  The $j = 3/2$ fermions in cubic crystals or cold atomic gases can form Cooper
pairs in both singlet ($J = 0$) and unconventional quintet ($J = 2$) $s$-wave
states. Our study utilizes analytical field theory to examine fluctuations in
these states within the framework of the Luttinger-Kohn model. We investigate
how collective modes evolve with varying spin-orbit coupling (SOC) strength. In
the singlet state, quintet Bardasis-Schrieffer modes soften at a finite
wavevector, hinting at Fulde-Ferrell-Larkin-Ovchinnikov physics. In the quintet
state, we identify additional gapless and gapped modes originating from the
partially broken symmetry due to SOC. Our results can be readily detected using
current experimental techniques.
