Self-heterodyne detection of the in situ phase of an atomic superconducting quantum interference device

We present theoretical and experimental analysis of an interferometric measurement of the in situ phase drop across and current flow through a rotating barrier in a toroidal Bose-Einstein condensate (BEC). This experiment is the atomic analog of the rf-superconducting quantum interference device (SQUID). The phase drop is extracted from a spiral-shaped density profile created by the spatial interference of the expanding toroidal BEC and a reference BEC after release from all trapping potentials. We characterize the interferometer when it contains a single particle, which is initially in a coherent superposition of a torus and reference state, as well as when it contains a many-body state in the mean-field approximation. The single-particle picture is sufficient to explain the origin of the spirals, to relate the phase-drop across the barrier to the geometry of a spiral, and to bound the expansion times for which the in situ phase can be accurately determined. Mean-field estimates and numerical simulations show that the interatomic interactions shorten the expansion time scales compared to the single-particle case. Finally, we compare the mean-field simulations with our experimental data and confirm that the interferometer indeed accurately measures the in situ phase drop.

Figure 1

Panel (a) shows a schematic of atoms in a target trap. The inner disk and the outer ring are the reference and science condensates, respectively. A blue-detuned laser forms a rotating weak link and is shown by the blue (gray) ellipse. Panel (b) is an in situ image from the experiment of atoms in a target trap.

Figure 2

Phase [panel (a)] and magnitude [panel (b)] of the single-particle ground-state angular wave function ${\phi }_S\left(\theta \right)$ as a function of $\theta$ for various values of rotation rate $\kappa$. The wave function is calculated in the frame rotating with a delta-function potential of strength $U_0=1$ located at $\theta =0$. For $\kappa \approx 1/2$, a sharp change in the phase occurs in the $\kappa$-dependent region $\theta \in (-{\theta }_0,{\theta }_0)$. The figure also shows the phase drop $\gamma$, defined in the text, for $\kappa =0.51$.

Figure 3

Numerical simulation of the integrated particle density $n(r,\theta ,t)$ of a single particle, with winding number equal to 1, expanding in the rotating frame after release from a target trap. Panel (a) shows $n(r,\theta ,t)$ with spirals at an early expansion time $t=0.25{\tau }_C$, evaluated at $r=r_S$. Panel (b) shows a later time $t=1.25{\tau }_C$, where the spirals are superimposed with circles due to self-interference of the toroidal wave function. The density near the center has been truncated for better contrast. The trap parameters are ${\sigma }_R=0.025r_S,{\sigma }_S=0.05r_S,U_0=1$, and $\kappa =0.51$. The lengths of the sides in panels (a) and (b) correspond to $5.12r_S$ and $12.8r_S$, respectively. The parameters are chosen such that the overlap between the expanding science and reference wave functions is sufficient to show the spiral over a large range of radii.

Figure 4

Schematic of a spiral-like contour (solid line) in the integrated density for $\kappa$ slightly larger than $1/2$, so that the winding number $n=1$. The contour has a constant phase $\xi (r,\theta )={\xi }_0$. The phase of ${\phi }_S\left(\theta \right)$ varies rapidly in the wedge $\theta \in (-{\theta }_0,{\theta }_0)$. In addition, an Archimedean spiral (dashed line) with the same initial angular velocity as the solid line is shown. Its parameters as well as the lengths $\delta$ and $\mathrm{\Delta }$ are defined in the text.

Figure 5

(a) Grayscale images of the interference pattern in the density profile after a 17-ms expansion time for four rotation rates of the barrier. The atom number density increases from white to black with white corresponding to zero density. The top and bottom rows show images from our experiment and GPE simulations with the same trapping potentials and atom number, respectively. The extracted $\mathrm{\Delta }/\delta$ for each rotation rate is shown above the images. The winding number is 0 for all images. (b) Images of experimental (top row) and GPE (bottom row) density profiles for four expansion times of a nonrotating condensate released from a target trap without a barrier.

Figure 6

Radial fringe spacing, $\delta$, of the interference pattern as a function of time elapsed after the release of the target trap. The data are for a toroidal trap without a barrier and a BEC without winding. The experimental, GPE, and single-particle fringe spacings are shown by red dots with one-standard-deviation statistical error bars, blue squares, and a black line, respectively. The value of $r_S$ has an uncertainty, which is shown by the shaded region around the black line.

Figure 7

The ratio $\mathrm{\Delta }/\delta$ as a function of in situ phase drop $\gamma$ across the rotating barrier from the GPE simulations of Fig. 5 (blue dots) and the single-particle prediction (solid line). Error bars are one-standard-deviation uncertainties from the fit to the density profile.