature remain to be understood.This includes the relation between condensation and su-perfluidity, as well as their connection to density modu-lation and phase fluctuations across the phase diagramof a dipolar gas. In a seminal work [17], Leggett askshow solid is a supersolid, deriving an upper bound forthe superfluid fraction of a stationary supersolid state,which connects to the degree of localization, i. e. the den-sity modulation. In the recently observed dipolar super-solid [4–6], the situation might be more complex becauseof many-body out-of equilibrium phenomena. Indeed,the macroscopic phase of a supersolid might dynamicallydevelop variations in space, caused e. g. by the crossingof the BEC-SSP phase transition, or by thermal andquantum fluctuations [4, 5, 12, 18]. Such effects couldimpact the superfluid properties of the system, going be-yond Leggett’s original prediction.

Local phase variations are typically not readily acces-

sible in experiments. However, the study of the dynam-ical response of a physical system to a high-energy scat-tering probe has proven to contain key information onthe state properties. Such powerful scattering proto-cols have been widely used across different physical dis-ciplines, ranging from high-energy [19–22] to condensed-matter physics [23, 24]. For instance, scattering of fastneutrons from superfluid liquid helium has enabled thefirst measurement of the condensate fraction in a stronglyinteracting many-body system [25]. In the realm of ultra-cold quantum gases, a similar concept has been employedto reveal e. g. beyond-mean-field effects, to measure quan-tum depletion and coherent fractions, or Tan’s universalcontact parameter [26–32].

In this Letter, we experimentally study the dynamicalresponse of a dipolar supersolid to a high-energy scat-tering probe by performing two-photon Bragg excitationin the free-particle-excitation regime (high energy andhigh momentum). We observe that the system responsestrongly reduces in the supersolid regime before vanish-ing in the ID phase. By benchmarking our data withtheoretical models, we identify the role of the density-modulation contrast and the phase variations in the ob-served response. Our study reveals the importance of thecoherent phase dynamics induced by the crossing of theBEC-to-supersolid phase transition.

The dynamical response of an interacting many-bodysystem to a weak scattering probe can be describedwithin the linear-response theory. An essential quantityis the dynamic structure factor (DSF), S(k, ω), whichcharacterizes the density response of a system to a scat-tering probe of momentum, hk, and energy, hω [33]. Forweak interatomic interactions, the DSF can be directly

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related to the excitation spectrum via the Bogoliubovamplitudes, uj and vj , describing the excitation mode jof energy hωj . It reads

S(k, ω) =∑j

∣∣∣∣ ∫ dr(u∗j(r) + v∗j (r)

)eikrψ0(r)

∣∣∣∣2×× δ(hω − hωj), (1)

where we neglect the creation of multiple excitations.Here, ψ0 is the system’s macroscopic ground-state wavefunction and δ(·) is the Dirac delta function.

Equation (1) gives different information depending onthe momentum and energy ranges [33]: For low-k trans-fer, S(k, ω) is sensitive to the system’s collective re-sponse, whereas, in the high-k and high-energy regime,the DSF informs about the momentum distribution of thesystem, n(k). We explore the latter regime for our super-solid state, focusing on the response along the density-modulated direction, y, with k = (0, ky, 0). In theregime of free-particle excitations (uj → eikjy, vj → 0,ωj → hk2j/2m with m the atomic mass), the impulseapproximation becomes valid and we find [33–37]

S(ky, ω) =∑j

n(0, ky − kj , 0) δ(hω − hωj). (2)

On resonance, ω = ωj and ky = kj , the DSF be-comes S(kj) ≡ S(kj , ωj) ∝ n(k = 0) and is uniquely de-termined by the system’s momentum distribution atk = 0.

To identify the free-particle regime in our system, webriefly review the basic properties of the excitation spec-trum of a supersolid, considering the simpler case ofan infinitely elongated cigar-shaped trap. A more ex-tensive description, including calculations for the fullthree-dimensional (3D) confined case, which we use here-after for comparison with the experiments, can be foundin Refs. [18, 38–42]. As shown in Fig. 1 (a), the super-solid spectrum exhibits a periodic structure in momen-tum space with a period given by the reciprocal lat-tice vector kc. The state develops a density modula-tion along the axial direction with wavelength 2π/kc(see inset in Fig. 1 (b)). The two lowest branches cor-respond to the superfluid and crystal branches, respec-tively [43]. They have been recently investigated in ex-periments [18, 38, 39]. The upper branch, appearing athigh energy and showing a gapped parabolic dispersion,is the one of interest here for its free-particle character.In addition, the flat band at ω ≈ 1.25ωz corresponds toa single-droplet excitation (mainly transverse breathingmodes), which couples to the parabolic branch, openingsmall energy gaps. Figure 1 (b) shows the correspondingDSF values. Interestingly, we observe that the DSF doesnot reflect the periodicity of the energy spectrum, ande. g. for the upper branch it shows large values at the

0

0.5ω (ω

z)

0 1 2ky (kc)

0 1 2ky (kc)

1.0

1.5

53.2 53.6 54 54.4 54.80.6

0.8

1BECSSP

(b)(a)

2 3 4 >5||u||

1 1 2 >2.6S (k, ω) (S*)

0

104

2-2 00n

(μm

-1)

y (μm)

B || zn2D

xy

as (a0)

(c)

S(k)

(S * )

0

1

C

53.2 54 54.8as (a0)

FIG. 1. (a) Axial excitation spectrum of the transverselysymmetric modes and (b) corresponding DSF of an infinitelyelongated dipolar supersolid at as = 51 a0 in a harmonic trapwith ωx,y,z = 2π × (250, 0, 160) Hz. The color maps corre-spond to ‖u‖ and S(k, ω), respectively. The inset shows theintegrated axial density profile n(y) of the ground state withmean density 4.7 × 103µm−1. (c) S(k) for the 3D-trappedsystem with ωx,y,z = 2π × (250, 31, 160) Hz. S(k) is calcu-lated at k = 4.2µm−1 ≈ 1.8 kc (grey line) and normalizedby its value at the BEC-SSP phase transition, S∗. The atomnumber is varied with as to match the experimental condi-tions [42]. The red (blue) line shows the result from the SIA(DAA). (upper inset) Integrated density profile of the groundstate at as = 54.49 a0 and N = 5 × 104 atoms. (lower inset)Evolution of the ground state’s central contrast C. For theinfinite (3D-trapped) case, kc = 2.3(2.4)µm−1.

momenta that continuously connect to the free-particleexcitations in the ordinary BEC [42].

