  Superconductors lacking both inversion symmetry and time-reversal symmetry
have been attracting much attention as a platform for exotic superconducting
phases and anomalous phenomena, including the superconducting diode effect.
Recent studies revealed intrinsic phases with this symmetry, named anapole
superconductivity and monopole superconductivity, which are $PT$-symmetric
superconducting states with and without Cooper pairs' total momentum,
respectively. To explore characteristic phenomena in these states, we calculate
and predict the superconducting piezoelectric effect and superconducting diode
effect. A close relationship with the finite-$q$ pairing, asymmetric Bogoliubov
spectrum, and quantum geometry is discussed. This study reveals the
piezoelectric and diode effects as potential probes to elucidate exotic
superconducting phases.
