Birth, life, and death of a dipolar supersolid
Maximilian Sohmen, Claudia Politi, Lauritz Klaus, Lauriane Chomaz, Manfred J. Mark, Matthew A. Norcia, Francesca Ferlaino
BBirth, life, and death of a dipolar supersolid
Maximilian Sohmen,
1, 2
Claudia Politi,
1, 2
Lauritz Klaus,
1, 2
Lauriane Chomaz, Manfred J. Mark,
1, 2
Matthew A. Norcia, 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 (Dated: January 19, 2021)In the short time since the first observation of supersolid states of ultracold dipolar atoms, substan-tial progress has been made in understanding the zero-temperature phase diagram and low-energyexcitations of these systems. Less is known, however, about their finite-temperature properties,particularly relevant for supersolids formed by cooling through direct evaporation. Here, we explorethis realm by characterizing the evaporative formation and subsequent decay of a dipolar supersolidby combining high-resolution in-trap imaging with time-of-flight observables. As our atomic systemcools towards quantum degeneracy, it first undergoes a transition from thermal gas to a crystallinestate with the appearance of periodic density modulation. This is followed by a transition to asupersolid state with the emergence of long-range phase coherence. Further, we explore the role oftemperature in the development of the modulated state.
Supersolid states, which exhibit both global phase co-herence and periodic spatial modulation [1–7], have re-cently been demonstrated and studied in ultracold gasesof dipolar atoms [8–10]. These states are typically ac-cessed by starting with an unmodulated Bose–Einsteincondensate (BEC), and then quenching the strength ofinteratomic interactions to a value that favors a density-modulated state. In this production scheme, the super-fluidity (or global phase coherence) of the supersolid isinherited from the pre-existing condensate. However, adipolar supersolid state can also be reached by directevaporation from a thermal gas with fixed interactions,as demonstrated in Ref. [10].A thermal gas at temperatures well above condensa-tion has neither phase coherence nor modulation, so bothmust emerge during the evaporative formation process.This leads one to question whether these two features ap-pear simultaneously, or if not, which comes first. Further,because this transition explicitly takes place at finite tem-perature T , thermal excitations may play an importantrole in the formation of the supersolid, presenting a chal-lenging situation for theory. Moreover, in the case ofa dipolar supersolid, the non-monotonic dispersion rela-tion and the spontaneous formation of periodic densitymodulation lead to important new length- and energy-scales not present in contact-interacting systems, whichdramatically modify the evaporative formation process.While the ground state and dynamics of a zero-temperature dipolar quantum gas can be computed bysolving an extended Gross–Pitaevskii equation [8, 11–17](see also Fig. 1a), similar treatments are currently lack-ing for finite temperatures in the supersolid regime. Inprinciple, effects of finite temperature can be taken intoaccount by perturbatively including the thermal popula-tion of excited modes. This can be done either coher-ently, by adding them in a single classical field whichabides the Gross–Pitaevskii equation, as in Refs. [18–20], or incoherently, by iteratively computing mode popula-tions via a set of coupled Hartree–Fock–Bogoliubov equa-tions [9, 21, 22]. In order to accurately describe dynami-cal processes occurring at temperatures approaching thecritical temperature, both coherent excitations and in-coherent interactions with the background thermal gasmust be accounted for, requiring either more advancedc-field [18] or quantum Monte Carlo [23–27] techniques.So far, theories with realistic experimental parametershave not been developed to unveil the finite-temperaturedipolar phase diagram and to determine the propertiesof the thermal-to-supersolid phase transitions.In this work, we experimentally study the evaporativetransition into and out of a supersolid state in a dilutegas of dysprosium atoms. As the atoms cool down toquantum degeneracy, the number of condensed atomsincreases, leading to the birth of the supersolid state.Continued evaporation and collisional loss lead to a re-duction of atom number, and eventually the death ofthe supersolid. Such an evaporation trajectory, as illus-trated in Fig. 1a, passes through the little-understoodfinite-temperature portion of the supersolid phase dia-gram. During the evaporative birth of the supersolid,we discover that the system first establishes strong peri-odic density modulation of locally coherent atoms, andonly later acquires long-range phase coherence. Whencomparing the birth and death of the supersolid, whichoccur at different temperatures, we observe higher levelsof modulation during the birth, suggesting that thermalfluctuations may play an important role in the formationof density modulation.For our experiments, we first prepare an opticallytrapped gas of approximately 10 dysprosium atoms (iso-tope Dy), precooled via forced evaporation to temper-atures of several hundred nanokelvin, at which point thegas remains thermal. From here, we can apply furtherevaporation either by a nearly-adiabatic ramp-down of a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n N c Ta s iiiBEC SSS ID a b -50 -25 0 25 50 c FIG. 1.
