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arXiv:2504.18220

Quantum droplets in a beyond-mean-field density-dependent gauge theory

Matthew Edmonds, Patrik Öhberg

Published Apr 25, 2025• 0 citations
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Abstract

The beyond-mean-field corrections appropriate to a bosonic many-body system experiencing a density-dependent gauge potential are derived, and from this the dimensional hierarchy of quantum droplet solutions are explored. Non-stationary quantum droplet solutions are supported by a single interaction parameter characterising the strength of the gauge potential, while in one dimension the beyond-mean-field theory can be solved exactly to yield chiral quantum droplets and dark soliton-like excitations. Numerical simulations of single and pairs of chiral droplets indicate a rich dynamics in the beyond-mean-field regime.

DOI
10.48550/arxiv.2504.18220

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The beyond-mean-field corrections appropriate to a bosonic many-body system experiencing a density-dependent gauge potential are derived, and from this the dimensional hierarchy of quantum droplet solutions are explored. Non-stationary quantum droplet solutions are supported by a single interaction parameter characterising the strength of the gauge potential, while in one dimension the beyond-mean-field theory can be solved exactly to yield chiral quantum droplets and dark soliton-like excitations. Numerical simulations of single and pairs of chiral droplets indicate a rich dynamics in the beyond-mean-field regime.
Quantum liquidlike states of matter have been realized in an ongoing series of experiments with ultracold Bose gases. Using a combination of analytical and numerical methods we identify the specific criteria for the existence of dark solitons in beyond-mean-field binary condensates, revealing how these excitations exist for both repulsive and attractive interactions, the latter leading to dark quantum droplets with properties intermediate between a dark soliton and a quantum droplet. The phenomenology of these excitations is explored within the full parameter space of the model, revealing a spatial profile of the excitation that differs significantly from the Zakharov-Shabat soliton, leading to a negative effective mass that is enhanced in the presence of the quantum fluctuations. Finally the dynamics of pairs of the excitations are explored, showing nonintegrable dynamics and dark soliton bound states in the attractive regime.
We look into dark solitons in a quasi-1D dipolar Bose gas and in a quantum droplet. We derive the analytical solitonic solution of a Gross-Pitaevskii-like equation accounting for beyond mean-field effects. The results show there is a certain critical value of the dipolar interactions, for which the width of a motionless soliton diverges. Moreover, there is a peculiar solution of the motionless soliton with a nonzero density minimum. We also present the energy spectrum of these solitons with an additional excitation subbranch appearing. Finally, we perform a series of numerical experiments revealing the coexistence of a dark soliton inside a quantum droplet.
Recent theoretical and experimental results show that one-dimensional (1D) weakly interacting atomic Bose-Bose mixtures with repulsive interspecies mean field (MF) interaction are stabilized by attractive quadratic beyond-mean-field (BMF) effects into self-bound quantum droplet (QD) in free space. Here, we construct an exact analytical model to investigate the structure and dynamics of QDs in presence of external harmonic confinement by solving the 1D extended Gross-Pitäevskii equation (eGPE) with temporal variation of MF and BMF interactions. The model provides the analytical form of wavefunction, phase, MF and BMF nonlinearities. The generation of QDs and interesting droplet to soliton transition in presence of regular/expulsive parabolic traps by taking the comparable MF and BMF interactions are illustrated. We derive the phase diagram of the droplet-soliton phase transition between amplitude of MF, BMF interactions and harmonic oscillator frequency. The strength and form of oscillator frequency are identified as key parameter for tuning the compression, fragmentation and transport of droplets. Finally, the stability of the obtained solutions are confirmed from Vakhitov-Kolokolov (VK) criterion and are found stable.
Abstract Solitonic excitations of the one-dimensional quantum droplets are obtained, which smoothly connect vacuum with the flat-top droplet, akin to compactons in classical liquids. These solitons are of the kink type, necessarily residing on a constant pedestal, determined by the mean-field (MF) repulsion and beyond mean field quantum correction and having exactly one-third of the uniform condensate amplitude. Akin to the kinks, the propagating modes occur in pairs and are phase-locked with the background. The lowest chemical potential and condensate amplitude at the flat-top boundary matches with the self-trapped quantum droplet. More general excitations of analogous kind are obtained through the Möbius transform, which connect the required solutions to elliptic functions in general.
We study spin and density excitations of one-dimensional self-bound Bose-Bose droplets within Bogoliubov theory, and show that spin excitations come alive, especially as the interspecies coupling is made less attractive. We argue that spin excitations are particularly relevant in the one-dimensional droplet regime, where droplets are realized within the mean-field stability regime, as has been confirmed by the quantum Monte Carlo simulations. As the interspecies coupling strength increases within the mean-field stability regime, spin modes ultimately fall below the particle-emission threshold, thus becoming observable in the droplet spectrum. We analyze the Bogoliubov model for both pseudospinor and population-imbalanced scalar mixtures, encompassing both the density and spin sectors. We corroborate our findings through variational analysis of density- and spin-breathing modes, which offers physical insight into the mode structure and independently validates the spectrum, as well as through real-time dynamics. Additionally, we compare our results with both Petrov's original theory, which considers the Lee-Huang-Yang (LHY) correction at the attractive edge of the mean-field stability regime, and a beyond-LHY description of Bose-Bose mixtures.
Abstract A one-dimensional bosonic gas with strong contact repulsion and attractive non-local interactions may form a quantum droplet with a flat-top density profile. We focus on a system in the Tonks–Girardeau limit of infinitely strong contact repulsion. We show that the main system features are the same for a broad class of non-local interaction potentials. Then, we focus on a limiting case, the one of slowly varying density profiles, to find approximate formulas for the surface and bulk energies of a droplet. We further characterise the system by numerically finding the excitation spectrum. It consists of two families: phononic-like excitations inside droplets and scattering modes. Analysis within the linearised regime is supplemented with the full, nonlinear dynamics of small perturbations.