Observation of Roton Mode Population in a Dipolar Quantum Gas

L. Chomaz1, R. M. W. van Bijnen2, D. Petter1, G. Faraoni1,3, S. Baier1, J. H. Becher1, M. J. Mark1,2, F. Wachtler¨ 4, L. Santos4, F. Ferlaino1,2,∗ 1Institut fur¨ Experimentalphysik,Universitat¨ Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria 2Institut fur¨ Quantenoptik und Quanteninformation,Osterreichische¨ Akademie der Wissenschaften, 6020 Innsbruck, Austria 3Dipartimento di Fisica e Astronomia, Universita` di Firenze, Via Sansone 1, 50019 Sesto Fiorentino, Italy 4Institut fur¨ Theoretische Physik, Leibniz Universitat¨ Hannover, Appelstr. 2, 30167 Hannover, Germany

The concept of a roton, a special kind of elementary ex- the interparticle interactions - and correlations - are typically citation, forming a minimum of energy at finite momen- weak, meaning that classically their range of action is much tum, has been essential to understand the properties of su- smaller than the mean interparticle distance. Because of this perfluid 4He [1]. In quantum liquids, rotons arise from diluteness, roton excitations are absent in ordinary quantum the strong interparticle interactions, whose microscopic gases, i. e. in Bose-Einstein condensates (BECs) with contact description remains debated [2]. In the realm of highly- (short-range) interactions [12]. However, about 15 years ago, controllable quantum gases, a roton mode has been pre- seminal theoretical works predicted the existence of a roton dicted to emerge due to magnetic dipole-dipole interac- minimum both in BECs with magnetic dipole-dipole inter- tions despite of their weakly-interacting character [3]. actions (DDIs) [3] and in BECs irradiated by off-resonant This prospect has raised considerable interest [4–12]; yet laser light [15]. Following the lines of the latter proposal, roton modes in dipolar quantum gases have remained elu- a roton softening has been recently observed in BECs cou- sive to observations. Here we report experimental and pled to an optical cavity [16]. Here, the excitation arises theoretical studies of the momentum distribution in Bose- from the infinite-range photon-mediated interactions and the Einstein condensates of highly-magnetic erbium atoms, inverse of the laser wavelength sets the value of krot. Addi- revealing the existence of the long-sought roton mode. By tionally, roton-like softening has been created in spin-orbit- quenching the interactions, we observe the roton appear- coupled BECs [17, 18] and quantum gases in shaken optical ance of peaks at well-defined momentum. The roton mo- lattices [19] by engineering the single-particle dispersion re- mentum follows the predicted geometrical scaling with the lation. inverse of the confinement length along the magnetisation Our work focuses on dipolar BECs (dBECs). As in super- axis. From the growth of the roton population, we probe fluid 4He, the roton spectrum in such systems is a genuine the roton softening of the excitation spectrum in time and consequence of the underlying interactions among the parti- extract the corresponding imaginary roton gap. Our re- cles. However, in contrast to helium, the emergence of a mini- sults provide a further step in the quest towards superso- mum at finite momentum does not require strong inter-particle lidity in dipolar quantum gases [13]. interactions. It instead exists in the weakly-interacting regime Quantum properties of matter continuously challenge our and originates from the peculiar anisotropic and long-range intuition, especially when many-body effects emerge at a character of the DDI in real and momentum space [3–12]. De- macroscopic scale. In this regard, the phenomenon of su- spite the maturity achieved in the theoretical understanding, perfluidity is a paradigmatic case, which continues to reveal the observation of dipolar roton modes has remained so far fascinating facets since its discovery in the late 1930s [2, 12]. an elusive goal. For a long time, the only dBEC available in A major breakthrough in understanding superfluidity thrived experiments consisted of chromium atoms [20], for which the on the concept of quasiparticles, introduced by Landau in achievable dipolar character is hardly sufficient to support a 1941 [1]. Quasiparticles are elementary excitations of mo- roton mode. With the advent of the more magnetic lanthanide mentum k, whose energies ε define the dispersion (energy- atoms [21, 22], a broader range of dipolar parameters became momentum) relation ε(k). available, opening the way to access the regime of dominant To explain the special thermodynamic properties of super- DDI. In this regime, novel exciting many-body phenomena fluid 4He, Landau postulated the existence of two types of have been recently observed, as the formation of droplet states low-energy quasiparticles: phonons, referring to low-k acous- stabilised by quantum fluctuations [23–25], which may be- tic waves, and rotons, gapped excitations at finite k initially in- come self-bound [26]. Lanthanide dBECs hence open new terpreted as elementary vortices. The dispersion relation con- roads toward the long-sought observation of roton modes. arXiv:1705.06914v2 [cond-mat.quant-gas] 27 Mar 2019 tinuously evolves from linear at low k (phonons) to parabolic- Prior to this work, dipolar rotons have been mostly con- like with a minimum (roton) at a finite k = krot. Neutron nected to pancake-like geometries [3–6, 8–12]. Here, we ex- scattering experiments confirmed Landau‘s remarkable intu- tend the study of roton physics to the case of a cigar-like ge- 4 ition [14]. In liquid He, krot scales as the inverse of the inter- ometry with trap elongation along only one direction (y) trans- atomic distance. This manifests a tendency of the system to verse to the magnetisation axis (z) (Fig. 1a). The anisotropic establish a local order, which is driven by the strong correla- character of the DDI (Fig. 1b, inset) together with the tighter tions among the atoms [2]. confinement along z is responsible for the rotonization of the In the realm of low-temperature quantum physics, ultra- excitation spectrum along y (Fig. 1b). To illustrate this phe- cold quantum gases realise the other extreme limit for which nomenon, we consider an infinite cigar-shaped dBEC and 2

ñ (a.u.) z the system undergoes a roton instability and the population at 1.0 a c ky = 0 is transferred in ±krot at an exponential rate [5, 6]. The 0.8 population of the roton mode is then readily visible in the mo-

x y 0.6 kx mentum distribution of the gas (Fig. 1c-d). In the extensively- 0.4 studied pancake geometries, the roton population in k-space 0.2 spreads over a ring of radius k = k because of the radial 0 rot attraction k b y symmetry of the conﬁnement (Fig. 1d). Such a spread can be repulsion d avoided using a cigar geometry. Here, the roton population

focuses in two prominent peaks at k = ±k , enhancing the a y rot s visibility of the effect (Fig. 1c). We explore the above-described physics using strongly 166

Energy, ϵ (k) magnetic Er atoms. The experiment starts with a stable

kx dBEC in a cigar-shaped harmonic trap of frequencies νx,y,z, elongated along the y axis. The trap aspect ratio, λ = νz/νy, k Momentum, ky rot can be tuned from about 4 to 30, corresponding to νz rang- k y ing from 150 Hz to 800 Hz, whereas νy and νz/νx are kept b1 b2 constant at about 35Hz and 1.6, respectively (Methods). An external homogeneous magnetic field, B, fixes the dipole ori- z entation (magnetisation) with respect to the trap axes and sets the values of as through a magnetic Feshbach resonance, cen- dominantly repulsive DDI y dominantly attractive DDI tered close to B = 0G. In previous experiments, we precisely calibrated the B-to-a conversion for this resonance [25]. The Figure 1 — Roton mode in a dBEC. a, axially elongated ge- s BEC is prepared at ai = 61a (B = 0.4G) with transverse ometry with dipoles oriented transversely. b, real (solid lines) s 0 (z) magnetisation. The characteristic dipolar length, defined and imaginary (norm of the dotted line) parts of the disper- as a = µ µ2 m/12πh¯ 2, is 65.5a , with m the mass and µ sion relation of a dBEC in the geometry a, showing the emer- dd 0 m 0 m the magnetic moment of the atoms, h¯ = h/2π is the reduced gence of a roton minimum for decreasing a (dashed arrow). s Planck constant, µ the vacuum permeability and a the Bohr The DDI changes from repulsive (blue) to attractive (red) de- 0 0 radius. pending on the dipole alignment (inset). b1, b2, the dipole alignment (color code as in inset) associated to small- (b1) To excite the roton mode, we quench as to a desired lower value, af, and shortly hold the atoms in the trap for a time and large-ky (b2) density modulations. c, d, distributions on s th. We measure that as converges to its set value with a the kx ky-plane associated with the roton population in cigar a or pancake d geometries with an identical roton population characteristic time constant of 1ms during th. We then re- and colorscale. lease the atoms from the trap, change as back close to its initial value and let the cloud expand for 30ms. We probe the momentum distributionn ˜(kx,ky) by performing standard resonant absorption imaging on the expanded cloud (Meth- focus on its axial elementary excitations, of momentum ky. ods). The measurement is then repeated at various values of These excitations correspond in real space to a density modu- as < add in a fixed trap geometry. The momentum distribu- lation along y of wavelength 2π/ky. For low ky, the atoms sit tion shows a striking behaviour (Fig. 2). For large enough as, mainly side-by-side and the repulsive nature of the DDI pre- n˜(kx,ky) shows a single narrow peak with an inverted aspect vails, stiffening the phononic part of the dispersion relation ratio compared to the trapped gas, typical of a stable BEC [12] (Fig. 1b1). In contrast, for ky`z & 1, `z being the characteristic (Fig. 2a). We define the center of the distribution as the origin z confinement length, the excitation favours head-to-tail align- of k. In contrast, when further decreasing as below a criti- ∗ ments and the DDI contribution to ε(ky) eventually changes cal value as , we observe a sudden appearance of two sym- sign [3] (Fig. 1b2). The resulting softening of ε(ky) is counter- metric finite-momentum peaks, of similar shape and located balanced by the contributions of the repulsive contact interac- at ky = ±krot (Fig. 2b-c). By repeating the experiment sev- tion, and of the kinetic energy, which ultimately dominates at eral times, we observe that the peaks consistently appear at very large ky. For strong enough DDI, this competition gives the same locations, and they are visible in the averaged dis- rise to a roton minimum in ε(ky), occurring at a momentum tributions. To quantitatively investigate the peak structures, ky = krot set by the geometrical scaling krot ∼ 1/`z (see below we fit a sum of three Gaussian distributions to the central cuts and e.g. [3, 8, 9]). of the averagen ˜(kx,ky) (Fig. 2a1-c1). From the fit we extract Similar to the helium case, the roton energy gap, ∆ = the central momentum, krot, and the amplitude, A∗, of the side ε(krot), depends on the density and on the strength of the inter- peaks. actions. In ultracold gases, both quantities can be controlled. A major fingerprint of the roton mode in dBECs is its geo- In particular, the scattering length as, setting the strength of metrical nature, leading to a universal scaling krot ∼ 1/`z (see the contact interaction, can be tuned using Feshbach reso- e.g. [3, 4, 8, 9]). In addition, the dependency of krot on as close nances [12]. As as is reduced, ∆ decreases, vanishes, and to the instability is expected to be mild as krot remains mainly eventually becomes imaginary (Fig. 1b). In the latter case, set by its geometrical nature [3, 8]. We investigate both depen- 3

