Collective modes in an unconventional superconductor with $j=3\mathrm{/}2$ fermions

The $j=3\mathrm{/}2$ 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.