Borromean Droplet in Three-Component Ultracold Bose Gases

We investigate droplet formation in three-component ultracold bosons. In particular, we identify the formation of a Borromean droplet, where only the ternary bosons can form a self-bound droplet while any binary subsystems cannot, as the first example of Borromean binding due to a collective many-body effect. Its formation is facilitated by an additional attractive force induced by the density fluctuation of a third component, which enlarges the mean-field collapse region in comparison to the binary case and renders the formation of a Borromean droplet after incorporating the repulsive force from quantum fluctuations. Outside the Borromean regime, we demonstrate an interesting phenomenon of droplet phase separation due to the competition between ternary and binary droplets. We further show that the transition between different droplets and gas phase can be conveniently tuned by boson numbers and interaction strengths. The study reveals the rich physics of a quantum droplet in three-component boson mixtures and sheds light on the more intriguing many-body bound state formed in multicomponent systems.

Figure 1

Mean-field phase diagram for three-component bosons under couplings (5) and $g_{33}=g$. The gray area marks the mean-field stable region (“S”), smaller than that for binary subsystems (bounded by a red square). After incorporating the LHY repulsion, a Borromean droplet can take place in regions I and II(A). The dashed lines separating II(A) and II(B) are determined by $C_{\min }=0$ at the droplet-gas transition (see text).

Figure 2

Droplet region (colored) in the $N-C$ plane for three typical points (marked by *) on the vertical line in Fig. 1, which have a fixed $g^{\prime }/g=-\sqrt{2}/2$ and different $g_{12}/g=-0.2(a);-1.5(b);-5(c)$. Red dashed lines show $C_{\min }$ when the system has minimal total energy at given $N$. In (b), the yellow area marks the regime for droplet phase separation.

Figure 3

(a) $C_{\min }$ (solid line) as a function of $g_{12}/g$ at the critical number $N_{t,c}$, in comparison with $C_{\min }^{(0)}$ (dashed line) from Eq. (9). (b) Phase diagram in the $N-g_{12}$ plane. The Borromean droplet (“BD”) occurs in region I for $N>N_{t,c}$ and in II(A) for $N\in (N_{t,c},N_{b,c})$. For $N<N_{t,c}$ the system is in the gas phase; for $N>N_{b,c}$ the binary droplet can also exist. Dashed lines show the function fits of $N_{t,c}^{(0)}$ [Eq. (11)] and $N_{b,c}^{(0)}$ (from Ref. [23]). Here $g^{\prime }/g=-\sqrt{2}/2$.

Figure 4

Density profile displaying the phase separation between ternary (1&2&3) and binary (1&2) droplets. Here $g^{\prime }/g=-\sqrt{2}/2$, $g_{12}/g=-1.5$, $N={10}^5$, $C=0.2$, corresponding to the triangular point in Fig. 2. Horizontal lines show function fits to the equilibrium densities of ternary [Eq. (10)] and binary (from Ref. [23]) droplets. The length and density units are, respectively, $a$ and $1/a^3$ $[a=mg/(4\pi )]$.