How to Dress Radio-Frequency Photons with Tunable Momentum

We demonstrate how the combination of oscillating magnetic forces and radio-frequency (rf) pulses endows rf photons with tunable momentum. We observe velocity-selective spin-flip transitions and the associated Doppler shift. Recoil-dressed photons are a promising tool for measurements and quantum simulations, including the realization of gauge potentials and spin-orbit coupling schemes which do not involve optical transitions.

Figure 1

Illustration of our scheme for creating a tunable atomic recoil momentum with rf transitions using magnetic forces. Panels (a) and (b) show the experimental conditions for ${\varphi }_{\mathrm{rf}}=0$ and ${\varphi }_{\mathrm{rf}}=\pi /2$, respectively. The spin-dependent forces and velocities are shown (as thick solid lines) for the amplitude of the wave function which is transferred from spin-down (red) to -up (blue) by the rf pulse marked by the gray dashed line. For ${\varphi }_{\mathrm{rf}}=0$, the average velocities $⟨v_{\downarrow }⟩$ and $⟨v_{\uparrow }⟩$ are different, which implies a finite recoil associated with the spin flip. In contrast, $⟨v_{\downarrow }⟩=⟨v_{\uparrow }⟩$ for ${\varphi }_{\mathrm{rf}}=\pi /2$, and there is no recoil.

Figure 2

Observation of velocity-selective rf transitions. (a) Absorption images of the spin-flipped atoms (i.e., in the $|m_F=-1⟩$ state) after 12 ms of TOF with and without magnetic shaking. The yellow dashed ellipses have major and minor axes obtained as FWHM of Gaussian fits. After TOF, the thermal could expand by a factor of 2.13; thus, a single-velocity class is narrower than the thermal cloud by $1/2.13\approx 0.47$. The Fourier limit of our velocity selection increases this to 0.50, and inclusion of eddy currents further modifies it to 0.45 (dash-dotted line). The field of view is $1\times 1{\mathrm{mm}}^2$. (b) Integrated column density distribution obtained from absorption images like those in (a), for different rf detunings. The solid lines are Gaussian fits to the data points. The rf phase was at ${\varphi }_{\mathrm{rf}}=0$ to maximize Doppler sensitivity. The asymmetry between the $\pm 300\mathrm{Hz}$ is most likely caused by bias field drifts (estimated in Fig. 3 to be 70 Hz) or small residual magnetic field gradients.

Figure 3

Observation of rf transitions with Doppler shifts. (a),(b) Central velocities of the spin-flipped atomic distribution [as in Fig. 2] are shown as a function of rf detuning for ${\varphi }_{\mathrm{rf}}=0$ and ${\varphi }_{\mathrm{rf}}=\pi$, respectively. Shifting the rf phase changes the sign of the Doppler shift and therefore the direction of the recoil momentum. The solid line represents the predicted Doppler shifts based on the calibration of recoil momentum. The dashed line takes into account the effects of eddy currents (see Supplemental Material [13]). The error bars are purely statistical based on five data points and correspond to 1 standard deviation. The inferred $1\sigma$ fluctuations for the frequency are 70 Hz.