  We present our experimental investigation of an optical Raman transition
between the magnetic clock states of $^{87}$Rb in an atom chip magnetic trap.
The transfer of atomic population is induced by a pair of diode lasers which
couple the two clock states off-resonantly to an intermediate state manifold.
This transition is subject to destructive interference of two excitation paths,
which leads to a reduction of the effective two-photon Rabi-frequency.
Furthermore, we find that the transition frequency is highly sensitive to the
intensity ratio of the diode lasers. Our results are well described in terms of
light shifts in the multi-level structure of $^{87}$Rb. The differential light
shifts vanish at an optimal intensity ratio, which we observe as a narrowing of
the transition linewidth. We also observe the temporal dynamics of the
population transfer and find good agreement with a model based on the system's
master equation and a Gaussian laser beam profile. Finally, we identify several
sources of decoherence in our system, and discuss possible improvements.