ñ (103 µm2) ñ (103 µm2) ñ (103 µm2) our system and its dynamical population, we develop both an 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 analytical model and full numerical simulations and compare a b c the ﬁndings with our experimental data. Our analytical model 5 starts by calculating the roton spectrum of the stationary BEC, )

-1 generalizing the results of Ref. [3] to a non-radially symmet- 0

(µm ric conﬁguration. Since the roton wavelength is much smaller x k than the extension of our 3D BEC along y, this mode can be -5 evaluated using a local density approximation in y. Hence,

for our model, we consider a dBEC homogeneous along y, 2 0.8 of axial density n0, harmonically conﬁned along x and z. To ) 2 a1 b1 0.4 c1 analytically evaluate the roton spectrum, we approximate the µm 3 1 0.4 BEC wavefunction using the Thomas-Fermi (TF) approxima- 0.2 tion. For dominant DDI, εdd = add/as ≥ 1, we ﬁnd that ε(ky) ñ (10 indeed rotonizes (Methods). In the vicinity of the roton min- 0 0 0 imum and for ε ∼ 1, the computed dispersion acquires a -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 dd -1 -1 -1 ky(µm ) ky(µm ) ky(µm ) gapped quadratic form similar to that of helium rotons: 2 2 2 6 2 2 2h¯ krot h¯ 2

) d e ε(ky) ' ∆ + (ky − krot) . (1) -1 m 2m

(µm √ 2 2 1/4 rot 5 The roton momentum reads as krot = 2m(E − ∆ ) /h¯, q 0 2 2 2 with ∆ = E − EI , EI = 2gn0(εdd − 1)/3, g = 4πh¯ as/m, 4 0 2 h¯ 2 −2 −2 and E0 = 2gεddn0 2m X + Z . The TF radii verify 2 2 3 X ,Z ∝ gn0 so that E0 ∝ hνz, with a scale set by εdd and νz/νx but independent of gn0 [15]. Close to the instability (∆ ' 0),

Roton momentum, k 2 the roton momentum thus follows a simple geometrical scaling, krot = κ/`z with the geometrical factor κ depending on 1.5 2 2.5 3 3.5 35 40 45 50 νz/νx alone. For completeness, we have also performed full -1 z-confinement scale, 1/ℓz (µm ) Scattering length, as (a0) 3D numerical calculations of the static Bogoliubov spectrum of a finite trapped dBEC, confirming the existence and scaling Figure 2 — Observed roton peaks and characteristic scal- of the roton mode (Supplementary Information). ings. a-c,n ˜(kx,ky) obtained by averaging 15-to-25 absorp- The above stationary description accounts for the existence tion images for (νz,λ) = (456Hz,14.4), th = 3ms and: a, of the roton mode in the cigar geometry used in experiments, as = 54a0, b, as = 44a0 and c, as = 37a0. a1-c1, correspond- and predicts the scaling of krot and ∆ with the system param- ing cuts at kx ≈ 0 (dots) and their fits to three-Gauss distri- eters. However the quench of as introduces a dynamic, which butions (lines), from which we extract krot and A∗. d, Mea- is crucial to quantitatively reproduce the experimental obser- ∗ sured krot depending on 1/`z at as . as for νy ≈ 35Hz (cir- vations. The reduction of as decreases the contact interaction cles) and νy = 17(1)Hz (triangle). e, krot depending on as for and additionally induces a compression of the cloud. This (νz,λ) = (149Hz,4.3) (circles). In (d-e), error bars show the yields a dynamical modification of the local roton dispersion 95% confidence interval of the three-Gauss fits (see a1-c1). relation, and the roton may be destabilised during the evo- The squares (diamonds) show predictions from the SSM (NS) lution. This dynamical destabilisation is well accounted for in their range of validity (Methods). Lines are guides-to-the- by a self-similar model (SSM) describing the evolution of the eye. cloud shape after the quench [12]. In particular, we consider a 3D harmonic confinement, and, starting from the stationary i TF profile at as, we evaluate the evolution of the cloud along dencies in the experiment. In a first set of experiments, we re- the change of as assuming that the TF shape is maintained but peat the quench measurements for various trap parameters and with time-dependent TF radii (Methods). We then estimate extract k . We clearly observe the expected geometrical scal- the local (along y) instantaneous roton spectrum ε(k ,y,t ) us- rot p y h ing. krot shows a marked increase with 1/`z = 2π mνz/h, ing a local density approximation, i.e. evaluating Eq. (1) with matching well with a linear progression with a slope of 1.61(4) the experimentally calibrated as(th), and the n0(y,th), X(y,th), (Fig. 2d). Note that no dependence of krot on νy is observed. In Z(y,th) estimated from the 3D profile (Methods). We indeed a second set of experiments, we fix the trap geometry and ex- find that the roton gap decreases and eventually turns imagi- 2 plore the dependence of krot on as. We observe that, within our nary (∆(th) < 0) (Fig. 3a). When this occurs the population at experimental uncertainty, krot stays constant when decreasing ky ≈ krot exponentially grows with an instantaneous local rate as (Fig. 2e). This behaviour contrasts the one expected for 2Im[ε(ky,y,th)]/h¯, Im[.] indicating the imaginary part, giving a phonon-driven modulation instability that exhibits a strong rise to two symmetric side peaks in the axial momentum dis- R R th as-dependence [27]. tributionn ˜1(ky,th) ∝ exp 2 0 Im[ε(ky,y,t)]dt dy (Fig. 3b) To gain a deeper understanding of the roton excitations in (Methods). The center of the peaks, ±krot(th), also evolves 4

1 -6 ) 2 d ν ñ (μm) 2 0 1

(h -4

2 >6000 )

-1 -1 ∆ m -2 0a 1 2 3 4 5 6 -2 5000 ( µ 8 y 100 4000 10 0 6 1 3000 2 0.1 2000 0.01 Momentum, k 4 4 10-3 1000 0 1 2 3 4 5 6 t (ms) h 6 0 2 -10 e -1 n1(μm ) Renormalised roton population b 10000 0 -5 ) m) -1 3 8000 (µm rot 0 6000 2

Position, y ( µ 4000 1 5 2000 c 0 10 0 Roton momentum, k 0 1 2 3 4 5 6 0 2 4 6 8 Hold time, t (ms) Hold time, th(ms) h

f Figure 3 — Dynamics of the roton mode. Results depending on th after quenching to as = 50a0 in (νz,λ) = (149Hz,4.3). a-c, ∆2 at y = 0, amplitude and momentum of the roton peak, from the SSM (solid lines), the NS (dashed lines) and the experiments (circles) with their 95% confidence interval from the three-Gauss fit (error bars). Shaded areas identify the dynamical destabilization. For comparison, the amplitudes are renormalized to their th = 3ms-values and the NS results are translated by -3.7 ms. krot is reliably extracted from the experiments (NS) for th ≥3 ms (2.5 ms). Inset, same plot in log scale. d, e, Integrated 1D density profile, in momentumn ˜1(ky) and real space n1(y), from one run of the NS.

with th but quickly converges after a few ms to its final value, However, despite this delay, the growth rate of the roton pop- krot (Fig. 3c). The measured krot and the calculated values ulation in the NS matches both the experimental observations from our parameter-free theory are in remarkable agreement and the SSM predictions in the early dynamics (Fig. 3b). (Fig. 2d-e and 3c). In the experiment, we study the time evolution of the roton mode in a fixed geometry (νz,λ). In a first set of mea- Our SSM quantitatively explains the experimental observa- f tions and provides us with a physical understanding. For com- surements, we fix as and follow the dynamics by recording pleteness, we perform numerical simulations (NS) of the sys- the momentum distribution at various th. We observe that krot tem dynamics. We calculate the time evolution of the general- does not change significantly while the roton population ini- ized non-local Gross-Pitaevskii equation, which accounts also tially grows, in excellent agreement with the theories (Fig. 3b- for quantum fluctuations, three-body loss processes, and finite c). The growth rate is a particularly relevant quantity as it is temperature, not included in the SSM (Supplementary Infor- directly connected to the imaginary excitation energy in the mation). The first two contributions limit the peak density of Bogoliubov description (for the roton, its gap), as revealed by the atomic cloud and stabilise the dBEC against collapse [4– our theory. In a second set of experiments, we thus systematically study the growth rate of the roton population for various 6, 24, 25], whereas the latter term thermally seeds the initial f as (Fig. 4). Our data show that the roton mode begins to pop- roton-mode population. The NS confirm both the SSM results ∗ ulate from a critical value of the scattering length, as . For and the experimental observations. Indeed the calculations f as > 52a0, we do not observe the roton peaks at any time. For show that, few ms after the quench, the system develops ro- f ton peaks in momentum space (Fig. 3d) and short-wavelength as . 52a0, after a time delay, A∗ undergoes an abrupt increase. density modulations at the center of the BEC (Fig. 3e), show- At longer time, A∗ then saturates and eventually slowly de- ing the predicted roton confinement [9]. The extracted value creases while atoms are coincidentally lost (Fig. 4a). By further lowering af, the roton population exhibits a faster growth of krot and its geometrical scaling are in very good quantita- s tive agreement with both the experimental data and the SSM rate and shorter time delay. calculations (Fig. 2d-e and 3c). Interestingly, the timescales From the growth rate of the roton population we for the emergence of roton peaks in the NS are few ms longer now extract an overall roton gap ∆¯ (as) (Methods). In than those observed in the experiment. The origin of this time brief, the dynamical variation of the gap is determined f shift remains an open question, whose answer could e.g. re- by the leading time dependence of as and A∗(as,th) ∝ R th ¯ ¯ quire a refinement of current models of quantum fluctuations. exp 2 0 dt Im[∆(as(t))]/h¯ . To investigate the scaling ∆(as), 5

