**Magnetic Sublevel Coherences for Precise
Measurements of Atomic g Factor Ratios>**

We study the evolution of a spatially periodic superposition
of magnetic sublevels of the ground state of Rb vapor in the presence of static
magnetic fields. Two traveling wave pulses with orthogonal polarizations
detuned from the excited state are applied at an angle of ~10 mrad to create a
superposition of magnetic sublevels of the same hyperfine ground state.
Magnetic sublevels differing by either Δm = 1 or Δm = 2 are coupled
by the laser fields. The spatially periodic superposition dephases due to the
velocity distribution of the sample. The dephasing can be detected by applying
a traveling wave read out pulse along one of the excitation directions, and
detecting the coherently scattered light along the other excitation direction.
This coherent transient response is called magnetic grating free induction
decay (MGFID). In the absence of magnetic fields, the temperature of a laser
cooled sample inferred using the MGFID is in agreement with the temperature
obtained by imaging the ballistic expansion of the sample. We have developed a
theoretical description of the evolution of the MGFID in arbitrary static
magnetic fields. The formulation describes the effect of the magnetic field as
a time dependent rotation of the atomic system about the quantization axis
using a rotation matrix approach. The predictions for the functional form of
Larmor oscillations in the MGFID based on this approach are in excellent
agreement with data from both room temperature vapor and laser cooled atoms. We
are currently exploiting the properties of the MGFID for a precise
determination of the ratio of atomic g factors using ^{85}Rb and ^{87}Rb.
We have also used a rate equation approach to numerically simulate properties
of the magnetic grating echo (MGE) in magnetic fields. The MGE can be observed
by applying two sets of excitation pulses separated by t = T. The second excitation
pulse cancels the effect of Doppler dephasing in the sample so that an echo is
observed at t = 2T. In the
presence of magnetic fields, experiments show that the MGE signal envelope
develops Larmor oscillations. The MGE amplitude at t = 2T exhibits oscillations
that depend on the Rabi frequencies of the excitation pulses and the magnetic
field. These observations are consistent with simulations. Our results suggest
that the MGE can be used to achieve a significantly improved measurement of the
atomic g factor ratio.

**Key Papers**

Representation of evolution of coherences with J =
3.

We consider the situation in which the quantization
axis is along *z* (solid black
line) and the magnetic field is along *y *(dotted line) for an atomic system with *J *= 1/2 .

(a) Surface ρ_{1/2,1/2 }(θ, φ, *t*) for *t *= 0.

(b) ρ_{1/2,1/2 }(θ, φ, *t*) for *t *= 1/4*T _{L}
*

(c) ρ_{1/2,1/2 }(θ, φ, *t*) for *t *= 1/2*T _{L}
*

(d) ρ_{1/2,1/2 }(θ, φ, *t*) for *t *= 3/4*T _{L}
*.

Here, *T _{L}* = 2π/ω

** **

Evolution of the in phase
and quadrature parts of the MGFID in a sample of cold atoms. a) The
excitation pulses have orthogonal linear polarizations and the magnetic field
is directed at an angle of π/4 with respect to the polarization of k_{1}.
b) The excitation pulses have opposite circular polarizations with the
direction of the magnetic field at an angle of π/4 to the direction of k_{1}.
In both cases the excitation pulses widths were 2 μs, and the detuning was
40 MHz. The data are shown as dots and the solid lines are fits based on
theory.

(a) Ground-state
coherence signal from the dual isotope MOT. (b) Close up of the data displayed
in (a) showing multiple-frequency components in the signal. Only one eighth of
the points are displayed so that the fit (red line) and the data (black dots)
can be distinguished. From a single shot, the ratio of effective atomic *g *factors* *is typically
determined to a precision of∼1*.*5 ppm. (c) Seventy-five data sets acquired in ∼3 minutes. The error bars represent
the fit uncertainties. The mean value is *r *=1.4988871and
the (1*σ*) standard deviation
of the mean is 1.0 x 10^{-6} (0.69 ppm).