From single-particle excitations to sound waves in a box-trapped atomic Bose-Einstein condensate

We experimentally and theoretically investigate the lowest-lying axial excitation of an atomic Bose-Einstein condensate in a cylindrical box trap. By tuning the atomic density, we observe how the nature of the mode changes from a single-particle excitation (in the low-density limit) to a sound wave (in the high-density limit). Throughout this crossover the measured mode frequency agrees with Bogoliubov theory. Using approximate low-energy models we show that the evolution of the mode frequency is directly related to the interaction-induced shape changes of the condensate and the excitation. Finally, if we create a large-amplitude excitation, and then let the system evolve freely, we observe that the mode amplitude decays nonexponentially in time; this nonlinear behavior is indicative of interactions between the elementary excitations, but remains to be quantitatively understood.

Figure 1

Probing the lowest axial mode. (a) Illustration of the experimental protocol. We prepare a BEC of ${}^{87}\mathrm{Rb}$ in a cylindrical box trap of length $L$ and radius $R$. Initially, the trapping potential $V_{\text{trap}}$ has a flat bottom. We then pulse a magnetic field gradient corresponding to $\mathrm{\Delta }U=k_{\mathrm{B}}\times 1$ nK along the box length for $\mathrm{\Delta }t=20$ ms. After a time $\tau$ of in-trap evolution we switch off the trap and let the cloud evolve in free space for 140 ms before extracting its center of mass, which reflects the velocity on release from the trap, $v_z$. (b) $v_z\left(\tau \right)$ for a BEC of $N=13\left(1\right)\times {10}^3$ atoms. We determine the oscillation frequency $\omega$ using a decaying sinusoidal fit. Inset: $v_z\left(\tau \right)$ for a noncondensed sample just above $T_{\mathrm{c}}$, with $N=10\left(1\right)\times {10}^3$.

Figure 2

Angular frequency of the lowest axial mode as a function of atom number, $N$. ${\omega }_K$ is the mode frequency in an ideal gas, and ${\omega }_I$ is the frequency of Bogoliubov sound with speed ${(gn/m)}^{1/2}$ and wavelength $2L$. These calculations use trap dimensions $L=26$ µm and $R=16$ µm. The shaded band ${\omega }_B$ shows the results of numerical Bogoliubov diagonalization (see Sec. 3b) accounting for the uncertainty of $\pm 1$ µm in $L$ and $R$.

Figure 3

(a) Axial density profile of the BEC, for three interaction strengths $gn/\left(\hslash {\omega }_K\right)$. (b) Change in the axial density profile due to the coherently occupied axial mode, scaled by $\sqrt{\omega }$ so the amplitude for $gn/\left(\hslash {\omega }_K\right)\gg 1$ does not depend on $N$.

Figure 4

Crossover from single-particle excitations to sound waves. We plot $\omega /{\omega }_K$ as a function of the interaction strength, now parametrized by $gn/\left(\hslash {\omega }_K\right)$. Here we assume $L=26$ µm and $R=16$ µm for calculations. The $x-y$ error bar in the bottom right corner indicates the fractional systematic uncertainties in the experimental data due to the uncertainties in the box dimensions. ${\omega }^{\left(2\right)}$ is determined using Bogoliubov theory within a truncated basis of just the two lowest-energy single-particle eigenstates of zero angular momentum (see text). This scheme fails even for relatively small $gn/\left(\hslash {\omega }_K\right)$, as it does not allow for the interaction-induced changes in the shape of the condensate or the excitation mode. ${\omega }^{\left(5\right)}$, based on a truncated basis of five single-particle eigenstates, is the minimal model that allows for the shape changes. This simple model already captures most of the crossover, and using progressively larger truncated bases does not qualitatively change the result, as shown by ${\omega }^{\left(15\right)}$, which is based on 15 single-particle eigenstates. ${\omega }_I$ is the sound-wave frequency, approached in the limit of large $gn/\left(\hslash {\omega }_K\right)$.

Figure 5

Nonlinear damping of the lowest-lying axial mode in the interaction-dominated regime, with fixed density $n_0$ such that $g{n}_0=k_{\text{B}}\times 2.1$ nK [corresponding to $g{n}_0/\left(\hslash {\omega }_K\right)\approx 17$]. (a) Velocity oscillations following kicks with $\mathrm{\Delta }U=k_{\text{B}}\times 0.3$ nK (left) and $k_{\text{B}}\times 3.6$ nK (right). The solid lines show exponentially decaying sinusoidal functions obtained by fitting to the early-time data, $\tau <0.25$ s. For large $\mathrm{\Delta }U$ (right panel) the decay is clearly nonexponential. (b) Initial decay rate ${\mathrm{\Gamma }}_i$ as a function of the normalized kick amplitude $\mathrm{\Delta }U/\left(g{n}_0\right)$. The solid line is a linear fit to data with $\mathrm{\Delta }U/\left(g{n}_0\right)<1$ [the dashed line displays its extrapolation for $\mathrm{\Delta }U/\left(g{n}_0\right)>1$]. Inset: frequency of the damped oscillation $\omega$ normalized to the low-$\mathrm{\Delta }U$ value, ${\omega }_0$.