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Physical Review A

One-dimensional quantum droplets under a linear gravitational-like trap

Saurab Das, Jayanta Bera, Ajay Nath

Published Dec 3, 2025• 0 citations
Publisher

Abstract

We investigate the influence of a constant and time-dependent linear gravitational-like potential on one-dimensional quantum droplets, governed by an extended Gross-Pitaevskii equation incorporating a repulsive cubic effective mean-field (EMF) term and an attractive quadratic beyond-mean-field (BMF) correction. By constructing a tailored external confinement, we derive an exact analytical family of droplet wave functions and explicitly characterize the effective interaction contributions. A key result is that the droplet's c.m. follows a strictly Newtonian equation of the form ${\stackrel{\ifmmode \ddot{}\else \"{}\fi{}}{X}}_{\mathrm{c}.\mathrm{m}.}(t)=\ensuremath{-}a(t)$, demonstrating that its falling trajectory depends solely on the applied linear potential. Notably, this motion is completely independent of atom number and of the EMF-BMF balance, providing a clean classical baseline for dynamics in linear-potential environments. Temporal modulation of the potential induces clear deviations from static trajectories and leaves measurable imprints on internal coherence, captured through the Shannon entropy and the Wigner phase-space distribution, with the modulation strength controlling phase-space localization, deformation, and coherence degradation. Numerical simulations substantiate the stability of the analytical solutions, demonstrating their robustness. These findings suggest promising implications for quantum sensing and metrological applications using ultradilute quantum fluids.

