Observation of Roton Mode Population in a Dipolar Quantum Gas

Abstract

The concept of a roton, a special kind of elementary excitation, forming a minimum of energy at finite momentum, has been essential to understand the properties of superfluid ⁴He ¹. In quantum liquids, rotons arise from the strong interparticle interactions, whose microscopic description remains debated ². In the realm of highly-controllable quantum gases, a roton mode has been predicted to emerge due to magnetic dipole-dipole interactions despite of their weakly-interacting character ³. This prospect has raised considerable interest ^4–12; 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 magnetisation 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 gases ¹³.

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 since its discovery in the late 1930s ^{2, 12}. A major breakthrough in understanding superfluidity thrived on the concept of quasiparticles, introduced by Landau in 1941 ¹. 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 quasiparticles: 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 = k_rot. Neutron scattering experiments confirmed Landau’s remarkable intuition ¹⁴. In liquid ⁴He, k_rot 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 atoms ².

In the realm of low-temperature quantum physics, ultra-cold quantum gases realise 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. Because of this diluteness, roton excitations are absent in ordinary quantum gases, i. e. in Bose-Einstein condensates (BECs) with contact (short-range) interactions ¹². 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 light ¹⁵. Following the lines of the latter proposal, a roton softening has been recently observed in BECs coupled to an optical cavity ¹⁶. Here, the excitation arises from the infinite-range photon-mediated interactions and the inverse of the laser wavelength sets the value of k_rot. Additionally, roton-like softening has been created in spin-orbit-coupled BECs ^{17, 18} and quantum gases in shaken optical lattices ¹⁹ 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 inter-particle 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 space ^3–12. 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 atoms ²⁰, for which the achievable dipolar character is hardly sufficient to support a roton mode. With the advent of the more magnetic lanthanide atoms ^{21, 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 stabilised by quantum fluctuations ^23–25, which may become self-bound ²⁶. 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 geometries ^{3–6, 8–12}. 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 magnetisation 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. 1b1). In contrast, for k_yℓ_z ≳ 1, ℓ_z being the characteristic z confinement length, the excitation favours head-to-tail alignments and the DDI contribution to ε(k_y) eventually changes sign ³ (Fig. 1b2). 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 = k_rot set by the geometrical scaling k_rot ~ 1/ℓ_z (see below and e.g. 3, 8, 9).

Figure 1. Roton mode in a dBEC.

a, 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 a, showing the emergence of a roton minimum for decreasing a_s (dashed arrow). The DDI changes from repulsive (blue) to attractive (red) depending on the dipole alignment (inset). b1, b2, the dipole alignment (color code as in inset) associated to small- (b1) and large-k_y (b2) density modulations. 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 colorscale.

Similar to the helium case, the roton energy gap, Δ = ε(k_rot), depends on the density and on the strength of the interactions. In ultracold gases, both quantities can be controlled. In particular, the scattering length a_s, setting the strength of the contact interaction, can be tuned using Feshbach resonances ¹². As a_s is reduced, Δ decreases, vanishes, and eventually becomes imaginary (Fig. 1b). In the latter case, the system undergoes a roton instability and the population at k_y = 0 is transferred in ±k_rot at an exponential rate ^{5, 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 = k_rot 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 = ±k_rot, 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 v_x,y,z, elongated along the y axis. The trap aspect ratio, λ = v_z/v_y, can be tuned from about 4 to 30, corresponding to v_z ranging from 150 Hz to 800 Hz, whereas v_y and v_z/v_x are kept constant at about 35 Hz and 1.6, respectively (Methods). An external homogeneous magnetic field, B, fixes the dipole orientation (magnetisation) with respect to the trap axes and sets the values of a_s through a magnetic Feshbach resonance, centered close to B = 0G. In previous experiments, we precisely calibrated the B-to-a_s conversion for this resonance ²⁵. The BEC is prepared at $a_{\text{s}}^{\text{i}}=61a_0$ (B = 0.4G) with transverse (z) magnetisation. The characteristic dipolar length, defined as $a_{\text{dd}}={\mu }_0{\mu }_m^{2}m/12\pi {ħ}^2,$ is 65.5 a₀, with m the mass and µ_m the magnetic moment of the atoms, ħ = h/2π is the reduced Planck constant, µ₀ the vacuum permeability and a₀ the Bohr radius.

