Low-energy resonances and bound states of aligned bosonic and fermionic dipoles

The low-energy scattering properties of two aligned identical bosonic and identical fermionic dipoles are analyzed. Generalized scattering lengths are determined as functions of the dipole moment and the scattering energy. Near resonance, where a new bound state is being pulled in, all nonvanishing generalized scattering lengths diverge, with the $a_{00}$ and $a_{11}$ scattering lengths being dominant for identical bosons and identical fermions, respectively. Implications for the energy spectrum and the eigenfunctions of trapped two-dipole systems and for pseudopotential treatments are discussed.

Figure 1

(Color online) Scattering data [(a)–(d) for identical bosons and (e)–(g) for identical fermions]: Scaled scattering lengths (b) $a_{00}∕r_c$, (c) $a_{20}∕r_c$, (d) $a_{22}∕r_c$, (f) $a_{11}∕r_c$, and (g) $a_{31}∕r_c$ as a function of the scaled dipole length $D_{*}∕r_c$ for two dipoles interacting through $V_{\mathrm{m}}$ for three different scattering energies: $E_{\mathrm{sc}}=9.36\times {10}^{-8}{E}_{r_c}$ (solid lins), $E_{\mathrm{sc}}=9.36\times {10}^{-6}{E}_{r_c}$ (dashed lines), and $E_{\mathrm{sc}}=9.36\times {10}^{-5}{E}_{r_c}$ (dotted lines). In (a), crosses and squares indicate the positions of $B_1$ and $B_2$ resonances, respectively. In (e), crosses and squares indicate the positions of $F_1$ and $F_2$ resonances, respectively. The resonance positions are obtained by analyzing the WKB phase of the adiabatic potential curves (see text).

Figure 2

(Color online) Eigenenergies for two identical bosons in the vicinity of (a) a $B_1$ resonance and (b) a $B_2$ resonance and for two identical fermions in the vicinity of (c) a $F_1$ resonance and (d) a $F_2$ resonance. Solid lines and circles show the bound-state energies for two dipoles in free space interacting through $V_{\mathrm{m}}$ and $V_{\mathrm{ps}}$, respectively. Dashed lines and crosses show the bound-state energies for two dipoles under external harmonic confinement interacting through $V_{\mathrm{m}}$ and $V_{\mathrm{ps}}$, respectively. The oscillator length $a_{\mathrm{ho}}$ is given by $\sqrt{\hslash ∕(\mu \omega )}$, and the hard-core radius $r_c$ of $V_{\mathrm{m}}$ is $0.00306a_{\mathrm{ho}}$.

Figure 3

(Color online) Eigenfunctions for two identical bosons in the vicinity of (a) a $B_1$ resonance $(D_{*}=0.1061a_{\mathrm{ho}})$ and (b) a $B_2$ resonance $(D_{*}=0.07147a_{\mathrm{ho}})$ and for two identical fermions in the vicinity of (c) a $F_1$ resonance $(D_{*}=0.07976a_{\mathrm{ho}})$ and (d) a $F_2$ resonance $(D_{*}=0.07167a_{\mathrm{ho}})$ in a spherical harmonic trap interacting via $V_{\mathrm{m}}$ for $\theta =0°$ (solid lines), $\theta \approx 18.2°$ (dashed lines), $\theta \approx 36.4°$ (dotted lines), and $\theta \approx 54.6°$ (dash-dotted lines). In all panels, $r_c$ equals $0.00306a_{\mathrm{ho}}$. Note the logarithmic scale for $r$ and the different ranges of the $y$ axis.