  Optomechanics and electromechanics have made it possible to prepare
macroscopic mechanical oscillators in their quantum ground states, in
quadrature squeezed states, and in entangled states of motion. In addition to
coaxing ever larger and more tangible objects into a regime of quantum
behavior, this new capability has encouraged ideas of using mechanical
oscillators in the processing and communication of quantum information and as
precision force sensors operating beyond the standard quantum limit. But 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. Indeed, early optomechanical proposals noted
the possibility to escape this restriction by creating strong quadratic
coupling of motion to light. Although there have been experimental
demonstrations of quadratically coupled optomechanical systems, these have not
yet accessed nonclassical states of motion. Here we create nonclassical states
by quadratically coupling motion to the energy levels of a Cooper-pair box
(CPB) qubit. By monitoring the qubit's transition frequency, we detect the
oscillator's phonon distribution rather than its position. Through microwave
frequency drives that change both the state of 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.
