Pattern formation in driven condensates

TL;DR

Driven condensates exhibit spontaneous pattern formation in quantum fluids through parametric instabilities that parallel classical hydrodynamics, yet reveal uniquely quantum features such as superfluidity and quantized vortices. This review surveys bosonic and fermionic Bose-Einstein condensates under periodic driving, covering one- and two-dimensional Faraday patterns, time-crystal interpretations, granulation beyond mean-field, and surface as well as counterflow-induced instabilities. It highlights experimental milestones—from the first Faraday observations to stable two-dimensional patterns and counterflow-driven vortex dynamics—and discusses their unifying description via parametric amplification and mode competition, with connections to supersolid-like sound modes and quantum turbulence. The insights offer a framework for probing nonequilibrium quantum dynamics, turbulence precursors, and the emergence of complex order in driven many-body systems, with implications for controllable pattern formation in quantum fluids.

Abstract

Spontaneous pattern formation out of homogeneous media is one of the well-understood examples of hydrodynamic instabilities in classical systems, which naturally leads to the question of its manifestation in quantum fluids. Bose-Einstein condensates (BECs) of atomic gases have been an ideal platform for studying many-body quantum phenomena, such as superfluidity, and simultaneously providing an opportunity to broaden our understanding of classical hydrodynamics into quantum systems. In this review, we introduce a range of experimental studies on the pattern formation in quantum fluids of atomic gases under external driving, including Faraday waves in one and two dimensions, surface patterns, and counterflow instabilities in a mixture of superfluids. The pattern formation in the quantum system can be understood through the parametric amplification process, where an unstable dynamical mode can be exponentially amplified, similar to classical systems. Remarkably, the governing equations for surface excitations of trapped BECs can be mathematically equivalent to those of shallow water, indicating a universal description of the hydrodynamic instability across classical and quantum domains. However, the condensates, as superfluids, also possess fundamental quantum characteristics, such as quantized vorticity and a distinct dissipation channel. These unique features showcase many-body fragmentation under strong modulation and the generation of vortices in the nonlinear regime, which could offer a pathway to the study of quantum turbulence. Furthermore, the coexistence of long-range phase coherence and density modulation in driven condensates could provide unexplored features, such as those seen in supersolid-like sound modes, within nonequilibrium settings.

Figures (14)

• Figure 1: Various patterns in classical systems. (a) Polygonal structure observed in Sputnik Planum, Pluto. Adapted with permission from ref McKinnon2016, Springer Nature Limited. (b) Various Chladni patterns in oscillating square-shaped metal plate. Adapted from ref Tuan2018 under a Creative Common license https://creativecommons.org/licenses/by/4.0/. (c) oscillatory pattern in a ferroin catalyzed Belousov–Zhabotinsky reaction. Adapted from ref Howell2021 under a Creative Common license https://creativecommons.org/licenses/by/4.0/.
• Figure 2: Faraday pattern in Bose-Einstein condensates. Absorption images of trapped quasi one-dimensional BEC after modulating the radial trap frequency. Labels in each image represent the driving frequencies. Adapted with permission from Engels2007, APS.
• Figure 3: Faraday and resonant waves in one-dimensional Bose gases. (a) Absorption images of trapped Bose-Einstein condensate after modulating the scattering length at $\omega/2\pi=200$ Hz. (b) Fast-Fourier transform (FFT) spectrum of the line density. The spectrum shows two peaks, where the primary peak (marked by blue arrow) represents the Faraday waves and the second peak (marked by red arrow) indicate the resonant wave. (c) Modulation wavelength dependence on the driving frequency $\omega$. The solid lines are the wave lengths obtained from a 3D variational calculation Nicolin2011 for the Faraday waves (blue line) and resonance wave (red line), respectively. Labels in each image represent the driving frequencies. Adapted from Nguyen2019 under the terms of the https://creativecommons.org/licenses/by/4.0/.
• Figure 4: Granulation of a Bose-Einstein condensate, (a) Experimental images and (b) Gross-Pitaevskii (GP) simulations of density profiles at different modulation times $t_h$ with low modulation frequency $\omega/2\pi=70$ Hz. Adapted with permission from ref Nguyen2019 under a Creative Commons license https://creativecommons.org/licenses/by/4.0/.
• Figure 5: Long-term stability of space-time crystal. (a) Phase contrast images of a single condensate driven at $f_D$=183.2 Hz recorded within 1.1ms time interval. (b) Two-dimensional Fourier transform in space and time in logarithmic scale. It reveals a spatial period of $k_z=0.14 \rm{\mu m}^{-1}$ and a temporal period at half the driving frequency $f=91.6$ Hz. Adapted from Smits2020 under the terms of the https://creativecommons.org/licenses/by/4.0/.