Enhanced Atom Interferometer Readout through the Application of Phase Shear

We present a method for determining the phase and contrast of a single shot of an atom interferometer. The application of a phase shear across the atom ensemble yields a spatially varying fringe pattern at each output port, which can be imaged directly. This method is broadly relevant to atom-interferometric precision measurement, as we demonstrate in a 10 m ${}^{87}\mathrm{Rb}$ atomic fountain by implementing an atom-interferometric gyrocompass with 10 mdeg precision.

Figure 1

(a) Schematic diagram of the apparatus, showing beam-tilt phase shear readout. Atoms are cooled and launched upward into an interferometer region, which is not shown. Once they fall back to the bottom, the wave packets are overlapped, yielding an interference pattern (blue fringes) imaged by two perpendicular cameras (CCD1,2). An additional optical pulse separates the output ports ($F=1$ and $F=2$) by pushing the $F=2$ atoms downward. All atom-optics pulses are performed by lasers incident from above and retroreflected off of a piezoactuated mirror. Tilting this mirror by an angle $\delta \theta$ for the third atom-optics pulse yields a phase shear. (b) A fluorescence image of the atomic density distribution taken with CCD2 after interference. Spatial fringes result from a third-pulse tilt $\delta \theta =60\mu \mathrm{rad}$ about the $x$ axis. The pushed $F=2$ atoms are heated, yielding reduced apparent contrast, and we ignore the $F=2$ port in subsequent analysis.

Figure 2

Horizontal fringes resulting from beam-tilt PSR in a $2T=2.3\mathrm{s}$ interferometer. (a) Spatial fringes observed on CCD2 with third-pulse tilt angles $\delta \theta =-80$, $-40$, 0, $+40$, $+80\mu \mathrm{rad}$ (left to right). Red versus blue regions show anticorrelation in atom population. Each image is the second-highest variance principal component arising from a set of 20 fluorescence images [24]. (b) Measured fringe spatial frequency $|{\kappa }_H|$, resulting from images filtered using principal component analysis [24, 30]. We bin the images vertically and fit a Gaussian modulated by the interference term $P(\mathbf{r})$. The curve is a prediction with no free parameters that includes a reduction in the measured wavelength from heating during imaging [31].

Figure 3

Single-shot phase readout with a $2T=50\mathrm{ms}$ interferometer. (a) Measured phase versus the applied phase of the final atom-optics pulse for 96 shots. A line with unity slope is shown for reference. The measured phase is fit from images like those in (b). The measurement scatter is dominated by vibrations of the Raman laser beam delivery optics. (b) Five sample interferometer shots [open circles in (a)], separated in measured phase by $\sim \pi /2\mathrm{rad}$. All images are filtered with principal component analysis.

Figure 4

Gyrocompass using PSR. Each $\Delta \kappa$ point is the combination of 40 trials, 20 each at two applied tilt values ($\delta \theta =\pm 60\mu \mathrm{rad}$). The horizontal intercept of a linear fit gives the direction of true north.

Figure 5

Arbitrary control of spatial fringe direction. (a) Second-highest variance principal components from sets of 20 trials with timing asymmetry $\delta T=-240$, $-160$, 0, $+160$, $+240\mu \mathrm{s}$ (left to right). (b) Comparable images for trials with both a beam tilt $\delta \theta =40\mu \mathrm{rad}$ and $\delta T=-160$, $-80$, 0, $+80$, $+160\mu \mathrm{s}$. (c) Measured fringe spatial frequency extracted from fits to principal component filtered images with vertical fringes. (d) Measured fringe angle extracted from fits to images with tilted fringes. In both (c) and (d), the curves are predictions with no free parameters.