Few-body resonances of unequal-mass systems with infinite interspecies two-body $s$-wave scattering length

Two-component Fermi and Bose gases with infinitely large interspecies $s$-wave scattering length $a_s$ exhibit a variety of intriguing properties. Among these are the scale invariance of two-component Fermi gases with equal masses, and the favorable scaling of Efimov features for two-component Bose gases and Bose-Fermi mixtures with unequal masses. This paper builds on our earlier work [Phys. Rev. Lett. 105, 170403 (2010)] and presents a detailed discussion of our studies of small unequal-mass two-component systems with infinite $a_s$ in the regime where three-body Efimov physics is absent. We report on nonuniversal few-body resonances. Just like with two-body systems on resonance, few-body systems have a zero-energy bound state in free space and a diverging generalized scattering length. Our calculations are performed within a nonperturbative microscopic framework and investigate the energetics and structural properties of small unequal-mass two-component systems as functions of the mass ratio $\kappa$, and the numbers $N_1$ and $N_2$ of heavy and light atoms. For purely attractive Gaussian two-body interactions, we find that the $(N_1,N_2)=(2,1)$ and $(3,1)$ systems exhibit three-body and four-body resonances at mass ratios $\kappa =12.314(2)$ and $10.4(2)$, respectively. The three- and four-particle systems on resonance are found to be large. It seems feasible that the features discussed in this paper can be probed experimentally with present-day technology.

Figure 1

Effective hyperradial potential curve $V_{\nu ,\mathrm{eff}}(R)$, Eq. (7), as a function of $R$ for, from top to bottom, $s_{\nu }=1,3/4,1/2,9/20$ and $0$.

Figure 2

$s_0$ coefficient as a function of $\kappa$ for the $(2,1)$ system at unitarity with ZR interactions and $L^{\Pi }=0^{+}$ (dotted line), $L^{\Pi }=1^{-}$ (solid line), $L^{\Pi }=2^{+}$ (dashed line), and $L^{\Pi }=3^{-}$ (dash-dotted line). Three-body Efimov physics is absent if $s_0>0$, which corresponds to $\kappa \lesssim 13.607$ for the $L^{\Pi }=1^{-}$ system and to $\kappa \lesssim 76.0$ for the $L^{\Pi }=3^{-}$ system.

Figure 3

Noninteger quantum number $q$ as a function of $[L_{\nu q}(x_0)]{}^{-1}$ for (a) $s_{\nu }=3/5$, (b) $s_{\nu }=1/2$, and (c) $s_{\nu }=2/5$. In panels (a) and (c), the solid, dashed, and dotted lines correspond to $x_0=1\times {10}^{-3}$, $2\times {10}^{-3}$, and $3\times {10}^{-3}$, respectively. In panel (b), the solid, dashed, and dotted lines correspond to $x_0={10}^{-5}$, ${10}^{-3}$, and ${10}^{-1}$, respectively; the solid and dashed lines are indistinguishable on the scale shown. Note that the scale of the $x$ axis is different in all three panels.

Figure 4

Solid, dashed, dotted, and dash-dotted lines show the quantity $h(E,s_{\nu })$, Eq. (20), as a function of $s_{\nu }$ for $E=1\times {10}^{-2}\hslash \omega$, $1\times {10}^{-1}\hslash \omega$, $1\hslash \omega$, and $10\hslash \omega$, respectively.

Figure 5

Dashed lines show the noninteger quantum number $q$ of the trapped $N$-body system, obtained by solving Eq. (19) self-consistently, as a function of the generalized energy-dependent scattering length ${\mathcal{V}}_{s_{\nu }}(E)$ for (a) $s_{\nu }=3/5$ and (b) $s_{\nu }=2/5$. For comparison, solid lines show the $q$ values that result when $h(E,s_{\nu })$ is artificially set to zero.

