Optomechanical effects of two-level systems in a back-action evading measurement of micro-mechanical motion

We show that the two-level systems (TLS) in lithographic superconducting circuits act as a power-dependent dielectric leading to non-linear responses in a parametrically coupled electromechanical system. Driven TLS shift the microwave resonance frequency and modulate the mechanical resonance through the optical spring effect. By pumping with two tones in a back-action evading measurement, these effects produce a mechanical parametric instability which limits single quadrature imprecision to 1.4 xzp. The microwave resonator noise is also consistent to a TLS-noise model. These observations suggest design strategies for minimizing TLS effects to improve ground-state cooling and quantum non-demolition measurements of motion.