Letter | Published:

Observation of roton mode population in a dipolar quantum gas

Nature Physicsvolume 14pages442446 (2018) | Download Citation

Abstract

The concept of a roton, a special kind of elementary excitation forming a minimum of energy at finite momentum, has been essential for the understanding of the properties of superfluid 4He (ref. 1). In quantum liquids, rotons arise from the strong interparticle interactions, whose microscopic description remains debated2. In the realm of highly controllable quantum gases, a roton mode has been predicted to emerge due to magnetic dipole–dipole interactions despite their weakly interacting character3. This prospect has raised considerable interest4,5,6,7,8,9,10,11,12; yet roton modes in dipolar quantum gases have remained elusive to observations. Here we report experimental and theoretical studies of the momentum distribution in Bose–Einstein condensates of highly magnetic erbium atoms, revealing the existence of the long-sought roton mode. Following an interaction quench, the roton mode manifests itself with the appearance of symmetric peaks at well-defined finite momentum. The roton momentum follows the predicted geometrical scaling with the inverse of the confinement length along the magnetization axis. From the growth of the roton population, we probe the roton softening of the excitation spectrum in time and extract the corresponding imaginary roton gap. Our results provide a further step in the quest towards supersolidity in dipolar quantum gases13.

Main

Quantum properties of matter continuously challenge our intuition, especially when many-body effects emerge at a macroscopic scale. In this regard, the phenomenon of superfluidity is a paradigmatic case, which continues to reveal fascinating facets following its discovery in the late 1930s2,12. A major breakthrough in understanding superfluidity thrived on the concept of quasiparticles, introduced by Landau in 19411. Quasiparticles are elementary excitations of momentum k, whose energies ε define the dispersion (energy–momentum) relation ε(k).

To explain the special thermodynamic properties of superfluid 4He, Landau postulated the existence of two types of low-energy quasiparticle: phonons, referring to low-k acoustic waves, and rotons, gapped excitations at finite k initially interpreted as elementary vortices. The dispersion relation continuously evolves from linear at low k (phonons) to parabolic-like with a minimum (roton) at a finite k = krot. Neutron scattering experiments confirmed Landau’s remarkable intuition14. In liquid 4He, krot scales as the inverse of the interatomic distance. This manifests a tendency of the system to establish a local order, which is driven by the strong correlations among the atoms2.

In the realm of low-temperature quantum physics, ultracold quantum gases realize the other extreme limit for which the interparticle interactions—and correlations—are typically weak, meaning that classically their range of action is much smaller than the mean interparticle distance. As a result of this diluteness, roton excitations are absent in ordinary quantum gases, that is, in Bose–Einstein condensates (BECs) with contact (short-range) interactions12. However, about 15 years ago, seminal theoretical works predicted the existence of a roton minimum both in BECs with magnetic dipole–dipole interactions (DDIs)3 and in BECs irradiated by off-resonant laser light15. Following the lines of the latter proposal, a roton softening has recently been observed in BECs coupled to an optical cavity16. Here, the excitation arises from the infinite-range photon-mediated interactions and the inverse of the laser wavelength sets the value of krot. In addition, roton-like softening has been created in spin–orbit-coupled BECs17,18 and quantum gases in shaken optical lattices19 by engineering the single-particle dispersion relation.

Our work focuses on dipolar BECs (dBECs). As in superfluid 4He, the roton spectrum in such systems is a genuine consequence of the underlying interactions among the particles. However, in contrast to helium, the emergence of a minimum at finite momentum does not require strong interparticle interactions. It instead exists in the weakly interacting regime and originates from the peculiar anisotropic and long-range character of the DDI in real and momentum space3,4,5,6,7,8,9,10,11,12. Despite the maturity achieved in the theoretical understanding, the observation of dipolar roton modes has remained so far an elusive goal. For a long time, the only dBEC available in experiments consisted of chromium atoms20, for which the achievable dipolar character is hardly sufficient to support a roton mode. With the advent of the more magnetic lanthanide atoms21,22, a broader range of dipolar parameters became available, opening the way to access the regime of dominant DDI. In this regime, novel exciting many-body phenomena have been recently observed, as the formation of droplet states stabilized by quantum fluctuations23,24,25, which may become self-bound26. Lanthanide dBECs hence open new roads toward the long-sought observation of roton modes.

Prior to this work, dipolar rotons have been mostly connected to pancake-like geometries3,4,5,6,8,9,10,11,12. Here, we extend the study of roton physics to the case of a cigar-like geometry with trap elongation along only one direction (y) transverse to the magnetization axis (z) (Fig. 1a). The anisotropic character of the DDI (Fig. 1b, inset) together with the tighter confinement along z is responsible for the rotonization of the excitation spectrum along y (Fig. 1b). To illustrate this phenomenon, we consider an infinite cigar-shaped dBEC and focus on its axial elementary excitations, of momentum k y . These excitations correspond in real space to a density modulation along y of wavelength 2π/k y . For low k y , the atoms sit mainly side-by-side and the repulsive nature of the DDI prevails, stiffening the phononic part of the dispersion relation (Fig. 1b, bottom left panel). In contrast, for \({k}_{y}{\ell }_{z}\) 1, where \({\ell }_{z}\) is the characteristic z confinement length, the excitation favours head-to-tail alignments and the DDI contribution to ε(k y ) eventually changes sign3 (Fig. 1b, bottom right panel). The resulting softening of ε(k y ) is counterbalanced by the contributions of the repulsive contact interaction, and of the kinetic energy, which ultimately dominates at very large k y . For strong enough DDI, this competition gives rise to a roton minimum in ε(k y ), occurring at a momentum k y  = krot set by the geometrical scaling \({k}_{{\rm{rot}}} \sim 1{\rm{/}}{\ell }_{z}\) (see below and, for example, refs 3,8,9).

Fig. 1: Roton mode in a dBEC.
Fig. 1

a, An axially elongated geometry with dipoles oriented transversely. b, Real (solid lines) and imaginary (norm of the dotted line) parts of the dispersion relation of a dBEC in the geometry in a, showing the emergence of a roton minimum for decreasing as (dashed arrow). The DDI changes from repulsive (blue) to attractive (red) depending on the dipole alignment (inset). The bottom panels show the dipole alignment (colour code as in the inset) associated with small-k y (left panel) and large-k y (right panel) density modulations, respectively. c,d, Distributions on the k x k y  plane associated with the roton population in cigar (a) or pancake (d) geometries with an identical roton population and colour scale.

Similar to the helium case, the roton energy gap, Δ = ε(krot), depends on the density and on the strength of the interactions. In ultracold gases, both quantities can be controlled. In particular, the scattering length as, setting the strength of the contact interaction, can be tuned using Feshbach resonances12. As as is reduced, Δ decreases, vanishes and eventually becomes imaginary (Fig. 1b). In this last case, the system undergoes a roton instability and the population at k y  = 0 is transferred to ±krot at an exponential rate5,6. The population of the roton mode is then readily visible in the momentum distribution of the gas (Fig. 1c,d). In the extensively studied pancake geometries, the roton population in k-space spreads over a ring of radius k = krot because of the radial symmetry of the confinement (Fig. 1d). Such a spread can be avoided using a cigar geometry. Here, the roton population focuses in two prominent peaks at k y  = ±krot, enhancing the visibility of the effect (Fig. 1c).

