  We report on broad instantaneous bandwidth microwave spectrum analysis with
hot $^{87}\mathrm{Rb}$ atoms in a microfabricated vapor cell in a large
magnetic field gradient. The sensor is a MEMS atomic vapor cell filled with
isotopically pure $^{87}\mathrm{Rb}$ and $\mathrm{N}_2$ buffer gas to localize
the motion of the atoms. The microwave signals of interest are coupled through
a coplanar waveguide to the cell, inducing spin flip transitions between
optically pumped ground states of the atoms. A static magnetic field with large
gradient maps the $\textit{frequency spectrum}$ of the input microwave signals
to a position-dependent $\textit{spin-flip pattern}$ on absorption images of
the cell recorded with a laser beam onto a camera. In our proof-of-principle
experiment, we demonstrate a microwave spectrum analyzer that has $\approx$ 1
GHz instantaneous bandwidth centered at 13.165 GHz, 3 MHz frequency resolution,
2 kHz refresh rate, and a -23 dBm single-tone microwave power detection limit
in 1 s measurement time. A theoretical model is constructed to simulate the
image signals by considering the processes of optical pumping, microwave
interaction, diffusion of $^{87}\mathrm{Rb}$ atoms, and laser absorption. We
expect to reach more than 25 GHz instantaneous bandwidth in an optimized setup,
limited by the applied magnetic field gradient. Our demonstration offers a
practical alternative to conventional microwave spectrum analyzers based on
electronic heterodyne detection.
