One-dimensional asymmetrically interacting quantum droplets in Bose-Bose mixtures

TL;DR

The paper addresses the formation and dynamics of one-dimensional quantum droplets in asymmetric Bose–Bose mixtures, where intraspecies interactions are unequal. It combines an extended GPE framework with a variational approach and sum-rule analysis to characterize ground-state properties and four collective modes, across a range of asymmetry $\lambda$ and atom number $N$. A key finding is the Gaussian-to-flat-top density transition driven by $\lambda$, with three critical atom numbers $N_{c1}$, $N_{c2}$, and $N_{c3}$ delineating distinct regimes, and a rich spectrum of modes including dipole, breathing, spin-dipole, and spin-breathing that respond differently to $\lambda$, $N$, and trap strength $\kappa$. The results provide detailed, experimentally relevant insights for ultracold $^{39}$K systems and establish a robust framework for exploring beyond-mean-field quantum droplets in low dimensions.

Abstract

We theoretically investigate ground-state properties and collective excitations of one-dimensional quantum droplets in asymmetric Bose-Bose mixtures with unequal intraspin interactions. Using the extended Gross-Pitaevskii equation supported by variational and sum-rule methods, we show that the intraspin interaction ratio substantially alters the droplet's density profile, driving a transition from Gaussian-like to flat-top shapes. By examining two experimentally relevant parameter regions, we analyze density profiles, radii, peak densities, and excitation spectra to distinguish quantum phases and to depict phase diagrams in the space of asymmetric interaction ratio and total atom number. We carefully study the frequencies of both well-known dipole and breathing modes and less-explored spin dipole and spin breathing modes. The breathing mode frequency decreases monotonically with interaction ratio, approaching asymptotically the result of a conventional weakly interacting Bose gas. It varies non-monotonically with total atom number, peaking at a critical point that highlights the crucial role of quantum fluctuations. In contrast, spin modes display distinct temporal spin density distributions and reveal in-phase and out-of-phase relative dynamics between components. Their frequencies depend instead monotonically on the interaction ratio and atom number. Our results provide a comprehensive understanding of asymmetric quantum droplets and link to experimentally accessible regimes in ultracold $^{39}$K atomic gases.

Figures (10)

• Figure 1: Typical phase diagram in the parameter space of interaction strength ratio $\lambda$ and total atom number $N$. Subplots (a) and (b) denote the contour plots of the peak density of quantum droplets in the $\lambda$-$\log_{10}(N)$ plane at the parameter regions I and II described in the main text, respectively. The gradient color represents the scaled peak density $\log_{10}[n_\mathrm{peak}/\max(n_\mathrm{peak})]$. The dashed line marks the critical position $N_{c1}$ to distinguish the phase boundary between Gaussian-like and flat-top phases, corresponding to the saturation point of the peak density. Here, the hollow circle, square, and diamond in subplot (a) denote the positions of three typical density profiles in Fig. \ref{['fig:densityprofile']}(a).
• Figure 2: Breathing mode frequency in the parameter space of interaction strength ratio $\lambda$ and total atom number $N$. Subplots (a) and (b) denote the contour plots of the breathing mode frequency of quantum droplets in the $\lambda$-$\log_{10}(N)$ plane at the parameter regions I and II, respectively. The gradient color represents the scaled peak density $\log_{10}[n_\mathrm{peak}/\max(n_\mathrm{peak})]$. Here, the dash-dotted and dotted lines, i.e., $N_{c2}$ and $N_{c3}$, mark the minimum of the root-mean-square radius and the maximum of the breathing mode frequency, respectively.
• Figure 3: Total density distribution $n$ at various values of interaction ratio $\lambda=g_1/g_2$ and trapping potential strength $\kappa$. (a) density profiles without external potential (i.e., $\kappa=0$) at three typical values of $\lambda=0.99$ (red solid line), $1.01$ (blue dashed line), $1.03$ (yellow dotted line) in parameter region I. The corresponding results obtained from the variational approach are shown by the circles, squares, and diamonds. The inset shows the corresponding effective potentials owing to the mean-field and Lee-Huang-Yang energy contributions. (b, c) density profiles for $\lambda=0.99$ and $\lambda=1.03$ at three values of potential strength $\kappa=0.001$, $0.01$, and $0.1$, denoted by the solid, dashed, and dash-dotted curves, respectively.
• Figure 4: (a) The root-mean-square radius $\sqrt{\left< x^2 \right>}$ and (b) the peak density $n_\mathrm{peak}$ of two spin components at three values of trapping potential strength $\kappa=0.001$ (solid lines), $0.01$ (dashed lines), and $0.1$ (dotted lines), as functions of the interaction ratio $\lambda$ in parameter region II with $N=5000$. The hollow circles, squares, and diamonds show the corresponding results obtained from the variational approach. The red and blue colors in (b) denote the results for spin components 1 and 2, respectively.
• Figure 5: (a) The root-mean-square radius $\sqrt{\left< x^2 \right>}$ and (b) the peak density $n_\mathrm{peak}$ of two spin components at three values of trapping potential strength $\kappa=0.001$ (solid lines), $0.01$ (dashed lines), and $0.1$ (dotted lines), as functions of the total atom number $N$ with $\lambda=5.3$. The hollow circles, squares, and diamonds show the corresponding results obtained from the variational approach. The red and blue colors in (b) denote the results for spin components 1 and 2, respectively. Here, the critical atom numbers $N_{c1}$ and $N_{c2}$ are determined by the saturated peak density of the quantum droplet, and the minimum of the radius, respectively.