We explore the above-described physics using strongly magnetic 166Er atoms. The experiment starts with a stable dBEC in a cigar-shaped harmonic trap of frequencies νx,y,z, elongated along the y axis. The trap aspect ratio, λ = ν z /ν y , can be tuned from about 4 to 30, corresponding to ν z ranging from 150 to 800 Hz, whereas ν y and ν z /ν x are kept constant at about 35 Hz and 1.6, respectively (Methods). An external homogeneous magnetic field, B, fixes the dipole orientation (magnetization) with respect to the trap axes and sets the values of as through a magnetic Feshbach resonance, centred close to B = 0 G. In previous experiments, we precisely calibrated the B-to-as conversion for this resonance25. The BEC is prepared at \({a}_{{\rm{s}}}^{{\rm{i}}}=61{a}_{0}\) (B = 0.4 G) with transverse (z) magnetization. The characteristic dipolar length, defined as \({a}_{{\rm{dd}}}={\mu }_{0}{\mu }_{{\rm{m}}}^{2}m{\rm{/}}12\pi {\hslash }^{2}\), is 65.5a0, where m is the mass and μm is the magnetic moment of the atoms, ħ = h/2π is the reduced Planck constant, μ0 is the vacuum permeability and a0 is the Bohr radius.

To excite the roton mode, we quench as to a desired lower value, \({a}_{{\rm{s}}}^{{\rm{f}}}\), and shortly hold the atoms in the trap for a time th. We measure that as converges to its set value with a characteristic time constant of 1 ms during th. We then release the atoms from the trap, change as back close to its initial value and let the cloud expand for 30 ms. We probe the momentum distribution \({\tilde{ n}}\left({k}_{x},{k}_{y}\right)\) by performing standard resonant absorption imaging on the expanded cloud (Methods). The measurement is then repeated at various values of as < add in a fixed trap geometry. The momentum distribution shows a striking behaviour (Fig. 2). For large enough as, \({\tilde{n}}\left({k}_{x},{k}_{y}\right)\) shows a single narrow peak with an inverted aspect ratio compared to the trapped gas, typical of a stable BEC12 (Fig. 2a, top). We define the centre of the distribution as the origin of k. In contrast, when further decreasing as below a critical value \({a}_{{\rm{s}}}^{* }\), we observe a sudden appearance of two symmetric finite-momentum peaks, of similar shape and located at k y  = ±krot (Fig. 2b,c, top panels). By repeating the experiment several times, we observe that the peaks consistently appear at the same locations, and they are visible in the averaged distributions. To quantitatively investigate the peak structures, we fit a sum of three Gaussian distributions to the central cuts of the average \({\tilde{ n}}\left({k}_{x},{k}_{y}\right)\) (Fig. 2a–c, bottom panels). From the fit, we extract the central momentum, krot, and the amplitude, A*, of the side peaks.

Fig. 2: Observed roton peaks and characteristic scalings.
Fig. 2

ac, The top panels show \({\tilde{ n}}({k}_{x},{k}_{y})\) obtained by averaging 15–25 absorption images for (ν z , λ) = (456 Hz,14.4), th = 3 ms and as = 54a0 (a), as = 44a0 (b) and as = 37a0 (c). The bottom panels show the corresponding cuts at k x  ≈ 0 (dots) and their fits to three-Gauss distributions (lines), from which we extract krot and A*. d, Measured krot depending on \(1{\rm{/}}{\ell }_{z}\) at as\({a}_{{\rm{s}}}^{* }\) for ν y  ≈ 35 Hz (circles) and ν y  = 17(1) Hz (triangle). e, krot depending on as for (ν z , λ) = (149 Hz, 4.3) (circles). In d,e, the error bars show the 95% confidence interval of the three-Gauss fits (see the bottom panels in ac). The squares (diamonds) show predictions from the SSM (NS) in their range of validity (Methods). The lines are guides to the eye.

A major fingerprint of the roton mode in dBECs is its geometrical nature, leading to a universal scaling \({k}_{{\rm{rot}}} \sim 1{\rm{/}}{\ell }_{z}\) (see, for example, refs 3,4,8,9). In addition, the dependency of krot on as close to the instability is expected to be mild as krot remains mainly set by its geometrical nature3,8. We investigate both dependencies in the experiment. In a first set of experiments, we repeat the quench measurements for various trap parameters and extract krot. We clearly observe the expected geometrical scaling. krot shows a marked increase with \(1{\rm{/}}{\ell }_{z}=2\pi \sqrt{m{\nu }_{z}{\rm{/}}h}\), matching well with a linear progression with a slope of 1.61(4) (Fig. 2d). Note that no dependence of krot on ν y is observed. In a second set of experiments, we fix the trap geometry and explore the dependence of krot on as. We observe that, within our experimental uncertainty, krot stays constant when decreasing as (Fig. 2e). This behaviour contrasts with the one expected for a phonon-driven modulation instability that exhibits a strong as-dependence27.

To gain a deeper understanding of the roton excitations in our system and its dynamical population, we develop both an analytical model and full numerical simulations (NSs) and compare the findings with our experimental data. Our analytical model starts by calculating the roton spectrum of the stationary BEC, generalizing the results of ref. 3 to a non-radially symmetric configuration. Since the roton wavelength is much smaller than the extension of our three-dimensional (3D) BEC along y, this mode can be evaluated using a local density approximation in y. Hence, for our model, we consider a dBEC homogeneous along y, of axial density n0, harmonically confined along x and z. To analytically evaluate the roton spectrum, we approximate the BEC wavefunction using the Thomas–Fermi (TF) approximation. For dominant DDI, εdd = add/as ≥ 1, we find that ε(k y ) indeed rotonizes (Methods). In the vicinity of the roton minimum and for εdd ~ 1, the computed dispersion acquires a gapped quadratic form similar to that of helium rotons:

$$\varepsilon {\left({k}_{y}\right)}^{2}\simeq {{\it{\Delta }}}^{2}+\frac{2{\hslash }^{2}{k}_{{\rm{rot}}}^{{\rm{2}}}}{m}\frac{{\hslash }^{2}}{2m}{\left({k}_{y}-{k}_{{\rm{rot}}}\right)}^{2}$$
(1)

The roton momentum reads as \({k}_{{\rm{rot}}}=\sqrt{2m}{\left({E}_{0}^{2}-{{\it{\Delta }}}^{2}\right)}^{1/4}{\rm{/}}\hslash\), where \({\it{\Delta }}=\sqrt{{E}_{0}^{2}-{E}_{{\rm{I}}}^{2}}\), EI = 2gn0(εdd − 1)/3, g = 4πħ2as/m and \({E}_{0}^{2}=2g{\varepsilon }_{{\rm{dd}}}{n}_{0}\frac{{\hslash }^{2}}{2m}\left({X}^{-2}+{Z}^{-2}\right)\). The TF radii X,Z satisfy X2,Z2gn0, so that E0 z , with a scale set by εdd and ν z /ν x but independent of gn0 (ref. 15). Close to the instability (\({\it{\Delta }}\simeq 0\)), the roton momentum thus follows a simple geometrical scaling, \({k}_{{\rm{rot}}}=\kappa {\rm{/}}{\ell }_{z}\) with the geometrical factor κ depending on ν z /ν x alone. For completeness, we have also performed full 3D numerical calculations of the static Bogoliubov spectrum of a finite trapped dBEC, confirming the existence and scaling of the roton mode (Supplementary Information).