To excite the roton mode, we quench a_s to a desired lower value, $a_{\text{s}}^{\text{f}},$ and shortly hold the atoms in the trap for a time t_h. We measure that a_s converges to its set value with a characteristic time constant of 1 ms during t_h. We then release the atoms from the trap, change a_s back close to its initial value and let the cloud expand for 30 ms. We probe the momentum distribution ñ(k_x,k_y) by performing standard resonant absorption imaging on the expanded cloud (Methods). The measurement is then repeated at various values of a_s < a_dd in a fixed trap geometry. The momentum distribution shows a striking behaviour (Fig. 2). For large enough a_s, ñ(k_x,k_y) shows a single narrow peak with an inverted aspect ratio compared to the trapped gas, typical of a stable BEC ¹² (Fig. 2a). We define the center of the distribution as the origin of k. In contrast, when further decreasing a_s below a critical value $a_{\text{s}}^{*},$ we observe a sudden appearance of two symmetric finite-momentum peaks, of similar shape and located at k_y = ±k_rot (Fig. 2b-c). 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 ñ(k_x,k_y) (Fig. 2a1-c1). From the fit we extract the central momentum, k_rot, and the amplitude, A_*, of the side peaks.

Figure 2. Observed roton peaks and characteristic scalings.

a-c, ñ(k_x,k_y) obtained by averaging 15-to-25 absorption images for (v_z, λ) = (456 Hz, 14.4), t_h = 3 ms and: a, a_s = 54a₀, b, a_s = 44a₀ and c, a_s = 37a₀. a1-c1, corresponding cuts at k_x ≈ 0 (dots) and their fits to three-Gauss distributions (lines), from which we extract k_rot and A_*. d, Measured k_rot depending on 1/ℓ_z at $a_s\lesssim a_s^{*}$ for v_y ≈ 35 Hz (circles) and v_y = 17(1) Hz (triangle). e, k_rot depending on a_s for (v_z,λ) = (149 Hz, 4.3) (circles). In (d-e), error bars show the 95% confidence interval of the three-Gauss fits (see a1-c1). The squares (diamonds) show predictions from the SSM (NS) in their range of validity (Methods). 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_rot ~ 1/ℓ_z (see e.g. 3, 4, 8, 9). In addition, the dependency of k_rot on a_s close to the instability is expected to be mild as k_rot remains mainly set by its geometrical nature ^{3, 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 k_rot. We clearly observe the expected geometrical scaling. k_rot shows a marked increase with $1/{\ell }_z=2\pi \sqrt{m{v}_z/h},$ matching well with a linear progression with a slope of 1.61(4) (Fig. 2d). Note that no dependence of k_rot on v_y is observed. In a second set of experiments, we fix the trap geometry and explore the dependence of k_rot on a_s. We observe that, within our experimental uncertainty, k_rot stays constant when decreasing a_s (Fig. 2e). This behaviour contrasts the one expected for a phonon-driven modulation instability that exhibits a strong a_s-dependence ²⁷.

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 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 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, ε_dd = a_dd/a_s ≥ 1, we find that ε(k_y) indeed rotonizes (Methods). In the vicinity of the roton minimum and for ε_dd ~ 1, the computed dispersion acquires a gapped quadratic form similar to that of helium rotons:

$$\epsilon {\left(k_y\right)}^2\simeq {\Delta }^2+\frac{2{ħ}^2{k}_{\text{rot}}^2}{m}\frac{{ħ}^2}{2m}{\left(k_y-k_{\text{rot}}\right)}^2.$$ (1)

