Correlated dynamics of collective droplet excitations in a one-dimensional harmonic trap

We address the existence and dynamics of one-dimensional harmonically confined quantum droplets appearing in two-component mixtures by deploying a nonperturbative approach. We find that, in symmetric homonuclear settings, beyond-Lee-Huang-Yang correlations result in flat-top droplet configurations for either decreasing intercomponent attraction or larger atom number. Asymmetric mixtures feature spatial mixing among the involved components with the more strongly interacting or heavier one exhibiting flat-top structures. Applying quenches on the harmonic trap we trigger the lowest-lying collective droplet excitations. The interaction-dependent breathing frequency, being slightly reduced in the presence of correlations, shows a decreasing trend for stronger attractions. Semianalytical predictions are also obtained within the Lee-Huang-Yang framework. For relatively large quench amplitudes the droplet progressively delocalizes and higher-lying motional excitations develop in its core. Simultaneously, enhanced intercomponent entanglement and long-range two-body intracomponent correlations arise. In sharp contrast, the dipole motion remains robust irrespective of the system parameters. Species-selective quenches lead to a correlation-induced dephasing of the droplet or to irregular dipole patterns due to intercomponent collisions.

Figure 1

Ground state droplet densities of a symmetric mixture in a harmonic trap as obtained in the MB approach. The cases of (a) different $\delta g$ and fixed $N_A=N_B=N=20$ and (b) varying atom number $N$ for constant interaction $\delta g=0.01$ are presented. A transition from a Gaussian-type distribution to a FT one for increasing either $\delta g$ or $N$ occurs. In (b) a FT profile appears for $N=50$. Inset of (a): Density around the trap center within the MF, MB and the eGPE approach for $\delta g=0.08$. Notice that only the MB calculation captures the FT profile.

Figure 2

Density profiles of harmonically confined droplets within the MB approach for varying $\delta g$ in (a) an interaction-imbalanced ($g_B=2g_A=0.1$) and (b) a mass-imbalanced ($m_B=2.1m_A$) bosonic mixture. A tendency towards a FT droplet building upon the B component being either (a) more strongly interacting or (b) heavier takes place for larger $\delta g$. Intercomponent spatial mixing is also enhanced for increasing $\delta g$. In both cases $N_A=N_B=N=20$, while in the mass-imbalance setting $g_B=1.6g_A=0.08$.

Figure 3

Two-body reduced intracomponent density for species (a) $A$ and (b) $B$ as well as (c) the respective intercomponent density in the ground state of the mass and interaction-imbalanced mixture with $m_B=2.1m_A, g_B=1.6g_A=0.08, \delta g=0.05$ and $N_A=N_B=N=20$. An intraspecies antibunching behavior at the center of the droplet occurs [see the diagonal of ${\rho }_{AA}^{\left(2\right)}(x_1,x_2), {\rho }_{BB}^{\left(2\right)}(x_1,x_2)$], which is more prominent for the more strongly interacting $B$ component. Intercomponent mixing is identified in ${\rho }_{AB}^{\left(2\right)}(x_1,x_2)$.

Figure 4

Density evolution of a weakly attractive, symmetric mixture upon considering a sudden increase of the trap frequency, namely, ${\omega }^{\mathrm{f}}=4{\omega }^{\mathrm{i}}=0.04$ within (a) the ML-MCTDHX and (b) the perturbative eGPE approach. The droplet undergoes a breathing motion, while higher-lying motional excitations arise in the droplet for long evolution times ($t>1000$). Apparently the eGPE prediction slightly overestimates the delocalization of the droplet density profile and the peak density of the localized excitations around the trap center at long evolution times. The bosonic mixture contains $N_A=N_B=N=20$ atoms with $g_A=g_B=0.1$ and $\delta g=0.08$.

Figure 5

Density profiles of (a) the symmetric homonuclear droplet and (b)–(d) the respective orbitals $|{\mathrm{\Phi }}_i{\left(x\right)|}^2$ with $i=1,\cdots ,4$ (see legend) at different evolution times. Namely, at (b) $t=830$, (c) $t=1260$, and (d) $t=1845$. Evidently, higher-lying orbitals support the spatial delocalization and excitation patterns of the density, while the lowest orbital has the dominant contribution to the density. The breathing dynamics of the symmetric mixture, characterized by $g_A=g_B=0.1, N_A=N_B=N=20$, is triggered by a frequency quench where ${\omega }^{\mathrm{f}}=4{\omega }^{\mathrm{i}}=0.04$.

Figure 6

(a)–(d) Profiles of the two-body coherence $G^2(x_1,x_2)$ in the breathing dynamics of the symmetric mixture with $g_A=g_B=0.1, N_A=N_B=N=20$ at distinct time instants (see legend) after a quench to ${\omega }^{\mathrm{f}}=4{\omega }^{\mathrm{i}}=0.04$. Suppression of two-body correlations in the bulk $\left[G^2(x_1,x_2=x_1)\right]$ occurs for long evolution times while a correlated behavior is evident for bosons located at different edges of the droplet $\left[G^2(x_1,x_2=-x_1)\right]$.

