Soliton-to-droplet crossover in a dipolar Bose gas in one and two dimensions

We analyze a system of dipolar atoms confined in geometries of quasi-low-dimensionality. Due to the long-range and anisotropic nature of dipolar interactions, the system supports both stable solitons and quantum droplets. In quasi-one-dimensional geometries, the transition between these states is known to manifest either as a first-order phase transition, associated with bistability, or as a smooth crossover. We investigate this transition by calculating the structure factor and showing that the response of the breathing mode provides an experimentally accessible probe. In addition, we identify regions of both bistability and smooth crossover in quasi-two-dimensional geometries. Finally, we connect our findings to previous experimental results and delineate the conditions under which two-dimensional dipolar bright solitons can be realized.

💡 Research Summary

This paper investigates self‑bound states in a dipolar Bose‑Einstein condensate of 166Er atoms confined to quasi‑one‑dimensional (quasi‑1D) tube and quasi‑two‑dimensional (quasi‑2D) plane geometries. Because dipolar interactions are long‑range and anisotropic, they compete with short‑range contact interactions and with the Lee‑Huang‑Yang (LHY) quantum‑fluctuation term. The resulting extended Gross‑Pitaevskii equation (eGPE) contains kinetic energy, external harmonic confinement, contact nonlinearity (∝as), dipolar nonlinearity (∝add), and a beyond‑mean‑field term (∝γQF). The authors solve the eGPE numerically in imaginary time to obtain ground‑state wavefunctions, and they linearise the eGPE to derive Bogoliubov‑de Gennes (BdG) equations for collective excitations. In parallel, a variational model (VM) is introduced, employing a separable ansatz with Gaussian‑type transverse profiles and an exponential axial profile, characterised by widths σx, σy, σz and shape exponents rρ, rz. Minimising the total energy yields analytic insight into the energy landscape, allowing rapid identification of phase boundaries.

In the quasi‑1D case the VM predicts a tricritical point at (NT≈9×10³ atoms, as,T≈50 a0). For scattering lengths below this value (as<as,T) the energy functional exhibits two distinct minima: one corresponding to a bright soliton (high peak density, decreasing axial width with particle number) and another to a quantum droplet (saturated peak density, increasing axial width with particle number). This bistable region signals a first‑order phase transition. For as>as,T the two minima merge and the system undergoes a smooth crossover: the axial width varies continuously and the maximum density rises sharply only near the crossover. The authors compute the structure factor S(k,δn) for each BdG mode and find that the breathing mode (axial width oscillation) displays a pronounced peak in S at the crossover, providing a clear experimental signature accessible via Bragg spectroscopy or modulation of the interaction strength. Numerical eGPE results confirm the VM predictions but locate the crossover at slightly lower atom numbers, indicating that the VM slightly overestimates the critical N.

The paper then explores the bistable regime by fixing as=45 a0. Imaginary‑time propagation with carefully chosen initial states reveals both soliton and droplet solutions coexisting for N≈3×10³. Energy comparison shows the droplet to be the true ground state while the soliton is metastable, confirming the first‑order nature of the transition.

Extending the analysis to a quasi‑2D geometry (confinement in a single transverse direction) yields qualitatively similar behaviour. The anisotropic dipolar interaction permits self‑binding even when only one direction is tightly confined, leading to a broad bistable region in the (N,as) plane that is absent in non‑dipolar mixtures. The VM again predicts a tricritical point, and the eGPE confirms both first‑order and continuous transitions depending on as. Importantly, the authors identify concrete parameter windows for realizing bright dipolar solitons in 2D: (i) as must be slightly smaller than the dipolar length add (≈66.5 a0 for 166Er), (ii) the transverse trap frequency should be modest (≈2π×180 Hz), and (iii) atom numbers of a few thousand to ten thousand are required. Under these conditions the breathing‑mode structure factor exhibits a clear maximum, offering a direct probe of the soliton‑to‑droplet crossover in experiments.

Finally, the authors compare their theoretical phase diagrams with recent experimental observations of soliton‑droplet transitions in non‑dipolar mixtures (e.g., Cheiney et al.). They argue that the dipolar system provides a cleaner, single‑component platform where the same physics appears but with the added flexibility of tuning anisotropy via the dipole orientation. The paper concludes that dipolar Bose gases are uniquely suited for studying self‑binding phenomena, that the breathing‑mode response is a robust experimental diagnostic, and that the identified 2D parameter regime opens the path toward the long‑sought realization of stable, bright dipolar solitons in two dimensions.