The above stationary description accounts for the existence of the roton mode in the cigar geometry used in experiments, and predicts the scaling of krot and Δ with the system parameters. However, the quench of as introduces dynamics, which is crucial to quantitatively reproduce the experimental observations. The reduction of as decreases the contact interaction and additionally induces a compression of the cloud. This yields a dynamical modification of the local roton dispersion relation, and the roton may be destabilized during the evolution. This dynamical destabilization is well accounted for by a self-similar model (SSM) describing the evolution of the cloud shape after the quench12. In particular, we consider a 3D harmonic confinement, and, starting from the stationary TF profile at \({a}_{{\rm{s}}}^{{\rm{i}}}\), we evaluate the evolution of the cloud along the change of as assuming that the TF shape is maintained but with time-dependent TF radii (Methods). We then estimate the local (along y) instantaneous roton spectrum ε(k y , y,th) using a local density approximation, that is, evaluating equation (1) with the experimentally calibrated as(th), and the n0(y, th), X(y,th) and Z(y,th) estimated from the 3D profile. We indeed find that the roton gap decreases and eventually turns imaginary (Δ(th)2 < 0) (Fig. 3a). When this occurs the population at k y  ≈ krot exponentially grows with an instantaneous local rate 2Im[ε(k y ,y,th)]/ħ, with Im[.] indicating the imaginary part, giving rise to two symmetric side peaks in the axial momentum distribution \({{\tilde{ n}}}_{1}\left({k}_{y},{t}_{{\rm{h}}}\right)\propto \int {\rm{\exp }}\left(2{\int }_{0}^{{t}_{{\rm{h}}}}{\rm{Im}}\left[\varepsilon \left({k}_{y},y,t\right)\right]{\rm{d}}t\right){\rm{d}}y\) (Fig. 3b) (Methods). The centre of the peaks, ±krot(th), also evolves with th but quickly converges after a few milliseconds to its final value, krot (Fig. 3c). The measured krot and the calculated values from our parameter-free theory are in remarkable agreement (Figs. 2d,e and 3c).

Fig. 3: Dynamics of the roton mode.
Fig. 3

Results depending on th after quenching to \({a}_{{\rm{s}}}^{{\rm{f}}}=50{a}_{0}\) in (ν z , λ) = (149 Hz, 4.3). ac, Δ2 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). The shaded areas identify the dynamical destabilization. For comparison, the amplitudes are renormalized to their th = 3 ms 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 momentum \({{\tilde{ n}}}_{1}({k}_{y})\) and real space n1(y), from one run of the NS.

Our SSM quantitatively explains the experimental observations and provides us with a physical understanding. For completeness, we perform NSs of the system dynamics. We calculate the time evolution of the generalized non-local Gross–Pitaevskii equation (NLGPE), which accounts also for quantum fluctuations, three-body loss processes and finite temperature, not included in the SSM (Supplementary Information). The first two contributions limit the peak density of the atomic cloud and stabilize the dBEC against collapse4,5,6,24,25, whereas the last term thermally seeds the initial roton mode population. The NSs confirm both the SSM results and the experimental observations. Indeed, the calculations show that, a few milliseconds after the quench, the system develops roton peaks in momentum space (Fig. 3d) and short-wavelength density modulations at the centre of the BEC (Fig. 3e), showing the predicted roton confinement9. The extracted value of krot and its geometrical scaling are in very good quantitative agreement with both the experimental data and the SSM calculations (Figs. 2d,e and 3c). Interestingly, the time scales for the emergence of roton peaks in the NS are a few milliseconds longer than those observed in the experiment. The origin of this time shift remains an open question, whose answer could, for example, require a refinement of current models of quantum fluctuations. However, despite this delay, the growth rate of the roton population in the NS matches both the experimental observations and the SSM predictions in the early dynamics (Fig. 3b).

In the experiment, we study the time evolution of the roton mode in a fixed geometry (ν z , λ). In a first set of measurements, we fix \({a}_{{\rm{s}}}^{{\rm{f}}}\) and follow the dynamics by recording the momentum distribution at various th. We observe that krot does not change significantly while the roton population initially grows, in excellent agreement with the theories (Fig. 3b,c). The growth rate is a particularly relevant quantity as it is directly connected to the imaginary excitation energy in the Bogoliubov description (for the roton, its gap), as revealed by our theory. In a second set of experiments, we thus systematically study the growth rate of the roton population for various \({a}_{{\rm{s}}}^{{\rm{f}}}\) (Fig. 4). Our data show that the roton mode begins to become populated from a critical value of the scattering length, \({a}_{{\rm{s}}}^{* }\). For \({a}_{{\rm{s}}}^{{\rm{f}}} > 52{a}_{0}\), we do not observe the roton peaks at any time. For \({a}_{{\rm{s}}}^{{\rm{f}}}\) 52a0, after a time delay, A* undergoes an abrupt increase. At longer time, A* then saturates and eventually slowly decreases while atoms are coincidentally lost (Fig. 4a). By further lowering \({a}_{{\rm{s}}}^{{\rm{f}}}\), the roton population exhibits a faster growth rate and shorter time delay.

Fig. 4: Population growth and roton gap.
Fig. 4

Results for (ν z , λ) = (149 Hz, 4.3). a,b, A* depending on th and on the rescaled time Th, after quenching to \({a}_{{\rm{s}}}^{{\rm{f}}}=39{a}_{0}\), \(44{a}_{0}\), \(47{a}_{0}\), \(50{a}_{0}\) and \(52{a}_{0}\). The shaded areas show the 95% confidence interval from the three-Gauss fit. The black line shows an exponential fit to the full data set with Th < 3 ms. c, Extracted \({\rm{Im}}[\bar{{\it{\Delta }}}]\) depending on as and on th (inset) from the experiments (dashed line with pentagons) with the propagated errors (shaded area) from the time-rescaling analysis and exponential fit (see b), and from the SSM (solid line). The dotted line and the corresponding shaded area show the experimental \({a}_{{\rm{s}}}^{* }\) and the confidence interval of its fit. The inset shows the same configuration as Fig. 3 (\({a}_{{\rm{s}}}^{{\rm{f}}}=50{a}_{0}\)).

