Soliton-To-Droplet Crossover in Dipolar Bose Gas Reveals Bistability in One Dimension

Researchers are increasingly interested in understanding how quantum systems transition between different states of matter, and a new study details the crossover between soliton and droplet formation in dipolar Bose gases. Malte Schubert, Thomas Bland, and Manfred J Mark, from Lund University and the Institut f ur Experimentalphysik at the Universit at Innsbruck, alongside colleagues, demonstrate how these transitions occur in both one and two-dimensional systems. Their work is significant because it identifies experimentally accessible methods, such as monitoring the breathing mode, to observe this crossover and pinpoints the conditions necessary to create stable, two-dimensional bright solitons, furthering our understanding of quantum many-body physics and potentially enabling new avenues for quantum simulation.

Soliton-droplet transitions in dipolar gases via variational and numerical analysis reveal complex dynamics

Scientists have demonstrated the existence of both stable solitons and quantum droplets within a system of dipolar atoms confined to quasi-low-dimensional geometries. This research, focused on the transition between these two states, reveals that this shift can manifest either as a first-order phase transition, leading to bistability, or as a smooth crossover, depending on interaction strengths.
The team achieved a detailed understanding of this transition by calculating the structure factor and identifying that the response of the breathing mode serves as an experimentally accessible probe. This study investigates a system of dipolar atoms, where long-range and anisotropic interactions play a crucial role in stabilizing both solitons and droplets.

Researchers employed a variational model, alongside numerical calculations of the extended Gross-Pitaevskii equation, to map out the phase diagram for quasi-one-dimensional and quasi-two-dimensional setups. The analysis revealed regions of both bistability and smooth crossover, mirroring earlier findings from studies of Bose mixtures, but uniquely observed within a dipolar Bose-Einstein condensate.

Figures presented in the work display phase diagrams for both quasi-1D and quasi-2D configurations, with colour indicating the maximum density of the ground state. White regions within these diagrams denote bistable areas where two distinct minima exist in the energy landscape, corresponding to droplet and soliton states.

The research establishes that the soliton-to-droplet transition is reflected in the structure factor, offering a potential avenue for experimental verification of the transition. Experiments show that the transition, including the observed bistability, also persists in a quasi-2D trap, a finding unique to dipolar BECs.

By carefully tuning interactions, the team delineated the conditions under which two-dimensional dipolar bright solitons can be realized, addressing a long-standing goal in the field. This work opens possibilities for exploring the emergence of dipolar supersolids and provides a pathway towards achieving quasi-2D solitons experimentally, furthering our understanding of self-binding phenomena in quantum gases.

Modelling Soliton-Droplet Transitions via Extended Gross-Pitaevskii Equation Simulations reveals key dynamical features

Scientists employed a multifaceted theoretical and numerical approach to investigate the behaviour of dipolar atoms in quasi-low-dimensional geometries, specifically examining the transition between stable solitons and droplets. The study began with the extended Gross-Pitaevskii equation (eGPE), a beyond mean-field description valid for weakly interacting bosons at zero temperature, to model the system’s wavefunction Ψ(r, t).

This equation, incorporating contact interactions, a dipole-dipole potential Vdd(r) = gdd(1−3 cos2 θ)/|r|3, and a fifth-order quantum fluctuation term, was central to their calculations. The coefficient gdd, determining the strength of dipolar interactions, was set to 3ħ2add/m, with add = 66.5a0 for 166Er, and the angle θ defined the orientation of the dipoles.

Researchers solved the eGPE in imaginary time to compute the ground state Ψ0, utilising Fourier space to efficiently calculate the convolution term and mitigating aliasing with transverse and z-direction cut-offs, ρc and Zc, respectively, exploiting the cylindrical symmetry of the wavefunction. A semi-analytical expression for the dipolar potential in momentum space, detailed in Appendix A, further enhanced computational efficiency.

