Figure 1

Optical double-well potential. (a) Schematic diagram of the optical setup for the double-well potential. An acousto-optic modulator (AOM) was driven by two frequencies, $f_1$ and $f_2$, and diffracted a collimated beam into two beams. The AOM was placed in the focal plane of a lens of focal length $F$ so that the two beams propagated parallel to each other. The radial separation of the potential wells, $d$, was controlled by the frequency difference, $\Delta f=|f_1-f_2|$. The acceleration due to gravity, $\overrightarrow{g}$, points into the page. The absorption image shows two well-separated condensates confined in the double-well potential diagrammed in (c). The field of view is $70\times 300\mu \mathrm{m}$. Energy diagrams for (b) initial single-well trap with $d=6\mu \mathrm{m}$ and (c) final double-well trap with $d=13\mu \mathrm{m}$. In both (b) and (c), $U_0=h\times 5\mathrm{k}\mathrm{H}\mathrm{z}$ and the peak atomic mean field energy was $\sim h\times 3\mathrm{k}\mathrm{H}\mathrm{z}$. The potential “dimple” in (b) was $<h\times 500\mathrm{H}\mathrm{z}$ which was much less than the peak atomic mean field energy allowing the trap to be characterized as a single well. The potential “barrier” in (c) was $h\times 4.7\mathrm{k}\mathrm{H}\mathrm{z}$ which was larger than the peak atomic mean field energy allowing the resulting split condensates to be characterized as independent.Reuse & Permissions

Figure 2

Matter wave interference. (a) Absorption image of condensates released from the double-well potential in Fig. 1c immediately after splitting and allowed to overlap during 30 ms of ballistic expansion. The imaging axis was parallel to the direction of gravitational acceleration, $\overrightarrow{g}$. The field of view is $600\times 350\mu \mathrm{m}$. (b) Radial density profiles were obtained by integrating the absorption signal between the dashed lines, and typical images gave $>60\%$ contrast. The solid line is a fit to a sinusoidally modulated Gaussian curve from which the phase of the interference pattern was extracted (see text). This figure presents data acquired in a single realization of the experiment.Reuse & Permissions

Figure 3

Phase coherence of the separated condensates. (a) The spatial phase of the matter wave interference pattern is plotted versus hold time after splitting the condensate. Each point represents the average of eight measurements. The phase evolution was due to unequal trap depths for the two wells, which was determined from the linear fit to be $h\times 70\mathrm{H}\mathrm{z}$ or $\sim 1\%$ of the trap depth. (b) Standard deviation of eight measurements of the relative phase. A standard deviation $\sim 104°$ (dashed line) is expected for random relative phases. Matter wave interference patterns after 0 and 5 ms holding are displayed. The curvature of the interference fringes increased with hold time limiting the coherence time of the separated condensates to 5 ms.Reuse & Permissions

Figure 4

Trapped-atom interferometry. (a) ac Stark phase shifts were applied to either well exclusively (solid circles and open circles) or both wells simultaneously (crosses) by turning off the corresponding rf signal(s) driving the AOM for a duration ${\tau }_p$. The resulting spatial phase of the matter wave interference pattern scaled linearly with ${\tau }_p$ and hence the applied phase shift. Applying the ac Stark shift to the opposite well (solid versus open circles) resulted in an interference pattern phase shift with opposite sign. Applying ac Stark shifts to both wells (crosses) resulted in no phase shift for the interference pattern. These data were taken with a slightly modified experimental setup such that the trap depth of the individual potential wells was $U_0=h\times 17\mathrm{k}\mathrm{H}\mathrm{z}$, corresponding to a $270°$ phase shift for a $50\mu \mathrm{s}$ pulse. (b) A $50\mu \mathrm{s}$ pulse induced a $70°$ shift independent of the pulse delay, ${\tau }_d$. The experimental setup was as described in Fig. 1 ($U_0=h\times 5\mathrm{k}\mathrm{H}\mathrm{z}$). Solid and open circles have the same meaning as in (a). The insets show the time sequence of the optical intensity for the well(s) temporarily turned off.Reuse & Permissions