From the growth rate of the roton population, we now extract an overall roton gap \(\bar{{\it{\Delta }}}({a}_{{\rm{s}}})\) (Methods). In brief, the dynamical variation of the gap is determined by the leading time dependence of as and \({A}_{* }\left({a}_{{\rm{s}}}^{{\rm{f}}},{t}_{{\rm{h}}}\right)\propto {\rm{\exp }}\left(2{\int }_{0}^{{t}_{{\rm{h}}}}{\rm{d}}t{\rm{Im}}\left[\bar{{\it{\Delta }}}({a}_{{\rm{s}}}(t))\right]{\rm{/}}\hslash \right)\). To investigate the scaling \(\bar{{\it{\Delta }}}({a}_{{\rm{s}}})\), we first consider the analytical expression of the elementary local roton gap, Δ as a function of as (see equation (1)). Developing Δ in the vicinity of \({a}_{{\rm{s}}}={a}_{{\rm{s}}}^{* }\) yields \({\it{\Delta }}({a}_{{\rm{s}}})\propto {\left({a}_{{\rm{s}}}^{* }-{a}_{{\rm{s}}}\right)}^{1/2}\). Here we confirm this scaling by considering the generic power-law dependence \({\rm{Im}}[\bar{{\it{\Delta }}}({a}_{{\rm{s}}})]{\rm{/}}\hslash ={\rm{\Gamma }}\delta ({a}_{{\rm{s}}})\) with \(\delta ({a}_{{\rm{s}}})={\left(\frac{{a}_{{\rm{s}}}^{* }-{a}_{{\rm{s}}}}{{a}_{0}}\right)}^{\beta }\left({a}_{{\rm{s}}}\le {a}_{{\rm{s}}}^{* }\right)\). Consequently, A* grows as \({\rm{\exp }}\left(2{\it{\Gamma }}{\int }_{0}^{{t}_{{\rm{h}}}}\delta ({a}_{{\rm{s}}}(t)){\rm{d}}t\right)={\rm{\exp }}\left(2{\it{\Gamma }}{T}_{{\rm{h}}}\right)\). By rescaling the time variable as \({t}_{{\rm{h}}}\to {T}_{{\rm{h}}}={\int }_{0}^{{t}_{{\rm{h}}}}\delta ({a}_{{\rm{s}}}(t)){\rm{d}}t\), we determine the parameters \({a}_{{\rm{s}}}^{* }\) and β such that all of the curves of Fig. 4a fall on top of each other (Fig. 4b). The best overlap of the experimental curves is found for \({a}_{{\rm{s}}}^{* }=53.0(4){a}_{0}\) and β = 0.55(8). We then perform an exponential fit to all of the rescaled data for Th < 3 ms and extract Γ = 465(83) s−1. The same analysis, applied to the calculated population growth from the SSM for different \({a}_{{\rm{s}}}^{{\rm{f}}}\), gives \({a}_{{\rm{s}}}^{* }\simeq 52.8{a}_{0}\), \(\beta \simeq 0.55\) and \({\it{\Gamma }}\simeq 472{{\rm{s}}}^{-1}\), which are very close to the experimental values. This time-resolved study allows us to readily extract the imaginary roton gap, \({\rm{Im}}[\bar{{\it{\Delta }}}]\), from both the experiment and the SSM, as a function of as and th (Fig. 4c). Our observations show the softening of the roton mode at \({a}_{{\rm{s}}}={a}_{{\rm{s}}}^{* }\) and the expected increase of \({\rm{Im}}[\bar{{\it{\Delta }}}]\) for \({a}_{{\rm{s}}} < {a}_{{\rm{s}}}^{* }\), nicely grasping the dynamics during the growth of the roton population (inset of Fig. 4c). The observed behaviour shows again a remarkable agreement with the theory predictions.

To conclude, our work demonstrates the power of weakly interacting dipolar quantum gases to access the regime of large-momentum, yet low-energy, excitations dressed by interactions. This newly accessible regime, which is largely unexplored in ultracold gases, raises fundamental questions and opens novel directions. Future key developments are to study the impact of such low-lying excitations on the superfluid behaviour of a dBEC8,9,10,12, the interplay between phononic and rotonic modes in the stability diagram of the quantum gas4,7 and the possible role of roton excitations as the triggering mechanism of the recently observed instability leading to the formation of metastable droplet arrays in elongated traps24,28. Of particular interest is the prospect of creating a supersolid and striped ground states in dBECs13,28. Indeed, the short-wavelength density modulation tied to the roton softening together with the quantum stabilization mechanism can favour the formation of such an exotic phase of matter, in which crystalline order coexists with phase coherence. In contrast to recent experiments29,30, where the density modulation is imposed by external fields yielding a supersolid-like arrangement of infinite stiffness, a dipolar supersolid would be compressible. Hence, experiments on dBECs provide the exciting opportunity to unveil similarities and differences among complementary approaches to supersolidity.

Methods

Trapping geometries

The BEC is confined in a harmonic trapping potential \(V({\bf{r}})=2m{\pi }^{2}\left({\nu }_{x}^{2}{x}^{2}+{\nu }_{y}^{2}{y}^{2}+{\nu }_{z}^{2}{z}^{2}\right)\), characterized by the frequencies (ν x ,ν y ,ν z ). The trap results from the crossing of two red-detuned laser beams of 1,064 nm wavelength at their respective foci. One beam, called vODTb, propagates nearly collinear to the z axis and the other, denoted hODTb, propagates along the y axis. By adjusting independently the parameters of the vODTb and hODTb, we can widely and dynamically control the geometry of the trap (see Supplementary Information). In particular, ν y is essentially set by the vODTb power while ν z (and ν x ) is set independently by that of the hODTb. This yields an easy tuning of the trap aspect ratio, λ = ν z /ν y , relevant for our cigar-like geometry.

After reaching condensation (see Supplementary Information), we modify the beam parameters to shape the trap into an axially elongated configuration, favourable for observing the roton physics \(\left({\nu }_{y}\ll {\nu }_{x},{\nu }_{z}\right)\). The trapping geometries probed in the experiments, whose (λ,ν x ,ν y ,ν z ) are reported in Supplementary Table 1, are achieved by changing the hODTb power with the vODTb power set to 7 W, so that ν y and ν z /ν x are kept roughly constant. Only the green triangle in Fig. 2d is obtained in a distinct configuration, with a vODTb power of 2 W, leading to (ν x ,ν y ,ν z ) = (156, 17, 198)Hz and λ = 11.6. The (λ,ν x ,ν y ,ν z ) are experimentally calibrated via exciting and probing the centre-of-mass oscillation of thermal samples. We note that the final atom number N, BEC fraction f and temperature T after the shaping procedure depend on the final configurations, as detailed in Supplementary Table 1.


Quench of the scattering length a s

To control as, we use a magnetic Feshbach resonance between 166Er atoms in their absolute ground state that is centred around B = 0 G (ref. 31). The B-to-as conversion has previously been precisely measured via lattice spectroscopy, as reported in ref. 25. Errors on as, taking into account statistical uncertainties of the conversion and effects of magnetic field fluctuations (for example, from stray fields), are of 3–5a0 for the relevant as range 27–67a0 in this work. After the BEC preparation and to investigate the roton physics via an interaction quench, we suddenly change the magnetic field set value, Bset, twice. First, we perform the quench itself and abruptly change Bset from 0.4 G (\({a}_{{\rm{s}}}^{{\rm{i}}}=61{a}_{0}\)) to the desired lower value (corresponding to \({a}_{{\rm{s}}}^{{\rm{f}}}\)) at the beginning of the hold in the trap (th = 0 ms). Second, we prepare the time-of-flight (TOF) expansion and imaging conditions (see the section entitled Imaging procedure) and abruptly change Bset from the quenched value back to 0.3 G (as = 57a0) at the beginning of the TOF expansion (tf = 0 ms). Due to delays in the experimental set-up (for example, coming from eddy currents in our main chamber), the actual B value felt by the atoms responds to a change of Bset via B(t) = Bset(t) + τdB/dt (ref. 32). By performing pulsed-radiofrequency spectroscopy measurements (pulse duration 100 μs) on a BEC after changing Bset (from 0.4 to 0.2 G), we verify this law and extract a time constant τ = 0.98(5) ms. Hence, as is also evolving during th and tf on a similar timescale. This effect is fully accounted for in the experiments and simulations, and we report the roton properties as a function of the effective value of as at th. We use th ranging from 2 to 7 ms. The lower bound on th comes from the time needed for as to effectively reach the regime of interest. We then consider the initial evolution for which \({t}_{{\rm{h}}}{\rm{/}}{\nu }_{y},{t}_{{\rm{h}}}{\rm{/}}{\tau }_{{\rm{coll}}}\ll 1\), with 1/τcoll being the characteristic collision rate. We estimate that τcoll ranges typically from 40 to 90 ms in the initial BECs of Supplementary Table 1 at \({a}_{{\rm{s}}}^{{\rm{i}}}=61{a}_{0}\). Experimentally, we observe that the roton, if it ever develops, has developed within the considered range of th.


