Two-dimensional supersolidity in a dipolar quantum gas
Matthew A. Norcia, Claudia Politi, Lauritz Klaus, Elena Poli, Maximilian Sohmen, Manfred J. Mark, Russell Bisset, Luis Santos, Francesca Ferlaino
TTwo-dimensional supersolidity in a dipolar quantum gas
Matthew A. Norcia, ∗ Claudia Politi,
1, 2, ∗ Lauritz Klaus,
1, 2
Elena Poli, MaximilianSohmen,
1, 2
Manfred J. Mark,
1, 2
Russell Bisset, Luis Santos, and Francesca Ferlaino
1, 2, † Institut f¨ur Quantenoptik und Quanteninformation,¨Osterreichische Akademie der Wissenschaften, Innsbruck, Austria Institut f¨ur Experimentalphysik, Universit¨at Innsbruck, Austria Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Germany (Dated: February 11, 2021)Supersolidity — a quantum-mechanical phenomenon characterized by the presence of both su-perfluidity and crystalline order — was initially envisioned in the context of bulk solid helium, asa possible answer to the question of whether a solid could have superfluid properties [1–5]. Whilesupersolidity has not been observed in solid helium (despite much effort)[6], ultracold atomic gaseshave provided a fundamentally new approach, recently enabling the observation and study of super-solids with dipolar atoms [7–16]. However, unlike the proposed phenomena in helium, these gaseoussystems have so far only shown supersolidity along a single direction. By crossing a structural phasetransition similar to those occurring in ionic chains [17–20], quantum wires [21, 22], and theoreti-cally in chains of individual dipolar particles [23, 24], we demonstrate the extension of supersolidproperties into two dimensions, providing an important step closer to the bulk situation envisionedin helium. This opens the possibility of studying rich excitation properties [25–28], including vortexformation [29–31], as well as ground-state phases with varied geometrical structure [7, 32] in a highlyflexible and controllable system.
Ultracold atoms have recently offered a fundamentallynew direction for the creation of supersolids — ratherthan looking for superfluid properties in a solid systemlike He, ultracold atoms allow one to induce a crys-talline structure in a gaseous superfluid, a system whichprovides far greater opportunity for control and obser-vation. This new perspective has enabled supersolidproperties to be observed in systems with spin-orbit cou-pling [33] or long-range cavity-mediated interactions [34],though in these cases the crystalline structure is exter-nally imposed, yielding an incompressible state. In con-trast, dipolar quantum gases of highly magnetic atomscan spontaneously form crystalline structure due to in-trinsic interactions [11–13], allowing for a supersolid withboth crystalline and superfluid excitations [14–16]. Inthese demonstrations, supersolid properties have onlybeen observed along a single dimension, as a linear chainof phase-coherent “droplets”, i.e. regions of high densityconnected by low-density bridges of condensed atoms,confined within an elongated optical trap.The extension of supersolidity into two dimensions is akey step towards creating an ultracold gas supersolid thatis closer to the states envisioned in solid helium. Com-pared to previous studies of incoherent two-dimensionaldipolar droplet crystals [8, 35], we work with both a sub- stantially higher atom number N and relatively strong re-pulsive contact interactions between atoms. This leads tothe formation of large numbers of loosely bound droplets,enabling us to establish phase coherence in two dimen-sions. In our system, the repulsive dipolar interactionsbetween droplets facilitate a structural transition froma linear to a two-dimensional array, analogous to theCoulomb-interaction-mediated structural phase transi-tions observed with ions [17–20]. Unlike ions however,our droplets are compressible and result from the spon-taneous formation of a density wave, allowing for dynam-ical variation in both droplet number and size. Further,the exchange of particles between droplets enables thespontaneous synchronization of the internal phase of eachdroplet across the system, and the associated superfluidexcitations [14–16].Dipolar quantum gases exhibit a rich set of ground-and excited-state phenomena due to the competitionbetween many energetic contributions. These includemean-field interactions of both contact and dipolar na-ture, quantum fluctuations, and external confinement,parameterized by potentially anisotropic trapping fre-quencies f x,y,z . Such systems can be described withgreat accuracy by using an extended Gross–Pitaevskiiequation (eGPE) [36–39]. Even a fine variation of the a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b a.b. t A t o m nu m be r N - y (µm) -5 0 5 y (µm) -5 0 5 y (µm) x ( µ m ) FIG. 1.