In a contact-interacting BEC, the inverse healinglength provides the scale of the crossover from the collec-tive to the free-particle character of the excitations [33].This notion can not be simply exported to the case ofdipolar gases because of the momentum dependence ofthe dipole-dipole interaction. We thus follow the def-inition based on the Bogoliubov-de-Gennes (BdG) the-ory. A free-particle excitation is an elementary ex-citation whose wave function is well approximated bya plane wave. This is typically justified for excita-tions of high enough energy and single-particle character(‖uj‖ =

∫|uj(r)|2dr = 1 and ‖vj‖ = 0, see color map

Fig. 1 (a)) [33, 44]. For our parameters, we find that the

3

density profile of the mode, |uj(r)|2, shows a plane-wavecharacter for hω >∼ 0.6 hωz.

To quantitatively compare theory with experiments,we perform similar ground-state and BdG-spectrum cal-culations for the 3D-trapped case [38], and extractS(k) [42]. As shown in Fig. 1 (c), in the free particleregime (k ≈ 1.8kc), S(k) starts to decrease when en-tering the SSP and further reduces by lowering as. Si-multaneously, the ground-state density develops a spatialmodulation (upper inset), whose contrast C rapidly in-creases (lower inset). Note that C evolves faster withas than S(k). For instance, at as = 53 a0, C ≈1, whereas S(k) reduces only by about 35 %. Here,C = (nmax − nmin)/(nmax + nmin) with nmax (nmin) be-ing the central maximum (minimum) of the integrateddensity [42].

To gain an intuitive understanding of the density-response reduction, we develop a 1D model [37]. Us-ing two different wavefunction ansatzes, we evaluate S(k)in the weak and strong density-modulation regimes. Asdiscussed in Refs. [11, 16, 45], for weakly modulated su-persolids, with C � 1, the ground-state wave functioncan be approximated by a fully coherent sine-modulatedfunction on top of a uniform background. At leading or-der in C, it reads ψ(y) =

√n (1 + C sin(kcy)/2), with

n the mean density. Applying this sine ansatz (SIA)in Eq. (2), we find S(k) ∝ n(1 − C2/8). This resultshows that an increasing contrast directly causes a sup-pression of the DSF. We find a similar C-dependencefor the superfluid fraction derived from Leggett’s for-mula [17], fSF = 1 − C2/2. Therefore, in the weaklymodulated regime, the reduction of the high-energy scat-tering response connects to the reduction of the super-fluid fraction. We benchmark our SIA results with theBdG calculations by evaluating C from the full GPE so-lution [42]. As shown in Fig. 1 (c), despite its simplicity,the SIA scaling reproduces very well the full numericsup to C <∼ 40 %. For larger C, as expected, the modelbreaks down.

For large C, we employ a droplet-array ansatz (DAA),describing the system as an array of ND droplets, ψ(y) =∑ND

j=1 χ(y − jd)eiθj [6, 12]. Each droplet is describedby a Gaussian function, χ(y), of size σ, separated bya distance d > σ from its neighbours. Each dropletis allowed to have an independent, yet uniform, phaseθj . Within the DAA, the DSF shows the proportional-

ity S(k) ∝ n∣∣∣ 1ND

∑ND

j=1 eiθj

∣∣∣2 σ/d. It decreases with both

the density overlap between droplets, set by σ/d, and thephase variance along the array. The latter effect is notincluded in the BdG theory, which describes a state pos-sessing a uniform phase. To benchmark the DAA resultswith the BdG calculations, we thus set θj = 0 for allj [42]. We find a very good agreement for C > 80 %.

In the experiments, we access the density responseof a supersolid by performing high-energy Bragg scat-

0 1 2 3ħω (ħωz)

0

5

Nex

/N (%

)

BEC

SSP

ID

4

58.19(8) a0

56.0(1) a0

55.3(1) a0

54.7(1) a0

54.4(1) a054.0(1) a053.8(1) a0

FIG. 2. Fraction of Bragg-excited atoms as a function of ω forvarious as across the BEC-SSP-ID regimes (see labels). Thespectra are vertically offset for visibility. Here and throughoutthe Letter, the error bars correspond to one standard error.Solid lines show the Gaussian fits to the data.

tering on a 166Er dipolar quantum gas, confined in anaxially elongated harmonic trap. A transverse homoge-neous magnetic field orientates the atomic dipoles andsets as [6]. We initially prepare the system in the or-dinary BEC phase, and enter the SSP via interactiontuning by linearly lowering as below a critical value, a∗s ,for which the BEC-SSP phase transition occurs. Similarto previous experiments [6, 38], a∗s is extracted with aninterferometric technique. For the present trap and atomnumbers, N , we measure a∗s = 54.94+28

−13 a0; see [42].

For the Bragg excitation, we project on the atoms anoptical lattice potential of constant depth V for a du-ration τ = 7 ms. The lattice has a constant wave vec-tor k = 4.2(3)µm−1 along y and moves with a variablefrequency ω. After the Bragg excitation, we measurethe integrated momentum distribution, n(kx, ky), using atime-of-flight expansion of 30 ms. The number of excitedatoms Nexc is extracted in a narrow region of interestaround k [42]. For a fixed as, we find a clear resonancein Nexc/N as we vary ω. From a Gaussian fit we extractthe resonance peak’s amplitude, F . From linear responsetheory, we expect F ∝ V 2τS(k) [46]. For the relevant asrange, we have checked the scaling with τ and V [42].Figure 2 shows examples of the Bragg-excitation spec-trum for various as. In the BEC regime until the onsetof the SSP, we observe a downward shift of the resonancefrequency without a significant change in F [42]. In con-trast, as we enter into the SSP regime, F undergoes astark reduction. In the ID regime, the resonance peakcompletely vanishes.

