Heating and atom loss during upward ramps of Feshbach resonance levels in Bose-Einstein condensates

The production of pairs of fast atoms leads to a pronounced loss of atoms during upward ramps of Feshbach resonance levels in dilute Bose-Einstein condensates. We provide comparative studies on the formation of these bursts of atoms containing the physical predictions of several theoretical approaches at different levels of approximation. We show that, despite their very different description of the microscopic binary physics during the passage of a Feshbach resonance, all approaches lead to virtually the same prediction on the total loss of condensate atoms, provided that the ramp of the magnetic field strength is purely linear. We give the reasons for this remarkable insensitivity of the remnant condensate fraction to the microscopic physical processes and compare the theoretical predictions with recent Feshbach resonance crossing experiments on ${}^{23}\mathrm{Na}$ and ${}^{85}\mathrm{Rb}$.

Figure 1

Magnetic field dependence of the binding energy $E_{\mathrm{b}}\left(B\right)$ of the highest excited multichannel vibrational bound state ${\varphi }_{\mathrm{b}}$ of ${}^{85}\mathrm{Rb}_2$ (solid curve) and the energy $E_{\mathrm{res}}\left(B\right)$ of the closed channel resonance state ${\varphi }_{\mathrm{res}}\left(r\right)$ (dashed curve). $E_{\mathrm{res}}\left(B\right)$ depends linearly on $B$ with $h^{-1}d{E}_{\mathrm{res}}∕dB=-3.46\mathrm{MHz}∕\mathrm{G}$. The inset shows $E_{\mathrm{b}}\left(B\right)$ (solid curve) as compared to the universal asymptotic formula $-{\hslash }^2∕m{a}^2$ for the near resonant binding energy (dotted curve) and its first improvement [42] that accounts for the length scale set by the van der Waals interaction in a single channel picture (dot-dashed curve).

Figure 2

Magnetic field dependence of the modulus of the binding energy of the highest excited multichannel vibrational bound state of ${}^{23}\mathrm{Na}_2$ in the vicinity of the $907\mathrm{G}$ Feshbach resonance. The figure compares the binding energies supported by the two-channel low energy potential matrix of Ref. 23 and this work (solid curve) with the highly accurate two-channel model of Ref. 22 (filled circles). The dotted curve indicates the universal asymptotic formula $E_{\mathrm{b}}=-{\hslash }^2∕m{a}^2$ and the dashed curve is the Gribakin and Flambaum [42] estimate that accounts for the length scale set by the van der Waals interaction in a single channel picture. We note that the binding energies are given on a logarithmic scale.

Figure 3

Magnetic field dependence of the population of the closed channel component of the highest excited vibrational bound state ${\varphi }_{\mathrm{b}}$ associated with the ${}^{23}\mathrm{Na}$ $907\mathrm{G}$ and the ${}^{85}\mathrm{Rb}$ $155\mathrm{G}$ Feshbach resonances. The populations were determined from the two-channel Hamiltonian (4).

Figure 4

Loss of atoms from a ${}^{23}\mathrm{Na}$ condensate at the end of a linear magnetic field ramp across the $853\mathrm{G}$ and $907\mathrm{G}$ Feshbach resonances, respectively, as a function of the inverse ramp speed. The filled circles are the experimental data from Ref. 21. The diamonds are determined from Eq. (16) using a spherical trap potential with the mean experimental trap frequency. The solid curve represents the calculation in the local density approximation, solving Eq. (16) for uniform gases, with densities weighted according to the initial distribution in a cylindrical trap. The dashed curve shows the corresponding result from a single channel calculation as described in Refs. 23, 28. The triangles correspond to the predictions of the two-level mean field approach of Eqs. (28, 29) with a spherical trap potential.

Figure 5

Loss of atoms from a ${}^{85}\mathrm{Rb}$ condensate at the end of a linear ramp of the magnetic field strength across the $155\mathrm{G}$ Feshbach resonance as a function of the inverse ramp speed. The circles with error bars are the experimental data from Ref. 2. The squares are determined from Eq. (16) using a cylindrical trap potential with the experimental trap frequencies. The diamonds are obtained with a spherical trap potential that accounts only for the mean trap frequency, and the stars represent a local density approximation for the cylindrical trap. The triangles correspond to the two-level mean field approach of Eqs. (28, 29) in a spherical trap. The inset shows the dynamics of the loss of condensate atoms in the cylindrical trap for the $20\mu \mathrm{s}∕\mathrm{G}$ ramp.

Figure 6

Dissociation spectra of the highest excited vibrational bound state of ${}^{85}\mathrm{Rb}_2$ as a function of the relative energy of the fragments. The speeds of the linear ramps across the $155\mathrm{G}$ Feshbach resonance were varied between $50\mathrm{G}∕\mathrm{ms}$ (solid curve) and $1000\mathrm{G}∕\mathrm{ms}$ (double dashed dotted curve). The ramps started from the experimental evaporation magnetic field strength of $B_i=162.2\mathrm{G}$ [2] down to a final magnetic field strength $B_f<B_0=154.9\mathrm{G}$, which was chosen sufficiently small that at each ramp speed the spectrum was insensitive to variations of $B_f$. The exact calculations based on Eq. (30) were performed with the two-channel Hamiltonian (4). We note that the relative kinetic energies of the molecular fragments are given on a logarithmic scale.

Figure 7

Mean single particle kinetic energies of the molecular fragments associated with the exact dissociation spectra obtained from Eq. (30) in Fig. 6 (circles) in dependence on the ramp speed. The solid curve has been obtained from the asymptotic prediction in Eq. (32) for comparison. The single particle kinetic energy is half the energy of the relative motion of the fragments in Fig. 6.