Multicomponent one-dimensional quantum droplets across the mean-field stability regime

TL;DR

This work addresses self-bound quantum droplets in one-dimensional multicomponent Bose mixtures by deriving the Lee-Huang-Yang (LHY) energy corrections across the mean-field stability regime using a Bogoliubov framework. It yields exact closed-form LHY energies for two-, three-, and four-component configurations (including fully symmetric cases) and constructs corresponding extended Gross-Pitaevskii equations (eGPEs) that remain valid throughout MF stability. Comparisons with the traditional two-component LHY treatment reveal quantitative differences in saturation density and droplet width, and reveal qualitative phenomena such as early phase separation for repulsive intercomponent interactions and a variety of mixed-droplet states in three-component systems. The results provide a comprehensive theoretical toolkit for investigating exotic self-bound states and their nonequilibrium dynamics in multicomponent quantum fluids and set the stage for experimental tests and higher-dimensional extensions.

Abstract

The Lee-Huang-Yang (LHY) energy correction at the edge of the mean-field stability regime is known to give rise to beyond mean-field structures in a wide variety of systems. In this work, we analytically derive the LHY energy for two-, three- and four-component one-dimensional bosonic short-range interacting mixtures across the mean-field stability regime. For varying intercomponent attraction in the two-component setting, quantitative deviations from the original LHY treatment emerge being imprinted in the droplet saturation density and width. On the other hand, for repulsive interactions an unseen early onset of phase-separation occurs for both homonuclear and heteronuclear mixtures. Closed LHY expressions for the fully-symmetric three- and four-component mixtures, as well as for mixtures comprised of two identical components coupled to a third independent component are provided and found to host a plethora of mixed droplet states. Our results are expected to inspire future investigations in multicomponent systems for unveiling exotic self-bound states of matter and unravel their nonequilibrium quantum dynamics.

Figures (7)

• Figure 1: Ground state droplet densities for a symmetric mixture featuring $N_A=N_B=N=10$, $g_A = g_B = g$, and varying intercomponent interactions $g_{AB}$ (see legends). A plethora of different cases are shown ranging from (a) strong and (b) weak attractions, to (c) weak (nearly suppressed) couplings all the way towards (d) repulsive intercomponent interactions. In all cases, the ground state density obtained within the original given by Eq. (\ref{['ApproxeGPE']}) (dashed lines) and the exact (solid lines) eGPEs (\ref{['eGPETwoComponent']}) are depicted. Evidently, the original eGPEs consistently predict significantly more localized configurations when $\left| g_{AB} \right| \neq g$, while in some cases (e.g. $g_{AB} = - 0.6g$) it even fails to adequately capture the droplet's FT density profile. Note that the dashed blue and green lines in panels (a) and (d) respectively are hardly visible since they coincide with the solid ones. (e) Dependence of the real and imaginary parts (see legend) of the LHY energy coefficient ($I(p)$) in Eq. \ref{['Ifactor']} on the interspecies interaction strength. (f) The droplet saturation density in both the exact and original approaches (see legend) with respect to $g_{AB}$.
• Figure 2: Ground state density configurations of a homonuclear two-component mixture for fixed $N_A=N_B \equiv N=10$, $m_A=m_B \equiv m$, and $g_{B}=0.5g_{A}$. Different intercomponent coupling strengths are considered ranging from (a) attractive $g_{AB} = -0.4g_{A}$, to (b) decoupled $g_{AB} = 0$, and (c) repulsive $g_{AB} = 0.4g_{A}$ cases. The outcome of the exact [Eq. \ref{['eGPETwoComponent']}] and the original [Eq. \ref{['ApproxeGPE']}] eGPEs is illustrated as well as the one stemming from numerical integration of the sum of the LHY energy [Eq. \ref{['LHYTwoComponent']}]. As for the balanced mixture, see Fig. \ref{['fig:scangAB2Comp']}, the original eGPEs overestimate the peak density of the droplets and underestimate their width. Interestingly, as long as the intercomponent interaction becomes repulsive [panel (c)], an early onset of the miscible-to-immiscible transition, coexisting with droplet configurations, appears to take place, and being solely captured by the exact eGPEs.
• Figure 3: Ground state phases of heteronuclear droplet densities for (a)-(d) $m_B=2m_{A}$, (e) $m_B=0.5m_{A}$, and (f) $m_B=5m_{A}$, whilst $N_A=N_B \equiv N=10$, $g_{A}=g_{B} \equiv g$. The results are obtained within the original mass-balanced eGPEs [Eq. (\ref{['ApproxeGPE']})] marked by "Original", the exact mass-balanced eGPEs given by Eq. \ref{['eGPETwoComponent']} and indicated as "Exact", and the full numerically calculated heteronuclear eGPEs designated by "Num". In the first two cases, the mass of each component is plugged in the corresponding equation. The heteronuclear systems feature attractive (a) $g_{AB} = -0.8 g$, and (b), (e) $g_{AB} = -0.4g$, (c) decoupled, $g_{AB} = 0$, and (d), (f) repulsive $g_{AB} = 0.4g$ intercomponent couplings. Interestingly, there is an excellent agreement between the ground state droplet densities predicted by the mass-balanced exact eGPEs [Eq. \ref{['eGPETwoComponent']}] and the fully general numerical approach. Minor deviations are observed for small mass differences and repulsive interactions which are, for instance, hardly visible in panel (d). Conversely, the original eGPEs [Eq. (\ref{['ApproxeGPE']})] exhibit significant quantitative disagreement.
• Figure 4: (a) The behavior of the LHY energy for the symmetric mixture stemming from a reduction of the genuine two-, three- and four-component settings as a function of the intercomponent interaction $G$. (b)-(d) Ground state densities of a fully symmetric three-component mixture with $N=10$ for varying interaction $G$ (see legends). A transition from a Gaussian to a FT profile with increasingly larger spatial extent, upon reducing the attraction [panels (b), (c)] or increasing repulsion [panels (c), (d)] is evident.
• Figure 5: Droplet density configurations arising in a homonuclear three-component mixture for varying interspecies couplings $g_{AC}=g_{BC} \equiv G_C$. The latter range from the attractive (a) $G_C=-0.5g$, (b) $G_C = -0.2g$, (c) $G_C = -0.1g$, to (d) the decoupled $G_C = 0$, and (e) the repulsive $G_C = 0.1g$, (f) $G_C = 0.4g$ interaction regimes, thus covering the available parametric space. As it can be seen, mixed phases exhibiting a structurally deformed third component having both a droplet core and self-bound atoms at the tails [panel (c)] for weak intercomponent attractions or a configuration of two fully distinct droplets [panel (d)] in the decoupled regime occur. Also, an early onset of the miscible-to-immiscible transition, coexisting with droplet configurations, similarly to the two-component mixture takes place for repulsive intercomponent couplings. In all cases, the remaining system parameters are held fixed and in particular correspond to $N_A=N_B=N_C = 10$, $m_A=m_B=m_C \equiv m$, $g_{A}=g_{B}=g_{C} \equiv g$, and $g_{AB}=-0.5g$.