Thermal nonlinearity and readout-induced TLS heating in phononic crystal resonators

Micro-mechanical resonators for quantum processing are particularly susceptible to readout-power-induced heating, stemming from the competing requirements of isolating the mechanical mode of interest to preserve high coherence, while simultaneously maintaining strong thermal anchoring to a low-temperature bath to keep the resonator cold. This trade-off limits the performance of piezoelectric phononic crystal resonators (PCRs), a leading platform for circuit quantum acoustodynamics, and their optical counterpart, optomechanical crystal cavities. In both systems, acoustic bandgaps engineered in periodically-patterned suspended structures are employed to suppress phononic leakage through the resonator supports, enabling ultra-long phonon lifetimes ultimately limited by coupling to microscopic two-level system defects (TLSs).

We show that readout-power heating and coupling to TLSs can jointly give rise to a mixed reactive-dissipative nonlinear behavior in a micro-mechanical resonator at millikelvin temperatures. We present experimental evidence of such amplitude-frequency nonlinearity in a thin-film quartz phononic crystal resonator. The reactive effect, either softening or hardening, depends on the ratio of thermal to phonon energy and arises from broadband heating of a spectrally large population of TLSs by the dissipated readout power. Combining standard TLS theory with a thermal conductance model, we quantitatively reproduce the steady-state resonator response and identify readout-enhanced relaxation damping from off-resonant TLSs as the main limit to mechanical coherence. At low power, deviations from the predicted generalized Duffing behavior emerge as individual strongly-coupled TLSs dominate the measured response. We conclude with preliminary results from an extended model designed to capture the time-dependent dynamics associated with TLS heating.