Density Waves and Jet Emission Asymmetry in Bose Fireworks

A Bose condensate, subject to periodic modulation of the two-body interactions, was recently observed to emit matter-wave jets resembling fireworks [Nature (London) 551, 356 (2017)]. In this Letter, combining experiment with numerical simulation, we demonstrate that these “Bose fireworks” represent a late stage in a complex time evolution of the driven condensate. We identify a “density wave” stage which precedes jet emission and results from the interference of matter waves. The density waves self-organize and self-amplify without breaking long range translational symmetry. This density wave structure deterministically establishes the template for the subsequent patterns of the emitted jets. Moreover, our simulations, in good agreement with experiment, address an apparent asymmetry in the jet pattern, and show that it is fully consistent with momentum conservation.

Figure 1

The real space density distribution $n(\mathbf{r})$ as a comparison between experimental data and simulations. In both, the modulation frequency, $\omega /2\pi$, is 2 kHz and the dc and ac interaction energies, respectively, are $U_0{n}_0\approx h\times 40\mathrm{Hz}$ and $U_1{n}_0\approx h\times 480\mathrm{Hz}$, where $h$ is Planck’s constant (see the main text for detailed definitions). As a function of modulation time $t$, the system exhibits three phases: density waves in a confined condensate (blue box), near-field emission (orange box), and far-field emission (red box).

Figure 2

Experiment and simulation comparison for early-stage density waves (DW) with $|k|=k_f$. (a) The real-space DW oscillations inside the condensate. Theory (top) and experiment (bottom) show good qualitative agreement. The experiment exhibits additional static, long-wavelength density modulations due to trap imperfections. The experimental details are provided in the main text. (b) The amplitude of the density waves in the primary mode comparing simulations (red solid line) and experiments (blue dots with error bars). In addition to fast oscillations, both results show a consistent observation of an exponential growth of the envelope until the matter-wave jets are emitted from the condensate.

Figure 3

Connection between density waves before jet emission and the subsequent matter-wave jet pattern. (a) shows the azimuthal density structure factor $S(k_f)$ from a single iteration of the GP simulations at the resonant wave number $k_f$ at $t=10$ (blue), 13 (purple), and 15 (red) ms prior to jet emission. At each time, we observe the same shape with a growing amplitude, consistent with the expected amplification process of density waves. The dashed black curve is the real-space azimuthal population distribution of jets $N(\varphi )$ at $t=45\mathrm{ms}$. The scaling factor, $N_0$, is the total number of atoms in the system. The alignment of all maxima and minima between $S(k_f,\varphi )$ and $N(\varphi )$ shows the equivalence between the density waves and jets. (b) schematically shows that the early-time density waves with wave number $k_f$ leads to the emission of counter-propagating jets with the same wave number $k_f$ at the long time.

Figure 4

Time evolution and correlations of the emitted jets. (a) shows the calculated jet emission pattern evolving from the near- to far-field regimes. The calculation is based on identical initial noise seeding. (b) shows the real space azimuthal population of the four images in (a), identified by the same color. Note that, the $t=45\mathrm{ms}$ far-field curve is equivalent to that shown dashed in Fig. 3. Here, unlike in Fig. 3, the peaks and valleys are slightly displaced with time. Panel (c) probes the emission asymmetry in real space ${\eta }_r=g^{(2)}(\pi )-g^{(2)}(0)$ (brown circles) and the momentum-space analogue ${\eta }_k$ (green squares). The main figure shows that the $(0,\pi )$ asymmetry is always absent in momentum space (${\eta }_k$ is strictly zero within numerical precision) so that momentum is conserved. In real space, using (b), we find that this $(0,\pi )$ asymmetry decreases with increasing time. The inset indicates the correlation function $g^{(2)}(\varphi )$ at the same four times as in (a), along with an early time momentum correlation function at $t=20\mathrm{ms}$ (black curve). Again, inversion $0-\pi$ symmetry is broken at short times, but it recovers after a long time-of-flight and is fully preserved in momentum space. The solid line (brown) in (c) is an analytical fit to ${\eta }_r$ [34].