Figure 3 shows the evolution of F across the BEC-SSP-ID phase diagram. The as-extension of the three phases(see background colors), has been determined from inde-pendent measurements of the phase coherence and den-

4

0

1

2

3

4

F (%

)5

BECSSPID

54 54.5 55 55.5 56 56.5 57as (a0)

53.5

FIG. 3. Experimental F (circles) versus as across the BEC-SSP-ID phases (white, red, blue shadings). For the lowestthree as, we do not observe a resonance and plot the standarddeviation of the data as an error estimate. Horizontal errorbars correspond to uncertainties of the magnetic field [42].We show the excited fraction expected from the BdG calcula-tions on the corresponding ground states (gray line) and fromthe RTE simulations (connected dots, one RTE simulationrun). In the RTE we cannot extract clear resonances for thelowest as. We furthermore show the rescaled BdG calcula-tions, which include ∆Θ obtained from the RTE (blue line).The gray shading corresponds to the uncertainty in as of theexperimental phase transition (vertical line).

sity modulation of the states [6, 42]. When reducing asin the BEC phase, we find a slight increase of F . Onboth sides of the BEC-SSP phase transition, we observesimilar F values, indicating a continuous behavior acrossthe transition. Remarkably, as soon as we lower as fur-ther by ∼0.5 a0, F drastically reduces to <∼ 1 %, which isclose to our detection level. Finally, for as < 54 a0, wedo not observe any resonant response.

We compare the experimental results with our BdGtheory for the 3D-trapped gas. While in the BEC regime,experiment and theory show a good agreement, in theSSP they start to substantially deviate from each other.The data shows a much faster reduction of the system’sresponse than the one predicted from the BdG theory.This result suggests that an important ingredient is miss-ing in the theory. Our DAA model provides a first intu-itive explanation: It suggests that the presence of phasevariations across the system can yield a reduction of theDSF. We envision two sources of phase variations. First,quantum and thermal fluctuations, which are expected todominate in the ID regime, yield phase patterns varyingfrom shot to shot. Second, coherent dynamics, as e. g. in-duced by the crossing of the BEC-SSP phase transition,leading to reproducible phase patterns. Neither phenom-ena are accounted for in the BdG calculations.

To investigate these effects, we simulate the systemreal-time evolution (RTE) [47]. Our calculations repro-duce the experimental sequence and include the linear

ramp in as, the Bragg excitation, the three-body losses,and an initial population of BdG modes from quantumand thermal noises [42]. From the simulated momentumdistributions, we extract the excited fractions, followingthe same procedure as for the experimental ones. Asshown in Fig. 3, contrary to the BdG results, the RTEsimulations describe remarkably well the data both inthe BEC and SSP phase.

0

(rad

)-8 -4 0 4 8

y (μm)

0

1

n y (ar

b. u

.)

-π

Θ53.6 54.2 54.8

as (a0)

time (ms)0 7

1

0

(a)

(c)

(b)

FIG. 4. RTE simulations without Bragg excitation. (a) Timeevolution of the integrated in-situ density of the wave func-tion for as = 54.04 a0. (b) 〈C〉τ (triangles) and ∆Θ (squares)versus as. The grey line corresponds to the central contrastobtained from the ground-state theory. The solid blue lineis a smooth interpolation of ∆Θ, fixed to unity at the phasetransition point. The shadings give the standard deviation ob-tained from 5 simulation runs. The vertical line correspondsto the phase transition point. (c) Phase-cuts correspondingto the simulation shown in (a).

To highlight the role of the contrast and phase varia-tions, we perform RTE simulations without the Braggexcitation for different holding times. As shown inFig. 4 (a), the density profiles n(y) exhibit only a slightreduction of the contrast with time due to atom loss. Asexpected, the calculated 〈C〉τ , time-averaged over theBragg scattering duration, increase with decreasing as.However, for each as, we observe a 10-30 % lower con-trast than the one extracted from the ground-state cal-culations. Since a reduced contrast would mean an in-crease in F , we deduce that the varying contrast can notexplain the mismatch between the BdG theory and boththe experimental and RTE results; see Fig. 3.

We now study the phase variations and its dynamics.The RTE calculations reveals that the phase of the wavefunction, θ(y), develops a non-uniform profile. For in-stance at as = 54.04 a0, θ(y) exhibits a stair-like profilewith fairly constant values within the density peaks anddiscrete phase steps in between them; see Fig. 4 (c). Thisbehaviour suggests that each density peak acquires anindependent phase, despite their density links. We alsoobserve that the phase patterns is fairly reproducible be-

5

tween simulation runs and mainly induced by the coher-ent dynamics arising by the crossing of the phase transi-tion [42].

Following the DAA model, phase variations areexpected to reduce S(k) by a factor ∆Θ ≈| 1ND

∑ND

j=1〈eiθj 〉τ |2 [33, 42]. As shown in Fig. 4 (b), ∆Θis almost unity close to the BEC-SSP phase transitionand significantly drops when lowering as towards the IDregime, where starts to flatten. The standard deviationof ∆Θ relates to the shot-to-shot reproducibility of thephase pattern. In the SSP, the deviation remains small,confirming that the phase variations originate from co-herent dynamics. In contrast, the deviation increaseswhen reaching the ID regime, highlighting the increasingeffects of fluctuations. We empirically account for the ef-fect of phase variations in the BdG theory by scaling theDSF with ∆Θ over the whole SSP-ID regimes. As shownin Fig. 3, this simple inclusion of ∆Θ demonstrates thepronounced impact of the coherent phase variations forthe experimentally observed response.