0,4 ing the generic power-law dependence Im[∆¯ (as)]/h¯ = Γδ(as) a af = 39a0 b ∗ β ) a −as

2 s ∗ a = 44a with δ(as) = (as ≤ a ). Consequently, A∗ grows as f 0 a0 s µ m 0,3 t 3 af = 47a0 R h 0 exp 2Γ 0 δ(as(t))dt = exp(2ΓTh). By rescaling the time 1 a = 50a ( f 0 R th * variable as th → Th = δ(as(t))dt, we determine the param- A 0 af = 52a0 , 0,2 ∗ e exp fit eters as and β such that all the curves of Fig. 4a fall on top d u t

i of each other (Fig. 4b). The best overlap of the experimental l

p ∗ curves is found for a = 53.0(4)a0 and β = 0.55(8). We then 0,1 s A m perform an exponential ﬁt to all the rescaled data for Th < 3ms and extract Γ = 465(83)s−1. The same analysis, applied to the f 0,0 calculated population growth from the SSM for different as, 2 4 6 0,1 1 10 ∗ −1 gives a ' 52.8a0, β ' 0.55, and Γ ' 472s , which are very Hold time, t (ms) Rescaled time, T (ms) s h h close to the experimental values. This time-resolved study al- 0 ) c lows to readily extract the imaginary roton gap, Im[∆¯ ], from z

( H both the experiment and the SSM, as a function of as and th h /

] 100 (Fig. 4c). Our observations show the softening of the roton ∆ [ mode at a = a∗ and the expected increase of Im[∆¯ ] for a < a∗, m s s s s I

, 0 nicely grasping the dynamics during the growth of the roton p 200 a 50 g population (inset of Fig. 4c). The observed behaviour shows y

r 100

a again a remarkable agreement with the theory predictions.

n 300 i 150 g a

m 200 I 0 1 2 3 4 5 6 To conclude, our work demonstrates the power of weakly- 400 Hold time, th (ms) 40 45 50 55 60 interacting dipolar quantum gases to access the regime of

Scattering length, as (a0) large-momentum, yet low-energy, excitations dressed by interactions. This newly accessible regime, which is largely un- Figure 4 — Population growth and roton gap. Results explored in ultracold gases, raises fundamental questions and for (νz,λ) = (149Hz,4.3). a, b, A∗ depending on th and on opens novel directions. Future key developments are to study f the rescaled time Th, after quenching to as = 39a0 (hexagons), the impact of such low-lying excitations on the superfluid be- f f f as = 44a0 (triangles), as = 47a0 (diamonds), as = 50a0 (cir- havior of a dBEC [8–10, 12], the interplay between phononic f cles) and as = 52a0 (squares). Shaded areas show the 95% and rotonic modes in the stability diagram of the quantum confidence interval from the three-Gauss fit. The black line gas [4, 7], and the possible role of roton excitations as trig- shows an exponential fit to the full dataset with Th < 3ms. gering mechanism of the recently observed instability lead- ¯ c, Extracted Im[∆] depending on as and on th (inset) from the ing to the formation of metastable droplet arrays in elongated experiments (dashed line with pentagons) with the propagated traps [24, 28]. Of particular interest is the prospect of creating errors from the time-rescaling analysis and exponential fit (see a supersolid and striped ground-states in dBECs [13, 28]. In- b) (shaded area), and from the SSM (solid line). The dotted deed, the short-wavelength density modulation tied to the ro- line and the corresponding shaded area show the experimental ton softening together with the quantum stabilization mecha- ∗ as and its fit’s confidence interval. The inset shows the same nism can favor the formation of such an exotic phase of matter, f configuration as Fig. 3 (as = 50a0). in which crystalline order coexist with phase coherence. Con- trary to recent experiments [29, 30], where the density modulation is imposed by external fields yielding a supersolid-like we first consider the analytical expression of the elemen- arrangement of infinite stiffness, a dipolar supersolid would be tary local roton gap, ∆ as a function of as (see Eq.(1)). compressible. Experiments on dBECs provide hence the ex- ∗ Developing ∆ in the vicinity of as = as yields ∆(as) ∝ citing opportunity to unveil similarities and differences among ∗ 1/2 (as − as) . Here we confirm this scaling by consider- complementary approaches to supersolidity.

[1] Landau, L. D. The theory of superﬂuidity of helium II. J. Phys. aps.org/doi/10.1103/PhysRevLett.90.250403. (Moscow) 5, 71 (1941). [4] Ronen, S., Bortolotti, D. C. E. & Bohn, J. L. Radial and [2] Grifﬁn, A. Excitations in a Bose-condensed Liquid. angular rotons in trapped dipolar gases. Phys. Rev. Lett. Cambridge Classical Studies (Cambridge University Press, 98, 030406 (2007). URL https://link.aps.org/doi/10. 1993). URL https://books.google.com.cy/books?id= 1103/PhysRevLett.98.030406. skF9nhqPF2MC. [5] Bohn, J. L., Wilson, R. M. & Ronen, S. How does a [3] Santos, L., Shlyapnikov, G. V. & Lewenstein, M. Roton-maxon dipolar Bose-Einstein condensate collapse? Laser Physics spectrum and stability of trapped dipolar Bose-Einstein conden- 19, 547–549 (2009). URL http://dx.doi.org/10.1134/ sates. Phys. Rev. Lett. 90, 250403 (2003). URL http://link. S1054660X09040021. 6

[6] Parker, N. G., Ticknor, C., Martin, A. M. & O’Dell, D. 107, 190401 (2011). URL https://link.aps.org/doi/10. H. J. Structure formation during the collapse of a dipo- 1103/PhysRevLett.107.190401. lar atomic Bose-Einstein condensate. Phys. Rev. A 79, [22] Aikawa, K. et al. Bose-Einstein condensation of erbium. Phys. 013617 (2009). URL https://link.aps.org/doi/10. Rev. Lett. 108, 210401 (2012). URL https://link.aps. 1103/PhysRevA.79.013617. org/doi/10.1103/PhysRevLett.108.210401. [7] Martin, A. D. & Blakie, P. B. Stability and struc- [23] Kadau, H. et al. Observing the Rosensweig instability of ture of an anisotropically trapped dipolar Bose-Einstein con- a quantum ferrofluid. Nature 530, 194–197 (2016). URL densate: Angular and linear rotons. Phys. Rev. A 86, http://dx.doi.org/10.1038/nature16485. 053623 (2012). URL https://link.aps.org/doi/10. [24] Ferrier-Barbut, I., Kadau, H., Schmitt, M., Wenzel, M. & Pfau, 1103/PhysRevA.86.053623. T. Observation of quantum droplets in a strongly dipolar Bose [8] Blakie, P. B., Baillie, D. & Bisset, R. N. Roton spectroscopy in a gas. Phys. Rev. Lett. 116, 215301 (2016). URL http://link. harmonically trapped dipolar Bose-Einstein condensate. Phys. aps.org/doi/10.1103/PhysRevLett.116.215301. Rev. A 86, 021604 (2012). URL https://link.aps.org/ [25] Chomaz, L. et al. Quantum-fluctuation-driven crossover from a doi/10.1103/PhysRevA.86.021604. dilute Bose-Einstein condensate to a macrodroplet in a dipolar [9] Jona-Lasinio, M., Łakomy, K. & Santos, L. Roton confine- quantum fluid. Phys. Rev. X 6, 041039 (2016). URL https: ment in trapped dipolar Bose-Einstein condensates. Phys. Rev. //link.aps.org/doi/10.1103/PhysRevX.6.041039. A 88, 013619 (2013). URL https://link.aps.org/doi/ [26] Schmitt, M., Wenzel, M., Bottcher,¨ F., Ferrier-Barbut, I. & Pfau, 10.1103/PhysRevA.88.013619. T. Self-bound droplets of a dilute magnetic quantum liquid. [10] Wilson, R. M., Ronen, S. & Bohn, J. L. Critical super- Nature 539, 259–262 (2016). fluid velocity in a trapped dipolar gas. Phys. Rev. Lett. [27] Nguyen, J. H., Luo, D. & Hulet, R. G. Formation of matter- 104, 094501 (2010). URL https://link.aps.org/doi/10. wave soliton trains by modulational instability. Science 356, 1103/PhysRevLett.104.094501. 422–426 (2017). [11] Natu, S. S., Campanello, L. & Das Sarma, S. Dy- [28] Wenzel, M., Bottcher,¨ F., Langen, T., Ferrier-Barbut, I. & Pfau, namics of correlations in a quasi-two-dimensional dipolar T. Striped states in a many-body system of tilted dipoles. Phys. Bose gas following a quantum quench. Phys. Rev. A 90, Rev. A 96, 053630 (2017). URL https://link.aps.org/ 043617 (2014). URL https://link.aps.org/doi/10. doi/10.1103/PhysRevA.96.053630. 1103/PhysRevA.90.043617. [29] Li, J.-R. et al. A stripe phase with supersolid properties in spin– [12] Pitaevskii, L. & Stringari, S. Bose-Einstein condensation and orbit-coupled Bose–Einstein condensates. Nature 543, 91–94 superfluidity, vol. 164 (Oxford University Press, 2016). (2017). [13] Boninsegni, M. & Prokof’ev, N. V. Colloquium. Rev. Mod. [30]L eonard,´ J., Morales, A., Zupancic, P., Esslinger, T. & Donner, Phys. 84, 759–776 (2012). URL https://link.aps.org/ T. Supersolid formation in a quantum gas breaking a continuous doi/10.1103/RevModPhys.84.759. translational symmetry. Nature 543, 87–90 (2017). [14] Henshaw, D. G. & Woods, A. D. B. Modes of atomic motions [31] Chin, C., Grimm, R., Julienne, P. S. & Tiesinga, E. Feshbach in liquid helium by inelastic scattering of neutrons. Phys. Rev. resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286 121, 1266–1274 (1961). URL https://link.aps.org/doi/ (2010). 10.1103/PhysRev.121.1266. [32] Lahaye, T. et al. d-wave collapse and explosion of a dipolar [15] O’Dell, D. H. J., Giovanazzi, S. & Kurizki, G. Rotons in Bose-Einstein condensate. Phys. Rev. Lett. 101, 080401 (2008). gaseous Bose-Einstein condensates irradiated by a laser. Phys. [33] Giovanazzi, S. et al. Expansion dynamics of a dipo- Rev. Lett. 90, 110402 (2003). URL https://link.aps.org/ lar Bose-Einstein condensate. Phys. Rev. A 74, 013621 doi/10.1103/PhysRevLett.90.110402. (2006). URL https://link.aps.org/doi/10.1103/ [16] Mottl, R. et al. Roton-type mode softening in a quantum gas PhysRevA.74.013621. with cavity-mediated long-range interactions. Science 336, [34] Baranov, M. Theoretical progress in many-body physics 1570–1573 (2012). URL http://science.sciencemag. with ultracold dipolar gases. Physics Reports 464, 71 – 111 org/content/336/6088/1570. (2008). URL http://www.sciencedirect.com/science/ [17] Khamehchi, M. A., Zhang, Y., Hamner, C., Busch, T. & article/pii/S0370157308001385. Engels, P. Measurement of collective excitations in a spin- [35] Blakie, P., Bradley, A., Davis, M., Ballagh, R. & Gar- orbit-coupled Bose-Einstein condensate. Phys. Rev. A 90, diner, C. Dynamics and statistical mechanics of ultra-cold 063624 (2014). URL https://link.aps.org/doi/10. Bose gases using c-field techniques. Advances in Physics 1103/PhysRevA.90.063624. 57, 363–455 (2008). URL http://dx.doi.org/10.1080/ [18] Ji, S.-C. et al. Softening of roton and phonon modes in a Bose- 00018730802564254. Einstein condensate with spin-orbit coupling. Phys. Rev. Lett. [36] Frisch, A. et al. Narrow-line magneto-optical trap for erbium. 114, 105301 (2015). URL http://link.aps.org/doi/10. Phys. Rev. A 85, 051401 (2012). 1103/PhysRevLett.114.105301. [37] Ahmadi, P., Timmons, B. P. & Summy, G. S. Geometrical [19] Ha, L.-C., Clark, L. W., Parker, C. V., Anderson, B. M. & effects in the loading of an optical atom trap. Phys. Rev. A Chin, C. Roton-maxon excitation spectrum of Bose con- 72, 023411 (2005). URL https://link.aps.org/doi/10. densates in a shaken optical lattice. Phys. Rev. Lett. 114, 1103/PhysRevA.72.023411. 055301 (2015). URL https://link.aps.org/doi/10. [38]W achtler,¨ F. & Santos, L. Quantum filaments in 1103/PhysRevLett.114.055301. dipolar Bose-Einstein condensates. Phys. Rev. A 93, [20] Griesmaier, A., Werner, J., Hensler, S., Stuhler, J. & Pfau, 061603 (2016). URL http://link.aps.org/doi/10. T. Bose-Einstein condensation of chromium. Phys. Rev. Lett. 1103/PhysRevA.93.061603. 94, 160401 (2005). URL https://link.aps.org/doi/10. [39]W achtler,¨ F. & Santos, L. Ground-state properties and elemen- 1103/PhysRevLett.94.160401. tary excitations of quantum droplets in dipolar Bose-Einstein [21] Lu, M., Burdick, N. Q., Youn, S. H. & Lev, B. L. Strongly dipo- condensates. Phys. Rev. A 94, 043618 (2016). URL https: lar Bose-Einstein condensate of dysprosium. Phys. Rev. Lett. //link.aps.org/doi/10.1103/PhysRevA.94.043618. 7

