## 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 ^{4}He (ref. ^{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 their weakly interacting character^{3}. This prospect has raised considerable interest^{4,5,6,7,8,9,10,11,12}; 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 gases^{13}.

## 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 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 ^{4}He, 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* = *k*_{rot}. Neutron scattering experiments confirmed Landau’s remarkable intuition^{14}. In liquid ^{4}He, *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, 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) 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 recently been 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}. In addition, 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 ^{4}He, 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 space^{3,4,5,6,7,8,9,10,11,12}. 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 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 stabilized by quantum fluctuations^{23,24,25}, 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^{3,4,5,6,8,9,10,11,12}. 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 sign^{3} (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
} = *k*_{rot} set by the geometrical scaling ${k}_{\mathrm{rot}}~1\mathrm{\u2215}{\ell}_{z}$ (see below and, for example, refs ^{3,8,9}).

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 this last case, the system undergoes a roton instability and the population at *k*_{
y
} = 0 is transferred to ±*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 ^{166}Er atoms. The experiment starts with a stable dBEC in a cigar-shaped harmonic trap of frequencies *ν*_{x,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 *a*_{s} through a magnetic Feshbach resonance, centred close to *B* = 0 G. In previous experiments, we precisely calibrated the *B*-to-*a*_{s} conversion for this resonance^{25}. The BEC is prepared at ${a}_{\mathrm{s}}^{\mathrm{i}}=61{a}_{0}$ (*B* = 0.4 G) with transverse (*z*) magnetization. The characteristic dipolar length, defined as ${a}_{\mathrm{dd}}={\mu}_{0}{\mu}_{\mathrm{m}}^{2}m\mathrm{\u2215}12\pi {\hslash}^{2}$, is 65.5*a*_{0}, where *m* is the mass and *μ*_{m} is the magnetic moment of the atoms, *ħ* = *h*/2π is the reduced Planck constant, *μ*_{0} is the vacuum permeability and *a*_{0} is the Bohr radius.

To excite the roton mode, we quench *a*_{s} to a desired lower value, ${a}_{\mathrm{s}}^{\mathrm{f}}$, 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 $\stackrel{\u0303}{n}\left({k}_{x},{k}_{y}\right)$ by performing standard resonant absorption imaging on the expanded cloud (Methods). The measurement is then repeated at various values of *a*_{s} < *a*_{dd} in a fixed trap geometry. The momentum distribution shows a striking behaviour (Fig. 2). For large enough *a*_{s}, $\xf1\left({k}_{x},{k}_{y}\right)$ shows a single narrow peak with an inverted aspect ratio compared to the trapped gas, typical of a stable BEC^{12} (Fig. 2a, top). We define the centre of the distribution as the origin of *k*. In contrast, when further decreasing *a*_{s} below a critical value ${a}_{\mathrm{s}}^{*}$, we observe a sudden appearance of two symmetric finite-momentum peaks, of similar shape and located at *k*_{
y
} = ±*k*_{rot} (Fig. 2b,c, 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 $\stackrel{\u0303}{n}\left({k}_{x},{k}_{y}\right)$ (Fig. 2a–c, bottom panels). From the fit, we extract the central momentum, *k*_{rot}, and the amplitude, *A*_{*}, of the side peaks.

A major fingerprint of the roton mode in dBECs is its geometrical nature, leading to a universal scaling ${k}_{\mathrm{rot}}~1\mathrm{\u2215}{\ell}_{z}$ (see, for example, refs ^{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 $1\mathrm{\u2215}{\ell}_{z}=2\pi \sqrt{m{\nu}_{z}\mathrm{\u2215}h}$, matching well with a linear progression with a slope of 1.61(4) (Fig. 2d). Note that no dependence of *k*_{rot} on *ν*_{
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 with 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 (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 *n*_{0}, 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:

The roton momentum reads as ${k}_{\mathrm{rot}}=\sqrt{2m}{\left({E}_{0}^{2}-{\Delta}^{2}\right)}^{1\u22154}\mathrm{\u2215}\hslash $, where $\Delta =\sqrt{{E}_{0}^{2}-{E}_{\mathrm{I}}^{2}}$, *E*_{I} = 2*gn*_{0}(*ε*_{dd} − 1)/3, *g* = 4π*ħ*^{2}*a*_{s}/*m* and ${E}_{0}^{2}=2g{\epsilon}_{\mathrm{dd}}{n}_{0}\frac{{\hslash}^{2}}{2m}\left({X}^{-2}+{Z}^{-2}\right)$. The TF radii *X*,*Z* satisfy *X*^{2},*Z*^{2} ∝ *gn*_{0}, so that *E*_{0} ∝ *hν*_{
z
}, with a scale set by *ε*_{dd} and *ν*_{
z
}/*ν*_{
x
} but independent of *gn*_{0} (ref. ^{15}). Close to the instability ($\Delta \simeq 0$), the roton momentum thus follows a simple geometrical scaling, ${k}_{\mathrm{rot}}=\kappa \mathrm{\u2215}{\ell}_{z}$ 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 *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 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 quench^{12}. In particular, we consider a 3D harmonic confinement, and, starting from the stationary TF profile at ${a}_{\mathrm{s}}^{\mathrm{i}}$, 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, that is, evaluating equation (1) with the experimentally calibrated *a*_{s}(*t*_{h}), and the *n*_{0}(*y*, *t*_{h}), *X*(*y*,*t*_{h}) and *Z*(*y*,*t*_{h}) estimated from the 3D profile. 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*, with Im[.] indicating the imaginary part, giving rise to two symmetric side peaks in the axial momentum distribution ${\stackrel{\u0303}{n}}_{1}\left({k}_{y},{t}_{\mathrm{h}}\right)\propto \int \mathrm{exp}\left(2{\int}_{0}^{{t}_{\mathrm{h}}}\mathrm{Im}\left[\epsilon \left({k}_{y},y,t\right)\right]\mathrm{d}t\right)\mathrm{d}y$ (Fig. 3b) (Methods). The centre of the peaks, ±*k*_{rot}(*t*_{h}), also evolves with *t*_{h} but quickly converges after a few milliseconds 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 (Figs. 2d,e and 3c).

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 collapse^{4,5,6,24,25}, 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 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 (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 ${a}_{\mathrm{s}}^{\mathrm{f}}$ 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 ${a}_{\mathrm{s}}^{\mathrm{f}}$ (Fig. 4). Our data show that the roton mode begins to become populated from a critical value of the scattering length, ${a}_{\mathrm{s}}^{*}$. For ${a}_{\mathrm{s}}^{\mathrm{f}}>52{a}_{0}$, we do not observe the roton peaks at any time. For ${a}_{\mathrm{s}}^{\mathrm{f}}$ ≲ 52*a*_{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 ${a}_{\mathrm{s}}^{\mathrm{f}}$, the roton population exhibits a faster growth rate and shorter time delay.

