The authors theoretically describe the stabilization of crystalline structures in interaction-driven Bose-Einstein condensates by analytically deriving a complex-valued Ginzburg-Landau equation for pattern formation. The resulting equation captures the competition between linear instability induced by the drive and nonlinear suppression induced by interactions and, in agreement with recent experiments, explains the emergence of square grid density patterns as stable states.
The authors theoretically describe the stabilization of crystalline structures in interaction-driven Bose-Einstein condensates by analytically deriving a complex-valued Ginzburg-Landau equation for pattern formation. The resulting equation captures the competition between linear instability induced by the drive and nonlinear suppression induced by interactions and, in agreement with recent experiments, explains the emergence of square grid density patterns as stable states.
The formation of patterns in driven systems has been studied extensively, and their emergence can be connected to a fine balance of instabilities and stabilization mechanisms. While the early phase of pattern formation can be understood on the basis of linear stability analyses, the longtime dynamics can only be described by accounting for the interactions between the excitations generated by the drive. Here, we observe the stabilization of square patterns in an interaction-driven, two-dimensional Bose-Einstein condensate. These patterns emerge due to inherent high-order processes that become relevant in the regime of large phonon occupations. Theoretically, this can be understood as the emergence of a stable fixed point of coupled nonlinear amplitude equations, which include phonon-phonon interactions. We experimentally probe the predicted flows toward such a stable fixed point, as well as repulsion from a saddle fixed point, using the experimental control unique to quantum gases.
Abstract We investigate spatiotemporal periodic patterns in harmonically trapped Bose–Einstein condensates (BECs) driven by a periodic modulation of the interaction. Resonant with the breathing mode, we show the emergence of a square lattice pattern containing two orthonormal stripes. We study the time evolutions of the lattice patterns for different driving strengths and dissipations. We find that its spatial periodicity and temporal oscillating frequency match the Bogoliubov dispersion, which is the intrinsic property of the system and relevant to the parametric amplification of elementary excitations. In the circumstances of strong driving strength and low dissipation, we further observe the triad interaction and the resulting superlattice state, which are well explained by the nonlinear amplitude equation for superimposed stripes. These results shed light on unexplored nonlinear spatiotemporal dynamics of two-dimensional patterns in harmonically trapped BECs that can pave the way for engineering exotic patterns by state-of-the-art experiments.
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.
We investigate the dynamical formation of nonlinear patterns in one-dimensional ring condensates under bichromatic periodic modulation of the interaction strength. The stability phase diagram of the condensate's homogeneous density state is analytically derived through a suitable biharmonic variant of the Mathieu equation and computing the associated Floquet spectrum. It reveals the complex interplay between the driving parameters, i.e., amplitude, frequencies, and the so-called frequencies' mixing angle, which dictate the instability onset and the selective enhancement of higher-order resonance tongues, thus offering precise control over the excited modes. These results are in agreement with time-dependent mean-field simulations evidencing the emergence of density wave modulations of specific momenta, while enabling a deeper understanding of the nonlinear stage of the relevant instability. Further insights on the ensuing unstable nonlinear dynamics are provided through a reduced {five-mode} model which captures the instability onset, the oscillatory behavior of the mode populations and the phase-space dynamics, in agreement with the mean-field predictions. Our study highlights the versatility of bichromatic driving to generate and control complex nonlinear patterns that are within reach in present day ultracold atom experiments.
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.
In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose–Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross–Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent structures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review.