Dynamics of a Tunable Superfluid Junction

We study the population dynamics of a Bose-Einstein condensate in a double-well potential throughout the crossover from Josephson dynamics to hydrodynamics. At barriers higher than the chemical potential, we observe slow oscillations well described by a Josephson model. In the limit of low barriers, the fundamental frequency agrees with a simple hydrodynamic model, but we also observe a second, higher frequency. A full numerical simulation of the Gross-Pitaevskii equation giving the frequencies and amplitudes of the observed modes between these two limits is compared to the data and is used to understand the origin of the higher mode. Implications for trapped matter-wave interferometers are discussed.

Figure 1

(a) Schematic of atom chip double-well trap. Central $Z$ wire [34] carries static trapping current, $I_S=2\mathrm{A}$, which, with uniform external fields ${\mathbf{B}}_{\mathrm{ext}}=⟨2.2,0.11,0⟩\mathrm{mT}$, results in an Ioffe-Pritchard style trap with harmonic trapping frequencies $({\omega }_{x_0,z_0},{\omega }_{y_0})=2\pi \times (1300,10)\mathrm{Hz}$. Side wires are 1.58 mm from trap center and carry rf currents with amplitude $I_{\mathrm{rf}}$. This rf current produces a $z$-polarized field at the trap location with amplitude $B_{\mathrm{rf}}=23.6\pm 0.6\mu \mathrm{T}$ [peak Rabi frequency $\Omega =2\pi \times (82\pm 2\mathrm{kHz})$]. A levitation beam [gray (pink)] is positioned to provide a force-canceling gravity ($z$ direction) while compressing the sample along $y$. Atoms are trapped $190\mu \mathrm{m}$ from the chip surface. (b) A schematic one-dimensional cut at $t=-0.5\mathrm{ms}$ through trapping potential along $x$ (solid line) in the presence of linear bias (dashed line) and (c) balanced potential at $t=0$, with ${\mathcal{Z}}_0\ne 0$.

Figure 2

(a) Population imbalance, $\mathcal{Z}$, vs time for $\delta =2\pi \times (0.1\pm 0.5)\mathrm{kHz}$, $N=5900\pm 150$. The dashed line is a decaying two-frequency sinusoidal fit to the data, using two fixed frequencies from the FT (lower inset). Each point is the average of six repetitions of the experiment; error bars are statistical. (b) Averaged absorption image after separation and 1.3 ms time of flight, with right and left measurement regions (dashed boxes) indicated. (c) FT amplitude spectrum of data showing two distinct peaks at $268\pm 6$ and $151\pm 13\mathrm{Hz}$ rising above the noise floor (grey).

Figure 3

Frequency components of population imbalance vs rf detuning (measured) and barrier height to chemical potential ratio (calculated). Experimental points (white circles) represent the two dominant Fourier components at each detuning; error bars represent uncertainty contributed by noise in the FT from a single time series, but do not include shot-to-shot fluctuations. The spectral weight is represented through the color map, which has been linearly smoothed between discrete values of $V_b/\mu$ and darker colors indicate greater spectral weight. All calculations use $N=8000$ and ${\mathcal{Z}}_0=0.075$, and a single-parameter fit of the data to the GPE curves shifts all experimental points by ${\delta }_{\mathrm{shift}}=2\pi \times 5.1\mathrm{kHz}$ [19] to compensate for a systematic unknown in $B_S(\mathbf{0})$. Statistical vertical error bars are shown, while a typical horizontal statistical error bar is shown at $V_b/\mu \approx 0.5$. Dashed lines represent 3D GPE frequencies, the solid line the plasma oscillation frequency predicted by the Josephson model, ${\omega }_p$, and the dotted line the hydrodynamic approximation, ${\omega }_{\mathrm{HD}}/2\pi$. White bars at $V_b/\mu \sim 0.1$ indicate the bounds of the GPE simulation corresponding to the systematic plus statistical uncertainty in atom number. Inset: ratio of healing length, $\xi$, to interwell distance, $d$, as a function of $V_b/\mu$. $\xi$ is calculated at the center of the barrier.

Figure 4

Fraction of low-frequency mode in population dynamics. Dashed line shows the GPE simulation for 8000 atoms with initial imbalance $\mathcal{Z}(0)=0.075$. The grey shaded area represents the variation of the GPE calculations over the range of $\mathcal{Z}(0)=0.05$ to 0.10. The vertical error bars are statistical; the statistical uncertainty in $\delta$ is $2\pi \times 0.5\mathrm{kHz}$ (not shown). The GPE calculation gives $R_1=1$ when $V_b/\mu \simeq 1.1$.