The roton momentum reads as $k_{\text{rot}}=\sqrt{2m}{\left(E_0^2-{\Delta }^2\right)}^{1/4}/ħ,$ with $\Delta =\sqrt{E_0^2-E_I^2},E_I=2g{n}_0\left({\epsilon }_{dd}-1\right)/3,g=4\pi {ħ}^2{a}_{\text{s}}/m,$ and $E_0^2=2g{\epsilon }_{dd}{n}_0\frac{{ħ}^2}{2m}\left(X^{-2}+Z^{-2}\right).$ The TF radii X, Z satisfy X², Z² ∝ gn₀ so that E₀ ∝ hv_z, with a scale set by ε_dd and v_z/v_x but independent of gn₀ ¹⁵. Close to the instability (Δ ≃ 0), the roton momentum thus follows a simple geometrical scaling, k_rot = κ/ℓ_z with the geometrical factor κ depending on v_z/v_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 k_rot and Δ with the system parameters. However the quench of a_s introduces dynamics, which is crucial to quantitatively reproduce the experimental observations. The reduction of a_s 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 destabilised during the evolution. This dynamical destabilisation is well accounted for by a self-similar model (SSM) describing the evolution of the cloud shape after the quench ¹². In particular, we consider a 3D harmonic confinement, and, starting from the stationary TF profile at $a_{\text{s}}^{\text{i}},$ we evaluate the evolution of the cloud along the change of a_s 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,t_h) using a local density approximation, i.e. evaluating Eq. (1) with the experimentally calibrated a_s(t_h), and the n₀(y,t_h), X (y,t_h), Z(y,t_h) estimated from the 3D profile (Methods). We indeed find that the roton gap decreases and eventually turns imaginary (Δ(t_h)² < 0) (Fig. 3a). When this occurs the population at k_y ≈ k_rot exponentially grows with an instantaneous local rate 2Im[ε(k_y,y,t_h)]/ħ, Im[.] indicating the imaginary part, giving rise to two symmetric side peaks in the axial momentum distribution ${\tilde{n}}_1\left(k_y,t_{\text{h}}\right)\propto \int \text{exp}\left(2{\int }_0^{t_{\text{h}}}\text{Im}\left[\epsilon \left(k_y,y,t\right)\right]dt\right)dy$ (Fig. 3b) (Methods). The center of the peaks, ±k_rot(t_h), also evolves with t_h but quickly converges after a few ms to its final value, k_rot (Fig. 3c). The measured k_rot and the calculated values from our parameter-free theory are in remarkable agreement (Fig. 2d-e and 3c).

Figure 3. Dynamics of the roton mode.

Results depending on t_h after quenching to $a_{\text{s}}^{\text{f}}=50a_0$ in (v_z, λ) = (149 Hz, 4.3). a-c, 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). Shaded areas identify the dynamical destabilization. For comparison, the amplitudes are renormalized to their t_h = 3 ms-values and the NS results are translated by -3.7 ms. k_rot is reliably extracted from the experiments (NS) for t_h ≥ 3 ms (2.5 ms). Inset, same plot in log scale. d, e, Integrated 1D density profile, in momentum ñ₁(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 numerical simulations (NS) of the system dynamics. We calculate the time evolution of the generalized non-local Gross-Pitaevskii equation, 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 stabilise the dBEC against collapse ^{4–6, 24, 25}, whereas the latter term thermally seeds the initial roton-mode population. The NS confirm both the SSM results and the experimental observations. Indeed the calculations show that, a few ms after the quench, the system develops roton peaks in momentum space (Fig. 3d) and short-wavelength density modulations at the center of the BEC (Fig. 3e), showing the predicted roton confinement ⁹. The extracted value of k_rot and its geometrical scaling are in very good quantitative agreement with both the experimental data and the SSM calculations (Fig. 2d-e and 3c). Interestingly, the timescales for the emergence of roton peaks in the NS are a few ms longer than those observed in the experiment. The origin of this time shift remains an open question, whose answer could e.g. 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 (v_z, λ). In a first set of measurements, we fix $a_{\text{s}}^{\text{f}}$ and follow the dynamics by recording the momentum distribution at various t_h. We observe that k_rot 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 $a_{\text{s}}^{\text{f}}$ (Fig. 4). Our data show that the roton mode begins to become populated from a critical value of the scattering length, $a_{\text{s}}^{*}.$ For $a_{\text{s}}^{\text{f}}>52a_0,$ we do not observe the roton peaks at any time. For $a_{\text{s}}^{\text{f}}\underset{˜}{<}52a_0,$ 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 $a_{\text{s}}^{\text{f}},$ the roton population exhibits a faster growth rate and shorter time delay.

Figure 4. Population growth and roton gap.

Results for (v_z, λ) = (149 Hz, 4.3). a, b, A_* depending on t_h and on the rescaled time T_h, after quenching to $a_{\text{s}}^{\text{f}}=39a_0$ (hexagons), $a_{\text{s}}^{\text{f}}=44a_0$ (triangles), $a_{\text{s}}^{\text{f}}=47a_0$ (diamonds), $a_{\text{s}}^{\text{f}}=50a_0$ (circles) and $a_{\text{s}}^{\text{f}}=52a_0$ (squares). Shaded areas show the 95% confidence interval from the three-Gauss fit. The black line shows an exponential fit to the full dataset with T_h < 3 ms. c, Extracted Im[Δ̄] depending on a_s and on t_h (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 $a_{\text{s}}^{*}$ and its fit’s confidence interval. The inset shows the same configuration as Fig. 3 ($a_{\text{s}}^{\text{f}}=50a_0$).

