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

TL;DR

This work addresses the soliton-to-droplet transition in dipolar Bose gases confined to quasi‑1D and quasi‑2D geometries, where long-range anisotropic interactions compete with short-range forces and quantum fluctuations. By combining an extended Gross-Pitaevskii equation, Bogoliubov–de Gennes analysis, and a variational model, the authors map out regions of smooth crossover and bistability, identifying a tricritical point in 1D and demonstrating persistent bistability in 2D. A key finding is that the breathing mode provides a clear experimental signature of the 1D transition via its structure-factor response, while in 2D the transition manifests through a quadrupole mode and anisotropic soliton stability; 2D dipolar bright solitons remain experimentally challenging yet potentially realizable. The results connect to prior non-dipolar and dipolar droplet studies and offer concrete guidelines for observing soliton-like states and bistability in ultracold dipolar gases, including in setups resembling 3D traps and quasi-2D planes.

Abstract

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.

Figures (6)

• Figure 1: Soliton-to-droplet transition in a dipolar gas. Peak density of the VM ground state as a function of the s-wave scattering length $a_s$ and the particle number $\mathcal{N}$ for (a1) a quasi-1D and (b1) a quasi-2D geometry. White region denotes bistability. The trapping frequencies are $\omega_x=\omega_y=2\pi\times55\rm Hz$ in (a) and $\omega_x=2\pi\times180\rm Hz$ in (b). (a2),(b2) Energy surfaces in the $\sigma_{r,y}, \sigma_z$-plane, where $\sigma_r$ is the width in the transverse directions ($x,y$), for three points marked in the phase diagram showing a soliton ground state (diamond), a bistability (circle) and a droplet ground state (square). In the bistable region, the droplet (a3),(b3) and the soliton (a4),(b4) are both solution of the eGPE. The isosurfaces of the particle density (orange) are taken at values of half of the maximum density.
• Figure 2: Soliton-to-droplet crossover in a quasi-1D system for $a_s=50a_0$ and $\omega_x=\omega_y=2\pi\times55\,\text{Hz}$. (a) Comparison between the width $\sigma_z$ of the ground state from the solution of the eGPE (solid line) and the one obtained from the VM (transparent line). The lines are colored with the peak density. (b) Energy of the ground state. (c) Excitations, where each mode frequency is colored by the maximum value of the structure factor (normed to unity). The inset shows the maximum value of the structure factor for the lowest-lying mode, the breathing mode.
• Figure 3: First-order phase transition in a quasi-1D system between a soliton and a droplet for $a_s=45a_0$ and $\omega_x=\omega_y=2\pi\times55\,\text{Hz}$. (a) The width $\sigma_z$ calculated from the solution of the eGPE (solid line) and the VM (transparent line). Metastable states are indicated by the dashed line. The color of the curves corresponds to the peak density. (b) Energy as a function of $\mathcal{N}$. (c) Excitation spectrum, where each mode frequency is colored according to its contribution to the structure factor.
• Figure 4: (a1) Critical atom number for 2D-solitons as a function of $\epsilon_{\rm dd}$ in a cylindrical symmetric system (black line) and for an anisotropic soliton (blue line) obtained from the VM. We define $\epsilon_{\rm dd}=a_{\rm dd}/a_s$ for the anisotropic/dipolar case and $\epsilon_{\rm dd}=a_{\rm dd}/2 a_s$ for the isotropic/anti-dipolar case. (a2) Aspect ratio $\sigma_z/\sigma_y$ as a function of the particle number $\mathcal{N}$ for different values of the $a_s$ for the anisotropic soliton obtained from the VM. (b) Width $\sigma_z$ of the ground state as a function of $\mathcal{N}$ for the solution of the eGPE (solid line) and the VM (transparent line) for $a=50a_0$ and $\omega_x=2\pi\times180\,\text{Hz}$. The color indicates the value of the peak density. (c) Energy as a function of $\mathcal{N}$. (d) Excitation frequencies as a function of $\mathcal{N}$ with their contribution to the structure factor indicated by color.
• Figure 5: First-order phase transition from a soliton to a droplet in an infinite plane potential. (a) Width $\sigma_z$ of the ground state as a function of $\mathcal{N}$, results from the solution of the eGPE and the VM are shown as a solid and transparent line, respectively. The color of each curve is given by the value of the peak density of the state. The dashed lines correspond to metastable states. (b) Energy as a function of $\mathcal{N}$. (c) Excitation frequencies, each colored by the maximum value of the structure factor.