Difference between a Photon’s Momentum and an Atom’s Recoil

When an atom absorbs a photon from a laser beam that is not an infinite plane wave, the atom’s recoil is less than $\hslash k$ in the propagation direction. We show that the recoils in the transverse directions produce a lensing of the atomic wave functions, which leads to a frequency shift that is not discrete but varies linearly with the field amplitude and strongly depends on the atomic state detection. The same lensing effect is also important for microwave atomic clocks. The frequency shifts are of the order of the naive recoil shift for the transverse wave vector of the photons.

Figure 1

(a) An atom interacts with an electromagnetic field for a short time at $t_1$ and $t_2$. The spatial variation of the dipole energy for the two dressed states, depicted in the graph, deflects the atomic wave functions and acts as a positive lens on the dressed state $|2⟩$ and a negative lens on $|1⟩$. (b) The spatially dependent difference of dressed state populations leads to a frequency shift of the order of the recoil shift for the transverse wave vector.

Figure 2

Lensing frequency shift as a fraction of the transverse photon recoil frequency shift ${\nu }_R$ vs (a) the initial size of the atomic cloud $w_{0x}$ and (b) the aperture radius a for Cs atoms at $1\mu \mathrm{K}$ and 1 nK, with $t_1/t_2=0.22$. The dotted lines represent no aperture at time $t_1$ [Eq. (6)] and the solid lines apertures at $t_1$ and $t_2$ [Eq. (5)]. The text describes three effects manifested in the curves: detection uniformity, the variation of the average dipole force with cloud and aperture size, and the ensemble average as a function cloud size and temperature. Generally, the lensing frequency shift is of the order of the transverse photon recoil frequency shift.

Figure 3

Lensing frequency shift for Cs atoms at $1\mu \mathrm{K}$ as a function of $t_1$, the time between the release of the cloud of atoms and the first field interaction. For the solid lines, $t_1$ is changed by varying the distance to the first field interaction with a fixed interrogation time of $T=t_2-t_1=0.5\mathrm{s}$. For the dashed lines, the distance is fixed and the atomic-fountain launch velocity is varied, which changes both $t_1$ and $T$. Here, $t_1=0.14\mathrm{s}$ for $T=0.5\mathrm{s}$, and $T$ ranges from 1 to 0.1 s.