Atom Interferometry with Floquet Atom Optics

Floquet engineering offers a compelling approach for designing the time evolution of periodically driven systems. We implement a periodic atom-light coupling to realize Floquet atom optics on the strontium ${{}^{1}S}_0-{}^3{P}_1$ transition. These atom optics reach pulse efficiencies above 99.4% over a wide range of frequency offsets between light and atomic resonance, even under strong driving where this detuning is on the order of the Rabi frequency. Moreover, we use Floquet atom optics to compensate for differential Doppler shifts in large momentum transfer atom interferometers and achieve state-of-the-art momentum separation in excess of $400\hslash k$. This technique can be applied to any two-level system at arbitrary coupling strength, with broad application in coherent quantum control.

Figure 1

(a) Measured normalized ground state population for a sine-wave Floquet pulse (black points) and associated pulse envelope $\overline{\mathrm{\Omega }}(t)\equiv \mathrm{\Omega }(t)/{\mathrm{\Omega }}_0$ (bottom panel) with modulation offset $\delta =-2\pi \times 0.35\mathrm{MHz}$, detuning $\mathrm{\Delta }=\pm \delta {\omega }_D=\pm 2\pi \times 2\mathrm{MHz}$, and Rabi frequency ${\mathrm{\Omega }}_0=2\pi \times 5\mathrm{MHz}$. Numerical density matrix solutions are shown for both positive (blue) and negative (red) detuning with no free parameters. These state trajectories are also illustrated on the Bloch sphere for an ideal Floquet $\pi$ pulse without spontaneous emission. (b) Square-wave Floquet pulse with $\delta =2\pi \times 1.75\mathrm{MHz}$, $\mathrm{\Delta }=\pm 2\pi \times 2\mathrm{MHz}$, and ${\mathrm{\Omega }}_0=2\pi \times 5\mathrm{MHz}$. The blue and red dashed arrows show the torque vectors for the associated Doppler detunings, which flip from left to right at $t=T/2$ as suggested by the gray arrows.

Figure 2

(a) Experimental characterization of optimal Floquet $\pi$-pulse modulation offset $\delta =\beta -|\mathrm{\Delta }|$ as a function of laser detuning $\mathrm{\Delta }/2\pi$, for sine-wave (circles) and square-wave (squares) AM. (b) Experimentally optimized pulse duration $t_{\pi }$. The colored curves (blue, green, yellow, and red) are the analytic solutions for square-wave modulation, with the color corresponding to the number of half-periods $n$ required to reach optimal transfer ($n=2$, 3, 4, and 5, respectively). The brown theory curves are numerical solutions of the Schrödinger equation for sine-wave modulation. (c) Measured efficiency of square-wave modulated pulses. The dashed, colored curves are the expected pulse efficiency based on the pulse duration, extrapolated from the measured efficiency of resonant $\pi$ pulses (dotted gray line). The error bars and bands show one standard deviation of measurement uncertainty. (d) Example time evolution for $\mathrm{\Delta }=2\pi \times 7\mathrm{MHz}$ (left) and $2\pi \times 14\mathrm{MHz}$ (right) with associated numerical Schrödinger fits corresponding to $n=3$ (green) and $n=5$ (red). As the detuning increases, the dynamics converge toward Rabi oscillations at reduced frequency $(2/\pi ){\mathrm{\Omega }}_0$.

Figure 3

LMT interferometer visibility as a function of the momentum separation of the arms using Floquet atom optics (blue squares) and conventional $\pi$ pulses (gray diamonds). The dotted lines are a guide to the eye. The error bars are the standard deviation of the residuals of the sinusoidal fits. The inset is an example for a $201\hslash k$ interferometer, showing the normalized excited state population versus the interferometer phase for Floquet atom optics (blue) and conventional pulses (gray), as well as the associated fits (solid curves).