[40] Bisset, R. N., Wilson, R. M., Baillie, D. & Blakie, P. B. Ground- performed the real time simulations. LS derived the an- state phase diagram of a dipolar condensate with quantum fluc- alytical and SSM. LC, FF, RMWvB and LS wrote the tuations. Phys. Rev. A 94, 033619 (2016). URL https: paper with the contributions of all the authors. //link.aps.org/doi/10.1103/PhysRevA.94.033619. [41] Lima, A. R. P. & Pelster, A. Quantum fluctuations in dipolar Bose gases. Phys. Rev. A 84, 041604 (2011). URL http:// link.aps.org/doi/10.1103/PhysRevA.84.041604. METHODS [42] Lima, A. R. P. & Pelster, A. Beyond mean-field low- lying excitations of dipolar Bose gases. Phys. Rev. A A. Trapping geometries. 86, 063609 (2012). URL http://link.aps.org/doi/10. 1103/PhysRevA.86.063609. [43] Kagan, Y., Muryshev, A. E. & Shlyapnikov, G. V. Col- The BEC is confined in a harmonic trapping potential 2 2 2 2 2 2 2 lapse and Bose-Einstein condensation in a trapped Bose gas V(r) = 2mπ (νx x + νy y + νz z ), characterized by the fre- with negative scattering length. Phys. Rev. Lett. 81, 933– quencies (νx,νy,νz). The ODT results from the crossing of 937 (1998). URL http://link.aps.org/doi/10.1103/ two red-detuned laser beams of 1064nm wavelength at their PhysRevLett.81.933. respective focii. One beam, called vODTb, propagates nearly [44] Ronen, S., Bortolotti, D. C. E. & Bohn, J. L. Bogoliubov collinear to the z-axis and the other, denoted hODTb propa- modes of a dipolar condensate in a cylindrical trap. Phys. Rev. A gates along the y-axis. By adjusting independently the param- 74, 013623 (2006). URL https://link.aps.org/doi/10. eters of the vODTb and hODTb, we can widely and dynami- 1103/PhysRevA.74.013623. [45] Bisset, R. N., Baillie, D. & Blakie, P. B. Roton excitations cally control the geometry of the trap (see Supplementary In- in a trapped dipolar Bose-Einstein condensate. Phys. Rev. A formation). In particular, νy is essentially set by the vODTb 88, 043606 (2013). URL https://link.aps.org/doi/10. power while νz (and νx) independently by that of the hODTb. 1103/PhysRevA.88.043606. This yields an easy tuning of the trap aspect ratio, λ = νz/νy, [46] Zambelli, F., Pitaevskii, L., Stamper-Kurn, D. M. & relevant for our cigar-like geometry. Stringari, S. Dynamic structure factor and momentum After reaching condensation (see Supplementary Informa- distribution of a trapped Bose gas. Phys. Rev. A 61, tion), we modify the beam parameters to shape the trap into an 063608 (2000). URL https://link.aps.org/doi/10. axially elongated configuration, favourable for observing the 1103/PhysRevA.61.063608. roton physics (ν ν ,ν ). The trapping geometries probed [47] Brunello, A., Dalfovo, F., Pitaevskii, L., Stringari, S. & Zam- y x z in the experiments, whose (λ,ν ,ν ,ν ) are reported in Sup- belli, F. Momentum transferred to a trapped Bose-Einstein x y z condensate by stimulated light scattering. Phys. Rev. A 64, plementary Table 1, are achieved by changing the hODTb 063614 (2001). URL https://link.aps.org/doi/10. power with the vODTb power set to 7W so that νy and νz/νx 1103/PhysRevA.64.063614. are kept roughly constant. Only the green triangle in Fig. 2d is obtained in a distinct configuration, with a vODTb power of • We are particularly grateful to B. Blakie for many in- 2W, leading to (νx,νy,νz) = (156,17,198)Hz and λ = 11.6. spiring exchanges. We thank D. O’Dell, M. Baranov, The (λ,νx,νy,νz) are experimentally calibrated via exciting E. Demler, A. Sykes, T. Pfau, I. Ferrier-Barbut and H. and probing the center-of-mass oscillation of thermal samples. P. Buchler¨ for fruitful discussions, and G. Natale for We note that the final atom number N, BEC fraction f , and his support in the final stage of the experiment. This temperature T after the shaping procedure depend on the final work is dedicated in memory of D. Jin and her inspir- configurations, as detailed in Supplementary Table 1. ing example. The Innsbruck group is supported through a ERC Consolidator Grant (RARE, no. 681432) and a FET Proactive project (RySQ, no. 640378) of the EU B. Quench of the scattering length as. H2020. LC is supported within a Marie Curie Project (DipPhase, no. 706809) of the EU H2020. FW and LS To control as we use a magnetic Feshbach resonance be- thank the DFG (SFB 1227 DQ-mat). All the authors tween 166Er atoms in their absolute ground state, which is thank the DFG/FWF (FOR 2247). Part of the compu- centered around B = 0G [31]. The B-to-as conversion has tational results presented have been achieved using the been previously precisely measured via lattice spectroscopy, HPC infrastructure LEO of the University of Innsbruck. as reported in Ref. [25]. Errors on as, taking into account statistical uncertainties of the conversion and effects of mag- • Competing Interests: The authors declare that they netic field fluctuations (e.g. from stray fields), are of 3-to- have no competing financial interests. 5 a0 for the relevant as range 27-67 a0 in this work. After the • Author Information: Correspondence and requests BEC preparation and in order to investigate the roton physics for materials should be addressed to F.F. (email: via an interaction quench, we suddenly change the magnetic [email protected]). field set value, Bset, twice. First we perform the quench it- i self and abruptly change Bset from 0.4G (as = 61a0) to the f • Author contributions: FF, LC, DP, GF, MJM, JHB, desired lower value (corresponding to as) at the beginning of SB conceived and supervised the experiment and took the hold in trap (th = 0ms). Second we prepare the ToF ex- the experimental data. LC analysed them. RMWvB de- pansion and imaging conditions (see Method Imaging pro- veloped the BdG calculations. FW, RMWvB and LS cedure) and abruptly change Bset from the quenched value 8 back to 0.3G (as = 57a0) at the beginning of the ToF ex- very similar, and, in particular, the two extracted momenta as- pansion (tf = 0ms). Due to delays in the experimental setup, sociated with the roton signal agree within 5%. This confirms e.g. coming from eddy currents in our main chamber, the ac- that the interparticle interactions play little role during the ex- tual B value felt by the atoms responds to a change of Bset pansion and justifies the use of Eq. (2). via B(t) = Bset(t) + τdB/dt [32]. By performing pulsed- radio-frequency spectroscopy measurements (pulse duration 100 µs) on a BEC after changing Bset (from 0.4 to 0.2 G), we D. Fit procedure for the ToF images. verify this law and extract a time constant τ = 0.98(5)ms. Then as is also evolving during th and tf on a similar timescale. For each data point of Figs. 2-4, we record between 12 This effect is fully accounted for in the experiments and simu- and 25 ToF images. By fitting a two-dimensional Gaussian lations, and we report the roton properties as a function of the distribution to the individual images, we extract their origin a t t effective value of s at h. We use h ranging from 2 to 7 ms. (kx,ky) = (0,0) and recenter each image. From the recen- t a The lower bound on h comes from the time needed for s to tered images, we compute the averagedn ˜(kx,ky), from which effectively reach the regime of interest. We then consider the we characterise the linear roton developing along ky. To do initial evolution for which t / ; t / 1, 1/ being h νy h τcoll τcoll so, we extract a one-dimensional profilen ˜1(ky) by averag- the characteristic collision rate. We estimate that τcoll ranges −1 ingn ˜(kx,ky) over kx within |kx| ≤ km = 3.5µm :n ˜1(ky) = typically from 40 to 90 ms in the initial BECs of Supplemen- R km R km n˜(kx,ky)dkx/ dkx. To quantitatively analyse the ob- tary Table 1 at ai = 61a . Experimentally we observe that the −km −km s 0 served roton peaks, we fit a sum of three Gaussian distribu- roton, if it ever develops, has developed within the considered tions ton ˜1(ky). One Gaussian accounts for the central peak range of th. and its centre is constrained to k0 ∼ 0. Its amplitude is denoted A0. The two other Gaussians are symmetrically located ∗ at k0 ± ky , and we constrain their sizes and amplitudes to be C. Imaging procedure. ∗ identical, respectively equal to σ and A∗ . We define the roton population contrast via C = A∗/A0. We focus on the roton ∗ −1 ∗ −1 In our experiments, we employ ToF expansion measure- side peaks by constraining ky > 0.5 µm and σ < 3 µm ments, accessing the momentum distribution of the gas [12], (peak at finite momentum of moderate extension compared to to probe the roton mode population. We let the gas ex- the overall distribution). pand freely for tf = 30ms, which translates the spatial imag- In order to analyse the onset and evolution of the roton pop- ing resolution (∼ 3.7 µm) into a momentum resolution of ulation (see Fig.4), we perform a second run of the fitting pro- −1 ∗ ∼ 0.32 µm while our typical roton momentum is krot & cedure, in which we constrain the value of ky more strictly. 2 µm−1. After ToF expansion, we record 2D absorption pic- The interval of allowed values is defined for each trapping tures of the cloud by means of standard resonant absorption geometry based on the results of the first run of the fitting pro- f imaging on the atomic transition at 401 nm. The imaging cedure. We use the results of the (as, th)-configurations where ∗ beam propagates nearly vertically, with a remaining angle the peaks are clearly visible and we set the allowed ky -range of ∼ 15o compared to the z-axis within the xz-plane. Thus to that covered by the 95% confidence intervals of the first- ∗ the ToF images essentially probe the spatial density distribu- fitted ky in these configurations. This constraint enables that tion ntof (x,y,tf) in the xy plane. When releasing the cloud the fitting procedure estimates the residual background popu- (tf = 0ms), we change B back to B = 0.3G (Method Quench lation on the relevant momentum range for the roton physics ∗ of the scattering length as). This change enables constant even for as > as (see e.g. Fig.4a). and optimal imaging conditions with a fixed probing procedure, i. e. , a maximal absorption cross-section. In addition, the associated increase of as to 57a0 allows to minimize E. Time-rescaling analysis and roton gap estimate. the time during which the evolution happens in the small-as regime where roton physics develops, such that we effectively In Fig. 4a-b, we systematically analyse the time evolution probe only the short-time evolution of the gas. In this work, f of the roton population for various as and link it to the roton we use the simple mapping: spectrum in a quench picture. The roton population is em- bodied by the amplitude A∗ of the three-Gaussian fit (Method ht¯ 2 hk¯ t hk¯ t n˜(k ,k ) = f n x f , y f ,t , (2) Fit procedure for the ToF images), which measures the den- x y tof f f ∗ m m m sityn ˜1(krot). A∗ is observed to initially increase if as < as , its f ∗ growth rate increases for decreasing as and as depends on the which neglects the initial size of the cloud in trap and the trap geometry. effect of interparticle interactions during the free expansion. If dynamically unstable, the krot component is ex- Using real-time simulations (see Supplementary Information), pected to grow exponentially with an instantaneous rate we simulate the experimental sequence and are able to com- 2Im[ε(krot,th)]/h¯. We hence expect an initial growth of A∗ pute both the real momentum distribution from the in-trap of the form: wavefunction and the spatial ToF distribution 30ms after Z t switching off the trap. Using the mapping of Eq. (2) and our f h ¯ f A∗(as,th) ∝ exp 2 dt Im[∆(as,t)]/h¯ , (3) experimental parameters, the two calculated distributions are 0 9