From the growth rate of the roton population we now extract an overall roton gap Δ̄(a_s) (Methods). In brief, the dynamical variation of the gap is determined by the leading time dependence of a_s and $A_{*}\left(a_{\text{s}}^{\text{f}},t_{\text{h}}\right)\propto \text{exp}\left(2{\int }_0^{t_{\text{h}}}dt\text{Im}\left[\overline{\Delta }\left(a_{\text{s}}\left(t\right)\right)\right]/ħ\right).$ To investigate the scaling Δ̄(a_s), we first consider the analytical expression of the elementary local roton gap, Δ as a function of a_s (see Eq. (1)). Developing Δ in the vicinity of $a_{\text{s}}=a_{\text{s}}^{*}$ yields $\Delta \left(a_{\text{s}}\right)\propto {\left(a_{\text{s}}^{*}-a_{\text{s}}\right)}^{1/2}.$ Here we confirm this scaling by considering the generic power-law dependence Im[Δ̄(a_s)]/ħ = Γδ (a_s) with $\delta \left(a_{\text{s}}\right)={\left(\frac{a_{\text{s}}^{*}-a_{\text{s}}}{a_0}\right)}^{\beta }\left(a_{\text{s}}\le a_{\text{s}}^{*}\right).$ Consequently, A_* grows as $\text{exp}\left(2\Gamma {\int }_0^{t_{\text{h}}}\delta \left(a_{\text{s}}\left(t\right)\right)dt\right)=\text{exp}\left(2\Gamma T_{\text{h}}\right).$ By rescaling the time variable as $t_{\text{h}}\to T_{\text{h}}={\int }_0^{t_{\text{h}}}\delta (a_{\text{s}}\left(t\right))dt,$ we determine the parameters $a_{\text{s}}^{*}$ and β such that all the curves of Fig. 4a fall on top of each other (Fig. 4b). The best overlap of the experimental curves is found for $a_{\text{s}}^{*}$ = 53.0(4) a₀ and β = 0.55(8). We then perform an exponential fit to all the rescaled data for T_h < 3 ms and extract Γ = 465(83) s⁻¹. The same analysis, applied to the calculated population growth from the SSM for different $a_{\text{s}}^{\text{f}},$ gives $a_{\text{S}}^{*}\simeq 52.8a_0,$ β ≃ 0.55, and Γ ≃ 472s⁻¹, which are very close to the experimental values. This time-resolved study allows to readily extract the imaginary roton gap, Im[Δ̄], from both the experiment and the SSM, as a function of a_s and t_h (Fig. 4c). Our observations show the softening of the roton mode at $a_{\text{s}}=a_{\text{s}}^{*}$ and the expected increase of Im[Δ̄] for $a_{\text{s}}<a_{\text{s}}^{*},$ 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 behavior of a dBEC ^{8–10, 12}, the interplay between phononic and rotonic modes in the stability diagram of the quantum gas ^{4, 7}, and the possible role of roton excitations as triggering mechanism of the recently observed instability leading to the formation of metastable droplet arrays in elongated traps ^{24, 28}. Of particular interest is the prospect of creating a supersolid and striped ground-states in dBECs ^{13, 28}. Indeed, the short-wavelength density modulation tied to the roton softening together with the quantum stabilization mechanism can favor the formation of such an exotic phase of matter, in which crystalline order coexists with phase coherence. Contrary to recent experiments ^{29, 30}, where the density modulation is imposed by external fields yielding a supersolid-like arrangement of infinite stiffness, a dipolar supersolid would be compressible. Experiments on dBECs provide hence 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 $V\left(\mathbf{\text{r}}\right)=2m{\pi }^2\left(v_x^2{x}^2+v_y^2{y}^2+v_z^2{z}^2\right),$ characterized by the frequencies (v_x, v_y, v_z). The ODT results from the crossing of two red-detuned laser beams of 1064 nm wavelength at their respective focii. 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, v_y is essentially set by the vODTb power while v_z (and v_x) independently by that of the hODTb. This yields an easy tuning of the trap aspect ratio, λ = v_z/v_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 (v_y ≪ v_x, v_z). The trapping geometries probed in the experiments, whose (λ, v_x, v_y, v_z) are reported in Supplementary Table I, are achieved by changing the hODTb power with the vODTb power set to 7W so that v_y and v_z/v_x 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 (v_x, v_y, v_z) = (156, 17, 198) Hz and λ = 11.6. The (λ, v_x, v_y, v_z) 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 shaping procedure depend on the final configurations, as detailed in Supplementary Table I.