From the growth rate of the roton population, we now extract an overall roton gap $\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\right)$ (Methods). In brief, the dynamical variation of the gap is determined by the leading time dependence of *a*_{s} and ${A}_{*}\left({a}_{\mathrm{s}}^{\mathrm{f}},{t}_{\mathrm{h}}\right)\propto \mathrm{exp}\left(2{\int}_{0}^{{t}_{\mathrm{h}}}\mathrm{d}t\mathrm{Im}\left[\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\left(t\right)\right)\right]\mathrm{\u2215}\hslash \right)$. To investigate the scaling $\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\right)$, we first consider the analytical expression of the elementary local roton gap, *Δ* as a function of *a*_{s} (see equation (1)). Developing *Δ* in the vicinity of ${a}_{\mathrm{s}}={a}_{\mathrm{s}}^{*}$ yields $\Delta \left({a}_{\mathrm{s}}\right)\propto {\left({a}_{\mathrm{s}}^{*}-{a}_{\mathrm{s}}\right)}^{1\u22152}$. Here we confirm this scaling by considering the generic power-law dependence $\mathrm{Im}\left[\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\right)\right]\mathrm{\u2215}\hslash =\mathrm{\Gamma}\delta \left({a}_{\mathrm{s}}\right)$ with $\delta \left({a}_{\mathrm{s}}\right)={\left(\frac{{a}_{\mathrm{s}}^{*}-{a}_{\mathrm{s}}}{{a}_{0}}\right)}^{\beta}\left({a}_{\mathrm{s}}\le {a}_{\mathrm{s}}^{*}\right)$. Consequently, *A*_{*} grows as $\mathrm{exp}\left(2\Gamma {\int}_{0}^{{t}_{\mathrm{h}}}\delta \left({a}_{\mathrm{s}}\left(t\right)\right)\mathrm{d}t\right)=\mathrm{exp}\left(2\Gamma {T}_{\mathrm{h}}\right)$. By rescaling the time variable as ${t}_{\mathrm{h}}\to {T}_{\mathrm{h}}={\int}_{0}^{{t}_{\mathrm{h}}}\delta \left({a}_{\mathrm{s}}\left(t\right)\right)\mathrm{d}t$, we determine the parameters ${a}_{\mathrm{s}}^{*}$ 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 ${a}_{\mathrm{s}}^{*}=53.0\left(4\right){a}_{0}$ and *β* = 0.55(8). We then perform an exponential fit to all of 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 ${a}_{\mathrm{s}}^{\mathrm{f}}$, gives ${a}_{\mathrm{s}}^{*}\simeq 52.8{a}_{0}$, $\beta \simeq 0.55$ and $\Gamma \simeq 472{\mathrm{s}}^{-1}$, which are very close to the experimental values. This time-resolved study allows us to readily extract the imaginary roton gap, $\mathrm{Im}\left[\stackrel{\u0304}{\Delta}\right]$, 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 ${a}_{\mathrm{s}}={a}_{\mathrm{s}}^{*}$ and the expected increase of $\mathrm{Im}\left[\stackrel{\u0304}{\Delta}\right]$ for ${a}_{\mathrm{s}}<{a}_{\mathrm{s}}^{*}$, 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 dBEC^{8,9,10,12}, 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 the 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 favour the formation of such an exotic phase of matter, in which crystalline order coexists with phase coherence. In contrast 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. 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 $V\left(\mathbf{r}\right)=2m{\pi}^{2}\left({\nu}_{x}^{2}{x}^{2}+{\nu}_{y}^{2}{y}^{2}+{\nu}_{z}^{2}{z}^{2}\right)$, 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 *a*_{s}, we use a magnetic Feshbach resonance between ^{166}Er atoms in their absolute ground state that is centred around *B* = 0 G (ref. ^{31}). The *B*-to-*a*_{s} conversion has previously been 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 (for example, from stray fields), are of 3–5*a*_{0} for the relevant *a*_{s} range 27–67*a*_{0} 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, *B*_{set}, twice. First, we perform the quench itself and abruptly change *B*_{set} from 0.4 G (${a}_{\mathrm{s}}^{\mathrm{i}}=61{a}_{0}$) to the desired lower value (corresponding to ${a}_{\mathrm{s}}^{\mathrm{f}}$) at the beginning of the hold in the trap (*t*_{h} = 0 ms). Second, we prepare the time-of-flight (TOF) expansion and imaging conditions (see the section entitled Imaging procedure) and abruptly change *B*_{set} from the quenched value back to 0.3 G (*a*_{s} = 57*a*_{0}) at the beginning of the TOF expansion (*t*_{f} = 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 *B*_{set} via *B*(*t*) = *B*_{set}(*t*) + *τ*d*B*/d*t* (ref. ^{32}). By performing pulsed-radiofrequency 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 ${t}_{\mathrm{h}}\mathrm{\u2215}{\nu}_{y},{t}_{\mathrm{h}}\mathrm{\u2215}{\tau}_{\mathrm{coll}}\ll 1$, with 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 1 at ${a}_{\mathrm{s}}^{\mathrm{i}}=61{a}_{0}$. 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 (see the section entitled Quench of the scattering length *a*_{s}). 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 *a*_{s} to 57*a*_{0} allows us 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:

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. 2–4, 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 $\stackrel{\u0303}{n}\left({k}_{x},{k}_{y}\right)$, from which we characterize the linear roton developing along *k*_{
y
}. To do so, we extract a 1D profile ${\stackrel{\u0303}{n}}_{1}\left({k}_{y}\right)$ by averaging $\stackrel{\u0303}{n}\left({k}_{x},{k}_{y}\right)$ over *k*_{
x
} within $\u2223{k}_{x}\u2223\le {k}_{\mathrm{m}}=3.5$ μm^{−1}: ${\stackrel{\u0303}{n}}_{1}\left({k}_{y}\right)={\int}_{-{k}_{\mathrm{m}}}^{{k}_{\mathrm{m}}}\stackrel{\u0303}{n}\left({k}_{x},{k}_{y}\right)\mathrm{d}{k}_{x}\mathrm{\u2215}{\int}_{-{k}_{\mathrm{m}}}^{{k}_{\mathrm{m}}}\mathrm{d}{k}_{x}$. To quantitatively analyse the observed roton peaks, we fit a sum of three Gaussian distributions to ${\stackrel{\u0303}{n}}_{1}\left({k}_{y}\right)$. One Gaussian accounts for the central peak and its centre is constrained to *k*_{0} ~ 0. Its amplitude is denoted *A*_{0}. 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 (${a}_{\mathrm{s}}^{\mathrm{f}}$, *t*_{h}) 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 *a*_{s} > *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 ${a}_{\mathrm{s}}^{\mathrm{f}}$ 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 ${\stackrel{\u0303}{n}}_{1}\left({k}_{\mathrm{rot}}\right)$. *A*_{*} is observed to initially increase if ${a}_{\mathrm{s}}^{\mathrm{f}}<{a}_{\mathrm{s}}^{*}$, its growth rate increases for decreasing ${a}_{\mathrm{s}}^{\mathrm{f}}$ and ${a}_{\mathrm{s}}^{*}$ depends on the trap geometry.

If dynamically unstable, the *k*_{rot} component is expected to grow exponentially with an instantaneous rate 2Im[*ε*(*k*_{rot}, *t*_{h})]/*ħ*. We hence expect an initial growth of *A*_{*} of the form:

where $\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}^{\mathrm{f}},{t}_{\mathrm{h}}\right)$ is the instantaneous value of the overall roton gap, corresponding to an average over the cloud, after quenching *a*_{s} to ${a}_{\mathrm{s}}^{\mathrm{f}}$. 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 ${\stackrel{\u0304}{\Delta}}_{}^{}$ is determined by *a*_{s}(*t*) and, by monitoring ${A}_{*}\left({a}_{\mathrm{s}}^{\mathrm{f}},{t}_{\mathrm{h}}\right)$, one can readily extract the scattering-length-dependent gap $\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\right)$, $\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}^{\mathrm{f}},t\right)=\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\left(t\right)\right)$. To investigate the scaling $\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\right)$, 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 *n*_{0}, *X* and *Z* fixed, and using the fact that $\Delta \left({a}_{\mathrm{s}}={a}_{\mathrm{s}}^{*}\right)=0$, one can easily obtain that for *a*_{s} in the vicinity of ${a}_{\mathrm{s}}^{*}$, $\Delta \left({a}_{\mathrm{s}}\right)\propto {\left({a}_{\mathrm{s}}^{*}-{a}_{\mathrm{s}}\right)}^{1\u22152}$. Here we verify this scaling for the global and dynamical quantity $\stackrel{\u0304}{\Delta}\left({a}_{\mathrm{s}}\right)$, considering the generic power-law dependence Im[*Δ*(*a*_{s})]/*ħ* = *Γ**δ*(*a*_{s}) with $\delta \left({a}_{\mathrm{s}}\right)={\left(\frac{{a}_{\mathrm{s}}-{a}_{\mathrm{s}}^{*}}{{a}_{0}}\right)}^{\beta}\left({a}_{\mathrm{s}}\left(t\right)\le {a}_{\mathrm{s}}^{*}\right)$, in which the parameters Γ, ${a}_{\mathrm{s}}^{*}$ and *β* mainly depend on the trap geometry.

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

This defines a time rescaling ${t}_{\mathrm{h}}\to {T}_{\mathrm{h}}={\int}_{0}^{{t}_{\mathrm{h}}}\delta \left({a}_{\mathrm{s}}\left(t\right)\right)\mathrm{d}t$ 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 2*Γ*. The relevant values of ${a}_{\mathrm{s}}^{*}$, *β* and *Γ* are the ones that result in the best overlap of the data for the initial growth of *A*_{*} (minimal spread in *T*_{h}).