Figure 6

Circles, squares, and diamonds show the SV energies for the $(2,1)$ system with $L^{\Pi }=1^{-}$ at unitarity for $\kappa =1,6$ and 10, respectively. The unlike particles interact through $V_{\mathrm{g}}$. Solid lines show three-parameter fits of the form ${\sum }_{j=0}^2{c}_j{r}_0^j$ to the SV energies.

Figure 7

Circles, squares, diamonds, triangles, pluses, and crosses show the SV energies for the $(2,1)$ system with $L^{\Pi }=1^{-}$ at unitarity for $\kappa =12.25,12.3,12.3131,12.314,12.315$ and $12.316$, respectively. The unlike particles interact through $V_{\mathrm{g}}$. Solid lines show three- to five-parameter fits to the SV energies.

Figure 8

Energies of the $(2,1)$ system with $L^{\Pi }=1^{-}$ at unitarity. Circles, crosses, and pluses show the SV energies for $V_{\mathrm{g}}$, extrapolated to the ZR limit, for the three energetically lowest-lying states as a function of $\kappa$. Solid and dotted lines show $E_{f,0q}$ ($q=0$ and $1$) and $E_{g,0q}$ ($q=-s_0$ and $-s_0+1$), respectively. This figure has been adapted from Ref. [49].

Figure 9

Symbols show the SV energies for the $(2,1)$ system with $L^{\Pi }=1^{-}$ interacting through $V_{\mathrm{g}}$ at unitarity for (a) $\kappa =12$ and (b) $\kappa =12.7$, respectively, as a function of $r_0$. For $\kappa =12$, the energies of the ground state (circles) and first excited state (pluses) are shown. For $\kappa =12.7$, the energies of the first and second excited states (pluses and squares) are shown. Solid lines show fits to the SV energies. Horizontal lines show the energy $E_{f,0q}$ for $q=0$ and $1$.

Figure 10

Eigenenergy $E$ of the trapped system at unitarity, obtained from Eq. (14), as a function of $R_0$ for (a) $s_{\nu }\approx 0.5579$ (lowest allowed $q$ value), (b) $s_{\nu }=1/2$ (lowest allowed $q$ value), (c) $s_{\nu }\approx 0.4180$ (lowest allowed $q$ value), and (d) $s_{\nu }\approx 0.4180$ (second lowest allowed $q$ value). Dash-dash-dotted, dash-dotted, dotted, dashed, and solid lines correspond to (a) $L_{\nu q}(x_0)=10,1,0,-1,$ and $-10$, (b) $L_{\nu q}(x_0)=2,1,0,-1,$ and $-2$, (c) $L_{\nu q}(x_0)=10,1,0,-1,$ and $-2$, and (d) $L_{\nu q}(x_0)=10,1,0,-1,$ and $-10$.

Figure 11

Dash-dotted, dotted, solid, and dashed lines show the scaled pair distribution function $4\pi P_{\mathrm{hl}}(r)r^2$ for the heavy-light pair of the $(2,1)$ system at unitary with $L^{\Pi }=1^{-}$ for (a) $\kappa =1,6.7,11$, and $11.5$, and (b) $\kappa =12,12.3,12.314$, and $12.5$, respectively. The heavy-light particles interact through $V_{\mathrm{g}}(r)$ with $r_0=0.003a_{\mathrm{ho}}$. Note the different scales of the axis in panels (a) and (b).

Figure 12

Dash-dotted, dotted, solid, and dashed lines show the scaled pair distribution function $4\pi P_{\mathrm{hh}}(r)r^2$ for the heavy-heavy pair of the $(2,1)$ system at unitary with $L^{\Pi }=1^{-}$ for (a) $\kappa =1,6.7,11$, and $11.5$, and (b) $\kappa =12,12.3,12.314$, and $12.5$, respectively. The heavy-light particles interact through $V_{\mathrm{g}}(r)$ with $r_0=0.003a_{\mathrm{ho}}$. Note the different scales of the axis in panels (a) and (b).