¯ f ∆(as,th) being the instantaneous value of the overall roton restrict its relevance to Th ≤ 3ms so that, e.g., in the inset of gap, corresponding to an average over the cloud, after quench- Fig. 4c, Im[∆¯ ] is shown up to th = 4ms after which the roton f ing as to as. Our results show that the most relevant effect of population is observed to deviate from the exponential growth the quench on the roton spectrum is given by the reduction picture (see Figs. 3b and 4a). Note that the Im[∆¯ ] estimates at of as itself. Hence the time dependence of ∆¯ is determined the different th are not independent. f by as(t) and, by monitoring A∗(as,th), one can readily extract In Fig. 2, we report on the roton momentum as a function of ¯ ¯ f ¯ the scattering-length dependent gap ∆(as), ∆(as,t) = ∆(as(t)). the system characteristics (as, `z). Here we estimate krot from To investigate the scaling ∆¯ (as), we first consider the simpler a th value that is individually optimised for each as and `z in- case of the static and local roton gap ∆. From Eq. (1) of the vestigated (largest visibility). We point out that the selected th main text characterising this elementary gap, assuming n0, X correspond, in the A∗ dynamics, to the late stage of the expo- ∗ and Z fixed, and using the fact that ∆(as = as ) = 0, one eas- nential growth, close to the maximum (i.e. Th ∼ 3ms). Here, ∗ ∗ 1/2 ily obtain that for as in the vicinity of as , ∆(as) ∝ (as −as) . the atom loss remains at a few percents level. Here we verify this scaling for the global and dynamical quan- In Fig. 2d, we show the value of krot at the onset of the pop- ∗ ∗ tity ∆¯ (as), considering the generic power-law dependence ulation of the roton mode, i.e. at as = as . We estimate as ∗ β for each trap geometry by analysing the evolution of the roton as−as ∗ Im[∆(as)]/h¯ = Γδ(as) with δ(as) = (as(t) ≤ a ), in a0 s population with as. We employ a simplified approach with ∗ which the parameters Γ, as and β mainly depend on the trap respect to the one in Fig. 4 (see Supplementary Information), ∗ geometry. which we estimate to lead to a maximum underestimate for as f Our full set of data A∗(as,th) for the geometry (νz,λ) = of about 1.5a0, lying within our experimental uncertainty on (149Hz,4.3) enables us to assess this scaling. Indeed Eq. (3) as; see Supplementary Table 2. then reads: Z t f h A∗(as,th) ∝ exp 2Γ δ(as(t))dt . (4) F. Analytical dispersion relation for an infinite axially 0 elongated geometry.

