Biselective pulses for large-area atom interferometry

We present designs for the augmentation “mirror” pulses of large-momentum-transfer atom interferometers that maintain their fidelity as the wave-packet momentum difference is increased. These biselective pulses, tailored using optimal control methods to the evolving bimodal momentum distribution, should allow greater interferometer areas and hence increased inertial measurement sensitivity, without requiring elevated Rabi frequencies or extended frequency chirps. Using an experimentally validated model, we have simulated the application of our pulse designs to large-momentum-transfer atom interferometry using stimulated Raman transitions in a laser-cooled atomic sample of ${}^{85}\mathrm{Rb}$ at 1 $\mu \mathrm{K}$. After the wave packets have separated by 42 photon recoil momenta, our pulses maintain a fringe contrast of 90%, whereas, for adiabatic rapid passage and conventional $\pi$ pulses, the contrast is less than 10%. Furthermore, we show how these pulses may be adapted to be robust to laser intensity variations between pulses and to suppress the detrimental off-resonant excitation that limits other broadband pulse schemes.

Figure 1

Atomic state trajectories (left) as a function of position within an LMT interferometer sequence showing the case where the beam-splitter and mirror operations are extended by a sequence of four augmentation pulses ($18\hslash \mathrm{k}$ trajectory). The initial momentum distribution (right), represented by the red shaded region, is separated into two arms. The momentum distribution seen by each pulse throughout the LMT sequence for ${}^{85}\mathrm{Rb}$ atoms with a Maxwell-Boltzmann temperature of $\sim 1 \mu \mathrm{K}$ is shown by the blue shaded regions on the right. As more pulses are added to increase the interferometer area, the separation in the resonance conditions for the arms begins to exceed the velocity acceptance of a $\pi$ pulse, shown in the bottom right for a Rabi frequency of 200 kHz.

Figure 2

Biselective phase-modulated augmentation pulses optimized with the phase-sensitive fidelity ${\mathcal{F}}_{\mathrm{real}}$ for a temperature of 1 $\mu \mathrm{K}$. The resulting phase profiles for the $n=1,4,7,10$ augmentation pulses are shown in the left panel. The right panel shows the simulated transfer efficiency of each biselective pulse (blue solid curve), the ARP tanh/tan pulse of equivalent length (orange dashed curve), and the WALTZ composite pulse (purple dotted curve) as a function of the Raman detuning. The corresponding detuning distributions seen by each augmentation pulse for the primary interferometer arms are shown by the shaded regions. The range of detunings included in each optimization is shown by the hatched regions.

Figure 3

(a) Phase profile and (b) simulated transfer efficiency for the $n=10$ biselective pulse found optimizing the fidelity ${\mathcal{F}}_{\mathrm{square}}$ (blue solid line). (c) The transfer efficiency for the ARP tanh/tan pulse (orange dashed line for duration $4t_{\pi }$ and green dot-dashed line for duration $8t_{\pi }$) and the WALTZ pulse (purple solid line). The detuning distributions of the two primary interferometer arms, corresponding to a temperature of 1 $\mu \mathrm{K}$ during the $n=10$ augmentation pulse, are shown by shaded regions. The hatched regions represent the bands of detuning where the transfer efficiency is maximized using GRAPE.

Figure 4

Simulated interferometer contrast for a cloud of ${}^{85}\mathrm{Rb}$ at 1 $\mu \mathrm{K}$ as a function of momentum separation for different pulse sequences. The contrast is numerically averaged over Raman detunings due to a velocity distribution corresponding to a temperature of 1 $\mu \mathrm{K}$ and (a) a uniform distribution of static Rabi rate errors of $\pm 10$% or (b) a random temporal variation of Rabi rate errors of $\pm 10$% from pulse to pulse. Contrast values for biselective GRAPE pulses found with fidelities ${\mathcal{F}}_{\mathrm{square}}$ and ${\mathcal{F}}_{\mathrm{real}}$ (blue dotted line with diamonds and red solid line with diamonds, respectively) are shown. The results for $\pi$ augmentation pulses (black dotted line with rectangles) and the tanh/tan ARP pulse (dashed orange line with circles) are also shown.

Figure 5

(a) and (c) The phase and amplitude profiles for two biselective pulses maximizing ${\mathcal{F}}_{\mathrm{square}}$. The dashed blue curve shows the case where no suppression of off-resonant excitation is included in the optimization, and the solid red curve shows the case where off-resonant excitation is suppressed. (b) The transfer efficiency of these two pulses as a function of detuning and (d) the transfer efficiency of the WALTZ and tanh/tan pulses. The diagonally hatched regions coinciding with the shaded detuning distributions show the optimization bands of detuning during the $n=8$ augmentation pulse. The outer hatched regions show the range of the detunings for which the state transfer is suppressed (${\delta }_{\min },{\delta }_{\max }=500,2500$ kHz). The response is unconstrained outside the hatched regions.

Figure 6

Diagram of the first half of an $N=1$ LMT pulse sequence, shown up to the central mirror pulse, indicating how the momentum and atomic state vary in each arm of the interferometer during the sequence. The resonant Raman frequency depends upon the pair of states coupled at each stage. “A” represents an augmentation pulse. Single and double arrows represent the internal states $|g\rangle$ and $|e\rangle$, respectively. Solid arrows represent the primary interferometer arms, and dotted arrows represent the trajectories which are discarded in the simulation.