Imaging procedure

In our experiments, we employ TOF expansion measurements, accessing the momentum distribution of the gas12, to probe the roton mode population. We let the gas expand freely for tf = 30 ms, which translates the spatial imaging resolution (~ 3.7 μm) into a momentum resolution of ~0.32 μm−1 while our typical roton momentum verifies krot 2 μm−1. After TOF expansion, we record 2D absorption pictures of the cloud by means of standard resonant absorption imaging on the atomic transition at 401 nm. The imaging beam propagates nearly vertically, with a remaining angle of ~15° compared to the z axis within the xz plane. Thus, the TOF images essentially probe the spatial density distribution nTOF(x,y,tf) in the xy plane. When releasing the cloud (tf = 0 ms), we change B back to B = 0.3 G (see the section entitled Quench of the scattering length as). This change enables constant and optimal imaging conditions with a fixed probing procedure (that is, a maximal absorption cross-section). In addition, the associated increase of as to 57a0 allows us to minimize the time during which the evolution happens in the small-as regime where the roton physics develops, such that we effectively probe only the short-time evolution in this respect. In this work, we use the simple mapping:

$${\tilde{ n}}\left({k}_{x},{k}_{y}\right)={\left(\frac{\hslash {t}_{{\rm{f}}}}{m}\right)}^{2}{n}_{{\rm{TOF}}}\left(\frac{\hslash {k}_{x}{t}_{{\rm{f}}}}{m},\frac{\hslash {k}_{y}{t}_{{\rm{f}}}}{m},{t}_{{\rm{f}}}\right)$$
(2)

which neglects the initial size of the cloud in the trap and the effect of interparticle interactions during the free expansion. Using real-time simulations (see Supplementary Information), we simulate the experimental sequence and are able to compute both the real momentum distribution from the in-trap wavefunction and the spatial TOF distribution 30 ms after switching off the trap. Using the mapping of equation (2) and our experimental parameters, the two calculated distributions are very similar, and, in particular, the two extracted momenta associated with the roton signal agree within 5%. This confirms that the interparticle interactions play little role during the expansion and justifies the use of equation (2).


Fit procedure for the TOF images

For each data point in Figs. 24, we record between 12 and 25 TOF images. By fitting a 2D Gaussian distribution to the individual images, we extract their origin (k x , k y ) = (0, 0) and recentre each image. From the recentred images, we compute the averaged \({\tilde{ n}}({k}_{x},{k}_{y})\), from which we characterize the linear roton developing along k y . To do so, we extract a 1D profile \({{\tilde{ n}}}_{1}({k}_{y})\) by averaging \({\tilde{ n}}({k}_{x},{k}_{y})\) over k x within \(\left|{k}_{x}\right|\le {k}_{{\rm{m}}}=3.5\) μm−1: \({{\tilde{ n}}}_{1}({k}_{y})={\int }_{-{k}_{{\rm{m}}}}^{{k}_{{\rm{m}}}}{\tilde{ n}}({k}_{x},{k}_{y}){\rm{d}}{k}_{x}{\rm{/}}{\int }_{-{k}_{{\rm{m}}}}^{{k}_{{\rm{m}}}}{\rm{d}}{k}_{x}\). To quantitatively analyse the observed roton peaks, we fit a sum of three Gaussian distributions to \({{\tilde{ n}}}_{1}({k}_{y})\). One Gaussian accounts for the central peak and its centre is constrained to k0 ~ 0. Its amplitude is denoted A0. The two other Gaussians are symmetrically located at \({k}_{0}\pm {k}_{y}^{* }\), and we constrain their sizes and amplitudes to be identical, respectively equal to σ* and A*. We focus on the roton side peaks by constraining \({k}_{y}^{* } > 0.5\) μm−1 and σ* < 3 μm−1 (peak at finite momentum of moderate extension compared to the overall distribution).

To analyse the onset and evolution of the roton population (see Fig. 4), we perform a second run of the fitting procedure, in which we constrain the value of \({k}_{y}^{* }\) more strictly. The interval of allowed values is defined for each trapping geometry on the basis of the results of the first run of the fitting procedure. We use the results of the (\({a}_{{\rm{s}}}^{{\rm{f}}}\), th) configurations where the peaks are clearly visible and we set the allowed \({k}_{y}^{* }\) range to that covered by the 95% confidence intervals of the first-fitted \({k}_{y}^{* }\) in these configurations. This constraint enables the fitting procedure to estimate the residual background population on the relevant momentum range for the roton physics, even for as > a*s (see, for example, Fig. 4a).


Time-rescaling analysis and roton gap estimate

In Fig. 4a,b, we systematically analyse the time evolution of the roton population for various \({a}_{{\rm{s}}}^{{\rm{f}}}\) and link it to the roton spectrum in a quench picture. The roton population is embodied by the amplitude A* of the three-Gaussian fit (see the section entitled Fit procedure for the TOF images), which measures the density \({{\tilde{ n}}}_{1}({k}_{{\rm{rot}}})\). A* is observed to initially increase if \({a}_{{\rm{s}}}^{{\rm{f}}} < {a}_{{\rm{s}}}^{* }\), its growth rate increases for decreasing \({a}_{{\rm{s}}}^{{\rm{f}}}\) and \({a}_{{\rm{s}}}^{* }\) depends on the trap geometry.

If dynamically unstable, the krot component is expected to grow exponentially with an instantaneous rate 2Im[ε(krot, th)]/ħ. We hence expect an initial growth of A* of the form:

$${A}_{* }\left({a}_{{\rm{s}}}^{{\rm{f}}},{t}_{{\rm{h}}}\right)\propto {\rm{\exp }}\left(2{\int }_{0}^{{t}_{{\rm{h}}}}{\rm{d}}t{\rm{Im}}\left[\bar{{\it{\Delta }}}\left({a}_{{\rm{s}}}^{{\rm{f}}},t\right)\right]{\rm{/}}\hslash \right)$$
(3)

where \(\bar{{\it{\Delta }}}\left({a}_{{\rm{s}}}^{{\rm{f}}},{t}_{{\rm{h}}}\right)\) is the instantaneous value of the overall roton gap, corresponding to an average over the cloud, after quenching as to \({a}_{{\rm{s}}}^{{\rm{f}}}\). Our results show that the most relevant effect of the quench on the roton spectrum is given by the reduction of as itself. Hence, the time dependence of \({\bar{\it{\Delta}}}_{}^{}\) is determined by as(t) and, by monitoring \({A}_{* }\left({a}_{{\rm{s}}}^{{\rm{f}}},{t}_{{\rm{h}}}\right)\), one can readily extract the scattering-length-dependent gap \(\bar{{\it{\Delta }}}({a}_{{\rm{s}}})\), \(\bar{{\it{\Delta }}}({a}_{{\rm{s}}}^{{\rm{f}}},t)=\bar{{\it{\Delta }}}({a}_{{\rm{s}}}(t))\). To investigate the scaling \(\bar{{\it{\Delta }}}({a}_{{\rm{s}}})\), we first consider the simpler case of the static and local roton gap Δ. From equation (1) of the main text characterizing this elementary gap, assuming n0, X and Z fixed, and using the fact that \({\it{\Delta }}\left({a}_{{\rm{s}}}={a}_{{\rm{s}}}^{* }\right)=0\), one can easily obtain that for as in the vicinity of \({a}_{{\rm{s}}}^{* }\), \({\it{\Delta }}({a}_{{\rm{s}}})\propto {({a}_{{\rm{s}}}^{* }-{a}_{{\rm{s}}})}^{1/2}\). Here we verify this scaling for the global and dynamical quantity \(\bar{{\it{\Delta }}}({a}_{{\rm{s}}})\), considering the generic power-law dependence Im[Δ(as)]/ħ = Γδ(as) with \(\delta ({a}_{{\rm{s}}})={\left(\frac{{a}_{{\rm{s}}}-{a}_{{\rm{s}}}^{* }}{{a}_{0}}\right)}^{\beta }\left({a}_{{\rm{s}}}(t)\le {a}_{{\rm{s}}}^{* }\right)\), in which the parameters Γ, \({a}_{{\rm{s}}}^{* }\) and β mainly depend on the trap geometry.

