Collapse and revival of the monopole mode of a degenerate Bose gas in an isotropic harmonic trap

We study the monopole (breathing) mode of a finite temperature Bose-Einstein condensate in an isotropic harmonic trap recently developed by Lobser et al. [Nat. Phys. 11, 1009 (2015)]. We observe a nonexponential collapse of the amplitude of the condensate oscillation followed by a partial revival. This behavior is identified as being due to beating between two eigenmodes of the system, corresponding to in-phase and out-of-phase oscillations of the condensed and noncondensed fractions of the gas. We perform finite temperature simulations of the system dynamics using the Zaremba-Nikuni-Griffin methodology [J. Low Temp. Phys. 116, 277 (1999)], and find good agreement with the data, thus confirming the two mode description.

Figure 1

Oscillation frequency of the in-phase $\left(\varphi =0\right)$ and out-of-phase $\left(\varphi =\pi \right)$ modes as a function of temperature from the coupled-modes analysis (solid lines). The dashed lines represent the TF limit $\left(\sqrt{5}{\omega }_0\right)$ and ideal gas limit $\left(2{\omega }_0\right)$ for the monopole mode frequency of the condensate and noncondensate, respectively. Assumptions made in the coupled-modes analysis become invalid for $T\lesssim 0.2T_c$.

Figure 2

Magnitude of the response of the in-phase $\left(\varphi =0\right)$ and out-of-phase $\left(\varphi =\pi \right)$ modes to a trap frequency perturbation as a function of temperature. The amplitude of the trap frequency modulation is $\epsilon =0.01$.

Figure 3

Amplitude of the monopole mode oscillation (squares) for atom numbers of (a) $N=8.9\times {10}^5$, (b) $N=9.7\times {10}^5$, (c) $N=6.7\times {10}^5$, (d) $N=5.4\times {10}^5$, and (e) $N=7.9\times {10}^5$. Each frame is labeled with the condensate fraction $(N_c/N)$ and temperature $(T/T_c)$. Error bars represent the statistical uncertainty of multiple realizations of the experiment at each time point. The dashed lines are a fit of the data to Eq. (17), which represents the envelope function for the superposition of two sinusoids. From the fit it is found that the beat frequency is $\mathrm{\Delta }\omega /{\omega }_0\sim 0.13$ on average and the damping rate lies in the range ${\mathrm{\Gamma }}_{\text{e}}\sim 1.5\text{–}3.5{\text{s}}^{-1}$.

Figure 4

Amplitude of the monopole mode oscillation for experimental atom numbers of (a) $N=8.9\times {10}^5$, (b) $N=9.7\times {10}^5$, (c) $N=6.7\times {10}^5$, (d) $N=5.4\times {10}^5$, and (e) $N=7.9\times {10}^5$. Each frame is labeled with the condensate fraction $(N_c/N)$ and temperature $(T/T_c)$, and the legend denotes the different modulation amplitudes used in the ZNG simulations. Error bars on the experimental data represent the statistical uncertainty of multiple realizations of the experiment at each time point. (f) Condensed fraction vs temperature for the ideal Bose gas, $N_c/N=1-{(T/T_c)}^3$ (solid line), the equilibrium state of the ZNG simulations (blue points), and the experimental data (red crosses). All simulations are performed with $N=8\times {10}^5$ atoms.

Figure 5

Simulated mean-square radius of the condensate density for a trap frequency modulation amplitude of $\epsilon =0.03$ at (a) $T=0.4T_c$, (b) $0.6T_c$, and (c) $0.8T_c$ (black circles) and resulting fit of Eq. (29) (blue line). The density of simulated points has been reduced for clarity.

Figure 6

Damping rate of the in-phase (squares) and out-of-phase (triangles) mode as a result of fitting Eq. (29) to the simulated evolution of $〈R_c^2〉$ at each temperature for a trap frequency modulation amplitude of $\epsilon =0.03$. The solid lines are guides to the eye. Representative fits are shown in Fig. 5.

Figure 7

$c$-field condensate (blue circles) and thermal cloud (red squares) atom numbers as a function of reduced temperature. Note that a significant number of thermal cloud atoms with energies above the cutoff are missing at higher temperatures. The ideal gas condensate number for $N=5\times {10}^4$ atoms is shown as a dotted line.

Figure 8

Relaxation of the monopole mode as indicated by the mean-square radius of the Bose gas from $c$-field simulations at three different temperatures: (a) $T/T_c=0.4$; (b) $T/T_c=0.75$; (c) $T/T_c=0.9$. For these data the driving frequency was ${\omega }_D=(1+\sqrt{5}/2){\omega }_0$. Similar results are obtained for all other driving frequencies simulated. Blue dots are results from the simulations, whereas the red lines are a fit to the data from the sum of two exponentially decaying sinusoids.

Figure 9

Comparison of the eigenmode properties for the ZNG and $c$-field methodologies for $N=5\times {10}^4$ atoms. (a) Excitation frequencies as a function of the reduced temperature. Blue circles: condensate, $c$-field. Blue crosses: condensate, ZNG. Red crosses: thermal cloud, $c$-field. Red plusses: thermal cloud, ZNG. The hydrodynamic frequency for the monopole mode of the condensate at $T=0 \left(\sqrt{5}{\omega }_0\right)$, and the expected frequency for a thermal cloud above $T_c \left(2{\omega }_0\right)$ are both indicated as horizontal dashed lines. (b) Decay rates as a function of the reduced temperature. The legend is the same as for (a).

Figure 10

Particle is rotated from its initial trajectory along $r_i$ by an angle $\phi$ due to the off-axis components of ${\mathbf{p}}_i$. After the particle position is updated to $r_f$ the momentum components are rotated and realigned with the position vector. Due to spherical symmetry the azimuthal angle is not unique and is chosen randomly. This is represented by the area of revolution of ${\mathbf{p}}_f$ about the final position vector $r_f$.

Figure 11

Cartoon depiction of the phase-space binning process. Particles (black points) are binned in position space using constant volume shells with the radial width of each bin decreasing with $r$. Within each spatial bin the particles are further binned in momentum space using a 2D grid of equal area bins based on the magnitude of their momentum $p$ and trajectory $cos\theta =\widehat{\mathbf{p}}·\widehat{\mathbf{r}}$.