**Precision Measurements of Atomic Recoil Frequency**

** **

The experiment involves a direct and precise measurement of the energy imparted to atoms by laser fields. A brief standing wave pulse of light is used to diffract atomic deBroglie waves. During the interaction with the light field, atoms absorb momenta in quantized units by absorption and stimulated emission of photons and return to the same internal state. The quantized units of momenta are 2nħk where n is an integer, ħ is h/2pi and k=2π/λ is the wave vector of light. Thus each atom evolves into a superposition of momentum states following laser interaction.

The atomic wave function is modulated at the atomic recoil
frequency ω_{r }= ħbΔk^{2}/2m (Δk is the
momentum transferred by the laser field and m is the atomic mass) and the goal
of the experiment is to measure the recoil energy ħω_{r} from
a precise determination of the recoil period T_{r }= π/ω_{r}
(~ 32 micro sec). ω_{r} can be related to h/m and measurement of
h/m can be used to determine the atomic fine structure constant alpha much more
precisely than a separate determinations of h or m. A measurement of alpha is
of interest because it is a fundamental constant that governs the strength of
light matter interactions and atomic structure.

Since the standing wave light potential is spatially
modulated with a period λ/2, and the traveling wave components of this
field are detuned far above the atomic resonance in a dilute gas (in this case,
laser cooled Rb), atoms experience a channeling force toward the nodes of the
potential. As a result, a density grating or one-dimensional optical lattice is
formed in the sample. This is a transient effect that can be observed on a time
scale of the order of the coherence time of the de-Broglie waves (a few
μs). The relatively short coherence time is much smaller than the period
of recoil modulation in the wave function (T_{r}). The coherence time
can be increased by cooling the atomic sample well below the typical operating
temperature of 50μK, which is the equilibrium temperature of atoms in the
laser cooled sample. However, cooling the sample can increase the atomic
density and produce density dependent systematic shifts in the recoil period.
Consequently, we utilize a second standing wave pulse and the echo effect to
observe the recoil modulation.

The second standing wave pulse applied at t = T interacts
with the momentum states of each atom after they have drifted apart over
macroscopic distances (much larger than the deBroglie wavelength). The
interaction with the second pulse recombines the momentum states at t = 2T in a
manner reminiscent of a photon echo. The interference of momentum states
separated by 2ħk in the vicinity of t = 2T is associated with the
formation of a density grating of period λ/2. The grating contrast can be
detected by coherently backscattering a traveling wave from the cold sample.
The intensity of backscattered light (known as a grating echo) exhibits a
periodic modulation at ω_{r} as a function of T. The recoil period
T_{r} can be measured with high by recording data over a large time
interval and fitting the periodic function.

Since the atomic state internal state is a long lived ground
state, the time scale of the experiment should be limited only by the transit
time for cold atoms through the region of interaction. We have developed
several techniques for measuring ω_{r} with high precision in both
the time domain and the frequency domain. We are presently interested in
further improving the time scale in an atomic fountain.

**Key Papers**

a) Experimental setup using a single, retro-reflected beam.

b) Experimental setup using
two separate, counter-propagating beams.

** **

Billiard ball representation of the momentum states
associated with the interferometer. Only three possible trajectories (0,
±ħq) due to the first excitation pulse are shown. A subset of higher-order
processes (±2ħq, ±3ħq, etc.) can be rephased at t = 2T, but only
trajectories differing by ħq contribute to the signal at t = 2T. The echo
envelope is also shown in the vicinity of t = 2T and its temporal width is
inversely proportional to the velocity distribution. Here, the time interval
between extrema is ~2 μs, T~1–30 ms, and T_{recoil }=
π/ω_{q} ~32 μs.

** **

A typical recoil measurement with Δ
= 50 MHz. The measurement is carried out by fitting the data to theory and
consists of one recoil period recorded at T ~ 900 μs (a) and a second
recoil period recorded at T ~ 9000μs (b). The fit gives ωq = 97
008.5(4) s^{-1}.

New method of measuring
atomic recoil, showing effect of a decoherence pulse (DP) on the atom
interferometer (AI). Two interfering momentum states (solid black lines) are
perturbed by a DP applied at t = 2T − δT. The DP can affect both
arms of the interferometer resulting in interference between displaced
trajectories (dashed lines) that is phase shifted with respect to the original
interference pattern. Constructive interference is observed if δT = nT_{r}
where n is an integer.

Observed Ramsey fringe
pattern; solid line is data; dashed line is fit which yields ω_{r}
= 97.0 × 10^{3} s^{-1} and θ_{2 }= 1.8.

_{r}
using a single sw DP with duration τ = 150 ns. The fit gives ω_{r}
= 97.0032 ± 0.00016 kHz.

Recoil frequency measurement precise to 220 ppb using a time scale of 90 ms and a fringe width of ~2 ms. The data was acquired in 10 minutes. Analysis of 82 similar measurements recorded under the same conditions over 14 hours of interrogation time shows an average precision of 380 ppb. The resulting uncorrected value of the recoil frequency was measured with a precision of 37 pbb (standard deviation of the mean).

AI
signal lifetime for two-pulse and various three-pulse configurations in a glass
cell with a sample temperature of *T**
**∼** *20 mK.
The horizontal axis is the time of the read-out pulse, *T*_{RO}, relative to the time of trap turn-off,
*T*_{0}, which signifies
the start of the experiment. The three-pulse AI is less sensitive to B
gradients and vibrations than the two-pulse AI. Decoherence due to these
effects is limited by adjusting the pulse spacing *T _{21}*, which controls the separation between wave packets
so that the measurement time scale is comparable to the transit time.