We present measurements of the hyperfine structure (HFS) of the
nPJ Rydberg states for a large range of principal quantum number
n (n=41−−55) employing microwave spectroscopy in an ultracold cesium Rydberg ensemble. A microwave field with $30\text{\ensuremath{-}}\textmu{}\mathrm{s}$ duration couples the $nS\ensuremath{\rightarrow}nP$ transition, yielding a narrow linewidth spectroscopy that approaches the Fourier limit, which allows us to resolve the hyperfine structure of
nPJ states. By analyzing the hyperfine splittings of
nPJ states, we determine the magnetic dipole HFS coupling constant
AHFS,P1/2=3.760(26) GHz for the
P1/2 state, and
AHFS,P3/2=0.718(27) GHz and ${\overline{B}}_{\mathrm{HFS},{P}_{3/2}}=\ensuremath{-}0.084(102)$ GHz for the
P3/2 state. Systematic uncertainties caused by stray electromagnetic field, microwave field power, and Rydberg interaction are analyzed. This measurement is significant for the investigation of Rydberg electrometry and quantum simulation with dipole interaction involving the
nPJ state.
We measure the hyperfine structure of
nP1/2 Rydberg states for
n= 42, 43, 44, and 46 using mm-wave spectroscopy on an ensemble of laser-cooled
85Rb atoms. Systematic uncertainties in our measurement from the Zeeman splittings induced by stray magnetic fields and dipole-dipole interactions between two Rydberg atoms are factored in with the obtained statistical uncertainty. Our final measurement of the
nP1/2 hyperfine coupling constant is
AHFS=1.443(31) GHz. This measurement is useful for studies of long-range Rydberg molecules, Rydberg electrometry, and quantum simulation with dipole-dipole interactions involving
nP1/2 atoms.