In conclusion, we demonstrate that the supersolidstates, when created via a dynamical tuning of the in-teractions, develop important phase variations across thesystem, which have to be taken into account to under-stand the system behavior. Those phase variations occureven in presence of sufficiently strong density links be-tween the droplets. Our work provides first steps to amore complete vision of the dipolar supersolid, includingout-of-equilibrium phenomena, and opens the door forfuture exploration of critical phenomena induced by thedynamical crossing of the BEC-SSP phase transition [48–50].

We thank S. Stringari for insightful discussions andB. Yang for the careful reading of the manuscript.Part of the computational results presented have beenachieved using the HPC infrastructure LEO of the Uni-versity of Innsbruck. This work is financially sup-ported through an ERC Consolidator Grant (RARE,no. 681432), a DFG/FWF (FOR 2247/PI2790) and ajoint-project grant from the FWF (I 4426, RSF/Russland2019). L. C. acknowledges the support of the FWF viathe Elise Richter Fellowship number V792. A. R. andS. M. R. acknowledge support from Provincia Autonomadi Trento and the Q@TN initiative. We also acknowledgethe Innsbruck Laser Core Facility, financed by the Aus-trian Federal Ministry of Science, Research and Economy.

* Correspondence and requests for materials should beaddressed to Francesca.Ferlaino@uibk.ac.at.

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7

Supplemental Material

A. Preparation of the BECs

We prepare a BEC of 166Er by loading about 3 × 106

thermal atoms into a crossed optical dipole trap (ODT)and subsequent evaporative cooling, see Ref. [6, 51]. Dur-ing the evaporative cooling, a homogeneous magneticfield of 1.9 G is present to ensure high enough rethermal-ization rates to obtain ultracold temperatures. After theevaporation, we adiabatically modify the correspondingODTs laser powers and beam waists, to shape the con-fining potential Vtrap(r) = m(ω2

xx2 +ω2

yy2 +ω2

zz2)/2 to a

cigar-shaped geometry with harmonic trapping frequen-cies ωx,y,z = 2π × [250(1), 31.7(13), 156(2)] Hz. Consec-utively, the magnetic field is lowered to a value corre-sponding to 64.9 a0. After this preparation procedure,we obtain a BEC with a total atom number of 1.2× 105

atoms and a condensed fraction of 70 %. The temper-ature of 95(5) nK is obtained from time-of-flight (ToF)expansion measurements.

To enter the BEC-SSP-ID regimes, we lower down aslinearly in 20 ms to the corresponding values given in themain manuscript. We then let the system equilibrate for10 ms and consecutively apply a Bragg pulse of 7 ms du-ration. In order to access the momentum distributionof our atomic cloud, we perform a ToF expansion, byabruptly switching off all trapping potentials directly af-ter the Bragg excitation. After 30 ms of free expansion,we take an absorption image of the cloud along the dipoledirection. We note that, due to residual magnetic fielddrifts in the experiment (estimated to be ±2 mG), the un-certainty on as, during the Bragg pulse, ranges between±0.1 a0 and ±0.2 a0, increasing for lower as. This uncer-tainty is represented by the corresponding error bars onour data in the main manuscript.

B. Determination of experimental BEC-SSP phasetransition

In order to determine a∗s for our experimental param-eters, we perform a time-of-flight expansion of the sys-tem in the BEC or SSP regime after the equilibrationtime. Here, no Bragg pulse is applied. We find eitheran expanded ordinary BEC or an interference patternof the expanded supersolid, where a part of the atomsappear in two side peaks around ky ≈ ±kc. The atomnumber in these two side peaks is directly related to themodulation contrast of the in-situ cloud [4–6]. We mea-sure the fraction of atoms in a single side peak, fside,and monitor its evolution versus as; see Fig. S1. In theBEC regime, where no density modulation is present, wefind fside = 0 down to as,1 = 55.00 a0. After crossingthe BEC-SSP phase transition, we observe fside > 0 foras ≤ 54.88 a0 = as,2, which increases with lower as. Tak-

SSP BEC

53.8 54 54.2 54.4 54.6 54.8 55 55.20

0.5

1

1.5

2

2.5

3

f side

(%)

as (a0)

Δas,P*Δas,B*

Δas*

55.4

FIG. S1. Fraction of atoms in one side peak of the atomiccloud’s interference pattern across the BEC-SSP phase tran-sition. Error bars denote one standard error obtained fromabout 30 measurements. The vertical line shows the obtainedphase transition point. The different grey shadings corre-spond to the different uncertainties that are taken into ac-count to obtain the total uncertainty, ∆a∗s , of a∗s (see text).

ing the mean, (as,1−as,2)/2, we find a∗s = 54.94 a0 with anuncertainty of ∆a∗s,P = 0.05 a0, coming from our resolu-tion in as; see shadings in Fig. S1. We include a magneticfield uncertainty corresponding to 2 mG (±0.12 a0 at a∗s ),which increases the uncertainty to ∆a∗s,B = ±0.13 a0.Furthermore, we take a finite resolution of fside ≈ 0.2 %into account and obtain the final estimate of the criticalpoint of the BEC-SSP phase transition a∗s = 54.94+28

−13 a0.

C. Transition from SSP to ID

0

2

x102

A Ф, A M

54 55as (a0)

BECSSPID

54.5

FIG. S2. Amplitudes Aφ (red circles) AM (blue squares)versus as. Error bars denote the standard error from about 30experimental realizations [6]. Non-zero values of AM enableus to identify modulated states and confirms the BEC-SSPtransition point (vertical line, gray shaded area). The SSP isidentified by AM ≈ Aφ > 0 and extends down to as = 53.9 a0.For lower as an ID state is observed (Aφ < AM > 0).