Quench of the scattering length a_s

To control a_s we use a magnetic Feshbach resonance between ¹⁶⁶Er atoms in their absolute ground state, which is centered around B = 0G ³¹. The B-to-a_s conversion has been previously precisely measured via lattice spectroscopy, as reported in Ref. 25. Errors on a_s, taking into account statistical uncertainties of the conversion and effects of magnetic field fluctuations (e.g. from stray fields), are of 3-to-5 a₀ for the relevant a_s range 27-67 a₀ in this work. After the BEC preparation and in order to investigate the roton physics via an interaction quench, we suddenly change the magnetic field set value, B_set, twice. First we perform the quench itself and abruptly change B_set from 0.4 G $\left(a_{\text{s}}^{\text{i}}=61a_0\right)$ to the desired lower value (corresponding to $a_{\text{s}}^{\text{f}})$ at the beginning of the hold in trap (t_h = 0 ms). Second we prepare the ToF expansion and imaging conditions (see Method Imaging procedure) and abruptly change B_set from the quenched value back to 0.3 G (a_s = 57 a₀) at the beginning of the ToF expansion (t_f = 0 ms). Due to delays in the experimental setup, e.g. coming from eddy currents in our main chamber, the actual B value felt by the atoms responds to a change of B_set via B(t) = B_set(t) + τdB/dt ³². By performing pulsed-radio-frequency spectroscopy measurements (pulse duration 100 µs) on a BEC after changing B_set (from 0.4 to 0.2 G), we verify this law and extract a time constant τ = 0.98(5) ms. Hence a_s is also evolving during t_h and t_f 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 a_s at t_h. We use t_h ranging from 2 to 7 ms. The lower bound on t_h comes from the time needed for a_s to effectively reach the regime of interest. We then consider the initial evolution for which t_h/v_y ; t_h/τ_coll ≪ 1, 1/τ_coll being the characteristic collision rate. We estimate that τ_coll ranges typically from 40 to 90 ms in the initial BECs of Supplementary Table I at $a_{\text{s}}^{\text{i}}=61a_0.$ Experimentally we observe that the roton, if it ever develops, has developed within the considered range of t_h.

Imaging procedure

In our experiments, we employ ToF expansion measurements, accessing the momentum distribution of the gas ¹², to probe the roton mode population. We let the gas expand freely for t_f = 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 k_rot ≳ 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 n_tof (x,y,t_f) in the xy plane. When releasing the cloud (t_f = 0 ms), we change B back to B = 0.3 G (Method Quench of the scattering length a_s). This change enables constant and optimal imaging conditions with a fixed probing procedure, i. e., a maximal absorption cross-section. In addition, the associated increase of a_s to 57 a₀ allows to minimize the time during which the evolution happens in the small-a_s 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:

$$\tilde{n}\left(k_x,k_y\right)={\left(\frac{ħt_{\text{f}}}{m}\right)}^2{n}_{\text{tof}}\left(\frac{ħk_x{t}_{\text{f}}}{m},\frac{ħk_y{t}_{\text{f}}}{m},t_{\text{f}}\right),$$ (2)

which neglects the initial size of the cloud in 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 Eq. (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 Eq. (2).

Fit procedure for the ToF images

For each data point of Figs. 2-4, we record between 12 and 25 ToF images. By fitting a two-dimensional Gaussian distribution to the individual images, we extract their origin (k_x,k_y) = (0, 0) and recenter each image. From the recentered images, we compute the averaged ñ(k_x,k_y), from which we characterise the linear roton developing along k_y. To do so, we extract a one-dimensional profile ñ₁(k_y) by averaging ñ(k_x,k_y) over k_x within $\left|k_x\right|\le k_{\text{m}}=3.5\mu {\text{m}}^{-1}:{\tilde{n}}_1\left(k_y\right)={\int }_{-k_{\text{m}}}^{k_{\text{m}}}\tilde{n}\left(k_x,k_y\right)d{k}_x/{\int }_{-k_{\text{m}}}^{k_{\text{m}}}d{k}_x.$ To quantitatively analyse the observed roton peaks, we fit a sum of three Gaussian distributions to ñ₁(k_y). One Gaussian accounts for the central peak and its centre is imposed 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 $k_y^{*}$ > 0.5µm⁻¹ and σ* < 3µm⁻¹ (peak at finite momentum of moderate extension compared to the overall distribution).

