Bragg Spectroscopy of a Strongly Interacting ${}^{85}\mathrm{Rb}$ Bose-Einstein Condensate

We report on measurements of the excitation spectrum of a strongly interacting Bose-Einstein condensate. A magnetic-field Feshbach resonance is used to tune atom-atom interactions in the condensate and to reach a regime where quantum depletion and beyond mean-field corrections to the condensate chemical potential are significant. We use two-photon Bragg spectroscopy to probe the condensate excitation spectrum; our results demonstrate the onset of beyond mean-field effects in a gaseous Bose-Einstein condensate.

Figure 1

Theoretical predictions for the peak of the Bragg resonance less the bare kinetic energy for an excitation of wave vector $k=2\frac{2\pi }{780\mathrm{nm}}$, in the case of homogenous density $7.6\times {10}^{13}{\mathrm{cm}}^{-3}$. The theory lines are described in more detail in the text and correspond to assuming the following quantities are small: $ka$, $\sqrt{8\pi n{a}^3}$, and $1/(k\xi )$ (solid green line); $ka$ and $\sqrt{8\pi n{a}^3}$ (dotted blue line); $ka$ (dash-dot-dotted magenta line); $\sqrt{8\pi n{a}^3}$ (dashed red line); and $1/(k\xi )$ (dash-dotted black line). For reference the top axis shows the size of the LHY correction, $\alpha$, relative to the condensate chemical potential.

Figure 2

Typical Bragg spectra at a scattering length of $100a_0$ (blue triangles), $585a_0$ (red circles), and $890a_0$ (black squares). The excitation fraction is determined from the measured momentum transferred to the BEC and plotted as a function of the frequency difference between the two Bragg beams. Lines are fits of the data as described in the text. Mean-field theory predicts a continuous increase in the line shift with increasing $a$; however, by $890a_0$ our data display a decreasing shift with stronger interactions.

Figure 3

(a) Bragg line shift and (b) width as a function of scattering length. In (a) the open circles are our observations. The solid circles are data corrected for a fitting systematic associated with the broad thermal atom background, and the error bars represent fit uncertainties. The theory lines and LHY correction are as in Fig. 1 except they are calculated for the trapped gas using a local density approximation for each of the corresponding data points. The mean BEC density ranges from $6.3\times {10}^{13}{\mathrm{cm}}^{-3}$ to $7.6\times {10}^{13}{\mathrm{cm}}^{-3}$. Error bars on the theory lines reflect uncertainty in these densities. Some of the error bars have been omitted for clarity. In (b) the solid black circles are the rms width of a Gaussian fit to the Bragg spectra. Black triangles are from a fit to a convolution of various contributions to the width calculated under the conditions of our measurements. The remaining symbols characterize constituent contributions to the convolution including the Lorentzian FWHM width due to collisions (blue squares), and the rms width of a Gaussian fitted to the contributions due to the inhomogeneous density (red diamonds) and the pulse duration (green circles). The largest contribution to the width comes from the pulse duration; because the jump to large $a$ initiates rapid expansion of the BEC, ever shorter pulses are used to obtain the spectra at larger $a$.