  When quasiparticles in a BCS superconductor recombine into Cooper pairs,
phonons are emitted within a narrow band of energies above the pairing energy
at 2$\Delta$. We show that a phonon bandgap restricting the escape of
recombination phonons from a superconductor can increase the quasiparticle
recombination lifetime by more than an order of magnitude. A phonon bandgap can
be realized and matched to the recombination energy with a phononic crystal, a
periodically-patterned dielectric membrane. We discuss in detail the
non-equilibrium quasiparticle and phonon distributions that arise in a
superconductor due to a phonon bandgap and a pair-breaking photon signal.
Although intrinsically a non-equilibrium effect, the lifetime enhancement in
the small-signal regime is remarkably similar to an estimate from an
equilibrium formulation. The equilibrium estimate closely follows
$\exp(\Omega_{bg}/k_BT_b)$, where $\Omega_{bg}$ is the phonon bandgap energy
bandwidth above 2$\Delta$, and $T_b$ is the phonon bath temperature of the
coupled electron-phonon system. We discuss the impact of a phononic bandgap on
the performance of superconducting devices, and propose a superconducting
microwave resonator circuit to measure the enhancement in the quasiparticle
lifetime.