We use the same analysis of Aφ and AM as in Ref. [6]to distinguish in the experiment the SSP and the ID

8

regime. In brief, Aφ relates to a reproducible interfer-ence pattern in time of flight and thus reveals a coherentand modulated state. AM relates only to the presenceof an in-situ density modulation (structures in the ToFimages). By combining both observables, one can dis-tinguish the SSP (Aφ ≈ AM > 0), the ID (Aφ < AM)and the ordinary BEC regimes (Aφ = AM = 0) in theexperiment, see Fig. S2. We find that for as < 53.9 a0the system is in the ID regime. The measurements areperformed directly after the equilibration time and with-out a Bragg excitation (same timings as in Sec. B). Wenote that the ratio Aφ/AM is mostly sensitive to phasefluctuations, which lead to different interference patternsin different experimental runs and is insensitive to repro-ducible phase variations in the system. The latter couldaffect the structure of the interference patterns, yet ina reproducible way. Therefore, Aφ/AM is an observablethat is adapted to describe the coherence of the systembut does not measure the phase variations, investigatedin the main manuscript, which are induced by diabaticdynamics.

D. Calibration of atom loss and atom number forBdG theory

53 54 55 56 57

2

3

4

5

6

7

N

x104

SSP BEC

as (a0)

FIG. S3. Atom number in the BEC versus as for differenttimes in the experiment, corresponding to the beginning (cir-cles), middle (triangles) and end (squares) of the Bragg pulse.Error bars denote the standard error from 5 measurements.The lines are spline interpolations to the corresponding data.The measurements were obtained without applying a Braggpulse. The vertical line indicates the measured phase transi-tion point.

Due to three-body recombination losses, the atomnumber, N , in the condensed part is decreasing duringthe 7 ms of the Bragg pulse by typically 10-30 %. There-fore, the atom number in the experiment varies withas, which we include in our BdG theory of the three-dimensionally trapped system. We note that we do notobserve additional atom loss due to the presence of theBragg excitation, as the wavelength of the used laser lightis far enough detuned from any atomic resonance (seeSec. E).

To extract N we perform an additional set of measure-ments in which we do not apply a Bragg pulse and, after agiven hold time in trap, take absorption images after a 30ms ToF expansion. From these images, the thermal com-ponent is fitted by an isotropic 2D Gaussian function andsubtracted. A final numerical integration over the imageyields N without the need of an additional fitting of thecondensed part itself. In Fig. S3, we show N across thephase diagram for different timings in the experiment,corresponding to the beginning, the middle and the endof the Bragg pulse. Each timing is interpolated with aspline fit. The fitted values of the intermediate timing(orange line in Fig. S3) is used as the atom number inour BdG theory.

E. Bragg spectroscopy

The Bragg excitation beams are realized holographi-cally with a digital-micromirror device (DMD), as de-tailed in Ref. [56]. In short, the setup uses a near-resonant laser light, red-detuned by 71(1) GHz from the401 nm transition of 166Er. These two Bragg beams in-terfere under an angle on the atoms’ position, givingrise to an interference pattern. In our setup this an-gle can be tuned, but for this current work we keep itfixed to obtain an interference pattern with a wave vec-tor k = 4.2(3)µm−1 along y. The value and uncertaintyon k is deduced from offline measurements of the anglebetween the two Bragg beams. To excite the system, theBragg scattering needs to supply energy, hω, which is in-troduced with a frequency difference, ω, between the twoBragg beams. Here, we use a sequence of holographicgratings that is uploaded on the DMD and continuouslyshifts the phase of one beam in 9 steps from 0 to 2π.Depending on the frame-rate of the uploaded sequence,we can vary ω from 0 Hz to 1000 Hz.

To calibrate the depth V of our Bragg potential,we perform Kapitza-Dirac-diffraction measurements [52].For these measurements, we tune the laser lightcloser to the atomic transition (20.6 GHz) and usethe maximally available power for our Bragg beams.By doing so, we achieve a maximum optical depthof Vmax/h = 430(50) Hz, corresponding to 3.2(4)Erec,where the recoil energy Erec = h2(k/2)2/(2m) = 135 Hz.In order to extract the potential depth V of our Bragg

9

-5 0 5

4

2

0

-2

-4 0

1

2

3 n (kx , ky ) (μm

2 )

x102

-5 0 5

6

kx (μm-1 ) kx (μm-1 )

k y (μ

m-1

)

FIG. S4. Examples of momentum distributions of a super-solid at as = 54.59(13) a0 after an applied Bragg excitation(a) on resonance at 1.7 hωz and (b) off resonance at 3.7 hωz.The two side peaks appearing around ky ≈ ±2.4µm−1 con-stitute, together with the central peak at k ≈ 0µm−1, theinterference pattern obtained when expanding a supersolidstate. The black box indicates the region of interest fromwhich Nexc is extracted. Each image is an average of 15 ex-perimental realizations.

scattering probe, we rescale this calibrated value ac-cording to the corresponding laser light detuning andpower used for the Bragg scattering [1]. We obtainV = 42(5) Hz, which is well inside the linear scatter-ing regime (see Sec. F and Fig. S6). The exact calibra-tion of the potential depth does not include systematiceffects, as for example inhomogeneities of V cross theatomic cloud or in-trap dynamics of the atoms duringthe Kapitza-Dirac-diffraction measurement. Neverthe-less, we note that the estimation of the linear responseregime is insensitive to the exact calibration of V . Wefurthermore note, that a direct comparison of F from theexperiment with the one from the RTE suggests that Vneeds to be rescaled by about 1.7.

Figure S4 gives examples of our images for a resonantand an off-resonant Bragg scattering frequency. For theresonant case (Fig. S4, left panel) we find scattered atomsat a high momentum around ky ≈ k. As they appear out-side of the interference patterns, observed from the un-perturbed system, they constitute a clean signal for theanalysis of the excited fraction. We count the numberof atoms, Nexc, in a region of interest (ROI), indicatedby the black boxes in Fig. S4. We note that we carefullychecked that neither F , nor ωk, changes within the un-certainties when increasing the ROI size by 30%.. Wemeasure the total atom number, N , for each measure-ment individually, by performing a similar count on arectangle of 12µm−1 by 14µm−1, covering all condensedand scattered atoms. By measuring Nexc/N for differentexcitation frequencies, we obtain a spectroscopy of theBragg scattering. We note that due to thermal atoms,present in the analyzed region of interest, all Bragg res-onances show a small offset, which is extracted from theoffset of the Gaussian fit to the resonance and then sub-tracted.

