Enhanced quasiparticle lifetime in a superconductor by selective blocking of recombination phonons with a phononic crystal

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.