Formation and quench of homonuclear and heteronuclear quantum droplets in one dimension

We study the impact of beyond Lee-Huang-Yang (LHY) physics, especially due to intercomponent correlations, in the ground state and the quench dynamics of one-dimensional quantum droplets with an ab initio nonperturbative approach. It is found that the droplet Gaussian-shaped configuration arising for intercomponent attractive couplings becomes narrower for stronger intracomponent repulsion and transits towards a flat-top structure either for larger particle numbers or weaker intercomponent attraction. Additionally, a harmonic trap prevents the flat-top formation. At the balance point where mean-field interactions cancel out, we show that a correlation hole is present in the few-particle limit of LHY fluids as well as for flat-top droplets. Introducing mass imbalance, droplets experience intercomponent mixing and excitation signatures are identified for larger masses. Monitoring the droplet expansion (breathing motion) upon considering interaction quenches to stronger (weaker) attractions, we explicate that beyond LHY correlations result in a reduced velocity (breathing frequency). Strikingly, the droplets feature two-body anticorrelations (correlations) at the same position (longer distances). Our findings pave the way for probing correlation-induced phenomena of droplet dynamics in current ultracold-atom experiments.

Figure 1

Ground-state densities ${\rho }^{\left(1\right)}\left(x\right)$ of a symmetric bosonic mixture for varying intracomponent interaction strength $g$ and particle number $N$ (see legends) representing various droplet configurations. (a), (b) ${\rho }^{\left(1\right)}\left(x\right)$ for different $g$ while keeping $\delta g/g=0.05$ fixed as obtained in the MB approach. The droplet profile features a gradual shrinking for larger $g$ and shows FT signatures around $g=0.1$. Comparison of ${\rho }^{\left(1\right)}\left(x\right)$ for (c) $N=40$ and (d) $N=10$ between different approaches (see legend) explicating the impact of beyond LHY correlations as imprinted in the density of the droplet. ${\rho }^{\left(1\right)}\left(x\right)$ for increasing particle number $N$ of (e) an LHY fluid ($\delta g=0$) and (f) a droplet for fixed $g=0.5$ and $\delta g/g=0.05$. The fluid experiences a gradual localization for increasing $N$. The system consists of $N_A=N_B\equiv N/2$ bosons of equal mass and intracomponent interactions $g_{AA}=g_{BB}\equiv g$ trapped in a box potential of length $L$.

Figure 2

Two-body density distributions of a symmetric bosonic mixture within the MB approach for different (a)–(d) intracomponent interactions $g$ while $\delta g/g=0.05$ and $N=40$. The distributions become spatially localized for increasing $g$ and a correlation hole emerges [(b), (c)] at interaction regimes where FT signatures appear in the density profile [see Figs. 1 and 1]. (e)–(h) The same as in (a)–(d) but for a varying particle number $N$ of an LHY fluid with $g=0.5$ and $\delta g=0$. The two-body correlation hole fades away for $N>10$.

Figure 3

Ground-state densities ${\rho }^{\left(1\right)}\left(x\right)$ of a symmetric bosonic mixture for fixed $g=0.05$ and varying $\delta g/g$ (see legend) for (a) $N=10$ and (b) $N=40$ within the MB approach. A transition from a Gaussian-shaped to a FT profile occurs for increasing $\delta g/g$. This transition is realized for smaller $\delta g/g$ values when $N$ is larger. Comparison of ${\rho }^{\left(1\right)}\left(x\right)$ for $N=40$ between different methods (see legend) for $g=0.05$ while (c) $\delta g/g=0.02$, and (d) $\delta g/g=0.2$. It becomes clear that beyond LHY correlations are suppressed for larger $\delta g/g$ since then $g_{AB}\to 0$. (e) ${\rho }^{\left(1\right)}\left(x\right)$ as predicted by the MGP for varying $N$ with $g=0.05$ and $\delta g/g=0.2$. The droplet profile becomes delocalized for increasing $N$ acquiring a FT profile which saturates for $N>500$. The symmetric bosonic mixture comprises of $N_A=N_B\equiv N/2$ bosons with the same mass and intracomponent interactions $g_{AA}=g_{BB}\equiv g$ while it experiences a box potential of length $L$.

Figure 4

Density maximum of the ground-state droplet with respect to $\delta g/g$ within the MB and the MGP frameworks as well as for different particle numbers, i.e., (a) $N=40$ and (b) $N=10$. Deviations caused by beyond LHY correlations range from $\sim 25\%$ ($\sim 11\%$) to $\sim 4\%$ ($\sim 5\%$) for $N=40$ ($N=10$). The mass-balanced bosonic mixture has intracomponent interactions $g_{AA}=g_{BB}\equiv g=0.05$.

Figure 5

Ground-state densities ${\rho }^{\left(1\right)}\left(x\right)$ of a harmonically trapped bosonic mixture with $N=40$ for different (a) $g$ with fixed $\delta g/g=0.05$ and (b) $\delta g/g$ with constant $g=0.1$ in the MB approach. The trap leads to localized density distributions experiencing a decreasing width for larger (smaller) values of $g$ ($\delta g/g$). (c) ${\rho }^{\left(1\right)}\left(x\right)$ for various frequencies $\omega$ of the harmonic trap (see legend) while $g=0.5$ and $\delta g/g=0.05$ are kept fixed. Evidently, a tighter trap results in the spatial localization of the droplet density distribution. (d) ${\rho }^{\left(1\right)}\left(x\right)$ for different particle numbers when $g=0.1$ and $\delta g/g=0.5$. The density profiles become wider for larger atom numbers. The dashed lines in (c) and (d) showcase specific MGP predictions which exemplify significant deviations from the MB case. The symmetric mixture comprises of $N_A=N_B\equiv N/2$ bosons of the same mass and intracomponent interactions $g_{AA}=g_{BB}\equiv g$ in a harmonic trap of frequency $\omega$.

