Long Phase Coherence Time and Number Squeezing of Two Bose-Einstein Condensates on an Atom Chip

We measure the relative phase of two Bose-Einstein condensates confined in a radio frequency induced double-well potential on an atom chip. We observe phase coherence between the separated condensates for times up to $\sim 200\mathrm{ms}$ after splitting, a factor of 10 longer than the phase diffusion time expected for a coherent state for our experimental conditions. The enhanced coherence time is attributed to number squeezing of the initial state by a factor of 10. In addition, we demonstrate a rotationally sensitive (Sagnac) geometry for a guided atom interferometer by propagating the split condensates.

Figure 1

Schematic of the atom chip interferometer. (a) Atoms were confined radially by the combined magnetic potential of a current-carrying wire and external bias field. Axial confinement in the $x$ direction was provided by a pair of end cap wires (not shown) [21]. By dressing the atoms with an oscillating rf field from a second wire, the single minimum in the magnetic trapping potential was deformed into a double well [13]. If the trapping position lies on the circle containing the trapping wire and centered on the rf wire, the splitting occurs in the horizontal plane. Condensates were placed $185\mu \mathrm{m}$ away from the chip surface. For the single well, the radial (axial) trap frequency was $f_r=2.1\mathrm{kHz}$ ($f_z=9\mathrm{Hz}$) and the Larmor frequency at the trap center was $\approx 190\mathrm{kHz}$ ($B_x\approx 0.27\mathrm{G}$). Splitting was performed over 75 ms by linearly ramping the frequency of the rf field from 143 to 225 kHz. Gravity points in the $+z$ direction. (b) Double-well potential. The separation $d$ between the two wells and the barrier height $U$ were controlled by adjusting the frequency or amplitude of the rf field. (c) Matter wave interference. For various hold times after splitting, absorption images of condensates released from the double-well potential were taken after 10 ms time of flight. The field of view is $260\times 200\mu \mathrm{m}$.

Figure 2

Phase evolution of the relative phase during three different time intervals. The evolution rate of the relative phase are determined from the linear fit to be (a) 191, (b) 198, and (c) 255 Hz. The data points represent the average of ten measurements for (a) and (b), and 15 for (c).

Figure 3

Long phase coherence of two separated condensates. Applying various phase shifts to the condensates at 2 ms after splitting, the shifts of the relative phase were measured at 7 and 191 ms, showing strong correlation. The dotted line denotes the ideal case of perfect phase coherence. Phase shifts were applied by pulsing an additional magnetic field in the $z$ direction for $50\mu \mathrm{s}$ with variable amplitude.

Figure 4

Phase diffusion and number squeezing. (a) The randomness probability of ten measurements of the relative phase is displayed up to 400 ms after splitting. The blue dotted curve (red dashed curve) shows a simulation for a phase-coherent state (number squeezed state with $s=10$), which have negligible initial phase uncertainty. The solid line includes an initial phase uncertainty of $0.28\pi$ (see text). The shaded region represents the window where ten data points from the sample with the given phase uncertainty would fall with 50% probability. (b) Contrast of the interference pattern. Since the end cap wires generate a field gradient $\frac{\partial B_z}{\partial x}$ as well as a field curvature $\frac{{\partial }^2{B}_x}{\partial x^2}$ at the position of the condensates, the two wells are not parallel to the trapping wire and consequently have slightly different axial trapping potential. This difference induces relative axial motion of the two condensates, which periodically reduces the contrast.

Figure 5

Confined atom interferometry with enclosed area. (a) A single trapped condensate at rest was accelerated by shifting the trap center by $\sim 430\mu \mathrm{m}$ in the axial direction ($t=0\mathrm{ms}$). The frequency of the rf field was linearly ramped from 154 to 204 kHz during 16 ms after launch, splitting the condensates and separating them by $5–6\mu \mathrm{m}$. (b) Phase measurements were done for up to 26 ms after the launch. The probability for random phases was determined for data sets of ten measurements. Until 16 ms, this probability was extremely low (less than ${10}^{-12}$) and the relative phase was constant, implying that the two condensates were still connected. For $t>16\mathrm{ms}$, the relative phase evolved (similar to Fig. 2), and the probability for a random phase distribution was smaller than 10%. This demonstrates that phase coherence was preserved after full splitting. The figure shows the interference pattern for $t=22\mathrm{ms}$. The field of view is $260\times 200\mu \mathrm{m}$.