Readout-delay-free Bragg atom interferometry using overlapped spatial fringes

A method for mitigating the readout delay characteristic of Bragg-based atom interferometry is presented, utilizing an asymmetric Mach-Zehnder interferometer sequence to generate spatial fringes that are read out while still overlapped. The time of flight after the final beamsplitter is engineered to facilitate constructive overlap of the fringes on the output states. A noise analysis performed using a precision atom interferometer is presented, comparing the traditional separated symmetric scheme with that of the separated and overlapped asymmetric scheme showing no significant increase in short-term phase noise.

Figure 1

(a) Diagram of a Mach-Zehnder atom-interferometer sequence. An atom source is prepared and released into free fall. A Bragg lattice is used to split this into two momentum states at $T_0$. After time $T_1$ a mirror pulse is applied, swapping the momentum states. After a subsequent time $T_2$, a second beamsplitter pulse is applied to recombine and interfere the atoms. Different output states are generated depending on $\delta T=T_1-T_2$. (b) When $\delta T=0$ the momentum states are output with the relative population providing phase information. (c) If $\delta T\ne 0$ a spatial interference pattern is observed with the phase of the interference providing information on the difference in paths. Parts (d), (e), and (f) illustrate the operation of the overlapped and separated spatial fringe methods. At the final beamsplitter (d), the two states are overlapped and out of phase, no spatial modulation observed. Waiting $t_{\pi /2}$ (e) results in the two states moving spatially into phase, with the resulting image showing fringes with increased signal-to-noise. (f) The two output states after fully separating, with (g) displaying example output data over 200 runs of a $T=50$ ms interferometer as the laser phase is scanned by $16\mathrm{deg}$ each run, with the fringe phase seen to be stable over many iterations of the experiment.

Figure 2

Scaling of the spatial fringe wavelength with the magnitude of the temporal asymmetry and the total time of flight. A short $T_1=1$ ms interferometer was performed $T_0=30\mathrm{ms}$ after release in order to determine the wavelength of the spatial fringes over a range of temporal asymmetries, $\delta T$. This was performed for both $T_{\text{TOF}}=218\mathrm{ms}$ (lower red) and $722\mathrm{ms}$ (upper blue) total time of flight. The 218 ms data was obtained using absorption imaging, with raw single-run images shown below the plot, while the 722 ms data was obtained using FMI, with single-run examples shown above the plot. The FMI data has a Savitzky-Golay filter applied. Each data point represents the mean measured wavelength for 100 runs, with the error bars indicating one standard deviation from the mean. The solid black line shows the analytic expression for the wavelength according to Eq. (7), dependent on both the temporal asymmetry and the time of flight from release, providing good agreement with the theoretical expression.

Figure 3

Comparison of the short-term phase noise for varying spatial frequencies, obtained by changing the value of the asymmetry $\delta T$ on a 1 ms interferometer observed using FMI. The phase noise, given by the Allan deviation at a 1 run averaging time, tends to increase with increasing asymmetry, indicating that lower values provide a more optimal operation of the interferometer. The contrast, indicated by the black points, is also seen to decrease as the asymmetry increases, limiting the extent to which the asymmetry can be increased. These features are illustrated by example fringes shown with their associated $\delta T$ on the top horizontal axis. The black line indicates the model, which includes a $5^{\circ }$ rotation of the imaging light relative to the Bragg lattice, suggesting a possible cause of the diminishing contrast. Longer wavelengths are restricted by the size of the cloud, limiting the ability to extract a phase value.

Figure 4

Comparison of the phase noise of the symmetric and asymmetric Mach-Zehnder interferometer as calculated through the Allan deviation. The phase for the symmetric case is obtained by scanning the laser phase then fitting a sinusoid to 20 points. The phase in the asymmetric case is obtained each run by fitting the sinusoidal modulation. The trend for various interferometer times, $\mathrm{T}=\{1,5,10,25,50,130\}\mathrm{ms}$, is seen to be the same between the two methods, with both integrating down according to a $1/\sqrt{\tau }$ trend, as would be expected for white noise. The extrapolated ${\left(\mathrm{Hz}\right)}^{-1/2}$ phase noise matches between the two methods. The $T=130\mathrm{ms}$ represents the phase from an overlapped spatial fringe.

Figure 5

Comparison of the phase noise for an overlapped and nonoverlapped asymmetric Mach-Zehnder interferometer for $T=\{1,50\}\mathrm{ms}$ in addition to an overlapped $T=130\mathrm{ms}$ asymmetric interferometer. The short-term phase noise is equivalent between the two methods, suggesting that the overlapped interferometer could be used to mitigate downtime without significantly limiting the phase noise.