From the Bragg excitation spectra, we extract theresonance peak’s amplitude, as discussed in the mainmanuscript, and a resonance frequency, ωk. The latteris shown in Fig. S5 as a function of as. We observe thatωk decreases monotonically from high to low as, acrossthe BEC-SSP phase transition. This beheviour is consis-tent with the extracted ωk from the RTE theory. Fur-thermore, we calculate the expected resonance frequencyfrom the BdG calculations, in which the resonance fre-quency is increasing again after crossing the BEC-SSPphase transition with lowering as. Therefore, the BdGtheory predicts a hardening of the measured excitationmodes, which is not observed neither in the experiment,nor in the RTE simulations. This qualitative differencemight stem from the increased phase variations that de-velop in the system, but further studies are needed toelucidate this point.

ωk (

ωz )

BECSSP

as (a0)54 55 56 57

1.5

2

2.5

FIG. S5. Extracted resonance frequencies, ωk, versus as (cir-cles) and their comparison with the expected resonance posi-tion from the BdG theory (gray line) and ωk from the RTE(connected dots). The vertical line indicates the BEC-SSPphase transition point and its shading the uncertainty on a∗s .

F. Variations of the excited fraction with the Braggpulse duration

In Figure S6, we present the measured evolution of Fwith the Bragg pulse duration from 0 to 7 ms for a fixed ω.Across the BEC-SSP-ID regimes, we find a linear scalingof F with τ , which is consistent with the expected scalingfrom BdG theory, F ∝ V 2τS(k) [46]. Furthermore, weprobe the quadratic scaling of F with V in the SSP andfind also here an agreement up to V = 80 Hz (see inset).

G. Evolution of the excitation spectrum with as forthe infinite cigar-shaped gas

We calculate the excitation spectrum and the dynamicstructure factor (DSF) of an infinitely elongated, cigar-

10

0 2 4 6τ (ms)

0

2

4

6F

(%)

0 40 80 1200

5

10

F (%

)

V/h (Hz)

FIG. S6. Evolution of F during the Bragg pulse for three ex-emplary as = [55.5(1), 54.0(1), 53.3(1)] a0 (black, red, blue),corresponding to the BEC, SSP, ID regimes, respectively.Each data point corresponds to an average of 5 to 15 mea-surements and its uncertainty to the standard error. Thesolid lines correspond to a linear fit and its shading to thefit’s 68 %-confidence bound. The inset shows the evolutionof F with the applied potential depth, V , of the Bragg pulsefor τ = 7 ms for a supersolid state at as = 54.28(14) a0. Thesolid line corresponds to a quadratic fit up to V = 80 Hz, thedashed line is the extension of the fitting.

shaped dipolar supersolid in Fig. 1(a, b) of the mainmanuscript. In Figure S7 we present the evolution ofthe excitation spectrum across the BEC-SSP-ID regimes.Figure S7 (a1-a5) shows the integrated density profiles ofthe ground state along the unconfined direction for dif-ferent values of as and a fixed mean axial density of4.7× 103 µm−1. Figure S7 (b1-b5) shows the correspond-ing excitation spectrum. At large enough as, the groundstate has a uniform density along the unconfined direc-tion (a1, a2 - BEC phase) and its excitation spectrumshows the typical phonon-maxon-roton spectrum, firstpredicted in [53, 54]. When decreasing as below a crit-ical value, the ground state becomes density modulated(a3, b3 - SSP phase) with a modulation wave numberkc = 2.3µm−1 close to the BEC’s roton momentum (b2),underlying the connection between roton softening andcrystallization. The density modulation has a finite con-trast and its value increases when lowering as furtherdown (a4, a5).

When crossing the BEC-SSP phase transition, the ex-citation spectrum changes dramatically, becoming pe-riodic, with the appearance of two gapless Goldstonebranches associated with phase (lower energy branch)and density (higher energy branch) excitations, respec-tively [38, 43, 55]. In addition to these gapless branches,one observes gapped parabolic branches of excitationswith energetic minima at integer multiples of kc. Theone branch at ky = kc is the one investigated in the mainmanuscript. For decreasing as, the energy minimum ofthis parabolic branch increases towards the ID regime[Fig. S7 (b3-b5)].

As described in the main paper, we use the norm of

the calculated Bogoliubov amplitude ‖uj‖ =∫|uj(r)|2dr

to distinguish whether a mode j is a collective excitationor has a single-particle character [33, 44]. Collective ex-citations feature ‖uj‖ � 1 whereas single-particle excita-tions have ‖uj‖ ' 1. In Figure S7 (b1-b5), we color eachexcitation mode according to ‖u‖. We find that lowerenergy modes, such as the roton mode in the BEC andthe Goldstone modes in the SSP have a clear collectivenature. The energetically higher modes (hω >∼ 0.5 hωz),of the parabolic branch in the SSP and the ky > kc-branch in the BEC, have ‖uj‖ ' 1 across the BEC-SSPphase transition. We note that the condition ‖uj‖ ' 1does not directly identify an excitation mode as a free-particle. To obtain a free-particle excitation, the modeneeds to be of single particle character and additionallyits energy needs to be mainly given by the kinetic energy.Therefore, free-particle excitations have a wave functionthat is a plane wave [33]. We note that for our exper-imentally relevant energy regime, the probed excitationmodes are described well by a plane wave, as shown inSec. I. and Fig. S9 (b).

