Parametric pair production of collective excitations in a Bose–Einstein condensate

V. Gondret, R. Dias, C. Lamirault, L. Camier, A. Micheli, C. Leprince, Q. Marolleau, S. Robertson, D. Boiron, C. I. Westbrook

Comptes Rendus Physique (2025) · 10.5802/crphys.266 · arXiv:2508.01654 · PDF

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Abstract

By exciting the transverse breathing mode of an elongated Bose-Einstein condensate, we parametrically produce longitudinal collective excitations in a pairwise manner. This process also referred to as Faraday wave generation, can be seen as an analog to cosmological particle production. Building upon single particle detection, we investigate the early time dynamics of the exponential growth and compare our observations with a Bogoliubov description. The growth rate we observe experimentally is in very good agreement with theoretical predictions, demonstrating the validity of the Bogoliubov description and thereby confirming the smallness of quasiparticle interactions in such an elongated gas. We also discuss the presence of oscillations in the atom number, which are due to pair correlations and to the rate at which interactions are switched off.

Figures4

Four-panel figure: the experimental setup; the modulated dipole-trap laser power and the resulting BEC width oscillation over time, both with the hold time Δt marked; and a density map of the atom velocity distribution over hold time showing two growing sidebands. — **Figure 1.** **Experimental parametric excitation of quasiparticles.** (a) Diagram of the experimental setup. (b) By modulating the dipole trap laser power $\mathcal{P}_{\mathrm{las}}$ , the breathing mode of the BEC is excited. (c) The BEC width $\sigma$ oscillates as long as the cloud is kept in the trap for a duration $\Delta t$ ; the breathing mode parametrically excites two longitudinal excitations with opposite momenta. (d) Time evolution of the longitudinal density profile of the atom velocity distribution. The dark band at $v_z = 0$ is the condensate; after a hold time of about 2 ms, sidebands appear on either side of it.

Detected atom number in the two sidebands versus hold time on a log scale, growing exponentially with an oscillating modulation, with fitted curves for each sideband. — **Figure 2.** **Detected atom number in each sideband as a function of the hold time.** In this data set the laser power was modulated by ±15%. The atom number was counted in each voxel as defined in the text. The data was fitted to Eq. (2) in the exponential regime, between 0.5 and 4 ms. The solid blue and dashed orange lines show the fits for the negative and positive velocity sidebands, respectively. The fits give a reduced chi-squared of $\chi_\nu^2 = 1.0$ and $\chi_\nu^2 = 1.2$ , respectively.

Figure 3. Comparison between the experimental and homogeneous undamped theoretical rate. (a) The asymmetric breathing mode of the BEC, fitted with a sine function along both axes; the modulation parameters are the same as in Fig. 2. (b) The relative, effective 1D interaction strength. (c) The fitted growth rate for different modulation amplitudes, compared to the prediction of Eq. (4). The theoretical growth rate is also shown as a line of unit slope.

Two log-scale plots of atom number growing with oscillations over hold time, comparing a sudden trap shutoff to a slower ramp-down, for a numerical simulation and for the experimental data. — **Figure 4.** **A more adiabatic mapping of the quasiparticles to the atoms.** (a) Numerical solution and (b) experimental plot of the evolution of the atom number when the transverse laser power, shown in the inset, is abruptly turned off (solid line and round markers, red) or ramped down over 1.5 ms (dashed line and square markers, green).

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Conclusion

In this paper, we have studied the production of quasiparticles in a parametrically driven BEC. Our observations of exponential growth and oscillation of the particle number are in agreement with Bogoliubov theory as applied in previous work. The data clearly shows the parametric nature of the process and the fact that the excitations are generated in a pairwise manner.

The measured growth rate is in very good agreement with the predicted value, even in the presence of a trap inhomogeneity. The precision of our measurements cannot isolate and measure the small effect of quasiparticle interactions, which should reduce the growth rate. Nevertheless, our results do confirm the smallness of this decay, if present. Future work will aim to improve the experimental procedure in order to increase the signal-to-noise ratio and further isolate this effect. Higher experimental precision may require a closer correspondence between the experiment and the model, especially if we wish to compare the value of the damping of the collective excitations with the prediction of previous work. Thus we envision repeating the above experiments in a square potential. Alternatively, the model could be improved by including the density inhomogeneities. Also, exciting the gas with a Feshbach resonance would allow to excite other $k$ modes while our excitation method is restricted to modes satisfying $\omega_k = \omega_\perp$ .

The observed oscillations are well understood. If we could precisely measure their amplitude and estimate $u_k$ and $v_k$ , we would be able to compare the quasiparticle population to their anomalous correlation. This comparison would demonstrate the (non)separability of the two-mode state without looking at many-body correlation functions.

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