Our full set of data \({A}_{* }\left({a}_{{\rm{s}}}^{{\rm{f}}},{t}_{{\rm{h}}}\right)\) for the geometry (ν z , λ) = (149 Hz, 4.3) enables us to assess this scaling. Indeed, equation (3) then reads:

$${A}_{* }({a}_{{\rm{s}}}^{{\rm{f}}},{t}_{{\rm{h}}})\propto {\rm{\exp }}\left(2{\it{\Gamma }}{\int }_{0}^{{t}_{{\rm{h}}}}\delta ({a}_{{\rm{s}}}(t)){\rm{d}}t\right)$$
(4)

This defines a time rescaling \({t}_{{\rm{h}}}\to {T}_{{\rm{h}}}={\int }_{0}^{{t}_{{\rm{h}}}}\delta ({a}_{{\rm{s}}}(t)){\rm{d}}t\) along which all our experimental data A*(as,th) should collapse in a unique curve, marked by an initial exponential growth of rate 2Γ. The relevant values of \({a}_{{\rm{s}}}^{* }\), β and Γ are the ones that result in the best overlap of the data for the initial growth of A* (minimal spread in Th).

To determine \({a}_{{\rm{s}}}^{* }\) and β, we then plot our full data set as a function of Th for various trial values of these parameters and evaluate the relevance of the trial couple \(({a}_{{\rm{s}}}^{* },\beta )\). Precisely, we assess the dispersion in Th of the full data set for a few fixed \({A}_{* }={A}_{* }^{i}\). We use a panel of 10 values of \({A}_{* }^{i}\) within [80, 200] μm2, corresponding to the range of the initial growth of A* for the geometry of Fig. 4a,b. We interpolate the experimental data A*(Th) for each \({a}_{{\rm{s}}}^{{\rm{f}}}\) using piecewise cubic polynomial interpolation, and extract the corresponding set of \({T}_{{\rm{h}}}^{i}\) at which \({A}_{* }({T}_{{\rm{h}}}^{i})={A}_{* }^{i}\). For each i, we evaluate the spread of the \({T}_{{\rm{h}}}^{i}({a}_{{\rm{s}}}^{{\rm{f}}})\) by two complementary quantities: the square root of their variance and the discrepancy between the maximal and minimal value of the set. We finally estimate the accuracy of a unified dependence A*(Th) for a given (\({a}_{{\rm{s}}}^{* },\beta\)) via the geometrical average of these two complementary quantities for all of the ten \({A}_{* }^{i}\) values. We fit the relevant \({a}_{{\rm{s}}}^{* }\) and β by minimizing this averaged spread. For the experimental data of Fig. 4a,b, this results in \({a}_{{\rm{s}}}^{* }=53.0(4){a}_{0}\) and β = 0.55(8).

Using these values of \({a}_{{\rm{s}}}^{* }\) and β, we observe that the initial growth of A* extends for Th ≤ 3 ms (before saturating and decreasing for longer Th). We estimate Γ by performing an exponential fit on the full set of data A*(Th) with Th ≤ 3 ms. This gives Γ = 465(83) s−1. Note that Fig. 4a,b show only 5 values of \({a}_{{\rm{s}}}^{{\rm{f}}}\) while the reported analysis considers all available experimental data (that is, 11 values between 31a0 and 52a0).

From the formula with the estimated \({a}_{{\rm{s}}}^{* }\), β and Γ, we can then compute the global roton gap \(\bar{{\it{\Delta }}}\). The extracted value is meaningful only for \({\bar{{\it{\Delta }}}}^{2} < 0\). For the experiments, we also restrict its relevance to Th ≤ 3 ms so that, for example, in the inset of Fig. 4c, \({\rm{Im}}[\bar{{\it{\Delta }}}]\) is shown up to th = 4 ms, after which the roton population is observed to deviate from the exponential growth picture (see Figs. 3b and 4a). Note that the \({\rm{Im}}[\bar{{\it{\Delta }}}]\) estimates at the different th are not independent.

In Fig. 2, we report on the roton momentum as a function of the system characteristics (as and \({\ell }_{z}\)). Here we estimate krot from a th value that is individually optimized for each as and \({\ell }_{z}\) investigated (largest visibility). We point out that the selected th corresponds, in the A* dynamics, to the late stage of the exponential growth, close to the maximum (that is, Th ~ 3 ms). Here, the atom loss remains at the level of a few per cent.

In Fig. 2d, we show the value of krot at the onset of the population of the roton mode (that is, at \({a}_{{\rm{s}}}={a}_{{\rm{s}}}^{* }\)). We estimate \({a}_{{\rm{s}}}^{* }\) for each trap geometry by analysing the evolution of the roton population with as. We employ a simplified approach with respect to the one in Fig. 4 (see Supplementary Information), which we estimate to lead to a maximum underestimate for \({a}_{{\rm{s}}}^{* }\) of about 1.5a0, lying within our experimental uncertainty on as (see Supplementary Table 2).


Analytical dispersion relation for an infinite axially elongated geometry

Equation (1) in the main text results from a similar procedure to that used in ref. 3 for rotons in infinite pancake traps. We consider a dBEC homogeneous along y but harmonically confined with frequencies ν x and ν z along x and z. For sufficiently strong interactions, the BEC is in the TF regime on the xz plane, in which the BEC wavefunction acquires the form \({\psi }_{0}(\rho )=\sqrt{n(\rho )}\), with n(ρ) = n0(1 − (x/X)2 − (z/Z)2), where X and Z are the TF radii, and ρ = (x,z). The calculation of n0, X and Z is detailed at the end of this section.

