Generation of dipolar supersolids through a barrier sweep in droplet lattices

Abstract: We propose a dynamical protocol to generate supersolids in dipolar quantum gases by sweeping a repulsive Gaussian barrier through an incoherent quasi-one-dimensional droplet array. Supersolidity is inferred by monitoring the ensuing dynamics of the density, momentum distribution, center-of-mass motion, and superfluid fraction within the framework of the extended Gross-Pitaevskii equation with quantum corrections. A persistent superfluid background arises, atop which the crystals oscillate in unison, indicating the establishment of phase coherence. This process is accompanied by energy redistribution and the gradual transfer of higher-lying momenta toward the zero momentum mode. The dependence of the superfluid fraction on the barrier velocity and height is also elucidated evincing the parametric regions which facilitate the rise of a superfluid background. Our results pave the way for engineering supersolid generation using experimentally accessible protocols.

• The paper demonstrates that a moving Gaussian barrier effectively triggers supersolidity in dipolar Bose gases by inducing inelastic collisions and phase coherence.
• It employs the extended Gross-Pitaevskii equation with Lee-Huang-Yang corrections to simulate dynamical stages, including collision-induced tunneling and collective oscillations.
• Results reveal a tunable parameter regime where optimal barrier velocity and height maximize supersolid nucleation, even under realistic three-body loss conditions.

Dynamical Generation of Dipolar Supersolids via Barrier Sweep in Droplet Lattices

Introduction and Theoretical Context

This work examines a nonequilibrium protocol for generating supersolid states in dipolar Bose gases by implementing a moving Gaussian barrier through an initially incoherent droplet lattice. Supersolidity in dipolar quantum gases manifests as coexisting superfluid and crystalline order—arising from the interplay between contact and dipole-dipole interactions, stabilized by quantum fluctuations described via the Lee-Huang-Yang correction. The study is grounded in the extended Gross-Pitaevskii equation (eGPE), which incorporates both contact and dipolar interactions as well as beyond-mean-field quantum corrections, and focuses on the experimentally relevant regime of highly magnetic $^{164}$Dy atoms.

Prior approaches for supersolid generation typically involve interaction quenches or sympathetic mechanisms such as heating, mixture composition, and the introduction of impurities. Here, a direct dynamical symmetry breaking is realized by externally perturbing an incoherent droplet array using a controlled barrier sweep. The resulting pathway towards supersolidity is distinct from both adiabatic and quench-based protocols, enabling experimental control via well-established optical and trapping techniques.

Protocol and Dynamical Response

The protocol consists of initializing a quasi-1D array of laser-polarized $^{164}$Dy droplets well within the droplet regime ($\epsilon_{\rm dd} > 1$) and subsequently applying a repulsive Gaussian barrier, which is swept along the elongated axis at a tunable velocity and amplitude. The eGPE formalism is solved numerically in three dimensions, allowing explicit simulation of density, phase, and energy redistribution throughout the dynamical process.

Three distinct dynamical stages are identified:

1. Pre-collision regime: The barrier is distant from the droplet array; droplets are stationary.
2. Collision and excitation regime: The barrier interacts with the outermost droplet, generating a cascade of inelastic droplet–droplet collisions mediated by both direct impact and self-evaporation. These interactions induce particle tunneling between droplets and lead to atom emission, analogously reminiscent of complex collisional dynamics observed in strongly correlated many-body systems.
3. Supersolid nucleation regime: At long times after the barrier has traversed the array, a persistent, coherent superfluid background forms, coexisting with crystalline density peaks. The droplets oscillate collectively in-phase atop this newly established coherent substrate. This dynamical phase exhibits clear signatures of increased long-range phase coherence and mode softening.

Diagnostics of Supersolidity

Several complementary observables confirm the onset of supersolidity:

• Momentum distribution: Post-collision, there is a marked transfer of occupation from higher-momentum sidebands to a dominant zero-momentum peak, evidencing global phase coherence and the nucleation of a superfluid background.
• Center-of-mass motion and collective excitations: The system exhibits low-frequency, in-phase oscillations with frequencies substantially below the bare trap frequency, characteristic of collective (i.e., Goldstone-type) modes unique to the supersolid regime. The decay of these oscillations and their relatively low frequency further delineate this phase from conventional BEC or isolated droplet regimes.
• Superfluid fraction ($f_s^u$): Employed as an order parameter, the time-dependent superfluid fraction grows significantly following the barrier sweep, attaining values commensurate with ground-state supersolids in analogous geometries. There exists a nontrivial parametric window of barrier velocities and heights which maximize supersolid formation; both the too-adiabatic (very slow) and diabatic (very fast) regimes suppress or eliminate supersolid nucleation.

These observations are robust with respect to system size (varying droplet numbers), and, crucially, supersolid nucleation persists even with realistic three-body loss rates. Losses naturally reduce atom number and inhibit long-term coherence, but do not prevent the fundamental dynamical mechanism from operating within experimental timescales.

Parameter Dependence and Numerical Results

The numerical results highlight several significant findings:

• Velocity and height window: Supersolidity is most efficiently nucleated when the barrier's kinetic energy is comparable to the droplet array's chemical potential, ensuring strong but non-destructive perturbations. For barrier velocities $v_0 \gtrsim 6\,\mu$m/ms, the system is largely unperturbed; for intermediate velocities ($v_0\sim v_g$), maximal supersolid fractions are achieved.
• Energy redistribution: The chemical potential increases during droplet-barrier collisions and asymptotically settles into the energy window associated with supersolid ground states (but above the initial droplets due to the non-adiabaticity).
• Crystalline structure persistence: Despite strong energy injection and atom redistribution, the modulated density profile is preserved, supporting long-lived in-sync oscillations, a hallmark of the supersolid regime.

Specific numerical details, such as grid size, convergence properties, and the use of split-step Crank-Nicolson methods, ensure the accuracy and reproducibility of the simulation results.

Implications and Future Directions

This work demonstrates that external, experimentally accessible time-dependent perturbations can catalyze the emergence of supersolidity in dipolar quantum gases, circumventing the need for tunable interactions or multimodal quantum engineering. The findings have clear implications for the controllable formation and dynamics of nonequilibrium supersolids:

• Experimental realization: The protocol employs standard optical elements (focused lasers as barriers) and can be immediately implemented with magnetic lanthanides in elongated geometries.
• Dynamical phase engineering: Beyond ground-state phase diagrams, the protocol opens avenues for studying nonequilibrium phase transitions, defect formation, and collective excitation spectra associated with supersolid order.
• Higher-dimensional generalizations: The method can be extended to 2D and 3D geometries, where richer lattice structures and additional collective modes are available. Sweeping or rotating barriers may lead to vortex nucleation, turbulence, and complex pattern formation.
• Connection to quantum turbulence and decoherence: By adjusting protocol parameters or introducing additional disorder, the relationship between supersolid order, phase coherence, and thermalization kinetics can be systematically explored.

Conclusion

The controlled barrier sweep protocol provides a versatile route for the real-time dynamical generation of dipolar supersolids in ultracold gases. Theoretical analysis via the eGPE augmented with quantum fluctuations establishes the microscopic mechanism: inter-droplet inelastic collisions mediated by the barrier result in atom transfer, symmetry breaking, and phase coherence. Supersolidity emerges robustly for a range of parameters, as verified by momentum-space, real-space, and collective dynamic diagnostics. The approach enables new experimental and theoretical studies of nonequilibrium phases of matter, quantum phase transitions, and collective quantum dynamics in strongly interacting dipolar systems.