R th This defines a time rescaling th → Th = δ(as(t))dt along 0 Equation (1) in the main text results from a similar proce- which all our experimental data A∗(as,th) should collapse in a unique curve, marked by an initial exponential growth of dure as that used in Ref. [3] for rotons in infinite q2D traps. ∗ We consider a dBEC homogeneous along y but harmonically rate 2Γ. The relevant values of as , β and Γ are the one that result in the best overlap of the data for the initial growth of confined with frequencies νx and νz along x and z. For suf- ficiently strong interactions the BEC is in the TF regime on A∗ (minimal spread in Th). ∗ the xz plane, in which the BEC wavefunction acquires the In order to determine as and β, we then plot our full dataset p 2 2 form ψ0(ρ) = n(ρ), with n(ρ) = n0 1 − (x/X) − (z/Z) , as a function of Th for various trial values of these parameters ∗ where X and Z are the TF radii, and ρ = (x,z). The calculation and evaluate the relevance of the trial couple (as ,β). Pre- of n0, X and Z is detailed at the end of this section. cisely, we assess the dispersion in Th of the full dataset for i i Due to the axial homogeneity, the elementary excitations few fixed A∗ = A∗. We use a panel of 10 values A∗ within 2 of the Bogoliubov-de Gennes spectrum have a defined axial [80,200]µm , corresponding to the range of the initial growth ik y−iεt/h¯ momentum ky, and take the form δψ(r,t) = u(ρ)e y − of A∗ for the geometry of Fig. 4a-b. We interpolate the exper- f v(ρ)e−ikyy+iεt/h¯ , where u,v denote the amplitudes of the spa- imental data A∗(Th) for each as using piecewise cubic poly- i tial modes oscillating in time with characteristic frequency nomial interpolation, extract the corresponding set of Th at i i ε/h¯ (see Supplementary Information for more details). We which A∗(Th) = A∗. For each i, we evaluate the spread of the T i(af) by two complementary quantities: (a) the square-root consider the standard GPE without LHY correction and three- h s (r,t) = of their variance and (b) the discrepancy between the maxi- body losses, and insert the perturbed solution ψ −iµt/h¯ mal and minimal value of the set. We finally estimate the ac- (ψ0(ρ) + ηδψ(r,t))e , where µ is the chemical poten- ∗ tial associated to ψ0 and η 1. After linearization we obtain curacy of a unified dependence A∗(Th) for a given (as ,β) via i the BdG equations for f±(ρ) = u(ρ) ± v(ρ): the geometrical average of (a) and (b) for all the ten A∗ val- ∗ ues. We fit the relevant as and β by minimizing this averaged ε f−(ρ) = Hkin f+(ρ), (5) spread. For the experimental data of Fig. 4a-b, this results in ∗ ε f+(ρ) = Hkin f−(ρ) + Hint[ f−(ρ)], (6) as = 53.0(4)a0 and β = 0.55(8). ∗ Using these values of as and β, we observe that the ini- where tial growth of A∗ extends for Th ≤ 3ms (before saturating and 2 2 h¯ 2 2 ∇ ψ0 decreasing for longer Th). We estimate Γ by performing an Hkin f±(ρ) = −∇ + ky + f±(ρ), (7) 2m ψ0 exponential fit on the full set of data A∗(Th) with Th ≤ 3ms. −1 Z 0 = ( ) 3 0 0 −iky(y−y ) This gives Γ 465 83 s . Note that Fig. 4a-b only show 5 H [ f−(ρ)] = 2 d r U(r − r )e f int values of as while the reported analysis considers all available 0 0 experimental data, i.e. 11 values between 31a0 and 52a0. ψ0(ρ)ψ0(ρ ) f−(ρ ). (8) ∗ From the formula with the estimated as , β and Γ, we can ¯ where U(r) is the binary interaction potential for two parti- then compute the global roton gap ∆. The extracted value is 2 2 r U(r) = g (r) + 3εdd 1−3cos θ only meaningful for ∆¯ < 0. For the experiments, we also cles separated by . δ 4π |r|3 includes 10 both contact and dipolar interactions (see Supplementary in- G. Self similar model and instantaneous roton spectrum. formation). Employing f+(ρ) = W(ρ)ψ0(ρ), and for ky 1/X,1/Z, The dynamics of the condensate during and after the quench we obtain the following equation for the function W(ρ): is crucial to understand the resulting momentum peaks and 2 2 their direct relation to the growth of roton excitations. Due 2 2 1 ∂ W 1 ∂ W 0 = 2gn 1 − x˜ − z˜ + to its large characteristic momentum, the roton spectrum may 0 X2 ∂x˜2 Z2 ∂z˜2 be evaluated at any time using local density approximation. 1 ∂W 1 ∂W On the other side, the evolution of the local density is deter- − gn0(1 + 2εdd) x˜ + z˜ X2 ∂x˜ Z2 ∂z˜ mined by the global dynamics of the condensate induced by the quench. Hence, interestingly, the analysis of the effect 2m 2 2 1 1 + (ε − E(ky) ) − 2gεddn0 + of the quench on the roton spectrum may be performed by h¯ 2 X2 Z2 combining a self-similar theory of the global dynamics [33], 4m 2 2 describing the evolution at low ky, and the model of the roton − 2 gn0(1 − εdd)E(ky) 1 − x˜ − z˜ , (9) h¯ spectrum developed in the Method Analytical dispersion re- 2 2 lation for an infinite axially elongated geometry, account- wherex ˜ = x/X,z ˜ = z/Z, E(ky) = h¯ ky /2m. For εdd = 1 the last term of Eq. (9) vanishes. In that case, the lowest-energy ing for the high-ky region of interest. We note that this analysis solution is given by W = 1, whose eigen-energy builds, as a approximates the real time evolution provided by the standard NLGPE [12, 34, 35]. This treatment is valid for moderate- function of ky, the dispersion ε0(ky) with enough quenches so that each of the descriptions applies (see 2 2 2 ε0(ky) = E(ky) + E0 , (10) below for an in-depth discussion) and as long as the high-k modes population only minimally impact the BEC dynamics, 2 E2 = g n h¯ 1 + 1 = that is for short-enough time scales. with 0 2 εdd 0 2m X2 Z2 . In the vicinity of εdd 1, the effect of the last term in Eq. (9) may be evaluated pertur- We assume that the condensate preserves its TF shape dur- batively, resulting in the dispersion ing the evolution: 2 2 " 2 2 2# ε(ky) ' ε0(ky) − 2EIE(ky), (11) x y z n(r,t) = n0(t) 1 − − − , 2 X(t) Y(t) Z(t) with EI = gn (ε − 1). 3 0 dd (16) This expression for the dispersion√ presents a roton mini- 1 where X(t) = bx(t)X0, Y(t) = by(t)Y0, and Z(t) = bz(t)Z0 mum for εdd > 1 at krot = h¯ 2mEI. Expanding Eq. (11) 2 2 are the re-scaled TF radii, with bx,y,z(t) the scaling coeffi- in the vicinity of the roton minimum, ε(ky) ' ε(krot) + 2 2 cients (bx,y,z(0) = 1). Their evolution after the quench can be 1 h d ε (ky) i 2 dk2 , we obtain Eq. (1) of the main text, with deduce from solving the hydrodynamics equations (see Sup- y ky=krot q plementary information). Prior to the quench of as, the sta- 2 2 ∆ = ε(krot) = E0 − EI . At the instability, ∆ = 0, and tionary TF solution is obtained from the self-consistent equa- √ ωx p ωy p 1 tions χx(0) = ω C(0)/A(0) and χy(0) = ω C(0)/B(0). krot = h¯ 2mE0. z z Employing a similar procedure as in Ref. [15] we obtain Solving these equations and using normalisation provides that the BEC aspect ratio χ = Z/X fulfills: X(0), Y(0), Z(0), and n(0) for known numbers of atoms N and trap frequencies ωx,y,z. We use this stationary solution as 2 2 (1 − εdd)(1 + χ) + 3εdd 2 the initial condition at the start of the quench (t = 0), and solve χ 2 2 = λ⊥, (12) (1 + 2εdd)(1 + χ) − 3εdd χ the system of differential equations resulting from the hydrodynamics evolution to obtain the scaling coefficients. An ex- with λ⊥ = νx/νz and ample for the relevant parameters of Fig. 3 is shown in Sup- 2 plementary Fig. 3. 2 gn0 3εdd χ Z = (1 + 2εdd) − (13) For a given position y we then evaluate the local TF 2π2mν2 (1 + χ)2 2 2 z profile: n(r,t) = n0(y,t) 1 − (x/X(y,t)) − (z/Z(y,t)) , 2 These two equations fully determine the TF solution for given where n0(y,t) = n0(t) 1 − (y/Y(t)) , X(y,t) = 2 2 1/2 1/2 εdd, gn0, and λ⊥. By inserting the expressions of X and Z X(t)1 − (y/Y(t))2 , and Z(y,t) = Z(t)1 − (y/Y(t))2 , E ' in 0, we find for εdd 1: with n0(t) = n0(0)/bxbybz. We may then employ a local den- 2 2 sity approximation and evaluate the local (in y) instantaneous 2 h νz 2 2 1 E0 = (1 + χ) λ⊥ + (14) roton spectrum using the results of the Method Analyti- 6 1 + 2χ cal dispersion relation for an infinite axially elongated p geometry: whereas χ simplifies into χ = λ⊥(1 + 1 + 1/λ⊥). As a result, at the instability krot`z depends only on the transverse 2 4 ε (ky,y,t) ky 8π 2 confinement aspect ratio λ⊥, giving the geometrical factor κ: 2 4 ' − n0(y,t)(add − as(t))ky (h¯ωz) lz 4 3 1/4 1/4 2 p 2 1 1 1 κ = k `z = 1 + χ λ + . (15) + 4πn (y,t)a + . (17) rot 3 ⊥ 1 + 2χ 0 dd X(y,t)2 Z(y,t)2 11

The roton population grows exponentially when the spectrum SUPPLEMENTARY INFORMATION becomes imaginary, leading to the appearance of the corresponding momentum peaks at large momenta. From the local H. Production of 166Er BECs. instantaneous roton spectrum we may estimate the momentum peak as We prepare a 166Er BEC similarly to Refs. [22, 25]. From 7 166 Z h R t 0 0 i a narrow-line magneto-optical trap with 3 × 10 Er atoms, 2 dt Im[ε(ky,y,t )] n˜1(ky,t) ∼ dy e 0 − 1 . (18) automatically spin-polarized in their absolute lowest Zeeman sub-level [36], at about 10µK, we directly load the atomic gas where we assume equal seeding for all unstable modes. in a crossed optical dipole trap (ODT) with an efficiency of We then evaluate the total population of the peak N∗(t) = more than 30%. A uniform magnetic field, B, is permanently R dkyn˜1(ky,t), as well as the roton momentum hki = applied along the vertical z axis, fixing the dipole orientation, R dk k n˜1(ky,t) . while its value is varied during the experimental sequence, to y y N∗(t) The previously discussed SSM does not describe properly tune as (see Method Quench of the scattering length as). the evolution when the shrinking of the condensate, and hence We achieve condensation by means of evaporative cooling in the increase of its peak density, is too large. As the gas gets the crossed ODT at B = 1.9G (as = 80(2)a0). During the dense, quantum fluctuations, which are not considered in the evaporation procedure, we first change the power and then the ellipticity of one of the ODT beams (see Section I). The final SSM, introduce in our experiments an effective repulsion that 5 crucially prevents large densities or condensate collapse (see atom number, typically 10 , and condensed fraction, typically Supplementary Information). This limits the use of the SSM 70%, are assessed by fitting the time-of-flight (ToF) absorp- ∗ tion images of the gas to a bimodal distribution, sum of a TF to the vicinity of as . Moreover, for our tightest trap the SSM results are also unreliable, since the theory predicts conden- profile and a broad Gaussian background. sate collapse before the momentum peak develops. That said, as shown in the main text, the SSM provides not only a clear intuitive understanding of our measurements, but to a very I. Details on the trapping geometries. large extent also a very good quantitative agreement with both our experimental results and our NS based on a generalized 2 2 2 2 2 2 2 The harmonic potential V(r) = 2mπ (νx x + νy y + νz z ) GPE, which is detailed in Supplementary Information. 166 Data availability: The data that support the plots within trapping the Er atoms results from the crossing of two this paper and other findings of this study are available from red-detuned laser beams of 1064nm wavelength at their re- y the corresponding author upon reasonable request. spective focii; the hODTb propagating along the -axis and the vODTb, propagating nearly collinear to the z-axis. The vODTb has a maximum power of 7 W and an elliptical profile with waists of 110 and 55 µm along x and y respectively. The hODTb has a maximum power of 24 W, a vertical waist 0 wz = 18 µm, and a controllable horizontal waist, wx = λ wz. The ellipticity λ 0 can be tuned from 1.57 to 15 by time averaging the frequency of the first-order deflection of an Acousto- Optic Modulator [37]. By adjusting independently λ 0 and the powers of the vODTb and of the hODTb, we can control the geometry of the trap. Precisely, νy is essentially set by the vODTb power, νz by that of the hODTb, and νx is controlled by both the power and ellipticity of the hODTb, with 0 νz/νx ≈ λ . We use the tuning of the hODTb power and ellipticity to perform evaporative cooling to quantum degeneracy (see Sec- tion H). After reaching condensation, we again modify the beam parameters to shape the trap into an axially elongated configuration (νy νx,νz). The trapping geometries probed in the experiments, (λ,νx,νy,νz), are reported in Supplemen- tary Table 1. They are achieved by changing the hODTb power with λ 0 = 1.57 and the vODTb power set to its maximum so that νy and νz/νx are kept roughly constant. Only the green triangle in Fig. 2d is obtained in a distinct configuration, with a vODTb power of 2W, leading to (νx,νy,νz) = (156,17,198)Hz and λ = 11.6. The (λ,νx,νy,νz) are experimentally calibrated via exciting and probing the center-of- mass oscillation of thermal samples. We note that the final atom number N, BEC fraction f , and temperature T after the 12 shaping procedure depend on the final configurations, as de- 0,5 tailed in Supplementary Table 1. T is extracted from the evolution with the ToF duration tf of the size of the background Gaussian in the TF-plus-Gaussian bimodal fit to the corre- 0,4 sponding ToF images of the gas. The values of N = f N and C