Calculated phases of dipolar droplet array. a.
In-trap ground-state density profiles calculated using eGPEfor atom numbers N ∈ [3 . , . , . × in the dropletsand trap aspect ratios α t = f x /f y ∈ [0 . , . , .
39] (leftto right). The scattering length a = 88 a , where a is theBohr radius. Green dots depict the droplet positions obtainedfrom the variational model, assuming the same N and dropletnumber N D as the eGPE. Stars connect to experimentallyobserved density profiles in Fig. 2b. b. Phase diagram, ob-tained from our variational model, as a function of N and α t for f x = 33 Hz, f z = 167 Hz. Linear (two-dimensional) phaseswith N D droplets are labelled as 1D N D (2D N D ). strength of these energetic contributions can lead to dra-matic qualitative changes in the state of the system, forexample enabling a transition from a uniform conden-sate to a supersolid, or in our present case, from a linearsupersolid to a two-dimensional one.Fig. 1a shows ground-state density profiles calculatedacross this transition using the eGPE at zero temper-ature. These profiles feature arrays of high-density droplets, immersed in a low-density coherent “halo” thatestablishes phase-coherence across the system. As thetrap becomes more round, the initially linear chain ofdroplets acquires greater transverse structure, eventuallyforming a zig-zag state consisting of two offset linear ar-rays.Although the eGPE has remarkable predictive power,full simulations in three dimensions are numericallyintensive, making a global survey of the array propertiesas a function of our experimental parameters difficult.To overcome this limitation, we employ a variationalansatz that captures the key behavior of the system, andallows us to disentangle the competing energetic contri-butions. In this approach, we describe an array of N D droplets by the wavefunction ψ ( r ) = (cid:80) N D j =1 ψ j ( r ), wherethe j –th droplet is assumed to be of the form: ψ j ( r ) ∝ (cid:112) N j exp (cid:16) − (cid:16) | ρ − ρ j | σ ρ,j (cid:17) r ρ,j (cid:17) exp (cid:16) − (cid:16) | z − z j | σ z,j (cid:17) r z,j (cid:17) , in-terpolating between a Gaussian and a flat-top profilecharacteristic of quantum droplets [40]. For a given totalnumber of atoms N and droplet number N D , energyminimization provides the atom number N j in eachdroplet, as well as their widths σ ρ ( z ) ,j , exponents r ρ ( z ) ,j ,and positions ρ j = ( x j , y j ). Repeating this energyminimization as a function of N D gives the optimalnumber of droplets. This model provides a good quali-tative description of the overall phase diagram (Fig. 1b),revealing that the interplay between intra-dropletphysics and inter-droplet interaction results in a richlandscape of structural transitions as a function of theatom number and the trap aspect ratio α t = f x /f y .Several trends are immediately visible from the phasediagram. Larger N and higher α t generally producestates with larger numbers of droplets. Further, as withions, a large number of droplets favors a 2D configuration,while tighter transverse confinement (small α t ) favors 1D[17–20]. A transition from 1D to 2D is thus expectedwhen moving towards larger N or to higher α t . In starkcontrast to the case of ions, the number of droplets typi-cally increases across the 1D to 2D transition, implying afirst-order nature, while only narrow regions in the phasediagram may allow for a 1D-to-2D transition at constantdroplet number.The variational results are in excellent agreement withour eGPE numerics, in terms of predicting the qualitativestructure of droplet array patterns, as shown in Fig. 1a.Slight discrepancies exist between the two theories re-garding the predicted droplet positions and the locationof the 1D-to-2D transition. This is likely because of thepresence of the halo in the eGPE simulation (and pre-sumably in the experiment), visible in Fig. 1a, which isnot accounted for in the variational model. This halo ap-pears to accumulate at the ends of the trap, pushing thedroplets toward the trap center and likely increasing theeffective trap aspect ratio experienced by the droplets.To explore the 1D to 2D transition experimentally, we lower transverse confinement b. d.c.a. Number of droplets atomic aspect ratio B x yz FIG. 2.