Figure 6

Ground-state densities ${\rho }_{\sigma }^{\left(1\right)}\left(x\right)$ of a mass-imbalanced mixture ($M_B\approx 2.12M_A$) for (a), (b) distinct intracomponent coefficients $g$ and fixed $\delta g/g=0.05$ (see legends) and (c) $\delta g/g=0.02$, $g=0.05$. The density profiles become narrower (flattened) for an increasing $g$ ($\delta g/g$). (d) ${\rho }_{\sigma }^{\left(1\right)}\left(x\right)$ for a strongly mass-imbalanced setting with $M_B\approx 19M_A$ while $\delta g/g=0.02$ and $g=0.05$. Excitation signatures are captured in the distribution of the heavy component being identified by the deformations of its respective density. Due to the mass imbalance, the components are discernible (mixed). The mixture has $N_A=N_B=20$ bosons residing in a box trap of length $L$.

Figure 7

Time evolution of the single-particle density ${\rho }^{\left(1\right)}(x;t)$ of a box trapped symmetric bosonic mixture upon considering an interaction quench from ${(\delta g/g)}_{\mathrm{in}}=0.02$ to ${(\delta g/g)}_{\mathrm{f}}=0.8$ within (a) the MB and (b) the MGP framework. An expansion of the droplet, due to the reduction of the intercomponent attraction caused by the quench, towards the box edges occurs and is found to be less pronounced in the MB scenario. Inset in (a) presents density profiles at $t=50$ (blue line) and $t=150$ (red line) where FT signatures are evident. (c) The same as (a) but for a quench from ${(\delta g/g)}_{\mathrm{in}}=0.2$ to ${(\delta g/g)}_{\mathrm{f}}=0.01$. The mixture tends towards an LHY fluid due to the quench and it undergoes a breathing motion. The postquench intercomponent attraction is larger compared to its prequench value. (d) Same as (a) but for a harmonically trapped mixture. Breathing of the droplet is observed due to the existence of the external trap. The inset in (d) shows characteristic instantaneous density profiles depicting expansion $t=53$ (blue line), contraction $t=68$ (red line), and FT formation in-between $t=60$ (black line). Dynamics of the von Neumann entropy for different postquench interaction strengths (see legend) starting from (e) ${(\delta g/g)}_{\mathrm{in}}=0.02$ and (f) ${(\delta g/g)}_{\mathrm{in}}=0.2$. A dynamical reduction (enhancement) of the intercomponent entanglement is triggered for larger (smaller) postquench values of $\delta g/g$. The mixture consists of $N=40$ bosons of the same mass and intracomponent interactions $g_{AA}=g_{BB}\equiv g=0.05$ for the box while $g_{AA}=g_{BB}\equiv g=0.1$ for the trap.

Figure 8

Dynamics of the variance after an interaction quench of a symmetric bosonic mixture experiencing (a), (b) a box potential of length $L$ and (c) a harmonic trap of $\omega =0.1$ as obtained within the MB and the MGP approaches (see legends). The behavior of the variance quantifies (a) the expansion and (b), (c) the breathing dynamics of the mixture. Beyond LHY correlations lead to a pronounced reduction of the expansion rate (a) and a slight negative shift in the breathing frequency of the droplet (c), (d). The values of the prequench and postquench interactions ${(\delta g/g)}_{\mathrm{in}}$ and ${(\delta g/g)}_{\mathrm{f}}$ are shown in the legends. The bosonic mixture has $N=40$ atoms with equal mass and intracomponent interactions $g_{AA}=g_{BB}\equiv g=0.05$.

Figure 9

Quenched $\sigma$-component density ${\rho }_{\sigma }^{\left(1\right)}\left(x\right)$ of a mass-imbalanced mixture after a change from ${(\delta g/g)}_{\mathrm{in}}=0.15$ to ${(\delta g/g)}_{\mathrm{f}}=0.01$ within the MB approach. (c) Time evolution of the corresponding variances. The components perform a breathing dynamics showing a phase difference with the heavy component being delayed. The box trapped mass-imbalanced mixture has $N=40$ bosons with $M_B=2.1M_A$ and interactions $g_{AA}=g_{BB}\equiv g=0.05$.

Figure 10

Snapshots of the two-body correlation function following an interaction quench from ${(\delta g/g)}_{\mathrm{in}}=0.02$ to ${(\delta g/g)}_{\mathrm{f}}=0.8$ in (a)–(e) box potential of length $L$ and (f)–(j) harmonic trap with $\omega =0.1$. Anticorrelations develop at the same location of the droplet and a correlated behavior appears at distinct places. The symmetric mixture consists of $N=40$ mass-balanced bosons and intracomponent interactions (a)–(e) $g_{AA}=g_{BB}\equiv g=0.05$ while (f)–(j) $g_{AA}=g_{BB}\equiv g=0.1$.