c ,

T are used for the initial states ψi of our real-time simulations s t 0,3 r a

(see Section N). t n o 0,2 C Extended Data Table I — dBEC parameters for the experimental measurements (Figs. 2-4). The typical statistical un- 0,1 certainties on νx and νz are below 1%, and can be up to 10% for νy. The experimental repeatability results in 5-to-10% 0,0 shot-to-shot ﬂuctuations of N, f and T. 35 40 45 50 55

Scattering length, as(a0) 4 λ νx (Hz ) νy (Hz ) νz (Hz ) N (10 ) f (%) T (nK) Figure 5 — Onset of the roton population. Evolution of C 4.3 114 35 149 9 66 45 with as at th = 3ms, in the geometries (νz,λ) = (149Hz,4.3) 4.9 154 40 195 10 65 50 (green squares) and (νz,λ) = (456Hz,14.4) (blue circles). 10.2 183 30 306 11 62 104 The error bars correspond to the propagated errors from the 14.3 267 32 456 8.6 50 150 95% confidence interval on A∗,0 from the three-Gauss fit. We ∗,c 21.3 357 30 638 8.4 36 179 empirically fit a linear step function to identify as (line). 29.7 432 26 771 7 20 171

K. Generalized Non-local Gross-Pitaevskii equation.

Our theory is based on an extended version of the non-linear ∗ J. as for various trap geometries. Gross-Pitaevskii Equation (NLGPE) 2 2 Z ∗ ∂ψ(r,t) h¯ ∇ 0 0 0 We estimate as for each trap geometry of Fig. 2d by study- ih¯ = − +V(r) + dr U(r − r )n(r ) ing the roton population as a function of as. When decreas- ∂t 2m L ing as, the roton population and in particular its contrast + ∆µ[n] − ih¯ 3 n2 ψ(r,t) (19) C = A∗/A0 (see Method Fit procedure for the ToF images) 2 exhibits a sharp increase from an essentially zero and flat con- L3 2 ∗,c ≡ HˆGP[ψ] − ih¯ n ψ(r,t), (20) trast. We extract as as the value of the scattering length 2 corresponding to the onset of the increase of the contrast for governing the evolution of a macroscopically occupied wave- th = 3ms; see Supplementary Fig. 1 and Supplementary Ta- function ψ(r,t), with corresponding atomic density n(r,t) = ble 2. This is a simplified approach with respect to the one in |ψ(r,t)|2 at position r and time t. The standard dipolar NL- Fig. 4 (see Method Time-rescaling analysis and roton gap GPE includes the kinetic energy, external trap potential and estimate.), which we estimate to lead to a maximum under- ∗ the mean-field effect of the interactions [12, 34]. These corre- estimate for as of about 1.5a0, which lies within our experi- spond to the three first terms of Eq. (19), where the mean-field mental uncertainty on as. interaction potential takes the form of a convolution of n with the binary interaction potential Extended Data Table II — Roton softening threshold for the 2 geometries realized in the experiment (Table I), deduced from 3εdd 1 − 3cos θ ∗,c U(r) = g δ(r) + 3 , (21) the C fits for the experimental onset of the population (as ) 4π |r| and from imaginary time NS (a∗,st). The fit uncertainties on s for two particles separated by r [34]. The first term cor- the experimental values are typically of 0.2a , except for the 0 responds to contact interactions between the particles with last trap it is of 0.8 a0. 2 4πh¯ as strength g = m . The DDI gives rise to the second term, which depends on both the distance and orientation (angle θ) ∗,c ∗,st λ νz (Hz ) as (a0) as (a0) of the vector r compared to the polarisation axis (z axis) of the 4.3 149 51.4 50 dipoles. Most properties of dBECs are well captured by this 10.2 306 50.2 46 standard NLGPE (mean-field) [12, 34]. 14.3 456 49 43 Recent experimental and theoretical results, however, have established the importance of accounting for quantum fluc- 21.3 638 47.7 39 tuations in dBECs [24, 25, 38–40]. Their effect can be in- 29.7 771 45.2 30 cluded in the NLGPE in a mean field treatment through a Lee- Huang-Yang correction to the chemical potential, ∆µ[n] = 13

3/2 2 √ 32g(nas) (1 + 3εdd/2)/3 π, which is obtained under a lo- modulation is delocalised over the entire condensate (Sup- cal density approximation [41, 42]. The accuracy of this mean plementary Fig. 6c). The excited states shown in Supplemen- ∗ 2 field treatment has been established, e.g., in Refs. [38–40], tary Fig. 6b-e correspond to the density |ψ0 + η(u − v )| , for and has proven succesful in explaining recent experimental particular pairs of (u,v) corresponding to phonon and roton results [24, 25]. The final nonlinear term in the extended NL- modes (Supplementary Fig. 6c and e), and exemplary modes GPE accounts for three-body losses [43], with an experimen- at higher energies (Supplementary Fig. 6b and d). Even while tally determined loss parameter L3, which is dependent on as the amplitude η = 0.2 of the excited modes is taken to be −41 6 −1 and typically of the order L3 ' 10 m s , as reported in equal in Figs. 6b - 6e, the roton excitation (Supplementary Ref. [25]. Fig. 6e) leads to markedly larger local density modulations than the phonon excitation (Supplementary Fig. 6c). The ToF signatures in Supplementary Fig. 6 are computed by letting L. Bogoliubov-de Gennes spectrum. the wave function of the excitation, η(u − v∗), expand ballis- tically for 30ms, i.e. neglecting interactions during the expan- Collective excitations of the dBEC are obtained by linearis- sion. The resulting density |η(u − v∗)|2 is then plotted (Sup- ing the NLGPE (see Section K) around a stationary state plementary Fig. 6b1 - e1). ψ0, which can be obtained by imaginary time propagation Carrying out the numerical BdG spectrum calculation for −iµt/h¯ −iεt/h¯ (see Section N). We write ψ = e (ψ0 + η[ue − various trap parameters, scattering lengths and atom numbers ∗ +i t/h v e ε ¯ ]), where µ is the chemical potential associated with confirms the geometrical scaling krot ∼ 1/`z, as well as its state ψ0, and u,v are spatial modes oscillating in time with weak dependence on as close to the instability as expected characteristic frequency ε/h¯ and η 1 [44]. Inserting this from the analytical model. We note in particular a good quan- ansatz in the NLGPE, and retaining only terms up to linear titative agreement of krot`z in Supplementary Fig. 6 with the order in η we obtain the Bogoliubov-de Gennes (BdG) equa- (stationary) analytic factor κ = 1.3 for this same trap geom- tions etry (see Method Analytical dispersion relation for an in- ! ! ! finite axially elongated geometry). A qualitative agreement Hˆ [ψ ] + A −A u u of the BdG stationary spectrum with the experimental obser- GP 0 = ε , (22) A −HˆGP[ψ0] − A v v vations is however not fully reached because of the role of the dynamics. A treatment of this effect is provided by the SSM where the operator A, acting on a function f and evaluated at and NS as discussed in the main text (see also the correspond- point r, is defined as ing Method and Section N). Finally, an alternative way of visualizing the excitation Z 0 0 0 0 spectrum is achieved by computing the dynamical structure (A f )(r) = dr ψ0(r )U(r − r ) f (r )ψ0(r) factor at T = 0 for each mode (u,v) [8, 46], 16 3 + √ ga3/2 1 + ε2 |ψ (r)|3 f (r). (23) s dd 0 Z 2 π 2 0 ∗ ∗ ik·r 0 S(k,ω ) = ∑ dr[u (r) + v (r)]e ψ0(r) δ(ω − ω), The above equations constitute an eigenvalue problem, which j we solve numerically using the Arnoldi method to obtain (24) eigenmodes (u,v) and corresponding excitation energies ε. where ω = ε/h¯. The structure factor of the axial modes of our The equations presented here are a generalization of the BdG finite trapped system (Supplementary Fig. 6a) presents fea- equations for dipolar systems as derived in Ref. [44], to in- tures resembling a roton-maxon spectrum, its continuum limit clude the LHY correction accounting for quantum fluctua- being a single-valued spectrum as depicted in Fig. 1b. To en- tions. hance the visibility, the delta function in Eq. (24) is replaced In order to depict the spectrum as a quasi-dispersion rela- by a Gaussian with a small finite width in ω. Since the dy- tion even in the presence of an axial confinement, we asso- namical structure factor determines the response of the system ciate to each elementary excitation an effective momentum when probed at specific energies and momenta, such as e.g. (eff) 2 1/2 in Bragg spectroscopy experiments [8, 47], it is interesting to ky = hky i [45]. The spectrum is discrete with phonon- (eff) (eff) note the difference in amplitude between the roton modes and like collective modes at low ky . For higher ky , the spec- other parts of the spectrum. In particular, in our quench exper- trum flattens, but eventually bends upwards again due to the iments where the system is effectively driven at a range of en- dominant kinetic energy. Instead of developing a smooth min- ergies and momenta, one would expect the strongest response imum, roton excitations appear as isolated low-lying modes from the roton modes. at intermediate momenta that depart from the overall spectrum [45]. These so-called roton fingers are related to confinement of the roton modes in the inhomogeneous BEC of M. Self-similar dynamics of the BEC. profile n0(y) [9], see also discussion in the main text. The confinement is evident from the BdG calculations, in which the lowest roton mode forms a short-wavelength den- In our self-similar model we account for the dynamics of sity modulation localized at the trap center (Supplementary the BEC profile and its effect on the roton spectrum (see de- Fig. 6e). This contrasts with phonon modes for which the tails in the corresponding Method). To do so, we assume that 14

