  While steady empirical progress has been made in understanding the structure
and composition of hot planet atmospheres, direct measurements of velocity
signatures, including winds, rotation, and jets, have lagged behind.
Quantifying atmospheric dynamics of hot planets is critical to a complete
understanding of their atmospheres and such measurements may even illuminate
other planetary properties, such as magnetic field strengths. In this
manuscript we present the first detection of the Balmer lines H$\alpha$ and
H$\beta$ in the atmosphere of the ultra-hot Jupiter WASP-33 b. Using
atmospheric models which include the effects of atmospheric dynamics, we show
that the shape of the average Balmer line transmission spectrum is consistent
with rotational velocities in the planet's thermosphere of $v_\text{rot} =
10.1^{+0.8}_{-1.0}$ km s$^{-1}$. We also measure a low-significance
day-to-night side velocity shift of $-4.6^{+3.4}_{-3.4}$ km s$^{-1}$ in the
transmission spectrum which is naturally explained by a global wind across the
planet's terminator. In a separate analysis the time-resolved velocity
centroids of individual transmission spectra show unambiguous evidence of
rotation, with a best-fit velocity of $10.0^{+2.4}_{-2.0}$ km s$^{-1}$,
consistent with the value of $v_\text{rot}$ derived from the shape of the
average Balmer line transmission spectrum. Our observations and analysis
confirm the power of high signal-to-noise, time-resolved transmission spectra
to measure the velocity structures in exoplanet atmospheres. The large
rotational and wind velocities we measure highlight the need for more detailed
3D global climate simulations of the rarefied upper-atmospheres of ultra-hot
gas giants.
