Universal Four-Body States in Heavy-Light Mixtures with a Positive Scattering Length

The number of four-body states known to behave universally is small. This work adds a new class of four-body states to this relatively short list. We predict the existence of a universal four-body bound state for heavy-light mixtures consisting of three identical heavy fermions and a fourth distinguishable lighter particle with a mass ratio $\kappa \gtrsim 9.5$ and short-range interspecies interaction characterized by a positive $s$-wave scattering length. The structural properties of these universal states are discussed, and finite-range effects are analyzed. The bound states can be experimentally realized and probed by utilizing ultracold atom mixtures.

Figure 1

Scaled energies as a function of $r_0/a_s$. The dashed line shows $E_2/|E_2^{\mathrm{ZR}}|$. The symbols show $E_3/|E_2^{\mathrm{ZR}}|$ for $\kappa =8.25–10.5$ (top to bottom), in steps of 0.25; $E_3$ is determined by the stochastic variational approach. Dotted lines show three-parameter fits. Inset: Symbols show $E_3^{\mathrm{ZR}}/|E_2^{\mathrm{ZR}}|$ as a function of $\kappa$. The solid line shows a four-parameter fit.

Figure 2

Scaled energies as a function of $r_0/a_s$ for (a) $\kappa =9.5$, (b) $\kappa =9.75$, and (c) $\kappa =10$. The dashed lines show $E_2/|E_2^{\mathrm{ZR}}|$, while the crosses (stochastic variational energies) and solid lines (fit) show $E_3/|E_2^{\mathrm{ZR}}|$ (these energies are also shown in Fig. 1). Circles and triangles show $E_4/|E_2^{\mathrm{ZR}}|$ for the energetically lowest-lying and second-lowest-lying four-body states, respectively. Dotted lines serve as a guide to the eye.

Figure 3

(a) Solid, dashed, dotted, and dash-dotted lines show $P(R)$ of the energetically lowest-lying state for $n=3$, $\kappa =8.5$, and $r_0/a_s=0.004$, 0.005, 0.006, and 0.015, respectively. Inset: Dotted, dash-dotted, and solid lines show $P(R)$ for $n=3$ ($\kappa =8.5$ and $r_0/a_s=0.004$), $n=3$ ($\kappa =9.75$ and $r_0/a_s=0.004$), and $n=4$ ($\kappa =9.75$ and $r_0/a_s=0.004$), respectively. (b) Solid, dashed, dotted, and dash-dotted lines show $P(R)$ for $n=4$, $\kappa =9.75$, and $r_0/a_s=0.004$, 0.005, 0.006, and 0.015, respectively. For $r_0/a_s=0.004$, 0.005, and 0.006, the energetically lowest-lying state is considered. For $a_s/r_0=0.015$, the energetically second-lowest-lying state is considered. Inset: Solid and dotted lines show $n_{\mathrm{rad}}(r_j)$ for the heavy and light particles, respectively, for $n=4$, $\kappa =9.75$, $r_0/a_s=0.004$, and $1^{+}$ symmetry.