1.5 1.5 1 1 d (a.u.) 0.5 S (arb.u.) (a.u.) 0.5 n 1.0 n 0 0 -20 -10 0 10 20 -20 -10 0 10 20 y ( m) n~ (a.u.) 0 1 y ( m) ~ ) n (a.u.) )

-1 -5 1 0.8 -1 -5 1

m d1 0 m 0 ( ( x 5 0 x 5 k ) 0 k

-5 0 5 z -5 0 5 -1 0.6 -1 k y ( m ) k ( m )

h y ( 1.5 E 1.5 1 c 0.4 1 e

(a.u.) 0.5 (a.u.) 0.5 n 0 n 0 -20 -10 0 10 20 -20 -10 0 10 20 y ( m) n~ (a.u.) 0.2 y ( m) n~ (a.u.) ) )

-1 1 -1 1 -5 c1 -5 e1 m 0 m 0

( 0 ( x 5 0 x 5 0 k -5 0 5 0 0.5 1 1.5 2 k -5 0 5 k ( m -1 ) k(eff) k ( m -1 ) y y z y

Figure 6 — BdG excitation spectrum. a, Excitation spectrum of the ground state of a BEC with N = 50.000 166Er atoms in a trap with (νx,νy,νz) = (267,32,456)Hz and scattering length as = 43.75a0, obtained by numerically solving the BdG equations. Roton modes appear as isolated modes lying below the main branch of the spectrum, forming a ’roton finger’. The dynamic structure factor S corresponding to each of the modes is indicated with colored shading. Highlighted in green are several exemplary modes, with panels b - e showing the corresponding excited state density modulation (black dashed line indicating the ground state, red solid line the excited state), and panels b1 - e1, the corresponding momentum distribution from 30 ms ToF expansion (Methods). b, b1 intermediate mode, c, c1 phonon mode, d, d1 single particle mode, e, e1 roton mode. the condensate preserves its TF shape during the evolution: u2)/2. Substituting Eq. (27) into Eq. (26) we obtain a closed " # set of equations for the scaling parameters: x 2 y 2 z 2 n(r,t) = n0(t) 1 − − − , ¨ X(t) Y(t) Z(t) 1 bx 1 A(t) 2 2 = −bx + 2 , (31) (25) (2π) νx bxbybz A(0) where X(t) = bx(t)X0, Y(t) = by(t)Y0, and Z(t) = bz(t)Z0 1 b¨ 1 B(t) y = −b + , (32) are the re-scaled TF radii, with bx,y,z(t) the scaling coeffi- 2 2 y 2 (2π) ν bxb bz B(0) cients (b (0) = 1). The corresponding hydrodynamic equa- y y x,y,z ¨ tions reduce to: 1 bz 1 C(t) = −bz + . (33) (2π)2 ν2 b b b2 C(0) m X¨ Y¨ Z¨ z x y z ∇ x2 + y2 + z2 = −∇µ(r,t), (26) 2 X y Z Prior to the quench of as, the stationary TF solution is obtained νx p from the self-consistent equations χx(0) = C(0)/A(0) R 3 0 0 0 νz where µ(r,t) = V(r) + gn(r,t) + d r Vdd(r − r )n(r ,t) is νy p and χy(0) = C(0)/B(0). Solving these equations and the local chemical potential. The latter acquires the form: νz using normalisation provides X(0), Y(0), Z(0), and n(0) for x 2 y 2 z 2 known numbers of atoms N and trap frequencies ν . We use µ(r,t) = V(r) + gn D − A − B −C , x,y,z 0 X Y Z this stationary solution as the initial condition at the start of (27) the quench (t = 0), and solve the system of differential equa- where D(t) = 1 + εddF1, A(t) = 1 + εdd(F1 − F2 + F3), B(t) = tions (31), (32) and (33) to obtain the scaling coefficients. An 1 + εdd(F1 − F2 − F3), and C(t) = 1 + εdd(F1 + 2F2). Here we example for the relevant parameters of Fig. 3 is shown in Sup- have introduced the functions: plementary Fig. 3. " # Z 1 3u2 F1(χx, χy)= du − 1 , (28) p 2 2 0 α −β N. Numerical simulations of the evolution. " # Z 1 3u2−1 3u2 F2(χx, χy)= du p − 1 , (29) Our simulations of the NLGPE are performed using a 0 2 α2−β 2 split operator technique. The evolution operator over a time "p # Z 1 9u2(1−u2) α2−β 2 − α ∆t (∆t → i∆t for imaginary time evolution) may be approxi- F3(χx, χy)= du p , (30) −iHˆ ∆t/h¯ −iTˆ∆t/h¯ −iVˆ ∆t/h¯ 2 0 4 β α2−β 2 mately split as e = e e + O(∆t ). In this expression, Tˆ is the kinetic energy term, and Vˆ the poten- with χx(t) = Z(t)/X(t), χy(t) = Z(t)/Y(t), α(χx, χy,u) = tial energy. The effective potential energy for the evolution 2 2 2 2 2 2 (χx + χy )(1 − u )/2 + u , and β(χx, χy,u) = (χx − χy )(1 − is given by the sum of external potential, interparticle interac- 15

as(t) (Method Quench of the scattering length as). More- ) 0 60 1.2 over, for the value of the three-body loss coefﬁcient L3 we use (a s 1.1 a linear ﬁt of the experimentally determined values [25]. From 58 1 the simulated evolution, we obtain the 3D wavefunction of the 56 0.9 gas as a function of th, ψ(r,th), from which we can extract the 54 X 0.8 spatial and momentum distributions. Y 0.7 52 Z, The simulation of the ToF expansion is performed in two 0.6 n , a parameters Rescaled steps. First we use a multi-grid analysis in order to rescale

Scattering length, a 50 0 s 0.5 the size of the numerical box as the cloud expands during the 0 1 2 3 4 5 6 t (ms) ToF expansion. After some expansion time the density drops h significantly, and the subsequent evolution can be readily cal- −iTtˆ /h¯ Figure 7 — Self-similar dynamics in quench. SSM re- culated via e . Our NS show clearly that the effect of sults after a quench from ai = 61a to af = 50a in the trap nonlinearity is small during the first stages of the evolution, s 0 s 0 and hence that the ToF expansion indeed may be employed (νz,λ) = (149Hz,4.3) and using the experimental cloud characteristics (configuration of Fig. 3). For reference, we plot the to image the momentum distribution of the condensate at the time in which the trap is opened. experimentally known as(th) (circles, left axis). We show the 3D TF radii X (down triangles), Y (diamonds), Z (up triangles) We evaluate the integrated momentum distribution R 2 and n0 (squares) renormalized by their th = 0-values (right n˜(ky,th) = dkxdkz|ψ˜ (k,th)| with ψ˜ (k,th) the Fourier axis). The compression mainly occurs along X. The subse- transform of ψ(r,th). After th of a few ms,n ˜(ky,th) shows quent increase of n0 is less than 2 % when ∆(y = 0) touches clear roton peaks. The exact th value for the peak emergence zero in this configuration (Fig. 3a). This demonstrates that the depends on the gas characteristics (in particular T) and on as. dominant effect on the roton spectrum at the instability comes We evaluate the roton momentum as the mean value of the f momentum in the roton peak. from the reduction of as itself, even in the case of as relatively ∗ close to as . From the imaginary-time evolution simulations, we are also ∗,st able to predict the as = as threshold for the mean-field instability of the BEC, which corresponds to the absence of a tions, local LHY correction, and three-body losses (see Sec- mean-field stable solution [39, 40]. To find this instability tion K). boundary, we proceed by steps. We start by calculating the We first evaluate using imaginary time evolution the ini- ground-state solution for a given as that we know to be well tial BEC wavefunction, ψ0(r), prior to the quench of as. The within the stable regime. We then reduce as in small steps initial wavefunction for the subsequent real-time evolution is and successively calculate the corresponding ground-state so- then constructed via ψi = ψ0 + ∆ψ, where ∆ψ accounts for lution using the solution of the previous step as starting con- thermal fluctuations, which we simulate by populating the ex- dition. We do so until no mean-field stable solution can be ∗,st cited states of the system as described in Ref. [35]. The ex- found. The predicted as are reported in Supplementary Ta- cited states used in this procedure are obtained from the full ble II. For the theory predictions shown in Fig. 2d (NS and BdG calculation detailed in Section L. Starting with this ini- SSM; see Method Analytical dispersion relation for an in- tial wavefunction ψi, we mimic as close as possible the condi- finite axially elongated geometry), we use quenched as val- tions of our experiments, including ramping, holding, and ToF ues such that the instability boundary is just slightly crossed, ∗,st times. In particular, we include the experimentally calibrated as = as − 1a0.