In order 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 based on the results of the first run of the fitting procedure. We use the results of the $\left(a_{\text{s}}^{\text{f}},t_{\text{h}}\right)$-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 that, for $a_{\text{s}}>a_{\text{s}}^{*},$ the fitting procedure estimates the residual background population on the relevant momentum range for the roton physics (see e.g. 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 $a_{\text{s}}^{\text{f}}$ 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 (Method Fit procedure for the ToF images), which measures the density ñ₁(k_rot). A_* is observed to initially increase if $a_{\text{s}}^{\text{f}}<a_{\text{s}}^{*},$ its growth rate increases for decreasing $a_{\text{s}}^{\text{f}}$ and $a_{\text{s}}^{*}$ depends on the trap geometry.

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

$$A_{*}\left(a_{\text{s}}^{\text{f}},t_{\text{h}}\right)\propto \text{exp}\left(2{\displaystyle {\int }_0^{t_{\text{h}}}dt\text{Im}\left[\overline{\Delta }\left(a_{\text{s}}^{\text{f}},t\right)/ħ\right]}\right),$$ (3)

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

Our full set of data $A_{*}\left(a_{\text{s}}^{\text{f}},t_{\text{h}}\right)$ for the geometry (v_z, λ) = (149 Hz, 4.3) enables us to assess this scaling. Indeed Eq. (3) then reads:

$$A_{*}\left(a_{\text{s}}^{\text{f}},t_{\text{h}}\right)\propto \text{exp}\left(2\Gamma {\displaystyle {\int }_0^{t_{\text{h}}}\delta \left(a_{\text{s}}\left(t\right)\right)dt}\right).$$ (4)

This defines a time rescaling $t_{\text{h}}\to T_{\text{h}}={\int }_{\text{0}}^{t_{\text{h}}}\delta \left(a_{\text{s}}\left(t\right)\right)dt$ along which all our experimental data A_*(a_s,t_h) should collapse in a unique curve, marked by an initial exponential growth of rate 2Γ. The relevant values of $a_{\text{s}}^{*},$ β and Γ are the one that result in the best overlap of the data for the initial growth of A_* (minimal spread in T_h).

In order to determine $a_{\text{s}}^{*}$ and β, we then plot our full dataset as a function of T_h for various trial values of these parameters and evaluate the relevance of the trial couple $(a_{\text{s}}^{*},\beta ).$ Precisely, we assess the dispersion in T_h of the full dataset for few fixed $A_{*}=A_{*}^i.$ We use a panel of 10 values $A_{*}^i$ 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_*(T_h) for each $a_{\text{s}}^{\text{f}}$ using piecewise cubic polynomial interpolation, extract the corresponding set of $T_{\text{h}}^i$ at which $A_{*}(T_{\text{h}}^i)=A_{*}^i.$ For each i, we evaluate the spread of the $T_{\text{h}}^i(a_{\text{s}}^{\text{f}})$ by two complementary quantities: (a) the square-root of their variance and (b) the discrepancy between the maximal and minimal value of the set. We finally estimate the accuracy of a unified dependence A_*(T_h) for a given $(a_{\text{s}}^{*},\beta )$ via the geometrical average of (a) and (b) for all the ten $A_{*}^i$ values. We fit the relevant $a_{\text{s}}^{*}$ and β by minimizing this averaged spread. For the experimental data of Fig. 4a-b, this results in $a_{\text{s}}^{*}=53.0(4)a_0$ and β = 0.55(8).

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

From the formula with the estimated $a_{\text{s}}^{*},$ β and Γ, we can then compute the global roton gap Δ̄. The extracted value is only meaningful for Δ̄² < 0. For the experiments, we also restrict its relevance to T_h ≤ 3 ms so that, e.g., in the inset of Fig. 4c, Im[Δ̄] is shown up to t_h = 4 ms after which the roton population is observed to deviate from the exponential growth picture (see Figs. 3b and 4a). Note that the Im[Δ̄] estimates at the different t_h are not independent.

In Fig. 2, we report on the roton momentum as a function of the system characteristics (a_s, ℓ_z). Here we estimate k_rot from a t_h value that is individually optimised for each a_s and ℓ_z investigated (largest visibility). We point out that the selected t_h correspond, in the A_* dynamics, to the late stage of the exponential growth, close to the maximum (i.e. T_h ~ 3 ms). Here, the atom loss remains at a few percents level.