Linear to zig-zag transition in an anisotropic trap. a.
We confine and condense dipolar
Dy atoms withinan anisotropic optical dipole trap (ODT) formed by the intersection of two laser beams. By tuning the aspect ratio of thetrap in the x - y plane ( α t ), perpendicular to an applied magnetic field B , we induce a transition between linear and zig-zagconfigurations of droplets. b. Single-trial images of the in-trap density profile of atoms at different α t , showing structuraltransition from linear to zig-zag states, as well as an increase in droplet number for higher α t . Stars indicate values α t and N corresponding to the eGPE calculations of Fig. 1a. c. Atomic aspect ratio α a versus trap aspect ratio α t . α a is the ratio ofminor to major axes of a two-dimensional Gaussian fit to the imaged in-trap density profile (inset). For the supersolid dropletarray (black markers) we see an abrupt change in α a at the critical trap aspect ratio α ∗ t , extracted from the fit (gray line, seemethods). The shape of the transition agrees well with eGPE prediction (green diamonds, see methods). For an unmodulatedcondensate (white markers), no abrupt change is evident. d. Distribution of droplet number versus α t , showing a distinctincrease in droplet number at the transition of linear to zig-zag configurations. use a condensate of highly magnetic Dy atoms con-fined within an anisotropic optical dipole trap with in-dependently tunable trap frequencies f x,y,z . The trap,shown in Fig. 2a, is shaped like a surf-board with thetight axis along gravity and along a uniform magneticfield that orients the atomic dipoles and allows tuningof the contact interaction strength. Typically, we per-form evaporation directly into our state of interest atour desired final interaction strength, as demonstratedin Refs. [13, 41]. A combination of in-trap and time-of-flight (TOF) imaging provides us with complementaryprobes of the density profile of our atomic states, andthe phase coherence across the system.We begin by studying the transition from one to twodimensions by changing the strength of transverse con-finement provided by the trap. Our optical setup allowsus to tune f y from roughly 75 to 120 Hz, while leaving f x , f z nearly constant at 33(2), 167(1) Hz, and thus tovary the trap aspect ratio α t in the plane perpendicu-lar to the applied magnetic field and our imaging axis. For small α t , the atoms are tightly squeezed transversely,and form a linear-chain supersolid (as seen in in-trap im-ages of Fig. 2b). As we increase α t above a critical value α ∗ t = 0 . α t , the 2D structure extends to two offset lines of dropletsin a zig-zag configuration. The observed patterns matchwell with the ground-state predictions from the eGPEcalculations when we globally fix the scattering length to88 a .We obtain higher atom numbers in the more oblatetraps (higher α t ), giving N = 6 . × at α t = 0 . N = 2 . × at α t = 0 .
28. This further facil-itates the crossing of the 1D to 2D transition, by favor-ing states with larger numbers of droplets in the broadertraps. In the zig-zag regime, two-dimensional modula-tion is clearly visible for durations beyond one second.Further, the droplet configuration patterns are fairly re-peatable, with clear structure visible in averaged images a. -40 -20 0 20 40 y position (µm) -40-2002040 x po s i t i on ( µ m ) b. -40 -20 0 20 40 y position (µm) -40-2002040 x po s i t i on ( µ m ) c. -40 -20 0 20 40 y position (µm) -40-2002040 x po s i t i on ( µ m ) FIG. 3.
Coherence in linear and zig-zag states.
