Phonon decoherence produced by two-level tunneling states

Phonon modes within pristine crystalline resonators now routinely reach the quantum ground state. Such systems are attractive for quantum information science applications, as advanced fabrication and processing can enable relatively long quantum coherence times, and precision control can be realized through optical, electrical, or qubit coupling. In many state-of-the-art systems, the phonon lifetime is limited by disorder. In particular, native oxides or damaged `dead layers' at surfaces can host two-level tunneling states that lead to a particularly problematic form of dissipation that increases at lower temperatures. As mechanical losses are driven down in systems such as micro-fabricated bulk acoustic wave resonators, tunneling states are expected to emerge as the dominant mechanism for phonon decoherence. A quantitative description of these mesoscopic systems therefore requires a framework that captures interactions between a selected phonon mode and a large ensemble of TLS. Here, we derive a quantum master equation for this coupled system, permitting the phonon decoherence produced by two-level tunneling states to be calculated. As an example, we estimate the lifetime of a variety of quantum states within quartz micro-resonators hosting a thin surface layer of tunneling states. We find that the phonon coherence time is maximized at low temperatures, in spite of increased mechanical dissipation, and that phonon-TLS coupling can be reduced for modes with strain nodes at the surfaces.