Non-classical energy squeezing of a macroscopic mechanical oscillator

Optomechanics and electromechanics have made it possible to prepare macroscopic mechanical oscillators in their quantum ground states¹, in quadrature-squeezed states² and in entangled states of motion³. However, the effectively linear interaction between motion and light or electricity precludes access to the broader class of quantum states of motion, such as cat states or energy-squeezed states. Strong quadratic coupling of motion to light could allow a way around this restriction^4–6. Although there have been experimental demonstrations of quadratically coupled optomechanical systems^5,7,8, these have not yet accessed non-classical states of motion. Here we create non-classical states by quadratically coupling motion to the energy levels of a Cooper-pair box qubit. Through microwave-frequency drives that change the state of both the oscillator and qubit, we then dissipatively stabilize the oscillator in a state with a large mean phonon number of 43 and sub-Poissonian number fluctuations of approximately 3. In this energy-squeezed state, we observe a striking feature of the quadratic coupling: the recoil of the mechanical oscillator caused by qubit transitions, closely analogous to the vibronic transitions in molecules^9,10.