Evaporation trajectory through the finite-temperature phase diagram. a. At T = 0 (bottomplane), the phase diagram for a gas of dipolar atoms isspanned by the s-wave scattering length a s and the conden-sate atom number N c . In an elongated trap it features aBEC (white) and independent droplet (ID, black) phases, sep-arated in places by a supersolid state (SSS, gray-scale). Theplotted lightness in the T = 0 phase diagram represents thedroplet link strength across the system (cf. [16]). Away from T = 0, the phase diagram is not known. We explore thisregion through evaporation into (near i) and out of (near ii)the SSS, along a trajectory represented schematically by thered arrow. b. Single-shot in-situ image of the density dis-tribution in trap. Here, a system of four “droplets” withinthe SSS region is shown, together with its projected densityprofile. c. Single-shot matter-wave interference pattern after35 ms TOF expansion, and the corresponding integrated pro-file. The background clouds of thermal atoms present are notvisible in the color scale of subfigures b, c. the trap depth (“slow ramp”), or by a rapid reductionof the trap depth followed by a hold time at fixed depth(“fast ramp”) to further lower the temperature and in-duce condensation into the supersolid state. The “slowramp” protocol yields a higher number of condensedatoms ( N c ∼ × ; see next paragraph for defini-tion), and lower shot-to-shot atom number fluctuations,whereas the “fast ramp” protocol ( N c ∼ ) allows tofollow the evolution of the system in a constant trap, dis-entangling the system dynamics from varying trap pa-rameters. In contrast to protocols based on quenchingthe interactions in a BEC [8–10], we hold the magneticfield (and hence the contact interaction strength) fixedduring the entire evaporation process at 17.92 G, wherethe system ground state at our N c is a supersolid (scat-tering length ∼ a ).For the present work, we have implemented in-situ Fara-day phase contrast imaging [28, 29], which allows usto probe the in-trap density of our quantum gas atmicron-scale resolution. During the formation of thedensity-modulated state, the translation symmetry isbroken along the long (axial) direction of our cigar-shaped trap [30], typically giving rise to a chain of threeto six density peaks, which we call “droplets.” Thesedroplets have a spacing of roughly three microns, clearly visible in our in-situ images (Fig. 1b). As in our pre-vious works [10, 16], we also image the sample after atime-of-flight (TOF) expansion using standard absorp-tion imaging. These TOF images include a spatiallybroad contribution which we attribute to thermal atoms,whose number N th and temperature T we estimate by2D-fitting of a Bose-enhanced Gaussian function [31], ex-cluding the cloud centre. Surplus atoms at the cloudcentre (compared to the broad Gaussian) are at leastlocally coherent, or “(quasi-)condensed” in the sense ofRefs. [32–34]. With the total number of atoms N mea-sured by pixel count, we define N c = N − N th to bethe number of these (at least locally) coherent atoms.During TOF, matter-wave interference between the ex-panding droplets gives rise to a characteristic interfer-ence pattern (Fig. 1c). The high contrast of the inter-ference pattern is visible in single TOF images and in-dicates that each individual droplet is by itself a phasecoherent many-body object. The stability of the inter-ference fringes within the envelope over multiple exper-imental realisations encodes the degree of phase coher-ence between droplets (cf. Refs. [10, 16] and discussionbelow). The combination of in-situ and TOF diagnos-tics provides complementary information allowing us tomeasure both density modulation and its spatial extent(number of droplets), as well as phase coherence.Figure 2 shows the birth of the supersolid. Start-ing from a thermal sample, we apply the “fast ramp”(225 ms) evaporation protocol to the desired final trapdepth, too fast for the cloud to follow adiabaticallyand intermediately resulting in a non-thermalized, non-condensed sample. Simply holding the sample at con-stant trap depth for a time t h , plain evaporation andcollisions lead to cooling and thermalization towards thesystem ground state. In Figure 2a we plot the average ax-ial in-situ density profile (cf. Fig. 1b) versus t h , for about20 images per time step without any image recentering.At early t h the atoms are primarily thermal, and showup as a broad, low-density background in our images.For t h < ∼
150 ms, inspection of single-shot images revealsan increasing, though substantially fluctuating number ofdroplets appearing out of the thermal cloud. After thistime, the droplet number stabilizes to its final value.To better quantify the growth of the modulated statewe consider the density-density correlator C (cid:48) ( d ) for thein-situ density profiles over distances d (see supplemen-tary material for details). We find that C (cid:48) ( d ) is well de-scribed by a cosine-modulated Gaussian, and define thedensity correlation length L (Fig. 2b) as its fitted width.This method provides a way to determine the extent overwhich density modulation has formed. Figure 2c shows L for the data set of Fig. 2a versus the number of coher-ent atoms N c , which we extract from TOF absorptionimages in separate experimental trials with identical pa-rameters. Interestingly, despite the strongly modulatedstructure of the supersolid state, the density correlation hold time t h (s) -10-50510 p o s i t i o n ( m ) a coherent atoms N c (1000) c o rr e l a t i o n l e n g t h L ( m ) c -10 -5 0 5 10-0.050.00.050.1 C ( d ) b -10 -5 0 5 10 distance d ( m) -0.20.00.20.4 C ( d ) FIG. 2.