Upper panels show averaged images of experimental TOF interferencepatterns, along with projections along horizontal and vertical directions of average (solid black lines) and individual images(gray lines). The vertical projection is calculated between the dashed lines. Lower panels show interference patterns calculatedfor the pictured in-trap droplet configurations (green outlines). a. Linear chain of phase-coherent droplets, showing uniaxialmodulation persisting in averaged image (26 trials). b. Zig-zag configuration of phase-coherent droplets, showing modulationalong two directions that persists in averaged image (51 trials), and hexagonal structure. The spacing of rows in the simulationwas adjusted to approximate the observed aspect ratio of TOF image. The image outlined in blue shows the average momentumdistribution calculated from a series of 20 variational calculations converging to slightly different droplet configurations, showingthe tendency of such fluctuations to broaden features in the interference pattern while maintaining the underlying structure. c. Zig-zag configuration of phase-incoherent droplets. Modulation remains in single images, as evidenced by the spread of graytraces in projection, but washes out in average (43 trials). as shown in the inset of Fig. 2c, which is an average of 23trials taken over roughly two hours.The transition from 1D to 2D is immediately visiblewhen plotting the atomic aspect ratio α a versus α t , asshown in Fig. 2c. We find that α a undergoes a rapidchange at α ∗ t , as the single linear chain develops two-dimensional structure. For comparison, we plot α a mea-sured for an unmodulated BEC, formed at a differentmagnetic field, which does not feature the sharp kinkpresent for the supersolid state.In Fig. 2d, we show the number of droplets present fordifferent α t . In the 1D regime, we typically see betweenfive and six droplets. This number abruptly jumps up byapproximately one droplet for 2D states near the tran-sition point, and then increases up to an average valueof eight droplets as α t is further increased. The changein droplet number indicates that the transition that weobserve is not of simple structural nature, but is also accompanied by a reconfiguration of atoms within thedroplets, as expected from theory (see Fig. 1).The measurements of in-trap density presented aboveinform us about the structural nature of the transition,but not about phase coherence, which is the key distin-guishing feature between an incoherent droplet crystaland a supersolid. Previous observations of 2D dropletarrays [35] were performed in traps where the groundstate is a single droplet [8], and the observed dropletcrystal was likely a metastable state lacking inter-dropletphase coherence. In contrast, we expect from our theo-retical calculations that the 2D array is the ground stateof our surfboard-shaped trap (for α t > α ∗ t ), facilitatingthe formation of a phase-coherent, and therefore super-solid state for our experimental parameters.We experimentally demonstrate the supersolid natureof our 2D modulated state using a matter-wave interfer-ence measurement, as previously used in linear supersolidchains [11–13], (Fig. 3a). In this measurement, an arrayof uniformly spaced droplets creates an interference pat-tern with spatial period proportional to the inverse of thein-trap droplet spacing. The relative internal phase of thedroplets determines both the contrast and spatial phaseof the interference pattern [42]. When averaging overmany interference patterns, obtained on separate runsof the experiment, clear periodic modulation persists forphase-coherent droplets, but averages out if the relativedroplet phases vary between experimental trials. Thus,the presence of periodic modulation in an average TOFimage provides a clear signature of supersolidity in oursystem, as it indicates both periodic density modulationand phase coherence.Figure 3a shows an example of such an averaged inter-ference pattern for a linear chain. Uniaxial modulationis clearly present along the direction of the chain, indi-cating a high degree of phase coherence. For comparison,we also show the expected interference pattern calculatedfor a linear array of four droplets from free-expansion cal-culations, showing similar structure.For conditions where in-trap imaging shows a 2D zig-zag structure, the averaged interference pattern exhibitsclear hexagonal symmetry (Fig. 3b). This is consistentwith our expectation, and is indicative of the triangularstructure of the underlying state. To confirm that theobserved modulation is not present without phase coher-ence, we repeat the measurement of Fig. 3b at a mag-netic field corresponding to independent droplets, andalso compute averaged interference pattern for a zig-zagstate with the phases of the individual droplets random-ized between simulated trials (Fig. 3c). In both cases, theaveraged image does not show clear periodic modulation.By exploiting the transition between linear and zig-zag states, we have accessed a regime where the super-solid properties of periodic density modulation and phasecoherence exist along two separate dimensions. Futurework will focus on further understanding the spectrumof collective excitations in the full two-dimensional sys-tem [26–28, 43], where both the crystalline structure andthe exchange of particles between droplets will play animportant role. Further investigations may elucidate inmore detail the nature of the phase transitions and ex-pected configurations in a wider range of trap aspect ra-tios, as well as the role that defects play in the 2D system,either as phase-slips in the zig-zag patterns [44, 45], or asvortices trapped between droplets of the array [29–31].We thank the Innsbruck Erbium team and Blair Blakiefor discussions. We acknowledge R. M. W. van Bijnen fordeveloping the code for our eGPE ground-state simula-tions. Author Contributions:
M.A.N, C.P., L.K., M.S.,M.J.M and F.F. contributed experimental work. E.Pand R.B. performed eGPE calculations. L.S. contributedvariational model. All authors contributed to interpreta-tion of results and preparation of manuscript.