Due to the axial homogeneity, the elementary excitations of the Bogoliubov–de Gennes spectrum have a defined axial momentum k y , and take the form \(\delta \psi (r,t)=u(\rho ){{\rm{e}}}^{i{k}_{y}y-i\varepsilon t/\hslash }-v(\rho ){{\rm{e}}}^{-i{k}_{y}y+i\varepsilon t/\hslash }\), where u and v denote the amplitudes of the spatial modes oscillating in time with characteristic frequency ε/ħ (see Supplementary Information for more details). We consider the standard NLGPE without beyond mean-field correction and three-body losses, and insert the perturbed solution ψ(r,t) = (ψ0(ρ) + ηδψ(r,t))eiμt/ħ, where μ is the chemical potential associated with ψ0 and \(\eta \ll 1\). After linearization, we obtain the Bogoliubov–de Gennes equations for f±(ρ) = u(ρ) ± v(ρ):

$$\varepsilon {f}_{-}(\rho )={H}_{{\rm{kin}}}\,{f}_{+}(\rho )$$
(5)
$$\varepsilon {f}_{+}(\rho )={H}_{{\rm{kin}}}\,{f}_{-}(\rho )+{H}_{{\rm{int}}}[{f}_{-}(\rho )]$$
(6)

where

$${H}_{{\rm{kin}}}\,{f}_{\pm }(\rho )=\frac{{\hslash }^{2}}{2m}\left(-{\nabla }^{2}+{k}_{y}^{2}+\frac{{\nabla }^{2}{\psi }_{0}}{{\psi }_{0}}\right){f}_{\pm }(\rho )$$
(7)
$$\begin{array}{lll}{H}_{{\rm{int}}}\,[{f}_{-}(\rho )] & = & 2\int {\rm d}^{3}r^{\prime} U(r-r^{\prime} ){{\rm{e}}}^{-{\rm{i}}{k}_{y}(y-y^{\prime} )}\\ & & {\psi }_{0}(\rho ){\psi }_{0}(\rho ^{\prime} ){f}_{-}(\rho ^{\prime} )\end{array}$$
(8)

where \(U({\bf{r}})=g\left(\delta ({\bf{r}})+\frac{3{\varepsilon }_{{\rm{dd}}}}{4\pi }\frac{1-3{{\rm{\cos }}}^{2}\theta }{{\left|{\bf{r}}\right|}^{3}}\right)\) is the binary interaction potential for two particles separated by r including both contact and dipolar interactions and \(g=\frac{4\pi {\hslash }^{2}{a}_{{\rm{s}}}}{m}\) (see Supplementary Information). θ is the angle between r and the magnetization axis (z).

Employing f+ (ρ) = W(ρ)ψ0(ρ), and for \({k}_{y}\gg 1{\rm{/}}X,1{\rm{/}}Z\), we obtain the following equation for the function W(ρ):

$$\begin{array}{lll}0 & = & 2g{n}_{0}\left(1-{\widetilde{x}}^{2}-{\widetilde{z}}^{2}\right)\left[\frac{1}{{X}^{2}}\frac{{\rm{\partial} }^{2}W}{{\rm{\partial} } {\widetilde{x}}^{2}}+\frac{1}{{Z}^{2}}\frac{{{\rm{\partial} }}^{2}W}{{\rm{\partial} }{\widetilde{z}}^{2}}\right]\\ & - & g{n}_{0}(1+2{\varepsilon }_{{\rm{dd}}})\left[\frac{1}{{X}^{2}}\widetilde{x}\frac{\partial W}{{\rm{\partial} } \widetilde{x}}+\frac{1}{{Z}^{2}}\widetilde{z}\frac{{\rm{\partial} } W}{{\rm{\partial} } \widetilde{z}}\right]\\ & + & \left(\frac{2m}{{\hslash }^{2}}\left({\varepsilon }^{2}-E{({k}_{y})}^{2}\right)-2g{\varepsilon }_{{\rm{dd}}}{n}_{0}\left(\frac{1}{{X}^{2}}+\frac{1}{{Z}^{2}}\right)\right)\\ & - & \frac{4m}{{\hslash }^{2}}g{n}_{0}(1-{\varepsilon }_{{\rm{dd}}})E({k}_{y})\left(1-{\widetilde{x}}^{2}-{\widetilde{z}}^{2}\right)\end{array}$$
(9)

where \(\widetilde{x}=x{\rm{/}}X\), \(\widetilde{z}=z{\rm{/}}Z\) and \(E({k}_{y})={\hslash }^{2}{k}_{y}^{2}{\rm{/}}2m\). For εdd = 1, the last term of equation (9) vanishes. In that case, the lowest-energy solution is given by W = 1, whose eigen energy builds, as a function of k y , the dispersion ε0(k y ) with

$${\varepsilon }_{0}{({k}_{y})}^{2}=E{({k}_{y})}^{2}+{E}_{0}^{2}$$
(10)

with \({E}_{0}^{2}=2g{\varepsilon }_{{\rm{dd}}}{n}_{0}\frac{{\hslash }^{2}}{2m}\left(\frac{1}{{X}^{2}}+\frac{1}{{Z}^{2}}\right)\). In the vicinity of εdd = 1, the effect of the last term in equation (9) may be evaluated perturbatively, resulting in the dispersion

$$\varepsilon {({k}_{y})}^{2}\simeq {\varepsilon }_{0}{({k}_{y})}^{2}-2{E}_{{\rm{I}}}E({k}_{y})$$
(11)

with

$${E}_{{\rm{I}}}=\frac{2}{3}g{n}_{0}({\varepsilon }_{{\rm{d}}d}-1)$$

This expression for the dispersion presents a roton minimum for εdd > 1 at \({k}_{{\rm{rot}}}=\frac{1}{\hslash }\sqrt{2m{E}_{{\rm{I}}}}\). Expanding equation (11) in the vicinity of the roton minimum, \(\varepsilon {({k}_{y})}^{2}\simeq \varepsilon {({k}_{{\rm{r}}ot})}^{2}+\frac{1}{2}{\left[\frac{{{\rm{d}}}^{2}{\varepsilon }^{2}({k}_{y})}{{\rm{d}}{k}_{y}^{2}}\right]}_{{k}_{y}={k}_{{\rm{rot}}}}\), we obtain equation (1) of the main text, with \({\it{\Delta }}=\varepsilon ({k}_{{\rm{rot}}})=\sqrt{{E}_{0}^{2}-{E}_{{\rm{I}}}^{2}}\). At the instability, Δ = 0, and \({k}_{{\rm{rot}}}=\frac{1}{\hslash }\sqrt{2m{E}_{0}}\).

Employing a similar procedure as in ref. 15, we obtain that the BEC aspect ratio χ = Z/X fulfils:

$${\chi }^{2}\left[\frac{(1-{\varepsilon }_{{\rm{dd}}}){(1+\chi )}^{2}+3{\varepsilon }_{{\rm{dd}}}}{(1+2{\varepsilon }_{{\rm{dd}}}){(1+\chi )}^{2}-3{\varepsilon }_{{\rm{dd}}}{\chi }^{2}}\right]={\lambda }_{\perp }^{2}$$
(12)

with λ = ν x /ν z and

$${Z}^{2}=\frac{g{n}_{0}}{2{\pi }^{2}m{\nu }_{z}^{2}}\left[(1+2{\varepsilon }_{{\rm{dd}}})-\frac{3{\varepsilon }_{{\rm{dd}}}{\chi }^{2}}{{(1+\chi )}^{2}}\right]$$
(13)

These two equations fully determine the TF solution for the given εdd, gn0 and λ. By inserting the expressions of X2 and Z2 into E0, we find for \({\varepsilon }_{{\rm{dd}}}\simeq 1\):

$${E}_{0}^{2}=\frac{{h}^{2}{\nu }_{z}^{2}}{6}{(1+\chi )}^{2}\left({\lambda }_{\perp }^{2}+\frac{1}{1+2\chi }\right)$$
(14)

whereas χ simplifies to \(\chi ={\lambda }_{\perp }\left(1+\sqrt{1+1{\rm{/}}{\lambda }_{\perp }}\right)\). As a result, at the instability, \({k}_{{\rm{rot}}}{\ell }_{z}\) depends only on the transverse confinement aspect ratio λ, giving the geometrical factor κ:

$$\kappa ={k}_{{\rm{rot}}}{\ell }_{z}={\left(\frac{2}{3}\right)}^{1/4}\sqrt{1+\chi }{\left({\lambda }_{\perp }^{2}+\frac{1}{1+2\chi }\right)}^{1/4}$$
(15)