To determine ${a}_{\mathrm{s}}^{*}$ and *β*, we then plot our full data set as a function of *T*_{h} for various trial values of these parameters and evaluate the relevance of the trial couple $\left({a}_{\mathrm{s}}^{*},\beta \right)$. Precisely, we assess the dispersion in *T*_{h} 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] μm^{2}, 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 ${a}_{\mathrm{s}}^{\mathrm{f}}$ using piecewise cubic polynomial interpolation, and extract the corresponding set of ${T}_{\mathrm{h}}^{i}$ at which ${A}_{*}\left({T}_{\mathrm{h}}^{i}\right)={A}_{*}^{i}$. For each *i*, we evaluate the spread of the ${T}_{\mathrm{h}}^{i}\left({a}_{\mathrm{s}}^{\mathrm{f}}\right)$ 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*_{*}(*T*_{h}) for a given (${a}_{\mathrm{s}}^{*},\beta $) via the geometrical average of these two complementary quantities for all of the ten ${A}_{*}^{i}$ values. We fit the relevant ${a}_{\mathrm{s}}^{*}$ and *β* by minimizing this averaged spread. For the experimental data of Fig. 4a,b, this results in ${a}_{\mathrm{s}}^{*}=53.0\left(4\right){a}_{0}$ and *β* = 0.55(8).

Using these values of ${a}_{\mathrm{s}}^{*}$ 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 show only 5 values of ${a}_{\mathrm{s}}^{\mathrm{f}}$ while the reported analysis considers all available experimental data (that is, 11 values between 31*a*_{0} and 52*a*_{0}).

From the formula with the estimated ${a}_{\mathrm{s}}^{*}$, *β* and *Γ*, we can then compute the global roton gap $\stackrel{\u0304}{\Delta}$. The extracted value is meaningful only for ${\stackrel{\u0304}{\Delta}}^{2}<0$. For the experiments, we also restrict its relevance to *T*_{h} ≤ 3 ms so that, for example, in the inset of Fig. 4c, $\mathrm{Im}\left[\stackrel{\u0304}{\Delta}\right]$ 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 $\mathrm{Im}\left[\stackrel{\u0304}{\Delta}\right]$ 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} and ${\ell}_{z}$). Here we estimate *k*_{rot} from a *t*_{h} value that is individually optimized for each *a*_{s} and ${\ell}_{z}$ investigated (largest visibility). We point out that the selected *t*_{h} corresponds, in the *A*_{*} dynamics, to the late stage of the exponential growth, close to the maximum (that is, *T*_{h} ~ 3 ms). Here, the atom loss remains at the level of a few per cent.

In Fig. 2d, we show the value of *k*_{rot} at the onset of the population of the roton mode (that is, at ${a}_{\mathrm{s}}={a}_{\mathrm{s}}^{*}$). We estimate ${a}_{\mathrm{s}}^{*}$ 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 ${a}_{\mathrm{s}}^{*}$ of about 1.5*a*_{0}, lying within our experimental uncertainty on *a*_{s} (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 ${\psi}_{0}\left(\rho \right)=\sqrt{n\left(\rho \right)}$, with *n*(*ρ*) = *n*_{0}(1 − (*x*/*X*)^{2} − (*z*/*Z*)^{2}), where *X* and *Z* are the TF radii, and *ρ* = (*x*,*z*). The calculation of *n*_{0}, *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 $\delta \psi \left(r,t\right)=u\left(\rho \right){\mathrm{e}}^{i{k}_{y}y-i\epsilon t\u2215\hslash}-v\left(\rho \right){\mathrm{e}}^{-i{k}_{y}y+i\epsilon t\u2215\hslash}$, 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*) = (*ψ*_{0}(*ρ*) + *ηδψ*(*r*,*t*))e^{−iμt/ħ}, where *μ* is the chemical potential associated with *ψ*_{0} and $\eta \ll 1$. After linearization, we obtain the Bogoliubov–de Gennes equations for *f*_{±}(*ρ*) = *u*(*ρ*) ± *v*(*ρ*):

where

where $U\left(\mathbf{r}\right)=g\left(\delta \left(\mathbf{r}\right)+\frac{3{\epsilon}_{\mathrm{dd}}}{4\pi}\frac{1-3{\mathrm{cos}}^{2}\theta}{{\u2223\mathbf{r}\u2223}^{3}}\right)$ is the binary interaction potential for two particles separated by **r** including both contact and dipolar interactions and $g=\frac{4\pi {\hslash}^{2}{a}_{\mathrm{s}}}{m}$ (see Supplementary Information). *θ* is the angle between **r** and the magnetization axis (*z*).

