We investigate Faraday instabilities in three-dimensional parametrically driven trapped dipolar quantum gases within the framework of the extended Gross-Pitaevskii equation. In the superfluid regime, periodic modulation of the short-range interactions induces the resonant excitation of discrete polygonal surface modes displaying sub-harmonic response. The resonance frequencies and the parametric windows of instability are accurately captured by an appropriate Mathieu equation, further corroborating our simulations. It is also shown that dipolar interactions in superfluids shift the resonance frequencies to lower values compared to their non-dipolar counterparts. Near the superfluid-to-supersolid transition, intrinsic density undulations associated with the softened roton mode accelerate pattern formation, yielding hybrid surface and bulk excitations. The bulk patterns prevail deeper in the supersolid regime, emerging from droplet collisions with the superfluid background. Our results reveal a crossover from surface collective modes to hybrid surface-bulk excitations and demonstrate how parametric driving dictates pattern formation in long-range interacting quantum fluids.
We investigate Faraday instabilities in three-dimensional parametrically driven trapped dipolar quantum gases within the framework of the extended Gross-Pitaevskii equation. In the superfluid regime, periodic modulation of the short-range interactions induces the resonant excitation of discrete polygonal surface modes displaying sub-harmonic response. The resonance frequencies and the parametric windows of instability are accurately captured by an appropriate Mathieu equation, further corroborating our simulations. It is also shown that dipolar interactions in superfluids shift the resonance frequencies to lower values compared to their non-dipolar counterparts. Near the superfluid-to-supersolid transition, intrinsic density undulations associated with the softened roton mode accelerate pattern formation, yielding hybrid surface and bulk excitations. The bulk patterns prevail deeper in the supersolid regime, emerging from droplet collisions with the superfluid background. Our results reveal a crossover from surface collective modes to hybrid surface-bulk excitations and demonstrate how parametric driving dictates pattern formation in long-range interacting quantum fluids.
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.
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.
Faraday and resonant density waves emerge in Bose-Einstein condensates as a result of harmonic driving of the system. They represent nonlinear excitations and are generated due to the interaction-induced coupling of collective oscillation modes and the existence of parametric resonances. Using a mean-field variational and a full numerical approach, we studied density waves in dipolar condensates at zero temperature, where breaking of the symmetry due to anisotropy of the dipole-dipole interaction (DDI) plays an important role. We derived variational equations of motion for the dynamics of a driven dipolar system and identify the most unstable modes that correspond to the Faraday and resonant waves. Based on this, we derived the analytical expressions for spatial periods of both types of density waves as functions of the contact and the DDI strength. We compared the obtained variational results with the results of extensive numerical simulations that solve the dipolar Gross-Pitaevskii equation in 3D, and found a very good agreement.
We study Faraday patterns in quasi-one-dimensional dipolar Bose-Einstein condensates with parametrically driven dipolar interactions. We show that in the presence of a roton minimum in the excitation spectrum, the emergent Faraday waves differ substantially in two- and one-dimensional geometries, providing a clear example of the key role of confinement dimensionality in dipolar gases. Moreover, Faraday patterns constitute an excellent tool to study nonlocal effects in polar gases, as we illustrate with two parallel quasi-one-dimensional dipolar condensates. Nonlocal interactions between the condensates give rise to an excitation spectrum characterized by symmetric and antisymmetric modes, even in the absence of hopping. We show that this feature, absent in nondipolar gases, results in a critical driving frequency at which a marked transition occurs between correlated and anticorrelated Faraday patterns in the two condensates. Interestingly, at this critical frequency, the emergent Faraday pattern stems from a spontaneous symmetry-breaking mechanism.
Supersolidity, combining superfluid and crystalline orders, has been realized in dipolar Bose-Einstein condensates by tuning interatomic interactions. Here we show that supersolidity can also emerge from mode coupling at a superfluid-solid interface, without modifying bulk interactions and for a broad class of superfluids. Using an analytical and numerical treatment of the coupled superfluid and phonon fields, we derive the criterion for a density-modulation instability driven by interfacial coupling and dependent on dimensionality. In superfluid helium, the instability first appears at the roton mode, while in a Bose-Einstein condensate with contact interactions it occurs at the lowest accessible wave vector set by the system size. Beyond the threshold, the ground state acquires an interfacial density modulation while the bulk remains superfluid, forming a hybrid superfluid-supersolid phase. Our results identify interfacial mode coupling as a promising route to supersolidity, enabling the simultaneous exploitation of interfacial supersolid and bulk superfluid quantum properties.
We observe experimentally the spontaneous formation of star-shaped surface patterns in driven Bose-Einstein condensates. Two-dimensional star-shaped patterns with l-fold symmetry, ranging from quadrupole (l=2) to heptagon modes (l=7), are parametrically excited by modulating the scattering length near the Feshbach resonance. An effective Mathieu equation and Floquet analysis are utilized, relating the instability conditions to the dispersion of the surface modes in a trapped superfluid. Identifying the resonant frequencies of the patterns, we precisely measure the dispersion relation of the collective excitations. The oscillation amplitude of the surface excitations increases exponentially during the modulation. We find that only the l=6 mode is unstable due to its emergent coupling with the dipole motion of the cloud. Our experimental results are in excellent agreement with the mean-field framework. Our work opens a new pathway for generating higher-lying collective excitations with applications, such as the probing of exotic properties of quantum fluids and providing a generation mechanism of quantum turbulence.