Funding:
The experimental team is financially sup-ported through an ERC Consolidator Grant (RARE,No. 681432), an NFRI grant (MIRARE, No. ¨OAW0600)of the Austrian Academy of Science, the QuantERAgrant MAQS by the Austrian Science Fund FWFNo I4391-N. L.S and F.F. acknowledge the DFG/FWFvia FOR 2247/PI2790. M.S. acknowledges support bythe Austrian Science Fund FWF within the DK-ALM(No. W1259-N27). L.S. thanks the funding by theDeutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) under Germany’s Excellence Strategy– EXC-2123 QuantumFrontiers – 390837967. M.A.N. hasreceived funding as an ESQ Postdoctoral Fellow fromthe European Union’s Horizon 2020 research and innova-tion programme under the Marie Sk(cid:32)lodowska-Curie grantagreement No. 801110 and the Austrian Federal Min-istry of Education, Science and Research (BMBWF).M.J.M. acknowledges support through an ESQ Discov-ery Grant by the Austrian Academy of Sciences. We alsoacknowledge the Innsbruck Laser Core Facility, financedby the Austrian Federal Ministry of Science, Researchand Economy. Part of the computational results pre-sented have been achieved using the HPC infrastructureLEO of the University of Innsbruck. ∗ M. A. N. and C. P. contributed equally to this work. † Correspondence should be addressed to
[email protected] [1] E. P. Gross, Unified theory of interacting bosons, Phys.Rev. , 161 (1957).[2] E. P. Gross, Classical theory of boson wave fields, Annalsof Physics , 57 (1958).[3] A. F. Andreev and I. M. Lifshitz, Quantum theory ofdefects in crystals, Sov. Phys. JETP , 1107 (1969).[4] G. V. Chester, Speculations on Bose–Einstein condensa-tion and quantum crystals, Phys. Rev. A , 256 (1970).[5] A. J. Leggett, Can a solid be “Superfluid”?, Phys. Rev.Lett. , 1543 (1970).[6] M. H.-W. Chan, R. Hallock, and L. Reatto, Overview onsolid 4 he and the issue of supersolidity, Journal of LowTemperature Physics , 317 (2013).[7] Z.-K. Lu, Y. Li, D. S. Petrov, and G. V. Shlyapnikov, Sta-ble dilute supersolid of two-dimensional dipolar bosons,Phys. Rev. Lett. , 075303 (2015).[8] D. Baillie and P. B. Blakie, Droplet crystal ground statesof a dipolar bose gas, Phys. Rev. Lett. , 195301 (2018).[9] S. M. Roccuzzo and F. Ancilotto, Supersolid behaviorof a dipolar bose-einstein condensate confined in a tube,Phys. Rev. A , 041601 (2019).[10] M. Boninsegni and N. V. Prokof’ev, Colloquium: Super-solids: What and where are they?, Rev. Mod. Phys. ,759 (2012).[11] L. Tanzi, E. Lucioni, F. Fam`a, J. Catani, A. Fioretti,C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno,Observation of a dipolar quantum gas with metastablesupersolid properties, Phys. Rev. Lett. , 130405(2019).[12] F. B¨ottcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn,M. Guo, T. Langen, and T. Pfau, Transient supersolidproperties in an array of dipolar quantum droplets, Phys.Rev. X , 011051 (2019).[13] L. Chomaz, D. Petter, P. Ilzh¨ofer, G. Natale, A. Traut-mann, C. Politi, G. Durastante, R. M. W. van Bijnen,A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino,Long-lived and transient supersolid behaviors in dipolarquantum gases, Phys. Rev. X , 021012 (2019).[14] M. Guo, F. B¨ottcher, J. Hertkorn, J.-N. Schmidt,M. Wenzel, H. P. B¨uchler, T. Langen, and T. Pfau, Thelow-energy goldstone mode in a trapped dipolar super-solid, Nature , 386 (2019).