Growth and spread of density modulationduring evaporation. a.
Averaged density profiles (no re-centering, approximately 20 shots per timestep) along thelong trap axis as a function of hold time t h after the “fastramp” reduction of trap depth (see main text). b. The den-sity correlator C (cid:48) ( d ) (solid black line) is fitted by a cosine-modulated Gaussian function (dashed red line) to extract thecorrelation length L . Gray regions are strongly influenced byimaging noise and excluded from fits. Correlators are dis-played for t h = 50 ms (upper) and t h = 300 ms (lower). c. Density-density correlation length L versus N c , for the sametimesteps shown in a. Horizontal error bars are the standarddeviation over repetitive shots, vertical error bars reflect thecorrelator fit uncertainty, the red points correspond to thecorrelators of subfigure b. The dashed line indicates the sim-ple atom-number scaling of the Thomas–Fermi radius of aharmonically trapped BEC, ∝ N / c . length L closely follows a scaling ∝ N / c , just as theThomas–Fermi radius of a harmonically trapped BEC,suggesting a dominant role of interactions over kineticenergy.While in-situ images provide information about den-sity modulation (diagonal long-range order), they donot carry direct information about phase coherence (off-diagonal long-range order), either within, or betweendroplets. For this, we use TOF imaging and address thequestion of whether the formation of density modulationprecedes global (i. e. interdroplet) phase coherence dur-ing evaporative formation of the supersolid, or the otherway round. crop time t c (s) ( a r b . u . ) T ( n K ) N c ( ) , N ( ) a b c A M A FIG. 3.
Development of modulation and coherencewhile evaporating into the supersolid state. a,
Sampletemperature T (left ordinate, bullets), total ( N , right ordi-nate, dashed red line) and coherent atom number ( N c , solidred line) as a function of the ramp crop time t c . The shadingsreflect the respective confidence intervals. b, The phasors P i (black dots), representing the magnitude and phase coherenceof modulation for selected t c (dotted lines; same radial scalefor all polar plots). The red shading reflects mean and vari-ance of the distribution. c, Evolution of the Fourier amplitudemeans A M (filled markers) and A Φ (open markers). For this study, we perform a “slow” (500 ms) finalforced evaporation ramp of constant slope that is nearlyadiabatic, and terminate the ramp at selected crop times t c [35]. After t c , we immediately release the atoms andperform TOF imaging. Figure 3a shows the observedevolution of the total ( N ) and (quasi-)condensed ( N c )atom number as well as the sample temperature ( T ) ver-sus t c . We expand on the observed evolution by mea-suring coherence properties. Following Refs. [10, 16],for each measurement i we extract a complex phasor P i = ρ i exp ( − iΦ i ), i. e. the Fourier component corre-sponding to the modulation wavelength in the TOF in-terference profile. For systems with a small number ofdroplets (but at least two), the magnitude of the pha-sor ρ i encodes the modulation strength and also the (lo-cal) degree of coherence within each of the individualdroplets. Meanwhile, the phase Φ i depends primarily onthe relative phase between the droplets (cf. [36]).We plot the phasors for different evaporation times onthe polar plane in Fig. 3b, where two effects become ap-parent. First, the modulus of the phasors grows duringthe evaporation, indicating that the degree of modula-tion increases. Second, the distribution of phases Φ i isinitially uniform, and then narrows down over t c . Todetermine the time sequence of these two effects, wecalculate the incoherent and coherent amplitude means, A M = (cid:104)| P { i } |(cid:105) , encoding modulation strength and localphase coherence, and A Φ = |(cid:104) P { i } (cid:105)| , encoding the de-gree of global phase coherence across the system [10, 16].Plotting A M and A Φ against t c (Fig. 3c), we notice atime lag of around 40 ms between the increase of A M and A Φ , indicating that during evaporation into a super-solid the translational and the phase symmetry are notbroken simultaneously [37]. Rather, density modulationand local phase coherence appear before global phase co-herence, consistent with predictions from Monte Carlosimulations, cf. e. g. [27].This observation suggests the transient formation of aquasi-condensate crystal – a state with local but not long-range coherence [32–34], whose increased compressibilityrelative to a thermal gas allows for the formation of den-sity modulation [38] – prior to the formation of a super-solid with phase coherence between droplets. The lack ofglobal phase coherence could be attributed to a Kibble–Zurek-type mechanism [39], in which different regions ofthe sample condense independently, or to the thermalpopulation of collective modes (which reduce long-rangecoherence) at finite temperature. As the evaporation pro-cess does not allow independent control of temperatureand