From our simulations, we also calculate the DSF. Inthe BEC phase [Fig. S7 (c1, c2)] the DSF is dominatedby the roton mode at ky = kc. Moreover, the deeperthe roton minimum, the stronger is its response to smalldensity perturbations. We note that, in the BEC phase,this affects also the density response even for momentahigher then the roton momentum. After crossing thephase transition into the SSP, we find that the DSF ofthe parabolic branch [Fig. S7 (c3)] smoothly connects tothe free-particle branch of the BEC phase. For decreasingas, the DSF of the free particle branch becomes smaller[Fig. S7 (c4, c5)].

H. BdG theory for the three-dimensional trappedgas

For the current manuscript, we employ similar BdGand ground state calculations as already described inRefs. [38, 47, 56]. In this theory the gas is trapped inall three dimensions. For calculating the ground states,we use the experimentally extracted atom number atthe intermediate timing of the Bragg pulse (presented inFig. S3, triangles). The radially integrated density pro-files of the ground states in the SSP regime are presentedin Fig. S8. We note that in this theory, the BEC-SSPphase transition lies at 3.79 a0 below the experimentallydetermined one. This shift in as between theory and ex-periment has also been found in Refs. [6, 56, 57]. There-fore, throughout the manuscript, the BdG theory and theground state calculations are presented with an up-shiftof 3.79 a0 for as.

The presence of an axial trapping potential, leads todiscrete excitation modes in the spectrum (typical energyspacing ∼ h× 20 Hz). Furthermore, each mode is broad-

11

momentum (kc)

0

0.5

1.5

ω (ω

z)

0 0.5 1 1.5 2ky (kc)

ω (ω

z)

0 0.5 1 1.5 2 0 0.5 1 1.5 2

y (μm)

0 0.5 1 1.5 2ky (kc) ky (kc) ky (kc)

0 0.5 1 1.5 2ky (kc)

-2 0 20

1.5n

(x10

4 μm

-1)

y (μm)-2 0 2

y (μm)-2 0 2

y (μm)-2 0 2

a2 a3 a4 a5

b2 b3 b4 b5

c2 c3 c4 c5

y (μm)-4 -2 0 2 4

a1

b1

c1

1.0

0

0.5

1.5

1.0

0 0.5 1 1.5 2ky (kc)

0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2ky (kc) ky (kc) ky (kc)

0 0.5 1 1.5 2ky (kc)

1 3 >5||u||

1 2 >2.6S (k, ω)

0

FIG. S7. Axial excitation spectra of the infinitely, extended gas across the BEC-SSP-ID regimes in a ωx,y,z = 2π(250, 0, 160) Hztrap with fixed axial density n0 = 4.7 × 103µm−1. (a1-a5) Integrated density profiles along the unconfined direction, foras = (52.00, 51.40, 51.25, 51.00, 49.75) a0. (b1-b5) Transverse symmetric modes of the corresponding excitation spectrumcolored according to ‖u‖. (c1-c5) The corresponding S(k, ω). For visibility, the DSF is broadened with a Gaussian function.

ened in ky. The finite duration of the Bragg pulse givesan energy broadening of each excitation mode (Fourierbroadening ∼ h × 130 Hz full width at half maximum)which is much bigger than the energy spacing betweenthe modes in the spectrum. Therefore, in the Braggspectroscopy only a single resonance is visible, which isconstituted of multiple excitation modes. To account forthis, we calculate S(k, ω) while broadening each modein energy according to the Fourier-broadening, expectedfrom a 7 ms Bragg pulse. After calculating the broad-ened S(k, ω) and evaluating it at the experimental k, wealso find in the BdG theory a single resonance in energy.To compare with the experiment, we extract S(k) froma Gaussian fit to this resonance; see also [56].

I. Free particle regime in the BdG theory for atrapped gas

To transfer the insights from the BdG calculations ofan infinitely extended system (see Sec. G) to the experi-mentally trapped case, we also analyze ‖u‖ and ‖v‖ of theexcitation modes obtained from the BdG calculations ofa three-dimensional trapped gas. Similar to the infinitely

extended system, we find that modes with hω >∼ 0.5 hωzhave ‖u‖ ≈ 1 and therefore a single-particle characteracross the BEC-SSP-ID regimes. This is exemplified inFig. S9 (a) where we show, for an exemplary state in theSSP, the norm of the obtained Bogoliubov amplitudesversus the energy of the corresponding mode.

As mentioned already in Sec. G, to further identify asingle particle excitation as a free particle one, the ex-citation’s wave function needs to be a plane wave. Toinvestigate this aspect, we study the excitation modes’density profiles and find for modes in the experimentallyrelevant energy regime a clear plane wave character. Fig-ure S9 (b) shows the radially integrated density profile ofan exemplary excitation mode of a supersolid and com-pares it to the integrated density of the ground state.The plane wave character is clearly visible as a modu-lation with k ≈ 4.1µm−1 across the whole system. Weonly find a mild reduction of the plane wave’s amplitudetowards the outer region of the system. As a comparison,we show in Figure S9 (c) the integrated density profile ofan excitation mode at lower energy, 0.55 hωz, which alsohas ‖u‖ ≈ 1, but is clearly not a plane wave.

12

y (μ

m)

as (a0)

n (x103 μm-1)

53.5 54 54.5

-10

-5

0

5

10

0 1 2 3 4 5

FIG. S8. Radially integrated in-situ density profiles versusas of the supersolid ground states in a trap with harmonicfrequencies ωx,y,z = 2π × (250, 31, 160) Hz. For each as, weuse the atom number measured in the experiment (see Fig. S3,triangles).