DOI
10.1103/mxq6-zc3f

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We investigate the influence of a constant and time-dependent linear gravitational-like potential on one-dimensional quantum droplets, governed by an extended Gross-Pitaevskii equation incorporating a repulsive cubic effective mean-field (EMF) term and an attractive quadratic beyond-mean-field (BMF) correction. By constructing a tailored external confinement, we derive an exact analytical family of droplet wave functions and explicitly characterize the effective interaction contributions. A key result is that the droplet's c.m. follows a strictly Newtonian equation of the form X\ifmmode¨\else¨\fic.m.(t)=\ensuremath−a(t){\stackrel{\ifmmode \ddot{}\else \"{}\fi{}}{X}}_{\mathrm{c}.\mathrm{m}.}(t)=\ensuremath{-}a(t)X\ifmmode¨\else¨\fi​c.m.​(t)=\ensuremath−a(t), demonstrating that its falling trajectory depends solely on the applied linear potential. Notably, this motion is completely independent of atom number and of the EMF-BMF balance, providing a clean classical baseline for dynamics in linear-potential environments. Temporal modulation of the potential induces clear deviations from static trajectories and leaves measurable imprints on internal coherence, captured through the Shannon entropy and the Wigner phase-space distribution, with the modulation strength controlling phase-space localization, deformation, and coherence degradation. Numerical simulations substantiate the stability of the analytical solutions, demonstrating their robustness. These findings suggest promising implications for quantum sensing and metrological applications using ultradilute quantum fluids.
We investigate the dynamics of harmonically confined quantum droplets in a binary Bose Einstein condensate within the one dimensional extended Gross Pitaevskii framework, including LHY corrections, under a constant linear (gravitational like) potential. By analyzing the COM and width dynamics, we show that the monopole (breathing) mode remains governed by the harmonic confinement, with a frequency asymptotically insensitive to the linear perturbation, demonstrating the robustness of internal collective excitations against uniform external forcing. In contrast, the COM exhibits interaction-dependent transport, with weak confinement producing large susceptibility and rapid displacement, whereas strong confinement suppresses transport even under large forcing. The COM response decreases monotonically with increasing trap frequency. We further characterize the evolving quantum state through the quantum Fisher information and Wigner quasi-probability distributions, showing that the linear potential enables controlled generation of states with enhanced metrological sensitivity over finite times, while stronger confinement shifts the onset of the high-sensitivity regime to larger forcing strengths. Numerical simulations based on the split-step Fourier method confirm the dynamical stability of the obtained solutions. These results illustrate the impact of linear gravitational like trap on the collective excitations, transport, and quantum metrological properties of 1D ultradilute quantum fluids.
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Abstract We explore the dynamical behavior of one-dimensional quantum droplets (QDs) governed by the extended Gross–Pitaevskii equation, under harmonic confinement supplemented by a static or time-dependent Gaussian spike (Gs) potential. Employing both variational analytical techniques and numerical simulations, we investigate the evolution of the root-mean-square (RMS) size, excitation spectrum, and phase-space dynamics of QDs. Our study reveals that while the harmonic trap sets the primary confinement, the Gs potential enables precise frequency tuning and control over droplet oscillations. A static Gs amplitude modifies the fundamental oscillation frequency depending on its sign, while a time-modulated Gs induces nonlinear dynamics, including higher harmonics and frequency mixing. Our analysis reveals that the resulting frequency spectrum is strongly influenced by inter- and intra-species interactions as well as by the parameters of the external trap. Notably, we establish a relationship between the frequency shift and the amplitude of the Gaussian spike. Wigner phase-space analysis further uncovers coherent rotational behavior, offering insights into hidden phase dynamics not apparent in real-space density profiles. These results highlight the role of time-dependent external potentials in tailoring the excitation landscape of quantum droplets, with implications for quantum simulation, precision control in atomtronics, and probing coherence in low-dimensional quantum fluids.
The nonlinear dynamics of quantum droplets in a quasi-one-dimensional BEC loaded in a two-core trap potential are investigated. The cases of standing and moving quantum droplets are considered. The frequencies of the Josephson oscillations and the condition for self-trapping are found. An analytical description is based on the variational approach to a system of coupled extended Gross-Pitaevskii (EGP) equations, including Lee-Huang-Yang correction terms. In the case of moving droplets, the system of equations describing the coupling of the relative motion of droplets in two cores with the dynamics of the relative phase and population imbalance (Josephson currents) is derived. Different regimes of droplet dynamics are found, including attraction and repulsion in dependence on the relative phase evolution. These phenomena are the realization of the Andreev-Bashkin effect for superfluids in a two-core trap potential. Analytical predictions of the model are confirmed by full numerical simulations of the coupled extended Gross-Pitaevskii equations.
We investigate the existence and dynamics of two-dimensional solitary waves in a quantum droplet environment described by the extended Gross-Pitaevskii equation featuring logarithmic mean-field and Lee-Huang-Yang interactions. In the modulationally stable regime of the background, we employ suitable multiscale asymptotic methods to derive effective nonlinear integrable models corresponding to the Kadomtsev-Petviashvili and Davey-Stewartson equations. Based on these reduced models, we construct approximate analytical solutions describing line solitons, algebraically localized lump solitons, ring solitons, and exponentially localized dromions embedded on the droplet background. The dynamical robustness of these solutions is monitored through numerical simulations. Line, lump and ring solitons stay closest to the theoretical predictions, although progressively deviate due to the emergence of small-amplitude radiation, while dromions depart from their analytical waveform the most, although they roughly maintain their shape. Our results unveil unprecedented multidimensional soliton solutions in models featuring the competition of mean-field and quantum fluctuations and as such are amenable to current ultracold atom experiments.
Recent observations of droplets in dipolar and binary Bose-Einstein condensate (BEC) motivates us to study the theory of droplet formation in details. Precisely, we are interested in investigating the possibility of droplet formation in a quasi one dimensional geometry. The recent observation have concluded that the droplets are stabilized by the competition between effective mean-filed and beyond mean-field interaction. Hence, it is possible to map the effective equation of motion to a cubic-quartic nonlinear Schr\odinger equation (CQNLSE). We obtain two analytical solutions of the modified Gross-Pitaevskii equation or CQNLSE and based on their stability we investigate the parameter regime for which droplets can form. The effective potential allows us to conclude about the regions of soliton domination and self bound droplet formations.