Photon Recoil Momentum in Dispersive Media

A systematic shift of the photon recoil momentum due to the index of refraction of a dilute gas of atoms has been observed. The recoil frequency was determined with a two-pulse light grating interferometer using near-resonant laser light. The results show that the recoil momentum of atoms caused by the absorption of a photon is $n\hslash k$, where $n$ is the index of refraction of the gas and $k$ is the vacuum wave vector of the photon. This systematic effect must be accounted for in high-precision atom interferometry with light gratings.

Figure 1

Kapitza-Dirac interferometer. The first pulse outcoupled a small fraction of atoms into the $|\pm 2n\hslash k⟩$ momentum states. The outcoupled atoms moved within the initial condensate. After a variable delay $\tau$, a second pulse was applied, and atoms outcoupled by the second pulse interfered with those outcoupled by the first pulse. The laser beam was applied perpendicular to the long axis of the condensate; the polarization, $\overrightarrow{E}$, was parallel to it and to the applied magnetic field bias, $\overrightarrow{B}$. The atoms were imaged after 38 ms of ballistic expansion. The field of view is $0.5\mathrm{mm}\times 1.5\mathrm{mm}$.

Figure 2

Interference fringes oscillating at the recoil frequency. (a) Absorption images for $\tau =10–50\mu \mathrm{s}$. The detuning was $\Delta /2\pi =+520\mathrm{MHz}$. The field of view is $0.5\mathrm{mm}\times 1.5\mathrm{mm}$. (b) Fraction of atoms in the $|0\hslash k⟩$ momentum state as a function of $\tau$. The fringes were fit using Eq. (1). The fitted frequency was $\omega =2\pi \times 15627(39)\mathrm{Hz}$ with decay constant ${\tau }_c=461(25)\mu \mathrm{s}$. The signal was normalized using the total atom number in all momentum states. The systematic scatter of the data from the fit indicates the reproducibility of the single shot measurements.

Figure 3

Recoil frequency as a function of detuning, $\Delta /2\pi$, showing the dispersive effect of the index of refraction. The average density of the condensate for the solid points was $1.14(4)\times {10}^{14}{\mathrm{cm}}^{-3}$, giving rise to a mean-field shift of 880 Hz. The shaded area gives the expected recoil frequency including the uncertainty in the density. The dashed line is at $\omega =4{\omega }_{\mathrm{rec}}+\rho U/\hslash$, the expected value without index of refraction effects. The dotted line is at $4{\omega }_{\mathrm{rec}}=15068\mathrm{Hz}$, the two-photon vacuum recoil frequency. The data shown as open diamonds had increased spontaneous light scattering due to ${\sigma }^{\pm }$ light contamination in the laser beam. The increased light scattering led to a lower initial density in the condensate, thus leading to a smaller mean-field shift. The ${\sigma }^{\pm }$ contamination allowed $\Delta m_F=\pm 1$ transitions, thus for small detunings the proximity to the $|1,-1⟩\to |0^{\prime },0⟩$ transition located at $\Delta /2\pi =-72\mathrm{MHz}$ resulted in higher spontaneous scattering rates. The open points have been scaled upward to correct for this lower density.