In Fig. 2d, we show the value of k_rot at the onset of the population of the roton mode, i.e. at $a_{\text{s}}=a_{\text{s}}^{*}.$ We estimate $a_{\text{s}}^{*}$ for each trap geometry by analysing the evolution of the roton population with a_s. 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 $a_{\text{s}}^{*}$ of about 1.5 a₀, lying within our experimental uncertainty on a_s; see Supplementary Table II.

Analytical dispersion relation for an infinite axially elongated geometry

Equation (1) in the main text results from a similar procedure as that used in Ref. 3 for rotons in infinite pancake traps. We consider a dBEC homogeneous along y but harmonically confined with frequencies v_x and v_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 $\delta \psi (r,t)=u(\rho )e^{i{k}_{y}y-i\epsilon t/ħ}-v(\rho )e^{-i{k}_{y}y+i\epsilon t/ħ},$ where u, 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 GPE without LHY correction and three-body losses, and insert the perturbed solution $\psi (r,t)=({\psi }_0(\rho )+\eta \delta \psi (r,t))e^{-i\mu t/ħ}$ where µ is the chemical potential associated to ψ₀ and η ≪ 1. After linearization we obtain the BdG equations for f_±(ρ) = u(ρ) ± v(ρ):

$$\epsilon f_{-}(\rho )=H_{\text{kin}}{f}_{+}(\rho ),$$ (5)

$$\epsilon f_{+}(\rho )=H_{\text{kin}}{f}_{-}(\rho )+H_{\text{int}}[f_{-}(\rho )],$$ (6)

where

$$H_{\text{kin}}{f}_{\pm }(\rho )=\frac{{ħ}^2}{2m}\left(-{\nabla }^2+k_y^2+\frac{{\nabla }^2{\psi }_0}{{\psi }_0}\right)f_{\pm }(\rho ),$$ (7)

$$\begin{array}{l}{H}_{\text{int}}[f_{-}(\rho )]=2{\displaystyle \int d^3{r}^{\prime }U(r-r^{\prime }){\text{e}}^{-\text{i}{k}_y(y-y^{\prime })}}\\ {\psi }_0(\rho ){\psi }_0({\rho }^{\prime })f_{-}({\rho }^{\prime }).\end{array}$$ (8)

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

Employing f₊(ρ) = W (ρ)ψ₀(ρ), and for k_y ≫ 1/X, 1/Z, we obtain the following equation for the function W (ρ):

$$\begin{array}{l}0=2g{n}_0\left(1-{\tilde{x}}^2-{\tilde{z}}^2\right)\left[\frac{1}{X^2}\frac{{\partial }^{2}W}{\partial {\tilde{x}}^2}+\frac{1}{Z^2}\frac{{\partial }^{2}W}{\partial {\tilde{z}}^2}\right]\\ -g{n}_0\left(1+2{\epsilon }_{dd}\right)\left[\frac{1}{X^2}\tilde{x}\frac{\partial W}{\partial \tilde{x}}+\frac{1}{Z^2}\tilde{z}\frac{\partial W}{\partial \tilde{z}}\right]\\ +\left(\frac{2m}{{ħ}^2}({\epsilon }^2-E{\left(k_y\right)}^2)-2g{\epsilon }_{dd}{n}_0\left(\frac{1}{X^2}+\frac{1}{Z^2}\right)\right)\\ -\frac{4m}{{ħ}^2}g{n}_0\left(1-{\epsilon }_{dd}\right)E\left(k_y\right)\left(1-{\tilde{x}}^2-{\tilde{z}}^2\right),\end{array}$$ (9)

where $\tilde{x}=x/X,\tilde{z}=z/Z,E(k_y)={ħ}^2{k}_y^2/2m.$ For ε_dd = 1 the last term of Eq. (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

$${\epsilon }_0{\left(k_y\right)}^2=E{\left(k_y\right)}^2+E_0^2,$$ (10)

with $E_0^2=2g{\epsilon }_{dd}{n}_0\frac{{ħ}^2}{2m}\left(\frac{1}{X^2}+\frac{1}{Z^2}\right).$ In the vicinity of ε_dd = 1, the effect of the last term in Eq. (9) may be evaluated perturbatively, resulting in the dispersion

$$\epsilon {\left(k_y\right)}^2\simeq {\epsilon }_0{\left(k_y\right)}^2-2E_{I}E\left(k_y\right),$$ (11)