Employing *f*_{+} (*ρ*) = *W*(*ρ*)*ψ*_{0}(*ρ*), and for ${k}_{y}\gg 1\mathrm{\u2215}X,1\mathrm{\u2215}Z$, we obtain the following equation for the function *W*(*ρ*):

where $\stackrel{\u0303}{x}=x\mathrm{\u2215}X$, $\stackrel{\u0303}{z}=z\mathrm{\u2215}Z$ and $E\left({k}_{y}\right)={\hslash}^{2}{k}_{y}^{2}\mathrm{\u2215}2m$. For *ε*_{dd} = 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 *ε*_{0}(*k*_{
y
}) with

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

with

$${E}_{\mathrm{I}}=\frac{2}{3}g{n}_{0}\left({\epsilon}_{\mathrm{d}d}-1\right)$$This expression for the dispersion presents a roton minimum for *ε*_{dd} > 1 at ${k}_{\mathrm{rot}}=\frac{1}{\hslash}\sqrt{2m{E}_{\mathrm{I}}}$. Expanding equation (11) in the vicinity of the roton minimum, $\epsilon {\left({k}_{y}\right)}^{2}\simeq \epsilon {\left({k}_{\mathrm{r}ot}\right)}^{2}+\frac{1}{2}{\left[\frac{{\mathrm{d}}^{2}{\epsilon}^{2}\left({k}_{y}\right)}{\mathrm{d}{k}_{y}^{2}}\right]}_{{k}_{y}={k}_{\mathrm{rot}}}$, we obtain equation (1) of the main text, with $\Delta =\epsilon \left({k}_{\mathrm{rot}}\right)=\sqrt{{E}_{0}^{2}-{E}_{\mathrm{I}}^{2}}$. At the instability, *Δ* = 0, and ${k}_{\mathrm{rot}}=\frac{1}{\hslash}\sqrt{2m{E}_{0}}$.

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

with *λ*_{⊥} = *ν*_{
x
}/*ν*_{
z
} and

These two equations fully determine the TF solution for the given *ε*_{dd}, *gn*_{0} and *λ*_{⊥}. By inserting the expressions of *X*^{2} and *Z*^{2} into *E*_{0}, we find for ${\epsilon}_{\mathrm{dd}}\simeq 1$:

whereas *χ* simplifies to $\chi ={\lambda}_{\perp}\left(1+\sqrt{1+1\mathrm{\u2215}{\lambda}_{\perp}}\right)$. As a result, at the instability, ${k}_{\mathrm{rot}}{\ell}_{z}$ depends only on the transverse confinement aspect ratio *λ*_{⊥}, giving the geometrical factor *κ*:

### 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 dynamics^{33}, 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 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* 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:

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 *a*_{s}, 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 *ν*_{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 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*) = *n*_{0}(*y*,*t*)[1 − (*x*/*X*(*y*,*t*))^{2} − (*z*/*Z*(*y*,*t*))^{2}], where *n*_{0}(*y*,*t*) = *n*_{0}(*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 *n*_{0}(*t*) = *n*_{0}(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:

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

where we assume equal seeding for all unstable modes. We then evaluate the total population of the peak ${N}_{*}\left(t\right)=\int \mathrm{d}{k}_{y}{\stackrel{\u0303}{n}}_{1}\left({k}_{y},t\right)$, as well as the roton momentum $\u27e8k\u27e9=\int \mathrm{d}{k}_{y}{k}_{y}\frac{{\stackrel{\u0303}{n}}_{1}\left({k}_{y},t\right)}{{N}_{*}\left(t\right)}$.

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 ${a}_{\mathrm{s}}^{*}$. 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.

### Data availability

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.

## Author information

### Affiliations

#### Institut für Experimentalphysik, Universität Innsbruck, Innsbruck, Austria

- L. Chomaz
- , D. Petter
- , G. Faraoni
- , S. Baier
- , J. H. Becher
- , M. J. Mark
- & F. Ferlaino

#### Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Innsbruck, Austria

- R. M. W. van Bijnen
- , M. J. Mark
- & F. Ferlaino

#### Dipartimento di Fisica e Astronomia, Università di Firenze, Sesto Fiorentino, Italy

- G. Faraoni

#### 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

### Supplementary Information

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