Localization and splitting of a quantum droplet with a potential defect

TL;DR

We study 1D harmonically trapped quantum droplets under a central defect that acts as a barrier or well, using the extended Gross-Pitaevskii equation with Lee-Huang-Yang corrections. The work analyzes ground-state existence and stability via Bogoliubov-de Gennes theory and examines non-equilibrium responses to quenches that induce droplet fragmentation or localization, accompanied by self-evaporation. A semiclassical energy picture explains fragmentation dynamics, while a projection onto single-particle eigenstates reveals selective excitation pathways tied to quantum superposition. The results are relevant for ultracold-atom experiments and suggest controllable droplet dynamics via external potentials, with avenues for lattice, beyond-LHY, and higher-dimensional extensions.

Abstract

We unravel the existence and nonequilibrium response of one-dimensional harmonically trapped droplet configurations in the presence of a central potential barrier or well. For fixed negative chemical potentials, it is shown that droplets fragment into two for increasing potential barrier heights, a process that occurs faster for larger widths. However, atoms from the droplet accumulate at the potential well, especially for wider ones, leading to a deformed droplet and eventually to the termination of the solution. Linearization analysis yields the underlying excitation spectrum which dictates stability and the behavior of the ensuing collective modes. Quenches in the potential height are used to demonstrate dynamical fragmentation of the droplet for potential barriers as well as self-evaporation along with droplet localization and eventual relaxation for longer evolution times in the case of potential wells. The presence of selective excitation processes emanating from quantum superposition in the induced droplet dynamics is explicated by evaluating the contribution of the participating single-particle eigenstates. Our results should be detectable by current ultracold atom experiments and may inspire engineered droplet dynamics with the aid of external potentials.

Figures (8)

• Figure 1: Schematic illustration of the external potential described by Eq. (\ref{['potential']}) corresponding to (a) a double-well with $V_0>0$ and (b) a harmonic trap with a central dip when $V_0<0$. The single-particle energy levels are marked by $\epsilon_n$, with $n=0,1,2,\dots$, obtained through exact diagonalization. The shaded areas in panel (a) designate the energy doublets. The respective eigenstates are provided in panels (c) [(d)] for $V_0>0$ [$V_0<0$]. In both cases, the employed potential characteristics correspond to $V_0=\pm 1$ and $\sigma=0.5$. (e), (f) The dependence of the underlying energy gaps among consecutive energy levels, $\Delta \epsilon_n= \epsilon_{n}-\epsilon_{n-1}$, with $n=1,2,\dots$ as a function of the potential depth $V_0$ (see legend). Inset in panel (d) shows a magnification at the center of $\left| \varphi_4 \right|^2$ to clearly visualize the existent "kink" of even eigenstates. In the double-well, $\Delta \epsilon_n$ among doublets increase for larger $V_0$, while in the potential well case additional bound states (BS) appear for increasing $V_0$ (shaded regions).
• Figure 2: Ground-state density configurations of a harmonically trapped droplet in the presence of (a)-(c) a potential well ($V_0<0$) and (d)-(f) a barrier ($V_0>0$) with respect to $V_0$ and different widths $\sigma$ (see legends). It is evident that the droplet shrinks in size under the influence of a potential well and eventually disappears, a process that occurs faster for wider potentials. A potential barrier leads to two droplet segments forming faster especially for larger widths. The chemical potential of the droplet and the strength of the LHY interaction are held fixed (see legends). The insets in panels (c) and (f) show the droplet density distributions when $\mu=-0.1$ and $\delta=0.7$ for the potential well and barrier respectively. A reduced LHY strength facilitates the occurrence of the droplets for larger negative values of $V_0$, while splitting of the structure takes place at smaller positive $V_0$.
• Figure 3: (a) Real part of the BdG excitation spectra for the harmonically trapped droplet under the influence of an external potential well as a function of its height, $V_0$. The behavior of the collective excitation branches, labeled $\omega^{(n)}_R$ with $n=0,1,2,\ldots$, is shown. Selected imaginary versus real parts of the ensuing BdG droplet spectra in the case of (b)-(d) potential well and (e)-(g) a potential barrier for different potential characteristics (see legends). The presence (absence) of finite imaginary parts in the case of a potential barrier (well) reflects the unstable (stable) nature of the droplet configurations. (h), (i) Dynamical evolution of the perturbed stationary configurations at selected $V_0$ values of the potential barrier. In all cases, the chemical potential, $\mu=-0.2$, and the LHY strength is $\delta=1$ (see legends).
• Figure 4: Density evolution of the droplet after ramping-up the central potential barrier of height $V_0=1$ and width (a) $\sigma=0.05$, (d) $\sigma=0.5$. (b) [(e)] Selected density profiles taken from (a) [(d)] at specific time-instants (see legends), along with the droplet fits (dashes lines) obtained through Eq. (\ref{['droplet_solution']}). The quench tends to produce two counterpropagating droplet fragments. The latter move further apart and their separation $D(t)$ increases both by (c) varying $V_0$ and fixing the width (see legend), and (f) varying $\sigma$ while $V_0=1$. All other parameters are the same as in Fig. \ref{['fig:spectra']}.
• Figure 5: Dynamics of the droplet density following a quench of the central potential well from height $V_0=0$ to $V_0=-1$ for various widths (a) $\sigma=0.05$ and (c) $\sigma=0.5$. Relative small amplitude quenches trigger a collective motion of the droplet [panel (a)]. However, larger amplitudes lead to self-evaporation along with droplet excitation at short times which eventually decays tending to a quasi-steady state at long times [panel (c)]. (b) [(d)] Instantaneous densities taken from (a) [(c)]. Panels (b), (d) demonstrate also fits of the analytical droplet solution described by Eq. (\ref{['droplet_delta']}) and Eq. (\ref{['droplet_solution']}) respectively, to the eGPE obtained wave functions at specific time-instants (see legend). Adequate agreement occurs, excluding the tails of the distributions. The chemical potential used is $\mu=-0,2$ and the strength of the LHY $\delta=1$.