Constructing a classical field for a Bose-Einstein condensate in an arbitrary trapping potential: Quadrupole oscillations at nonzero temperatures

We optimize the classical field approximation of the version described by M. Brewczyk, M. Gajda, and K. Rzążewski [J. Phys. B 40, R1 (2007)] for the oscillations of a Bose gas trapped in a harmonic potential at nonzero temperatures, as experimentally investigated by Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. Similar to experiment, the system response to external perturbations strongly depends on the initial temperature and the symmetry of perturbation. While for lower temperatures the thermal cloud follows the condensed part, for higher temperatures the thermal atoms oscillate rather with their natural frequency, whereas the condensate exhibits a frequency shift toward the thermal cloud frequency ($m=0$ mode) or in the opposite direction ($m=2$ mode). In the latter case, for temperatures approaching critical, we find that the condensate begins to oscillate with the frequency of the thermal atoms, as in the $m=0$ mode. A broad range of frequencies of the perturbing potential is considered.

Figure 1

Cuts of the total (thick solid line), condensate (dashed line), and thermal (thin solid line) densities as obtained by integrating a one-particle density matrix along the direction of imaging and then applying the splitting procedure described in Sec. 2a. Note the bimodal character of the density distribution (here, the condensate fraction equals $0.3$). The oscillatory unit of length is defined via the axial trap frequency, $\sqrt{\frac{\hslash }{m{\omega }_z}}$, and equals $0.565$ $\mu$m.

Figure 2

Relative populations of various harmonic oscillator states. The energy cutoff equals $p_{\max }^2/m$, where $p_{\max }$ is the momentum cutoff.

Figure 3

Relative populations multiplied by the state energies for various harmonic oscillator states. The figure clearly shows that for higher energy states the equipartition of an energy is established. Therefore, the higher energy states become quasiparticle modes. The fraction $k_{B}T/N$ equals approximately $4.62\times {10}^{-4}$.

Figure 4

Total density cuts for the self-consistent Hartree-Fock (thick line) and classical field (thin line) methods for a system with the total number of atoms $N=17\hspace{0.5em}306$ and the temperature $T=128.7\hspace{0.5em}$nK.

Figure 5

Widths of a condensate (a) and its thermal cloud (b) as a function of time for the $m=0$ mode. The widths [solid symbols in (a) and open symbols in (b)] are calculated from the formula (23) after the condensate and the thermal cloud are extracted out of the classical field. The solid lines are fits to an exponentially damped sine function: $A_{c,\mathrm{th}}\hspace{0.5em}\exp (-{\gamma }_{c,\mathrm{th}}t)\sin ({\omega }_{c,\mathrm{th}}t+{\vartheta }_{c,\mathrm{th}})+B_{c,\mathrm{th}}$. These fits allow us to obtain the frequencies and the damping rates of the oscillations of the condensate and the thermal cloud in response to the external perturbation. The initial temperature of the cloud and the driving frequency are $T^{\text{'}}=0.8$ and ${\omega }_d=1.75\hspace{0.5em}{\omega }_{\perp }$. Panel (c) shows more clearly the frequency and phase shifts between the condensate and the thermal cloud.

Figure 6

Condensate (top) and the thermal cloud (bottom) frequencies for the $m=0$ collective mode as a function of reduced temperature. Black (solid and open) symbols correspond to the numerical results obtained for various driving frequencies (according to the legend), whereas gray symbols (solid for the condensate and open for the thermal cloud) with error bars are taken from the experiment by Jin et al. [3].

Figure 7

Damping rates for the condensate (top) and the thermal cloud (bottom) for the $m=0$ mode as a function of reduced temperature. Black (solid and open) symbols correspond to the numerical results obtained for various driving frequencies (according to the legend), whereas gray symbols (solid for the condensate and open for the thermal cloud) with error bars are taken from the experiment by Jin et al. [3].

Figure 8

Condensate (solid symbols) and thermal cloud (open symbols) frequencies for the $m=2$ collective mode as a function of reduced temperature. Black symbols correspond to the numerical results obtained for various driving frequencies (according to the legend; here the shape of the symbol determines the driving frequency), whereas gray symbols with error bars are taken from the experiment by Jin et al. [3]. Note, however, that the only experimental point for the thermal cloud is marked with an open gray circle.

Figure 9

Damping rates for the condensate (top) and the thermal cloud (bottom) for the $m=2$ mode as a function of reduced temperature. Black (solid and open) symbols correspond to the numerical results obtained for various driving frequencies (according to the legend), whereas gray symbols (solid for the condensate and open for the thermal cloud) with error bars are taken from the experiment by Jin et al. [3]. Note that the only experimental point for the thermal cloud is marked with an open gray circle.