Figure 7

Time evolution of the von Neumann entropy in the course of the droplet breathing motion induced by various quenches of the trap frequency at distinct interspecies couplings (see legend). The symmetric bosonic mixture with $N_A=N_B=N=20$ experiences $g_A=g_B=0.1$ (see legend). The finite entropy evinces the presence of intercomponent entanglement, while its sharp increase for sufficiently large $\delta g$ and ${\omega }^{\mathrm{f}}$ signals the prominent excitation dynamics of the droplet.

Figure 8

Breathing mode frequency (${\omega }_{\mathrm{br}}/{\omega }^{\mathrm{f}}$) of the symmetric mixture with $g_A=g_B=0.1$ in units of the postquench trap frequency for varying interspecies attraction and atom number (see legend). The predictions of ${\omega }_{\mathrm{br}}/{\omega }^{\mathrm{f}}$ are given within the MB approach (solid lines), the MF (dashed lines) and variational (dotted lines) approximation. A monotonic decreasing behavior of ${\omega }_{\mathrm{br}}/{\omega }^{\mathrm{f}}$ is observed for increasing either $\delta g$ or $N_A=N_B=N$. The breathing frequency is (slightly) smaller in the MB approach as compared to the MF and the variational approximation.

Figure 9

Time evolution of the position variance of droplets within the MB method. (a) The case of a trap quenched (${\omega }^{\mathrm{f}}=4{\omega }^{\mathrm{i}}=0.04$) homonuclear mixture, characterized by $g_B=2g_A=0.1, N=20$, for varying interspecies attractions (see legend) is depicted. The collapse and revival pattern observed in the weakly attractive mixture constitutes an imprint of the droplet excitations during its breathing motion. (b), (c) Same as (a) but for a heteronuclear mixture with $m_B=2.1m_A, g_B=1.6g_A=0.08$, and $N=20$, following (b) a sudden increase of the trap frequency according to ${\omega }^f=2{\omega }^i$ and (c) a species-selective quench where ${\omega }_{\mathrm{B}}^{\mathrm{f}}=2{\omega }_{\mathrm{B}}^{\mathrm{i}}=0.02$ and ${\omega }_{\mathrm{A}}^{\mathrm{f}}={\omega }_{\mathrm{A}}^{\mathrm{i}}=0.015$. The dynamics change from being out of phase to phase locked among the two components for increasing attraction.

Figure 10

Density evolution of the $\sigma$ component (see legends) of a heteronuclear mass-imbalanced mixture with $m_B=2.1m_A, g_B=1.6g_A=0.08, N_A=N_B=N=20$ and strong intercomponent attraction $\delta g=0.01$. The dynamics induced by quenching the trap frequency, ${\omega }_{\sigma }^{\mathrm{f}}=4{\omega }_{\sigma }^{\mathrm{i}}$, is monitored within (a), (c) the MB and (b), (d) MF approaches. Evidently, the build-up of correlations leads to a dephasing of the breathing mode amplitude at long evolution times.

Figure 11

Time evolution of the $\sigma$-species position variance of a mass-imbalanced heteronuclear mixture characterized by $m_B=2.1m_A, g_B=1.6g_A=0.08, N_A=N_B=N=20$, and strong intercomponent attraction $\delta g=0.01$. The dynamics induced by a quench where ${\omega }_{\sigma }^{\mathrm{f}}=4{\omega }_{\sigma }^{\mathrm{i}}$ is monitored within the MB (solid lines) and MF (dashed lines) approach. A correlation-induced dephasing behavior is observed in the MB case in sharp contrast to the MF dynamics where an irregular breathing motion persists.

Figure 12

One-body density evolution of a droplet building upon a weakly attractive symmetric mixture with $N_A=N_B=N=20, g_A=g_B=0.1$, and $\delta g=0.08$ after an abrupt displacement of the trap center by $\delta x_{\sigma }$. (a) The displacement is performed in both components, where $\delta x_A=\delta x_B=10$, resulting in a stable dipole motion having a frequency equal to the one of the trap. (b), (c) A species-selective quench with $\delta x_A=10$ and $\delta x_B=0$ leads to an energy transfer from species $A$ to $B$ and consequent out-of-phase dipole oscillations of the individual components. All results were obtained within the MB approach.

Figure 13

Dynamics of the spatially averaged position of the $\sigma$-species center of mass $\langle X_{\sigma }\left(t\right)\rangle$ following a sudden displacement of the trap center by an amount $\delta x_{\sigma }$ within the MB approach. (a) The case of homonuclear interaction-imbalanced $g_B=2g_A=0.1$ (solid lines) and balanced with $g_B=g_A=0.1$ (dashed lines) bosonic mixtures subjected to quenches with $\delta x_A=\delta x_B=10$ for different interactions and atom numbers (see legends) is depicted. The emergent dipole motion is independent of the system parameters. Species-selective quenches with (b) $\delta x_A=10, \delta x_B=0$ and (d) $\delta x_A=-\delta x_B=10$ in a symmetric mixture characterized by $g_A=g_B=0.1, N_A=N_B=N=20$. (c) Dipole motion induced by $\delta x_A=\delta x_B=-10$ in a mass-imbalanced heteronuclear mixture where $m_B=2.1m_A, g_B=1.6g_A=0.08$, and $N_A=N_B=N=20$. Apparently, following either species-selective quenches or using mass-imbalanced mixtures gives rise to a more complex dipole dynamics, with a strong dependence on $\delta g$.