condensation rate without also changing density ortrap geometry, we cannot reliably determine the relativeimportance of these effects (or others) from the experi-ment. Dedicated theoretical studies at finite temperaturewill thus be needed to elucidate the impact of these typesof processes and to understand the exact formation pro-cess.After the birth of the supersolid state, both densitymodulation and global phase coherence persist for re-markably long times, exceeding one second. Figure 4shows the evolution of the coherent atom number N c andtemperature T versus the hold time t h under conditionssimilar to Fig. 2. Evaporative cooling first increases thecoherent atom number until, at long t h , atom losses be-come dominant and lead to a continuous decrease of N c ,eventually leading to the disappearance of the modulatedstate. However, this death of the supersolid is not a meretime-reversal of the birth. N c decreases, i. e. evolves inthe opposite direction, but more slowly and at lower tem-perature than for the birth. Thus, a comparison of thesetwo processes provides us with important clues to theimpact of temperature on the supersolid.We contrast the birth and death of the supersolid inFig. 4 by also plotting the observed in-situ density mod-ulation M , which is calculated by Fourier transformingthe in-situ density profiles and normalizing the Fouriercomponent corresponding to the modulation wavelengthto the zero-frequency Fourier component. By compar-ing M between times that have similar N c during the -3 -1 hold time t h (s) m odu l a t i on M -0.500.511.52 N C ( )
50 75 100 125 150T (nK) N C (10 ) FIG. 4.
Lifecycle of a supersolid state.
Density mod-ulation M (from in-situ images) during the evaporation pro-cess (left ordinate, bullets; the vertical error bars reflect thepropagated uncertainty returned by the fitting routine). Thesample temperature decreases during the hold time t h and isencoded by the color filling. N c (from TOF images) is thenumber of coherent atoms over t h (right ordinate, red line;the light-red shading reflects the measurement standard de-viation). At two times where N c ∼ . × (vertical dashedlines), but at which the atoms have different temperatures, M differs substantially. The corresonding averaged in-situimages below confirm a higher level of modulation at earlier t h . Inset: The observed modulation M plotted versus N c . birth and the death of the supersolid, respectively, wefind that the degree of modulation is higher during thebirth of the supersolid than during the death. Becausethe sample is hotter at shorter hold times, this suggeststhat the observed modulation is increased at higher tem-perature, perhaps due to thermal population of collectivemodes, or due to finite-temperature modifications to thedispersion relation [40], as predicted in Ref. [22]. Again,further development of finite-temperature theory will beneeded to conclusively determine the importance of sucheffects.The role of finite temperature in the formation of mod-ulation, as well as the mechanism by which phase vari-ations across the modulated state arise and then ulti-mately disappear, represent important future directionsfor theoretical investigations of dipolar supersolids awayfrom the relatively well understood T = 0 limit. Ex-perimentally, it would be of great interest to study theevaporative formation process in a larger and more uni-form system, where distinct domains may be observedto form, and a broader separation of length-scales maybe explored in correlation measurements. Such measure-ments, along with improved finite-temperature theory,could enable more precise statements as to the nature ofthe supersolid phase transition away from zero tempera-ture.We thank Russell Bisset, the Innsbruck Erbium team,Massimo Boninsegni, Philippe Chomaz, Thomas Pohl,Nikolay Prokof’ev and Boris Svistunov for insightful dis-cussions, and Gianmaria Durastante, Philipp Ilzh¨oferand Arno Trautmann for early contributions. Thiswork is financially supported through an ERC Con-solidator Grant (RARE, No. 681432), an NFRI grant(MIRARE, No. ¨OAW0600) of the Austrian Academy ofScience, the QuantERA grant MAQS by the AustrianScience Fund FWF No. I4391-N, and the DFG/FWFvia FOR 2247/PI2790. M.S. acknowledges support bythe Austrian Science Fund FWF within the DK-ALM(No. W1259-N27). M.A.N. has received funding as anESQ Postdoctoral Fellow from the European Union’sHorizon 2020 research and innovation programme underthe Marie Sk(cid:32)lodowska-Curie grant agreement No. 801110and the Austrian Federal Ministry of Education, Sci-ence and Research (BMBWF). M.J.M. acknowledgessupport through an ESQ Discovery Grant, by the Aus-trian Academy of Sciences. L.C. acknowledges supportthrough the FWF Elise Richter Fellowship No. V792. Wealso acknowledge the Innsbruck Laser Core Facility, fi-nanced by the Austrian Federal Ministry of Science, Re-search and Economy. ∗ Correspondence should be addressed to
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