Droplet-gas phases and their dynamical formation in particle imbalanced mixtures

Abstract

We explore the ground state phase diagram and nonequilibrium dynamics of genuine two-component particle-imbalanced droplets in both isotropic and anisotropic three-dimensional confinements. A gradual transition from mixed droplet-gas to gas configurations is revealed as the average intercomponent attraction decreases or the transverse confinement becomes tighter. Within the mixed structures, a specific majority fragment binds to the minority droplet, satisfying the density ratio locking condition, while the remaining atoms are in a gas state. Our extended Gross-Pitaevskii numerical results are corroborated by a suitable variational approximation capturing the shape and characteristics of droplet-gas fragments. The tunability of the relatively low gas fraction is showcased through parametric variations of the atom number, the intercomponent imbalance, the trap aspect ratio, or the radius of a box potential. To validate the existence and probe the properties of these exotic phases, we simulate the standard time-of-flight and radio frequency experimental techniques. These allow to dynamically identify the resilience of the droplet fragment and the expansion of the gas fraction. Our results, amenable to current experimental cold atom settings, are expected to guide forthcoming investigations aiming to reveal unseen out-of-equilibrium droplet dynamics.

Figures (9)

• Figure 1: Isosurfaces of the normalized densities of each component for the $N_1/N_2 = 80/20$ particle imbalanced Bose-Bose mixture with $N=2\times10^5$. Top panels (a1)-(a4) correspond to trap frequency aspect ratio $\omega_z/\omega_r = 1$ and bottom ones (b1)-(b4) refer to $\omega_z/\omega_r = 50$. Each column corresponds to different values of $\delta a$ [see labels at the top]. For every $[\omega_z/\omega_r, \delta a]$ combination the upper (lower) panels depict the normalized density of the minority (majority) component. Mixed droplet-gas configurations form upon increasing the mean-field attraction (see panels (a1), (a2), (b1), (b2)) giving their place to pure gas states for larger $\delta a$ (panels (a3), (a4), (b3), (b4)).
• Figure 2: (a1)-(a4) [(b1)-(b4)] Normalized density profiles $n_i(x,0,0)/ \max (n_i(x,0,0))$ of each component for an isotropic $\omega_z/\omega_r = 1$ [anisotropic, $\omega_z/\omega_r = 50$] 3D confinement. Each column refers to different values of $\delta a$ (see top row legends). The blue dotted lines indicate the edges of the planar harmonic oscillator length $\l_{r}=7\mu m$. The droplet core occurs always at distances smaller than $l_r$, while the gas portion extends beyond the oscillator length. For anisotropic traps, the gas fraction extends only over the plane, and therefore gets more enhanced, compare upper and lower panels. A transition from a mixed droplet-gas to a gaseous phase takes place for less negative $\delta a$. In all cases, the particle imbalance is 80/20 with total atom number $N=2 \times 10^5$.
• Figure 3: (a) Energy (total) per particle, $E/N$, as a function of $\delta a$ for different particle imbalances (see legend) and $\omega_z = \omega_r$. The inset focuses on $-4<\delta a<-1$ to highlight when the sign of $E/N$ changes from positive to negative. Stronger imbalances exhibit larger $E/N$ due to the presence of an enhanced fraction of excess particles. An excellent agreement occurs for $E/N$ between the system with 60/40 and the one obeying the density ratio locking (dashed lines). (b) $E/N$ with respect to the trap anisotropy $\omega_z / \omega_r$ and different interaction strengths (see legend). The energy per particle increases with $\omega_z / \omega_r$, specifically as the 2D regime is approached, due to the large potential energy in the $z$-direction quantified by $E_{HO}$ (dotted line) irrespective of $N_1 / N_2$. (c) Phase diagram of the droplet-gas and the pure gas phase in the plane ($\bar{N}_1/N_1 - \delta a$). The number of atoms populating the droplet fragment of the majority component, $\bar{N}_1 / N_1$, for varying $\delta a$ and different imbalances (see legend) is also depicted. A decreasing tendency is observed for larger imbalances. In all cases, an adequate estimate is given by employing the density ratio locking in the thermodynamic limit (dotted lines). The comparison pertains only to the case where a droplet core is present, since for large $\delta a$ a pure gas phase occurs as indicated by the light blue shaded region. The phase boundary is determined by TOF discussed in Sec. \ref{['time_of_flight_dyn']}.
• Figure 4: Comparison of the ground state two-component droplet densities (see legend) of a particle imbalanced setting computed via the VA (dashed lines) and the set of eGPEs (solid lines). The system features averaged mean-field interactions, $\delta a = -5.2 ~ a_0$, and fixed imbalance, $N_1/N_2 = 80/20$, while the total particle number, $N$, increases (see legend) and the external confinement is an (a) isotropic spherical trap and (b) a hard sphere. The insets in panels (a), (b) provide a magnification of the majority component gas fraction. The agreement between the VA and the eGPE results for both the droplet and the gas fragments gradually improves for larger atom numbers. By increasing the hard sphere radius, $R_s$, while holding $N=2 \times 10^5$ constant in panel (c), the majority droplet core remains unchanged, while the gas segment (see in particular the inset) expands occupying the available space all the way towards the boundaries. These results are obtained within the eGPEs but a similar behavior is predicted with the VA (not shown for better visualization). In panels (b) and (c) the vertical dashed lines mark the location of the radius of each hard sphere.
• Figure 5: Simulated TOF dynamics of the (a1)-(a3) majority and (b1)-(b3) minority component in the $N_1/N_2 = 80/20$ imbalanced configuration for different trap aspect ratios, $\omega_z / \omega_r$, and averaged mean-field interactions, $\delta a$, see legends. In all cases, the dynamics of the density profiles $n^{(1D)}_i(x)$ with $i=1,2$ (see main text) is presented. In the droplet-gas configuration [panels (a1), (b1)] the droplet segment quantified by the high-density central region remains self-bound in the course of the evolution, while the surrounding gas of the majority component expands [see e.g. panel (a1)]. In sharp contrast, within the gas phase [panels (a2), (b2)], both components expand after the trap release. For the anisotropically trapped droplet-gas state [panels (a3), (b3)] the induced evolution follows a similar trend to the isotropic setting [panels (a1), (b1)]. The density along the $z$ direction, $n^{(1D)}_i(z)$, remains intact as can be seen in the insets of (a3) and (b3). In all cases, the white dashed lines indicate the semi-analytical estimate of the expansion radius of a single-component Bose gas.