[15] G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Pet-ter, M. J. Mark, L. Chomaz, and F. Ferlaino, Excitationspectrum of a trapped dipolar supersolid and its experi-mental evidence, Phys. Rev. Lett. , 050402 (2019).[16] L. Tanzi, S. Roccuzzo, E. Lucioni, F. Fam`a, A. Fioretti,C. Gabbanini, G. Modugno, A. Recati, and S. Stringari,Supersolid symmetry breaking from compressional oscil-lations in a dipolar quantum gas, Nature , 382 (2019).[17] G. Birkl, S. Kassner, and H. Walther, Multiple-shellstructures of laser-cooled 24 mg+ ions in a quadrupolestorage ring, Nature , 310 (1992).[18] M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M.Itano, and D. J. Wineland, Ionic crystals in a linear paultrap, Phys. Rev. A , 6493 (1992).[19] S. Fishman, G. De Chiara, T. Calarco, and G. Morigi,Structural phase transitions in low-dimensional ion crys-tals, Phys. Rev. B , 064111 (2008).[20] E. Shimshoni, G. Morigi, and S. Fishman, Quantum zigzag transition in ion chains, Phys. Rev. Lett. ,010401 (2011).[21] W. K. Hew, K. J. Thomas, M. Pepper, I. Farrer, D. An-derson, G. A. C. Jones, and D. A. Ritchie, Incipient for-mation of an electron lattice in a weakly confined quan-tum wire, Phys. Rev. Lett. , 056804 (2009).[22] A. C. Mehta, C. J. Umrigar, J. S. Meyer, and H. U.Baranger, Zigzag phase transition in quantum wires,Phys. Rev. Lett. , 246802 (2013).[23] G. E. Astrakharchik, G. Morigi, G. De Chiara, andJ. Boronat, Ground state of low-dimensional dipolargases: Linear and zigzag chains, Phys. Rev. A , 063622(2008).[24] J. Ruhman, E. G. Dalla Torre, S. D. Huber, and E. Alt-man, Nonlocal order in elongated dipolar gases, Phys.Rev. B , 125121 (2012).[25] L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Roton-maxon spectrum and stability of trapped dipolar bose-einstein condensates, Phys. Rev. Lett. , 250403 (2003).[26] S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Radialand angular rotons in trapped dipolar gases, Phys. Rev.Lett. , 030406 (2007).[27] R. M. Wilson, S. Ronen, J. L. Bohn, and H. Pu, Mani-festations of the roton mode in dipolar bose-einstein con-densates, Phys. Rev. Lett. , 245302 (2008).[28] R. N. Bisset, D. Baillie, and P. B. Blakie, Roton excita-tions in a trapped dipolar bose-einstein condensate, Phys.Rev. A , 043606 (2013).[29] A. Gallem´ı, S. M. Roccuzzo, S. Stringari, and A. Recati,Quantized vortices in dipolar supersolid bose-einstein-condensed gases, Phys. Rev. A , 023322 (2020).[30] S. M. Roccuzzo, A. Gallem´ı, A. Recati, and S. Stringari,Rotating a supersolid dipolar gas, Phys. Rev. Lett. ,045702 (2020).[31] F. Ancilotto, M. Barranco, M. Pi, and L. Reatto,Vortex properties in the extended supersolid phaseof dipolar bose-einstein condensates, arXiv preprintarXiv:2012.15157 (2020).[32] Y.-C. Zhang, F. Maucher, and T. Pohl, Supersolidityaround a critical point in dipolar bose-einstein conden-sates, Phys. Rev. Lett. , 015301 (2019).[33] J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas,F. C¸ . Top, A. O. Jamison, and W. Ketterle, A stripe phase with supersolid properties in spin–orbit-coupledBose-–Einstein condensates, Nature , 91 (2017).[34] J. L´eonard, A. Morales, P. Zupancic, T. Esslinger, andT. Donner, Supersolid formation in a quantum gas break-ing a continuous translational symmetry, Nature , 87(2017).[35] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier,I. Ferrier-Barbut, and T. Pfau, Observing the rosensweiginstability of a quantum ferrofluid, Nature , 194(2016).