Pattern formation in driven condensates

Abstract

The onset of pattern formation in a spatially homogeneous system subjected to external driving is an important topic in various scientific fields. A celebrated classical example is the Faraday instability, where a vertically oscillated fluid surface undergoes a parametric resonance, giving rise to standing waves that self-organize into regular spatial patterns. Bose-Einstein condensates (BECs) provide an ideal quantum-mechanical platform for studying pattern-forming mechanisms due to their exceptional degree of experimental control. As a compressible state of quantum matter, a condensate responds sensitively to external perturbations, including time-periodic modulation of interactions, trapping potentials, or external fields. These features make BECs particularly well suited for exploring driven nonequilibrium phases and pattern formation. In this chapter, we review the remarkable progress achieved in this field over the past two decades. We begin with the first theoretical proposal predicting parametric instabilities and emergent Faraday waves in driven condensates. We then discuss key experimental and theoretical breakthroughs that confirmed these predictions and refined the understanding of the underlying mechanisms. This line of research has culminated in the recent observation of a stabilized square lattice pattern in a periodically driven BEC confined in a two-dimensional geometry. This driven superfluid state with superposed density modulation was shown to exhibit some features of a supersolid state.

Figures (13)

• Figure 1: Imaginary part of the exponent $\mathrm{Im}\,(\lambda)$ as a function of $a$ for $b = 0.25$. Notice the main lobe centered around $a = 1$. The second lobe, shown in the inset, is much smaller than the first one that it can hardly be seen on the same scale. The figure is taken from Ref. PhysRevA.76.063609.
• Figure 2: In-trap absorption images of Faraday waves in a BEC. Frequency labels for each image represent the driving frequency at which the transverse trap confinement is modulated. The figure is taken from Ref. PhysRevLett.98.095301.
• Figure 3: Faraday pattern from the simulations of GP Eq. (\ref{['eq:GPE']}). This case corresponds to the experiment of Ref. PhysRevLett.98.095301, namely, a cloud of $N=5 \times 10^5$$^{87}$Rb atoms, trapped by $\omega_{r} = 2\pi \times 160.5$ Hz,$\omega_z = 2\pi \times 7$ Hz with a 20% modulation of the radial confinement at a driving frequency $\omega/(2\pi)=321$ Hz. The panels show snapshots of the $y$-integrated density profile (i.e., the observable in the experiments) at (a) $t=17\mathrm{ms}$, (b) $t=18\mathrm{ms}$, (c) $t=19\mathrm{ms}$, (d) $t=20\mathrm{ms}$, (e) $t=21\mathrm{ms}$, and (f) $t = 22\mathrm{ms}$. The figure is taken from Ref. PhysRevA.76.063609.
• Figure 4: Spatial period vs. driving frequency $\omega$. The experimental data are indicated by filled squares, while the solid lines are the theory of Ref. PhysRevE.84.056202. The blue data points are the primary peak of the fast-Fourier transform of the line-density profiles, while the red data points correspond to a secondary peak, when one exists. The error bars here correspond to the standard error of the mean determined from ten different experimental runs for each point. The solid blue line is the calculated spatial period $\lambda_F = 2\pi/k_F$ of the Faraday mode from Eq. (\ref{['eq:kF']}), while the red is that of the resonant mode $\lambda_R = 2\pi/k_R$ from Eq. (\ref{['eq:kR']}). The figure is taken from Ref. PhysRevX.9.011052.
• Figure 5: Interaction dependence of $\lambda_F = 2\pi/k$. The experimental data are indicated by filled squares, while the solid line is the theory of Ref. PhysRevE.84.056202. The figure is taken from Ref. PhysRevX.9.011052.