Formation Dynamics of Quantum Droplets for Homonuclear and Heteronuclear Mixtures

TL;DR

This work investigates the real-time formation and dynamics of quantum droplets in one-dimensional two-component Bose mixtures, incorporating Lee-Huang-Yang corrections via an extended GPE derived from Bogoliubov theory. Using a tight-binding lattice and a Crank-Nicolson scheme with tanh-sinh quadrature for the LHY term, the authors compare homonuclear and heteronuclear droplets across varied initial conditions. A key finding is that LHY fluctuations dominantly determine binding, with deeper late-time binding at intermediate mass ratios around $m_2/m_1\approx 1.2$ and notably limited equilibration due to 1D constraints; Gaussian initial states promote rapid formation and robust coalescence regardless of mass ratio. The study also characterizes localization, overlap, and breathing behavior, providing practical criteria for droplet formation and offering guidance for experiments aiming to probe droplet lifetimes and stability in quasi-1D settings.

Abstract

Great effort has been invested over the past decade in studying the properties of quantum droplets, a phase of bosonic quantum matter that arises as a consequence of the fluctuating Lee-Huang-Yang correction. However, the dynamics of droplet formation for heteronuclear Bose mixtures is partially understood. Here, we numerically analyze the formation process for homonuclear and heteronuclear boson mixtures in one dimension using a tight-binding model and real-time evolution. A systematic sweep of interaction strengths, mass ratios, and initial conditions allows us to characterize quantitative criteria for droplet formation and equilibration. We find that the energy contribution of the LHY correction dominates the energetic profile of the droplets formed, with the deepest binding occurring for mass ratios $m_2/m_1 \in [1.2,2.0]$. Breathing oscillations are observed, and the low equilibration rate is consistent with the restricted nature of the phase space for one-dimensional systems; the oscillation frequency is found to have a very weak correlation to the interaction strengths. For the simulation, Gaussian over discrete initial conditions are clearly favorable to the formation of droplets. The results contained herein provide rich insight into the dynamical nature of quantum droplet physics.

Figures (15)

• Figure 1: Formation time analysis. (a,b) Formation time $t_{\text{form}}$ versus $\delta U$ for homonuclear and heteronuclear cases. (c,d) Formation time versus initial separation. Gaussian initial conditions (blue circles) achieve essentially instantaneous formation, while discrete initial conditions (red squares) require $t_{\text{form}} \approx 0.4$ (arbitrary units).
• Figure 2: Equilibration time analysis. (a,b) Equilibration time $t_{\text{eq}}$ versus $\delta U$ showing clustering around $t_{\text{eq}} \approx 87.5$.
• Figure 3: Late-time energy analysis. (a,b) Late-time energy $E_{\text{late-time}}$ versus $\delta U$ for homonuclear and heteronuclear cases, showing monotonic deepening with increasingly negative $\delta U$. (c,d) Energy time evolution with color indicating $\delta U$ magnitude.
• Figure 4: Late-time energy versus initial separation for (a) homonuclear and (b) heteronuclear droplets.
• Figure 5: Energy partitioning analysis. (a) Energy components versus time for a representative run. (b) Energy ratios at late-time versus $|\delta U|$, showing $E_{\text{LHY}}/E_{\text{total}} \approx 1$. (c) LHY fraction of late-time energy. (d) Kinetic versus interaction energy scatter plot.