J. Comparison of the finite-size BdG theory to theself-consistent SIA and DAA model

From the ground state profiles of the three-dimensionaltrapped BdG theory in Fig. S8, we numerically evaluateour SIA and DAA model on the corresponding groundstates. To self-consistently evaluate the SIA result fromthe ground state, we need to estimate the contrast of thedensity modulation, C = (nmax − nmin)/(nmax + nmin).We determine nmin from the minimum density at y = 0and nmax from the density of one of the two most-centraldensity peaks. To evaluate the DAA model, we numeri-cally extract the 1/e-size, σ, of the two central dropletsand the distance d between them. To estimate the den-sity, we calculate the mean density in the central regionbetween the two central droplets. Therefore, our modelcomparison takes only the central part of the system intoaccount and neglects the outer density regions. Further-more, as our models extract only the scalings of the DSFwith the ground state properties and not its absolutevalue, we renormalize the SIA and the DAA. For thecomparison in Fig. 1 of the main manuscript we show thefinite BdG and the SIA rescaled to unity for the point atthe phase transition. The DAA is rescaled directly on theBdG theory to match its values in the lower as regime.Over the investigated as range, we find that both, theSIA and the DAA, describe the BdG theory well for lowand large C, respectively (see main text), and for mo-menta k ≥ 4.0µm−1. This gives an estimate for which

0 0.5 1

1

energy (ħωz )

||u||,

||v||

10

0.1

freeparticlescollective

(a)

-10 -5 0 5 10

0

1

y (μm)

nex

c , n

GS (

arb.

u.)

(b)

0

1 (c) n

exc , n

GS (

arb.

u.)

FIG. S9. (a) Bogoluibov amplitudes, ‖v‖ (crosses) and ‖u‖(circles), for the excitation modes of a trapped supersolid atas = 54.49 a0. (b) Integrated density profile, nexc, of an exem-plary excitation mode with an energy of 1.7 hωz (red line) andk ≈ 4.1µm−1 and a comparison with the integrated groundstate density, nGS, (grey line). (c) The same as in (b) but foran excitation mode at 0.55 hωz.

momenta the impulse approximation becomes valid.

K. Real time evolution of the Bragg scattering

Our theory for the RTE simulations was already pre-sented in Ref. [47]. For the current manuscript, the sim-ulations start with the ground state wave function withN = 8.5 × 104 atoms at as = 60.9 a0. We add a ther-mal population, corresponding to a randomly drawn oc-cupation of the system’s excited states (incl. a ran-dom phase) with a Poisson distribution, whose mean isgiven by the Bose distribution (+1/2 to simulate quan-tum fluctuations) for the mode’s energy at a temperature

13

of 100 nK [58]. This increases the total atom number toabout 1×105 (similar to the experimental situation) andsimulates thermal and quantum fluctuations in the sys-tem.

In the time evolution, we reproduce the experimentalsequence, including a 20 ms long linear as ramp, followedby a 10 ms holding at the final as and a consecutive 7 msBragg excitation along y. In order to obtain the system’smomentum distribution, we perform a Fourier-transformof its wave function. On this momentum distribution, weperform the same analysis as on the experimental data,i. e. we analyse the fraction of excited atoms in a regionof interest around k ≈ 4.2µm−1 for various ω. The res-onances are fitted with a Gaussian function to obtain Ffrom the RTE simulations. For the same reasons as men-tioned for the BdG calculations (Sec. H.), the crossingof the BEC-SSP phase transition in the RTE happens3.34 a0 below the experimental phase transition. There-fore, throughout the manuscript, the RTE theory is up-shifted in as by 3.34 a0. We note that, when perform-ing the RTE directly on the corresponding ground statesfrom the BdG theory, i. e. we do not include thermalnoise, the as-ramp and atom loss, we recover an excitedfraction that is well described by the BdG theory. Thisindicates that the chosen analysis in ToF gives a reliablemeasurement of S(k) and in particular a consistent resultwith the 3D-trapped BdG calculations.

L. Real-time evolution and characteristics of thestate’s wave function

To study the time evolution of the contrast and thephase of the dynamically created supersolid states, weperform RTE simulations without applying a Bragg ex-citation and monitor the axial density and the phase pro-files from the calculated wave functions [see Fig. 4 (a, c)in main manuscript]. Typically, in the RTE we observeND = 4 − 6 droplets, containing a variable atom num-ber across the system. We evaluate the time-dependentcentral contrast, C, between the two central densitypeaks numerically. The phase-variation factor ∆Θ =| 1ND

∑ND

j=1〈eiθj 〉τ |2 is calculated by extracting the meanphase, θj , of each single density peak over its full-widthat half maximum.

Figure S10 (a) shows C(t) over the whole simulationtime. For early times, we observe a small, but finite Cdue to density noise in the simulations, which is com-ing from the included thermal fluctuations. During theholding time ([−10, 7] ms), for as < a∗s , we observe that

C first increases and consecutively slightly decreases dueto atom loss. Only for as < 54.2 a0 we find an oscillatingbehaviour of the contrast in time. We note that thereis a time delay between the development of the densitymodulation in the system and the timing of the as ramp(which occurs during [−30, −10] ms).

0

1

time (ms)

α (r

ad)

C0

2π

-30 -20 -10 0 7

ramp of as holdtiming of

Bragg pulse

(a)

(b)

53.94 a054.24 a054.54 a054.74 a054.84 a054.94 a055.14 a0

FIG. S10. RTE simulations with V = 0 (no Bragg excita-tion applied). (a) The time evolution of the extracted centralcontrast of the integrated in-situ density distributions for dif-ferent as (see legend). (b) The extracted phase incoherenceof a central cut through the wave function (see text). Theshadings represent the standard deviation from 5 simulationruns with different statistical draws of the thermal popula-tion. The data presented in Fig. 4 (a, b) corresponds to thetime window of [0, 7] ms.

To give another insight into the time evolution of thesystem’s phase profile, we extract the global phase vari-ation, α = 1

L

∫L|φ(0, y, 0) − 〈φ〉L |dy, of the wave func-

tion, which is extracted along a cut of φ along y. Here,〈φ〉L denotes the averaged phase over the central regionL = [−7, 7]µm, see also [4]. In Figure S10 (b), we showthe time evolution of α for the whole simulation time.For all as one sees a first local maximum in α (around−20 ms), coming from an axial breathing mode which isexcited due to the as ramp. At longer times, we find for54.5 a0 <∼ as ≤ a∗s , that α remains small while the densitycontrast is finite. For as < 54.5 a0, we find that α seemsto approach a constant value, which increases with loweras.