To probe collective excitations, the team linearised around the ground state, introducing perturbations to Ψ(r, t) and deriving the Bogoliubov, de Gennes (BdG) equations. Diagonalising these equations using standard eigenvalue solvers revealed the stable excitations and their corresponding energies, determined by the normalization condition ħωj = ∫ d3rfj(H −μ + 2X)fj.

The study also pioneered a variational model (VM) to directly minimise the energy functional, substituting an ansatz Ψ(r) = √Nφ(x, y)ψ(z) into the eGPE. This ansatz, comprising Gaussian functions for transverse and axial dimensions, incorporated variational parameters like widths σj and exponents rρ, rz to capture both soliton and droplet shapes.

By minimising the total energy with respect to these parameters, the ground state was determined, allowing for radial asymmetry and capturing flat-top profiles characteristic of droplets at high particle numbers. Numerical results for an infinite tube and plane potential were then compared with VM predictions, providing insights into collective excitations and the soliton-to-droplet transition, particularly around a predicted tricritical point at (NT, as,T) ≈ (9 × 103, 50a0).

Soliton-droplet transitions and bistability in quasi-low-dimensional dipolar gases

Scientists investigated a system of dipolar atoms confined to quasi-low-dimensional geometries, revealing the existence of both stable solitons and droplets within the system. The research focused on the transition between these states, identifying whether it manifests as a first-order phase transition with bistability or a smooth crossover.

Calculations of the structure factor demonstrated that the response of the breathing mode serves as an experimentally accessible probe for this transition. Experiments revealed regions of both bistability and smooth crossover in quasi-two-dimensional geometries, expanding understanding of these states beyond one-dimensional systems.

The team measured the peak density of the variational model ground state as a function of the s-wave scattering length and particle number, mapping out the phase diagram. White regions on the diagrams denote bistability, where both soliton and droplet states coexist as stationary solutions. Data shows that the energy surfaces in the σr,y, σz-plane exhibit distinct characteristics depending on the ground state, with unique profiles for soliton, bistable, and droplet states.

Specifically, the analysis of the extended Gross-Pitaevskii equation ground state confirmed the presence of a crossover transition and a bistable region in the quasi-1D setup, mirroring findings from Bose mixtures. Measurements confirm that this soliton-to-droplet transition also exists in a quasi-2D trap, a unique finding for dipolar Bose-Einstein condensates.

The team demonstrated that the transition, including the bistability, is reflected in the structure factor, providing a means to probe the transition experimentally. Figures presented display phase diagrams for both quasi-1D and quasi-2D setups, with colour indicating maximum ground state density.

Soliton-droplet transitions and bistability in dipolar atomic condensates are actively investigated

Scientists have investigated the transition between solitons and droplets in systems of dipolar atoms confined to quasi-one- and quasi-two-dimensional traps. Their calculations demonstrate the existence of both a bistable region, where soliton and droplet states can coexist, and a smooth crossover between these states, dependent on particle number and interaction strength.

Variational methods and imaginary time propagation were employed to model these transitions, with the variational method proving most accurate within the soliton and droplet regimes but less reliable near the transition point. This research connects previous work on transitions in non-dipolar Bose, Bose mixtures with established studies of dipolar bright solitons and droplets.

In quasi-one-dimensional systems, a maximum response in the structure factor was identified for the breathing mode, offering a potential experimental method for detecting the transition. Furthermore, the study revealed that bistability is not limited to quasi-one-dimensional traps but can also occur in quasi-two-dimensional geometries, accompanied by the emergence of an additional quadrupole mode.

The authors acknowledge that while evidence suggests a similar transition may have been observed in prior experiments with 164Dy atoms, further investigation is needed to definitively confirm the existence of genuine solitons in quasi-two-dimensional traps. The authors note limitations in the accuracy of the variational method near the transition region.

Future research could focus on refining these calculations or exploring the system’s behaviour with different experimental parameters to fully characterise the observed phenomena and potentially realise stable two-dimensional dipolar bright solitons. This work contributes to a better understanding of dipolar quantum systems and offers insights into controlling their properties for potential applications in quantum technologies.