[36] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, andT. Pfau, Observation of quantum droplets in a stronglydipolar bose gas, Phys. Rev. Lett. , 215301 (2016).[37] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. W¨achtler,L. Santos, and F. Ferlaino, Quantum-fluctuation-drivencrossover from a dilute bose-einstein condensate to amacrodroplet in a dipolar quantum fluid, Phys. Rev. X , 041039 (2016).[38] F. W¨achtler and L. Santos, Quantum filaments in dipo-lar bose-einstein condensates, Phys. Rev. A , 061603(2016).[39] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,Ground-state phase diagram of a dipolar condensate withquantum fluctuations, Phys. Rev. A , 033619 (2016).[40] L. Lavoine and T. Bourdel, 1d to 3d beyond-mean-field dimensional crossover in mixture quantum droplets(2020), arXiv:2011.12394 [cond-mat.quant-gas].[41] M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark,M. A. Norcia, and F. Ferlaino, Birth, life, and deathof a dipolar supersolid, arXiv preprint arXiv:2101.06975(2021).[42] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, andJ. Dalibard, Interference of an array of independentBose–Einstein condensates, Phys. Rev. Lett. , 180403(2004).[43] J.-N. Schmidt, J. Hertkorn, M. Guo, F. B¨ottcher,M. Schmidt, K. S. Ng, S. D. Graham, T. Langen,M. Zwierlein, and T. Pfau, Roton excitations in an oblatedipolar quantum gas, arXiv preprint arXiv:2102.01461(2021).[44] K. Pyka, J. Keller, H. Partner, R. Nigmatullin, T. Burg-ermeister, D. Meier, K. Kuhlmann, A. Retzker, M. B.Plenio, W. Zurek, et al. , Topological defect formation and spontaneous symmetry breaking in ion coulomb crystals,Nature communications , 1 (2013).[45] S. Ulm, J. Roßnagel, G. Jacob, C. Deg¨unther,S. Dawkins, U. Poschinger, R. Nigmatullin, A. Retzker,M. Plenio, F. Schmidt-Kaler, et al. , Observation of thekibble–zurek scaling law for defect formation in ion crys-tals, Nature communications , 1 (2013).[46] A. Trautmann, P. Ilzh¨ofer, G. Durastante, C. Politi,M. Sohmen, M. J. Mark, and F. Ferlaino, Dipolar quan-tum mixtures of erbium and dysprosium atoms, Phys.Rev. Lett. , 213601 (2018). Methods
Experimental apparatus and protocols:
Our ex-perimental apparatus has been described in detail inRef. [46]. Here, we evaporatively prepare up to N =6 . × condensed Dy atoms in a crossed opti-cal dipole trap formed at the intersection of two beamsderived from the same 1064 nm laser, although detunedin frequency to avoid interference. One beam (the staticODT) has an approximately 60 µ m waist. The second(the scanning ODT) has an 18 µ m waist, whose positioncan be rapidly scanned horizontally at 250 kHz to cre-ate a variably anisotropic time-averaged potential. Bytuning the power in each beam, and the scanning rangeof the scanning ODT, we gain independent control of thetrap frequencies in all three directions. The two trappingbeams propagate in a plane perpendicular to gravity, andcross at a 45 ° angle, which leads to the rotation of thezig-zag state at high α t visible in Fig. 2b.We apply a uniform magnetic field oriented along grav-ity and perpendicular to the intersecting dipole traps,with which we can tune the strength of contact interac-tions between atoms. This allows us to create unmod-ulated Bose-Einstein condensates, supersolid states, orstates consisting of independent droplets at fields of B =23.2 G, 17.92 G, and 17.78 G, respectively.Details of our imaging setup are provided in Ref. [41].In-trap and TOF images are performed along the verticaldirection (along B and gravity), using standard phase-contrast and absorption techniques, respectively. Theresolution of our in-trap images is approximately one mi-cron. We use a 36 ms TOF duration for imaging interfer-ence patterns. Atom number:
