Collective oscillations of a confined Bose gas at finite temperature in the random-phase approximation

We present a theory for the linear dynamics of a weakly interacting Bose gas confined inside a harmonic trap at finite temperature. The theory treats the motions of the condensate and of the noncondensate on an equal footing within a generalized random-phase approximation, which (i) extends the second-order Beliaev-Popov approach by allowing for the dynamical coupling between fluctuations in the thermal cloud, and (ii) reduces to an earlier random-phase scheme when the anomalous density fluctuations are omitted. Numerical calculations of the low-lying spectra in the case of isotropic confinement show that the present theory obeys with high accuracy the generalized Kohn theorem for the dipolar excitations and demonstrate that combined normal and anomalous density fluctuations play an important role in the monopolar excitations of the condensate. Mean-field theory is instead found to yield accurate results for the quadrupolar modes of the condensate. Although the restriction to spherical confinement prevents quantitative comparisons with measured spectra, it appears that the non-mean-field effects that we examine may be relevant to explain the features exhibited by the breathing mode as a function of temperature in the experiments carried out at JILA on a gas of ${}^{87}\mathrm{Rb}$ atoms.

Figure 1

Spectral responses (in arbitrary units) for the monopolar excitation as functions of frequency $\omega$ (in units of ${\omega }_0$) as calculated from the HF-RPA at $T∕T_c=0.5$, plotted for two values of $\eta$ (in units of ${\omega }_0$) as indicated in the panels. The three panels display the total spectral response (a) and the contributions of the condensate (b) and of the noncondensate (c).

Figure 2

Spectral response (in arbitrary units) for the monopolar excitation as a function of frequency $\omega$ (in units of ${\omega }_0$) as calculated from the HF-RPA at $T∕T_c=0.5$ and $\eta =0.005{\omega }_0$ with two kinds of basis set.

Figure 3

(a) Spectral response (in arbitrary units) for the dipolar excitation as a function of frequency $\omega$ (in units of ${\omega }_0$), as calculated from the HF-RPA at $T∕T_c=0.6$ with the choice $\eta =0.005{\omega }_0$. The density fluctuations at resonance (in arbitrary units) are plotted as functions of the radial coordinate $r$ [in units of $a_{ho}={(\hslash ∕m{\omega }_0)}^{1∕2}$] in (b) for the condensate and in (c) for the noncondensate (solid lines). In the same panels are also shown the corresponding results from the analytical expressions of the mode eigenvectors (circles).

Figure 4

The same as Fig. 3, but for the HFB-RPA.

Figure 5

Spectral response (in arbitrary units) as a function of frequency $\omega$ (in units of ${\omega }_0$) for the monopole mode (a) and the quadrupole mode (b), as calculated with $\eta =0.01{\omega }_0$ from the HFB-RPA at the temperatures indicated in the figure. The curves are progressively shifted upwards by one unit for clarity and the quadrupole response at $T∕T_c=0.1$ is reduced by a factor of 3. The dashed line in each panel indicates how the condensate resonance moves with temperature.

Figure 6

Amplitude of the HFB-RPA resonances (in arbitrary units) from Fig. 5 as a function of reduced temperature $T∕T_c$ for the monopole (a) and the quadrupole (b). The solid and empty circles refer to the condensate and to the noncondensate, respectively. The lines are guides to the eye.

Figure 7

Spectral response (in arbitrary units) as a function of frequency $\omega$ (in units of ${\omega }_0$) for the monopole mode (a) and the quadrupole mode (b), at $T∕T_c=0.6$ with $\eta =0.01{\omega }_0$ from the HFB-RPA (solid lines), the SOBP-RPA (dashed lines), and the HF-RPA (dot-dashed lines). The arrows in panel (a) point to the condensate resonance position given by each RPA theory. The SOBP-RPA spectra as defined in Eqs. (17, 18) do not include the contribution from the direct excitation of the noncondensate.

Figure 8

Monopole excitation frequency ${\omega }_M$ (in units of ${\omega }_0$) as a function of reduced temperature $T∕T_c$, as predicted by various theories: the HFB-Popov (solid line), the HFB-RPA (solid circles), the SOBP-RPA (empty circles), and the HF-RPA (stars). The lines connecting the symbols are guides to the eye.

Figure 9

Density fluctuations (in arbitrary units) as functions of the radial coordinate $r$ (in units of $a_{ho}$) for the monopole mode (a) and the quadrupole mode (b), as calculated from the HFB-RPA for $T∕T_c=0.6$ at the appropriate excitation frequency of the condensate. In both panels the condensate density fluctuation is reduced by a factor of 5 for clarity.