Collective Excitations of a Trapped Bose-Einstein Condensate in the Presence of a 1D Optical Lattice

We study low-lying collective modes of an elongated ${}^{87}\mathrm{R}\mathrm{b}$ condensate produced in a 3D magnetic harmonic trap with the addition of a 1D periodic potential which is provided by a laser standing wave along the axial direction. While the transverse breathing mode remains unperturbed, quadrupole and dipole oscillations along the optical lattice are strongly modified. Precise measurements of the collective mode frequencies at different heights of the optical barriers provide a stringent test of the theoretical model recently introduced [M. Krämer et al., Phys. Rev. Lett. 88, 180404 (2002)].

Figure 1

In the upper part we show absorption images taken after exciting the quadrupole mode of a condensate in the combined trap with $V_{\mathrm{o}\mathrm{p}\mathrm{t}}=3.2E_r$ and waiting different evolution times (10, 40, 60, 80, and 100 ms) before switching off the trap and letting the cloud expand for 29 ms. In the lower part we show the evolution of the aspect ratio ($R_z/R_{\perp }$) of the central interference peak obtained from the absorption images together with a sinusoidal fit to extract the frequency of the mode.

Figure 2

Frequency of the quadrupole mode of a condensate trapped in the combined potential ($\mathrm{\text{harmonic magnetic trap}}+1D$ optical lattice) as a function of the dipole mode frequency measured for different values of the optical lattice depth from $0E_r$ to $3.7E_r$. The frequencies of both the modes, characterized by a dynamics along the optical lattice, show a marked dependence on the optical potential depth. The line represents a linear fit with a slope of $1.57\pm 0.01$.

Figure 3

Effective mass values extracted from the dipole mode frequency (open circle) and from the quadrupole mode frequency (closed circle) as a function of the optical lattice height. The continuous line represents the theoretical curve from 16 obtained neglecting the role of the mean field interactions, while dashed and dotted lines correspond, respectively, to the values obtained in 22 numerically solving the Gross-Pitaevskii equation and evaluating the effective mass from the quadrupole and the dipole mode frequencies.

Figure 4

Frequency of the transverse breathing mode of the condensate in the combined trap as a function of the optical lattice depth $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$. The dashed line corresponds to the expected value $2{\nu }_{\perp }$.