Production of three-body Efimov molecules in an optical lattice

We study the possibility of associating metastable Efimov trimers from three free Bose atoms in a tight trap realized, for instance, via an optical lattice site or a microchip. The suggested scheme for the production of these molecules is based on magnetically tunable Feshbach resonances and takes advantage of the Efimov effect in three-body energy spectra. Our predictions of the energy levels and wave functions of three pairwise interacting ${}^{85}\mathrm{Rb}$ atoms rely upon exact solutions of the Faddeev equations and include the tightly confining potential of an isotropic harmonic atom trap. The magnetic field dependence of these energy levels indicates that it is the lowest-energetic Efimov trimer state that can be associated in an adiabatic sweep of the field strength. We show that the binding energies and spatial extents of the trimer molecules produced are comparable, in their magnitudes, to those of the associated diatomic Feshbach molecule. The three-body molecular state follows Efimov’s scenario when the pairwise attraction of the atoms is strengthened by tuning the magnetic field strength.

Figure 1

(Color online) Magnetic field dependence of the vibrational energy levels $E_{2\mathrm{B}}$ of a pair of ${}^{85}\mathrm{Rb}$ atoms in a ${\nu }_{\mathrm{ho}}=300\mathrm{kHz}$ trap (upper part) as compared to the binding energies of the highest excited vibrational state of the ${}^{85}\mathrm{Rb}_2$ dimer molecule in free space (lower part). The circles in the upper part indicate numerical solutions of the two-body Schrödinger equation for a microscopic interaction potential explicitly incorporating the exact scattering length as well as the exact asymptotic $-C_6∕r^6$ interaction energy, while the solid curves indicate calculations using the separable potential approach of Appendix . The diamonds in the lower part are experimental binding energies obtained from Ref. 46, and the squares indicate results of the full coupled-channels calculations of Kokkelmans [44]. The solid curve indicates the dimer binding energies obtained from the separable potential approach, while the dash-dotted curve corresponds to their near-resonant approximation of Eq. (5). The inset of the lower part of the figure shows the singularity of the scattering length at the magnetic field strength $B_0=155.041\mathrm{G}$ [46].

Figure 2

(Color online) Magnetic field dependence of the vibrational energy levels of ${}^{85}\mathrm{Rb}_3$ trimers relative to the binding energy $E_{\mathrm{b}}$ of the ${}^{85}\mathrm{Rb}_2$ Feshbach molecule (using $E_{\mathrm{b}}=0$ for $B<B_0$) in free space (cf. Fig. 1). The first Efimov states, whose energies are indicated by the solid and dashed curves, emerge at about 154.4 G and 155 G, respectively. The energies of the other Efimov states are not resolved even on the logarithmic energy scale. The second Efimov state (dashed curve) ceases to exist at about 155.5 G.

Figure 3

(Color online) Magnetic field dependence of the vibrational energy levels $E_{3\mathrm{B}}$ of three interacting ${}^{85}\mathrm{Rb}$ atoms in a low-frequency ${\nu }_{\mathrm{ho}}=200\mathrm{Hz}$ trap as compared to the energies of the free space Efimov states of Fig. 2. The energies of the trapped atoms are shown relative to $E_{2\mathrm{B}}+\frac{3}{2}h{\nu }_{\mathrm{ho}}$, where we have chosen $E_{2\mathrm{B}}$ as the lowest energy of a pair of trapped ${}^{85}\mathrm{Rb}$ atoms—i.e., the two-body level that correlates adiabatically—in the limit ${\nu }_{\mathrm{ho}}\to 0$, with the binding energy $E_{\mathrm{b}}$ of the Feshbach molecule of Fig. 1. In analogy, we have subtracted the dimer binding energy $E_{\mathrm{b}}$ from the Efimov trimer energies in free space for the purpose of comparison.

Figure 4

(Color online) Magnetic field dependence of the vibrational energy levels $E_{3\mathrm{B}}$ of three ${}^{85}\mathrm{Rb}$ atoms relative to the zero-point energy $\frac{6}{2}h{\nu }_{\mathrm{ho}}$ of three hypothetically noninteracting atoms in a tightly confining ${\nu }_{\mathrm{ho}}=300\mathrm{kHz}$ trap.

Figure 5

(Color online) Hyperradial probability densities $P\left(R\right)$ of the lowest-energetic vibrational states of three ${}^{85}\mathrm{Rb}$ atoms at $B=158.1\mathrm{G}$ for the trap frequencies ${\nu }_{\mathrm{ho}}=1\mathrm{MHz}$, ${\nu }_{\mathrm{ho}}=300\mathrm{kHz}$, and ${\nu }_{\mathrm{ho}}=50\mathrm{kHz}$. The legends show the associated energies of the interacting atoms relative to the zero-point energy of three hypothetically noninteracting atoms—i.e., $E_{3\mathrm{B}}-\frac{6}{2}h{\nu }_{\mathrm{ho}}$. The hyperradius is given on a logarithmic scale.

Figure 6

Jacobi coordinates of the relative motion of three atoms. The set of coordinates $\mathbf{\rho }$ and $\mathbf{r}$ is selected in such a way that it is suited to describe the hypothetical situation of an interacting pair of atoms (2,3) with atom 1 playing the role of a spectator.

Figure 7

The reduced kernel matrix ${\mathcal{K}}_{\kappa {\kappa }^{\prime }}\left(E_{3\mathrm{B}}\right)$ of Eq. (B29), at low vibrational excitations $\kappa ,{\kappa }^{\prime }<100$, for a ${\nu }_{\mathrm{ho}}=300\mathrm{kHz}$ trap and a magnetic field strength of $B=158.1\mathrm{G}$. The transition from the linear to the connecting exponential mesh at $\kappa =N_{\mathrm{e}}=40$ is illustrated by the logarithmic scale of the axes associated with the vibrational quantum numbers $\kappa$ and ${\kappa }^{\prime }$ [cf. Eq. (B75)]. In the linear part $(0⩽\kappa ,{\kappa }^{\prime }<N_{\mathrm{e}})$ the reduced kernel matrix was evaluated exactly using the results of Appendix while the approximation of Eq. (B60) was used in the exponential part. The logarithmic scaling of the axes associated with $\kappa$ and ${\kappa }^{\prime }$ prevents the row $\kappa =0$ and the column ${\kappa }^{\prime }=0$ from being displayed.

Figure 8

(Color online) Solutions $f_{\kappa }$ of the matrix equation (B29), using the reduced kernel matrix shown in Fig. 7, versus the vibrational quantum number $\kappa$. The legends show the energies $E_{3\mathrm{B}}-\frac{6}{2}h{\nu }_{\mathrm{ho}}$ of the associated three-body levels of interacting ${}^{85}\mathrm{Rb}$ atoms relative to the zero-point energy of hypothetically noninteracting atoms in the ${\nu }_{\mathrm{ho}}=300\mathrm{kHz}$ trap.