Spatial interference from well-separated split condensates

We use magnetic levitation and a variable-separation dual optical plug to obtain clear spatial interference between two condensates axially separated by up to $0.25$ mm—the largest separation observed with this kind of interferometer. Clear planar fringes are observed using standard (i.e., nontomographic) resonant absorption imaging. The effect of a weak inverted parabola potential on fringe separation is observed and agrees well with theory.

Figure 1

Schematic illustrating different BEC splitting geometries: (a) radial splitting using, e.g., rf dressed potentials [(red) arrow at left indicates imaging direction for fringe observation]; (b) axial splitting using a blue-detuned dipole beam; (c) the variable-separation axial splitting using dual dipole beams reported on in this paper.

Figure 2

The 658-nm dipole beam path for splitting the BEC. The acousto-optical modulator (AOM) was offset by a distance $d_1$ from the focal point of the $1\times$ beam expander to enable output beam deflection at the beam waist after the final lens. The modulated rf carrier frequency results in two rf sidebands and a suppressed carrier, resulting in two optical beams with spatial separation determined by the rf modulation frequency as shown in the experimental beam image series.

Figure 3

Interference patterns $(0.8\times 0.5$ mm${}^2$) for (a) 60-$\mu$m-separated BECs and (e) BECs split 60 $\mu$m–250 $\mu$m–60 $\mu$m over a 160-ms period. In both cases the pictures were taken after a further 135 ms of magnetic levitation, which, for (a), corresponds to the triangle in Fig. 4. The phases of the Fourier components of the fringes for the selected areas in the (red) boxes in (a) and (e) can be obtained [(blue) points in (b) and (f)] and fit with a sawtooth linear phase shift [(red) curves in (b) and (f)]. These sawtooth phase corrections can then be applied to the Fourier transform, before performing the inverse transform shown in (c) and (g). These corrected images can then be averaged over the image rows to obtain the (blue) circles in (d) and (h), with their sinusoidal fits [(blue) curves]. Absorption is measured using the natural logarithm. Each row or column (i.e., pixel) corresponds to $5\times 5$ $\mu {\mathrm{m}}^2$. The fringe period in (e) is shorter than in (a), as the condensates have a residual counterpropagating velocity after the 250 $\mu \mathrm{m}\to 60\mathrm{\hspace{0.5em}}\mathrm{\mu }$m separation phase.

Figure 4

Fringe spacing as a function of levitation time. Ballistic expansion theory for $d=60$ $\mu$m-separated BECs [diagonal (black) line] and experimental data (circles) are shown, as well as a [thick (blue)] $\sinh (\omega t)/\omega$ curve using $\omega =14$ rad/s (a geometrical ring property), which has a $d=60$ $\mu$m fit. The model aptly represents the fringe spacing in an inverted parabolic potential. The triangle was derived from the image Fig. 3a.

Figure 5

Relative theoretical probability distributions (fringes; with even symmetry about $z=0$) obtained when two Gaussian initial wave packets (black) are released for 150 ms in a potential: $U_z=0$ (red) and $U_z=-m{\omega }^2{z}^2/2$ (blue and purple curves). Interatomic repulsion is either absent (red and blue curves) or present (purple curve). Increasing the nonlinear term in the Schrödinger equation affects the width of the final distribution but not the fringe period.