with $E_I=\frac{2}{3}g{n}_0\left({\epsilon }_{dd}-1\right).$

This expression for the dispersion presents a roton minimum for ε_dd > 1 at $k_{\text{rot}}=\frac{1}{ħ}\sqrt{2m{E}_I}.$ Expanding Eq. (11) in the vicinity of the roton minimum, $\epsilon {\left(k_y\right)}^2\simeq \epsilon {\left(k_{\text{rot}}\right)}^2+\frac{1}{2}{\left[\frac{d^2{\epsilon }^2\left(k_y\right)}{d{k}_y^2}\right]}_{k_y=k_{\text{rot}}},$ we obtain Eq. (1) of the main text, with $\text{Δ}=\epsilon \left(k_{\text{rot}}\right)=\sqrt{E_0^2-E_I^2}.$ At the instability, Δ = 0, and $k_{\text{rot}}=\frac{1}{ħ}\sqrt{2m{E}_0}.$

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

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

with λ_⊥ = v_x/v_z and

$$Z^2=\frac{g{n}_0}{2{\pi }^{2}m{v}_z^2}\left[\left(1+2{\epsilon }_{dd}\right)-\frac{3{\epsilon }_{dd}{\chi }^2}{{\left(1+\chi \right)}^2}\right]$$ (13)

These two equations fully determine the TF solution for given ε_dd, gn₀, and λ_⊥. By inserting the expressions of X² and Z² in E₀, we find for ε_dd ≃ 1:

$$E_0^2=\frac{h^2{v}_z^2}{6}{\left(1+\chi \right)}^2\left({\lambda }_{\perp }^2+\frac{1}{1+2\chi }\right)$$ (14)

whereas χ simplifies into $\chi ={\lambda }_{\perp }\left(1+\sqrt{1+1/{\lambda }_{\perp }}\right).$ As a result, at the instability k_rotℓ_z depends only on the transverse confinement aspect ratio λ_⊥, giving the geometrical factor κ:

$$\kappa =k_{\text{rot}}{\ell }_z={\left(\frac{2}{3}\right)}^{1/4}\sqrt{1+\chi }{\left({\lambda }_{\perp }^2+\frac{1}{1+2\chi }\right)}^{1/4}.$$ (15)

Self similar model and instantaneous roton spectrum

The dynamics of the condensate during and after the quench is crucial to understand 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 side, 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 dynamics ³³, describing the evolution at low k_y, and the model of the roton spectrum developed in the Method 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 NLGPE ^{12, 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 modes population only minimally impact the BEC dynamics, that is for short-enough time scales.

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

$$n\left(\mathbf{\text{r}},t\right)=n_0\left(t\right)\left[1-{\left(\frac{x}{X\left(t\right)}\right)}^2-{\left(\frac{y}{Y\left(t\right)}\right)}^2-{\left(\frac{z}{Z\left(t\right)}\right)}^2\right],$$ (16)

where X(t), Y(t) and Z(t) are the re-scaled TF radii. Their evolution after the quench can be deduce from solving the hydrodynamics equations (see Supplementary information). Prior to the quench of a_s, the stationary TF solution is obtained from the self-consistent equations deduced from the hydrodynamics equations by cancelling all time derivative. Solving these equations and using normalisation provides X(0), Y(0), Z(0), and n(0) for known atom number N and trap frequencies v_x,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 re-scaled 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 Method Analytical dispersion relation for an infinite axially elongated geometry:

$$\begin{array}{l}\frac{{\epsilon }^2\left(k_y,y,t\right)}{{\left(h{v}_z\right)}^2{l}_z^4}\simeq \frac{k_y^4}{4}-\frac{8\pi }{3}{n}_0\left(y,t\right)\left(a_{dd}-a_s\left(t\right)\right)k_y^2\\ +4\pi n_0\left(y,t\right)a_{dd}\left[\frac{1}{X{\left(y,t\right)}^2}+\frac{1}{Z{\left(y,t\right)}^2}\right].\end{array}$$ (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

$${\tilde{n}}_1\left(k_y,t\right)\sim {\displaystyle \int dy}\left[e^{2{\int }_0^{t}d{t}^{\prime }\text{Im}\left[\epsilon \left(k_y,y,t^{\prime }\right)\right]}-1\right].$$ (18)

where we assume equal seeding for all unstable modes. We then evaluate the total population of the peak N_*(t) = ∫ dk_yñ₁(k_y,t), as well as the roton momentum $\langle k\rangle =\int d{k}_y{k}_y\frac{{\tilde{n}}_1\left(k_y,t\right)}{N_{\ast }\left(t\right)}.$

The previously discussed SSM does not describe properly 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 $a_{\text{S}}^{\ast }.$ 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 GPE, which is detailed in Supplementary Information.