Self-similar model and instantaneous roton spectrum

The dynamics of the condensate during and after the quench is crucial for the understanding of the resulting momentum peaks and their direct relation to the growth of roton excitations. Due to its large characteristic momentum, the roton spectrum may be evaluated at any time using local density approximation. On the other hand, the evolution of the local density is determined by the global dynamics of the condensate induced by the quench. Hence, interestingly, the analysis of the effect of the quench on the roton spectrum may be performed by combining a self-similar theory of the global dynamics33, describing the evolution at low k y , and the model of the roton spectrum developed in the section entitled Analytical dispersion relation for an infinite axially elongated geometry, accounting for the high-k y region of interest. We note that this analysis approximates the real-time evolution provided by the standard NLGPE12,34,35. This treatment is valid for moderate-enough quenches so that each of the descriptions applies (see below for an in-depth discussion) and as long as the high-k mode population only minimally impact the BEC dynamics, that is, for short-enough timescales.

Here, we assume that the condensate preserves its TF shape during the evolution:

$$n({\bf{r}},t)={n}_{0}(t)\left[1-{\left(\frac{x}{X(t)}\right)}^{2}-{\left(\frac{y}{Y(t)}\right)}^{2}-{\left(\frac{z}{Z(t)}\right)}^{2}\right]$$
(16)

where X(t), Y(t) and Z(t) are the rescaled TF radii. Their evolution after the quench can be deduced from solving the hydrodynamics equations (see Supplementary Information). Prior to the quench of as, the stationary TF solution is obtained from the self-consistent equations deduced from the hydrodynamics equations by cancelling all time derivatives. Solving these equations and using normalization provides X(0), Y(0), Z(0) and n(0) for known atom number N and trap frequencies νx,y,z. We use this stationary solution as the initial condition at the start of the quench (t = 0), and solve the system of differential equations resulting from the hydrodynamics evolution to obtain the rescaled radii. An example for the relevant parameters of Fig. 3 is shown in Supplementary Fig. 3.

For a given position y, we then evaluate the local TF profile: n(r,t) = n0(y,t)[1 − (x/X(y,t))2 − (z/Z(y,t))2], where n0(y,t) = n0(t)[1 − (y/Y(t))2], X(y,t) = X(t)[1 − (y/Y(t))2]1/2 and Z(y,t) = Z(t)[1 − (y/Y(t))2]1/2, with n0(t) = n0(0)X(0)Y(0)Z(0)/X(t)Y(t)Z(t). We may then employ a local density approximation and evaluate the local (in y) instantaneous roton spectrum using the results of the section entitled Analytical dispersion relation for an infinite axially elongated geometry:

$$\begin{array}{lll}\frac{{\varepsilon }^{2}({k}_{y},y,t)}{{(h{\nu }_{z})}^{2}{l}_{z}^{4}} & \simeq & \frac{{k}_{y}^{4}}{4}-\frac{8\pi }{3}{n}_{0}(y,t)({a}_{{\rm{dd}}}-{a}_{{\rm{s}}}(t)){k}_{y}^{2}\\ & + & 4\pi {n}_{0}(y,t){a}_{{\rm{dd}}}\left[\frac{1}{X{(y,t)}^{2}}+\frac{1}{Z{(y,t)}^{2}}\right]\end{array}$$
(17)

The roton population grows exponentially when the spectrum becomes imaginary, leading to the appearance of peaks at large momenta. From the local instantaneous roton spectrum, we may estimate the associated peak in the momentum distribution as

$${{\tilde{\it n}}}_{1}({k}_{y},t) \sim \int {\rm{d}}y\left[{{\rm{e}}}^{2{\int }_{0}^{t}{\rm{d}}t^{\prime} {\rm{Im}}\left[\varepsilon \left({k}_{y},y,t^{\prime} \right)\right]}-1\right]$$
(18)

where we assume equal seeding for all unstable modes. We then evaluate the total population of the peak \({N}_{* }(t)=\int {\rm{d}}{k}_{y}{{\tilde{\it n}}}_{1}\left({k}_{y},t\right)\), as well as the roton momentum \(\left\langle k\right\rangle =\int {\rm{d}}{k}_{y}{k}_{y}\frac{{{\tilde{\it n}}}_{1}({k}_{y},t)}{{N}_{* }(t)}\).

The previously discussed SSM does not properly describe the evolution when the shrinking of the condensate, and hence the increase of its peak density, is too large. As the gas gets dense, quantum fluctuations, which are not considered in the SSM, introduce in our experiments an effective repulsion that crucially prevents large densities or condensate collapse (see Supplementary Information). This limits the use of the SSM to the vicinity of \({a}_{{\rm{s}}}^{* }\). Moreover, for our tightest trap, the SSM results are also unreliable, since the theory predicts condensate 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 large extent also a very good quantitative agreement with both our experimental results and our NS based on a generalized NLGPE, which is detailed in the Supplementary Information.


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The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We are particularly grateful to B. Blakie for many inspiring exchanges. We thank D. O’Dell, M. Baranov, E. Demler, A. Sykes, T. Pfau, I. Ferrier-Barbut and H. P. Büchler for fruitful discussions, and G. Natale for his support in the final stage of the experiment. This work is dedicated to the memory of D. Jin and her inspiring example. The Innsbruck group is supported through an ERC Consolidator Grant (RARE, no. 681432) and a FET Proactive project (RySQ, no. 640378) of the EU H2020. L.C. is supported within a Marie Curie Project (DipPhase, no. 706809) of the EU H2020. F.W. and L.S. thank the DFG (SFB 1227 DQ-mat). All authors thank the DFG/FWF (FOR 2247). Part of the computational results presented have been achieved using the HPC infrastructure LEO of the University of Innsbruck.

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Affiliations

  1. Institut für Experimentalphysik, Universität Innsbruck, Innsbruck, Austria

    • L. Chomaz
    • , D. Petter
    • , G. Faraoni
    • , S. Baier
    • , J. H. Becher
    • , M. J. Mark
    •  & F. Ferlaino
  2. Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Innsbruck, Austria

    • R. M. W. van Bijnen
    • , M. J. Mark
    •  & F. Ferlaino
  3. Dipartimento di Fisica e Astronomia, Università di Firenze, Sesto Fiorentino, Italy

    • G. Faraoni
  4. Institut für Theoretische Physik, Leibniz Universität Hannover, Hannover, Germany

    • F. Wächtler
    •  & L. Santos

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Contributions

F.F., L.C., D.P., G.F., M.J.M., J.H.B. and S.B. conceived and supervised the experiment and collected the experimental data. L.C. analysed the data. R.M.W.v.B. developed the Bogoliubov–de Gennes calculations. F.W., R.M.W.v.B. and L.S. performed the real-time simulations. L.S. derived the analytical model and the SSM. L.C., F.F., R.M.W.v.B. and L.S. wrote the paper with contributions from all of the authors.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to F. Ferlaino.

Supplementary information

  1. Supplementary Information

    Supplementary Figure 1–3, Supplementary Table 1–2, Supplementary References

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DOI

https://doi.org/10.1038/s41567-018-0054-7