Colliding Bose-Einstein Condensates to Observe Efimov Physics

We explore the manifestation of Efimov physics through the collision energy dependence of the three-body scattering observables and propose that it can be measured by observing atom loss in collisions of Bose-Einstein condensates. Our study shows that log-periodic Efimov features in the scattering observables extend beyond the usual threshold regime to nonzero collision energies and result from two interfering pathways. Further, these oscillations have a one-to-one connection with the scattering length oscillations at zero energy and thus to Efimov states themselves.

Figure 1

Schematic potentials and pathways—#1 and #2—for recombination with $J=0$. Channel $\alpha$ represents three free atoms; channel $\beta$, the atom plus dimer; and $V_{\alpha \beta }$, the coupling between the two channels.

Figure 2

The $J=0$ recombination rate $K_3$ scaled as $K_3{E}^2/a^4$ [28] to emphasize the Efimov oscillations, especially the connection between the $a$ dependence and the $E$ dependence from Eqs. (2, 5), respectively. Equation (3) was used with $\mathrm{Cs}+\mathrm{Cs}+\mathrm{Li}$ masses, ${\Phi }^{\prime }=0$, and $r_0=15\mathrm{bohr}$. The amplitudes of the two interfering pathways were assumed equal to emphasize the oscillations.

Figure 3

Scaled three-body recombination rates $K_3(E)E^2$ for (a) $\mathrm{Cs}+\mathrm{Cs}+\mathrm{Cs}$ with $a_{\mathrm{Cs}+\mathrm{Cs}}=8000\mathrm{bohr}$ and (b) $\mathrm{Cs}+\mathrm{Cs}+\mathrm{Li}$ with $a_{\mathrm{Cs}+\mathrm{Li}}=5000\mathrm{bohr}$. The lowest two contributing angular momenta are shown along with their sum and its thermal average, $⟨K_3(T)⟩T^2$. The dotted lines are from Eq. (5). The expected lower and upper limits of universal behavior are indicated by the dashed lines at $E_{\mathrm{th}}\approx {\hslash }^2/2{\mu }_3{a}^2$ and $E_s\approx {\hslash }^2/2{\mu }_3{r}_0^2$, respectively. The rapid oscillations in the $J=1^{-}$ contribution in (b) are not related to the Efimov effect.

Figure 4

The $J=0^{+}$ contribution to $K_3(E)E^2$ for $\mathrm{Cs}+\mathrm{Cs}+\mathrm{Li}$ with different $a_{\mathrm{Cs}+\mathrm{Li}}$ (in bohr). At $a_{\mathrm{Cs}+\mathrm{Li}}=1000\mathrm{bohr}$, the two oscillations as $E$ decreases from $E_s$ correspond to the ground and first excited Efimov states, respectively. At $a_{\mathrm{Cs}+\mathrm{Li}}=5000\mathrm{bohr}$, the second excited Efimov state appears since the number of oscillations increases to three. The dotted line is from Eq. (5).

Figure 5

(a) The time-dependent fractional atom loss, $1-N(t)/N_0$, during the collision of a Li BEC with a Cs BEC at $E=4\mu \mathrm{K}$ along with snapshots of the atomic densities in the overall center-of-mass frame. Though the condensate dynamics is quite complicated, the loss is always smooth during the collision. Initially, there are ${10}^4$ atoms in each condensate and $a_{\perp }=\sqrt{\hslash /m_{\mathrm{Cs}}{\omega }_{\mathrm{Cs}}}$ with ${\omega }_{\mathrm{Cs}}=2\pi 800\mathrm{Hz}$. (b) The final loss fraction of Cs atoms multiplied by $E^{2+\gamma /2}$ (filled and open circles) to show the Efimov oscillations. Oscillations are seen in both cases, and their positions correspond to those seen in $K_3{E}^2$ (no symbols).