We extract the condensed atom number N from absorption imaging performed along a horizontaldirection in a separate set of experimental trials underotherwise identical experimental conditions. This allowsfor a larger field of view, and better fitting of thermalatoms. N is determined by subtracting the fitted thermalcomponent from the total absorption signal.For comparison between experiment and theory, andbetween the variational and eGPE theory methods, weassociate N with the number of atoms in the droplets,and not in the diffuse halo that surrounds the droplets.From simulation of TOF expansion, we find that the halo is repelled at early expansion times, and is likely indis-tinguishable from the thermal cloud in our TOF mea-surements. While it is possible that some of the halo iscounted in N , we neglect this possibility and assume that N includes only atoms within droplets. Scattering length:
The positions of phase boundariesbetween different droplet configurations are quite sensi-tive to the scattering length a , which is not known withhigh precision in our range of magnetic fields. For alltheory, we use a value of a = 88 a , where a is the Bohrradius, as this value provides good agreement betweenexperiment and theory for the 1D-to-2D transition point. Extracting critical aspect ratio:
The critical aspectratio α ∗ t is extracted from fit to the function α a = α for α t < α ∗ t , α a = (cid:112) α + b ( α t − α ∗ t ) for α t > α ∗ t , where α ∗ t , α , and b are fit parameters. The error bars reportedin Fig. 2c represent the standard error on the mean, andare smaller than the markers on most points. Interference patterns:
The predicted interference pat-terns of Fig. 3 are calculated by assuming free expansionof Gaussian droplets. In reality, the droplets are prob-ably not Gaussian, and interactions during TOF expan-sion may modify the interference pattern. However, thedroplet shape primarily effects the envelope of the inter-ference pattern, which is not our primary interest here,and from eGPE simulations, we expect the effects of in-teractions to be minor, provided that the droplets be-come unbound in a time short compared to the TOF,which we verify by both looking at shorter TOFs andcomparing the fringe spacing observed in TOF with thatexpected from the in-trap droplet spacing. The positionsand size of the droplets are tuned to provide illustrativeinterference patterns.
Droplet number:
We extract the droplet number fromour in-trap images using a peak-finding algorithm ap-plied to smoothed images. The algorithm finds the localmaxima above a threshold, which is chosen to be 40%of the overall peak value. Each in-trap density distribu-tion is classified as linear array or 2D zig-zag based onthe atomic aspect ratio. Finally, the counts with a givendroplet number are normalized by the total number oftrials to get the probability shown in Fig. 2d. Fluctua-tions in the number of atoms in a given trial can pushdroplets above or below the threshold value, contributingto the spread in extracted droplet number for a given α tt