Figure 13

Hyperradial density $P_{\mathrm{hyper}}(R)$ of the $(2,1)$ system with $L^{\Pi }=1^{-}$ at unitary. Symbols show the results from the SV calculation where the heavy-light particles interact through $V_{\mathrm{g}}(r)$ with $r_0=0.003a_{\mathrm{ho}}$ for (a) $\kappa =1$ (circles), $\kappa =6.7$ (squares), $\kappa =11$ (diamonds), and $\kappa =11.5$ (triangles), and (b) $\kappa =12$ (circles), $\kappa =12.3$ (squares), $\kappa =12.314$ (diamonds), and $\kappa =12.5$ (triangles). Lines show the hyperradial density $|F_{0q}(x)|{}^2$, Eq. (12). The $s_0$ entering into Eq. (12) is taken from the ZR model. $q$ is set to $0$ for $\kappa =1$ and $6.7$, and calculated according to Eq. (10) with $E$ taken from the SV calculations for the other $\kappa$. Note the different scales of the axis in panels (a) and (b).

Figure 14

Circles, squares, and diamonds show the SV energies for the $(2,1)$ system with $L^{\Pi }=1^{-}$ interacting through $V_{\mathrm{g}}$ with $r_0=0.005a_{\mathrm{ho}}$ as a function of $a_{\mathrm{ho}}/a_s$ for $\kappa =12$, $12.314$, and $12.5$, respectively. Dotted lines show fits to the SV energies (see text for details). For comparison, the solid line shows the energy for two trapped atoms interacting through $V_{\mathrm{zr}}$.

Figure 15

Circles, pluses, squares, and triangles show the SV energies for the $(3,1)$ system with $L^{\Pi }=1^{+}$ interacting through $V_{\mathrm{g}}$ at unitarity as a function of $r_0$ for (a) $\kappa =1,2,4$ and $8$, and (b) $\kappa =10,10.4,10.5$ and $10.6$. Dotted lines are shown as a guide to the eye.

Figure 16

Circles, squares, and diamonds show the extrapolated ZR energies as a function of $\kappa$ for the $(2,1)$ system with $L^{\Pi }=1^{-}$ symmetry, the $(3,1)$ system with $L^{\Pi }=1^{+}$ symmetry, and the $(4,1)$ system with $L^{\Pi }=0^{-}$ symmetry, respectively. As a guide to the eye, dotted lines connect data points. For the $(3,1)$ and $(4,1)$ systems, our FR SV energies allow for a reliable extrapolation to the $r_0\to 0$ limit only for relatively small $\kappa$.

Figure 17

Hyperradial density $P_{\mathrm{hyper}}(R)$ of the $(3,1)$ system with $L^{\Pi }=1^{+}$ at unitary. Symbols show the results from the SV calculation where the heavy-light particles interact through $V_{\mathrm{g}}(r)$ for (a) $\kappa =1$ and $r_0=0.03a_{\mathrm{ho}}$ (circles), $\kappa =4$ and $r_0\approx 0.044a_{\mathrm{ho}}$ (diamonds), and $\kappa =8$ and $r_0=0.04a_{\mathrm{ho}}$ (triangles); (b) $\kappa =10$ and $r_0\approx 0.054a_{\mathrm{ho}}$ (circles), $\kappa =10$ and $r_0\approx 0.040a_{\mathrm{ho}}$ (diamonds), and $\kappa =10$ and $r_0\approx 0.027a_{\mathrm{ho}}$ (triangles); and (c) $\kappa =10.4$ and $r_0\approx 0.041a_{\mathrm{ho}}$ (circles), $\kappa =10.5$ and $r_0\approx 0.027a_{\mathrm{ho}}$ (diamonds), and $\kappa =10.6$ and $r_0\approx 0.027a_{\mathrm{ho}}$ (triangles). In panel (a), lines show the hyperradial density $|F_{0q}(R)|{}^2$, Eq. (12), with $q$ set to zero and $s_0$ determined by fitting the SV density to Eq. (12). Note the different scales of the axis in panels (a)–(c).