Collective excitation of a Bose-Einstein condensate by modulation of the atomic scattering length

We excite the lowest-lying quadrupole mode of a Bose-Einstein condensate by modulating the atomic scattering length via a Feshbach resonance. Excitation occurs at various modulation frequencies, and resonances located at the natural quadrupole frequency of the condensate and at the first harmonic are observed. We also investigate the amplitude of the excited mode as a function of modulation depth. Numerical simulations based on a variational calculation agree with our experimental results and provide insight into the observed behavior.

Figure 1

Quadrupole oscillation excited by the modulation of $a$. A series of in situ polarization phase-contrast images of condensates taken during excitation at $\Omega =(2\pi )$ 10 Hz, separated in time by 15 ms. The change in peak density is nearly an order of magnitude from the most compressed to the most extended condensate. The horizontal scale of these images has been stretched by a factor of 2 for clarity. There is negligible excitation of the dipole mode.

Figure 2

Axial [(red) circles] and radial [(blue) squares] $1/e$ sizes of a condensate during and after excitation with $a_{\mathrm{av}}~3\hspace{0.5em}{a}_0$ and $\delta a~2\hspace{0.5em}{a}_0$, where (a) $\Omega =(2\pi )3$ Hz or (b) $\Omega =(2\pi )10$ Hz. The natural oscillation frequency is ${\omega }_Q=(2\pi )8.2(1)$ Hz. Note that the 1.8-Hz beat note is the cause of the diminished amplitude in (b) after the modulation drive is turned off. The solid lines are results of the variational calculation using the same parameters as in the experiment and contain no adjustable parameters. The dashed lines are representative of the driving excitation. The resolution of the optical imaging system is $~3$ $\mu$m, which limits accurate determination of the radial sizes. The frequency of the breathing mode is on the order of 470 Hz and therefore no effects of this mode are present.

Figure 3

Fractional amplitude of the drive [(red) circles] and quadrupole [(blue) squares] modes during excitation at frequency $\Omega$ with a modulation depth $\delta a~1\hspace{0.5em}{a}_0$, where $a_{\mathrm{av}}~3\hspace{0.5em}{a}_0$. The solid lines are results of the variational calculation with no adjustable parameters. The oscillation is notably asymmetric for fractional amplitudes of order 1 and larger, as shown in Fig. 2b.

Figure 4

Quadrupole oscillation driven by a large-amplitude excitation, $\delta a~3\hspace{0.5em}{a}_0$ where $a_{\mathrm{av}}~3\hspace{0.5em}{a}_0$, near resonance $\Omega =(2\pi )9$ Hz. The nonsinusoidal behavior leads to fragmentation of the condensate during the compression stages when the axial size becomes smaller than the axial harmonic oscillator size of $~17$ $\mu$m. The image at right was taken during the compression stage at $t=0.52$ s and shows fragmentation of the condensate; the field of view is $25\times 380$ $\mu {\text{m}}^2$.

Figure 5

Fractional amplitude of oscillation, when $\Omega \approx {\omega }_Q$, as a function of the fractional modulation depth $\delta a/a_{\mathrm{av}}$ after 1 s of excitation. The solid curve is the result of the variational calculation.