Optimal control of mirror pulses for cold-atom interferometry

Atom matterwave interferometry requires mirror and beam splitter pulses that are robust to inhomogeneities in field intensity, magnetic environment, atom velocity, and Zeeman substate. We present theoretical results which show that pulse shapes determined using quantum control methods can significantly improve interferometer performance by allowing broader atom distributions, larger interferometer areas, and higher contrast. We have applied gradient ascent pulse engineering (grape) to optimize the design of phase-modulated mirror pulses for a Mach-Zehnder light-pulse atom interferometer, with the aim of increasing fringe contrast when averaged over atoms with an experimentally relevant range of velocities, beam intensities, and Zeeman states. Pulses were found to be highly robust to variations in detuning and coupling strength and offer a clear improvement in robustness over the best established composite pulses. The peak mirror fidelity in a cloud of $\sim 80\mu \mathrm{K}{}^{85}\mathrm{Rb}$ atoms is predicted to be improved by a factor of 2 compared with standard rectangular $\pi$ pulses.

Figure 1

Phase profile ${\varphi }_L\left(t\right)$ for grape inversion pulse optimizing ${\mathcal{F}}_{\text{real}}^{\pi }$ subject to constrained total duration of $20\mu \mathrm{s}$ and fixed effective Rabi frequency of $2\pi \times 200\mathrm{kHz}$. The fidelity after 100 iterations was 0.99.

Figure 2

Excited state probability and phase profiles plotted against detuning for various composite inversion pulses and grape pulse of Fig. 1. The simulation assumes a single Rabi frequency of $2\pi \times 200\mathrm{kHz}$.

Figure 3

Simulation of inversion operation for multiple composite pulses and the grape pulse of Fig. 1 in a ${\sigma }^{+}-{\sigma }^{+}$ Raman arrangement, as the laser detuning ${\delta }_L$ (defined as ${\omega }_{12}-{\omega }_L$) is scanned. This simulation uses an effective Rabi frequency of $2\pi \times 360$ kHz, and accounts for both the coupling strengths of 5 $m_{\mathrm{F}}$ sublevels and the Stark shift. Both the simulation parameters and model are described elsewhere [8, 44]. The offset in peak is due to the Stark shift, and the magnitude of the peak gives an indication of a pulse's ability to excite atoms across the momentum distribution for all $m_F$ sublevels, which we assume are equally populated.

Figure 4

Final excited state population shown for a range of off-resonance and pulse-length errors for different mirror pulses: (a) rectangular $\pi$ pulse; (b) KNILL pulse; (c) WALTZ pulse; (d) GRAPE pulse from Fig. 1. The simulations were performed at the same effective Rabi frequency of $2\pi \times 200\mathrm{kHz}$. The grape mirror pulse demonstrates a clear improvement in robustness to variations in detuning and pulse amplitude. The contours are at 0.15, 0.3, 0.45, 0.6, 0.75, 0.9, and 0.95.

Figure 5

Contrast after Mach-Zehnder sequence integrated over thermal distributions of ${}^{85}\mathrm{Rb}$ atoms with various temperatures. The beam splitters in each case are rectangular $\pi /2$ pulses. There is a large increase in contrast due to the grape UR mirror pulse in Fig. 1, which improves the rectangular $\pi$ pulse by a factor of 1.76 at $100\mu \mathrm{K}$, approaching the theoretical limit of a sequence with a “perfect” $\pi$ pulse resonant for all atoms in the sample. Comparisons with different sequences are made at the same Rabi frequency of $2\pi \times 200\mathrm{kHz}$.