Bragg Spectroscopy of the Multibranch Bogoliubov Spectrum of Elongated Bose-Einstein Condensates

We measure the response of an elongated Bose-Einstein condensate to a two-photon Bragg pulse. If the duration of the pulse is long, the total momentum transferred to the condensate exhibits a nontrivial behavior which reflects the structure of the underlying Bogoliubov spectrum. It is thus possible to perform a spectroscopic analysis in which axial phonons with a different number of radial nodes are resolved. The local density approximation is shown to fail in this regime, while the observed data agree well with the results of simulations based on the numerical solution of the Gross-Pitaevskii equation.

Figure 1

$P_z$, in arbitrary units, as a function of $\omega$ for short pulses ($t_B=1\mathrm{m}\mathrm{s}$). Bottom, center, and top plots correspond to different $k$ values. Points with error bars are the measured values. Solid lines are the results of GP simulations. The dashed lines are the LDA predictions.

Figure 2

The time dependence of the momentum transferred during the Bragg pulse, for $k=3.1\mu {\mathrm{m}}^{\mathrm{-}\mathrm{1}}$. Solid curves are the results of GP simulations with $V_B=0.2\hslash {\omega }_{\perp }$ plotted at various times. Starting with the lowest solid curve, $t_B=1,2,3,4,5$, and 6 msec. The dashed curves are the corresponding LDA predictions. The arrows indicate the GP predictions for the frequencies of the normal modes of the condensate, having 0, 1, and 2 radial nodes and the same axial wave vector $k$.

Figure 3

$P_z$ in units of $N\hslash k$, as a function of $\omega$ for long Bragg pulses. For (a) and (b), $t_B=10\mathrm{m}\mathrm{s}\mathrm{e}\mathrm{c}$ and 6 msec, respectively, and $k=1.4\mu {\mathrm{m}}^{\mathrm{-}\mathrm{1}}$ and $3.1\mu {\mathrm{m}}^{\mathrm{-}\mathrm{1}}$, respectively. The circles are the measured values. The dashed line is a guide to the eye. The upper solid curve is the result of GP simulations with $V_B=0.35\hslash {\omega }_{\perp }$ and $0.2\hslash {\omega }_{\perp }$, for (a) and (b), respectively. The lower solid curve is the same but with $V_B$ 10 times smaller. For the lower curve, $P_z$ is multiplied by 10 and 20 for (a) and (b), respectively. The arrows on the $\omega$ axes indicate the normal mode frequencies, as in Fig. 2. (c) is the same as (b), but with greater intensity. For the GP simulation in (c), $V_B=0.26\hslash {\omega }_{\perp }$. The dotted curves are the LDA predictions.

Figure 4

Excitation frequencies vs $k$ for a trapped Bose-Einstein condensate. The open circles are the frequencies of the normal modes with $n_r=0,1,2,\dots$ nodes in the radial direction, as calculated from the GP equation. The filled circles are the position of the maxima of the main peaks in $P_z$ for the long-pulse measurements of Fig. 3. The dashed lines are the prediction for an infinite cylinder with $k\ll {\xi }^{-1}$. The dotted line indicates the inverse healing length.