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

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