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^4,5,6. 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.

光机械学和机电学使制备处于量子基态¹、正交压缩态²和运动纠缠态³的宏观机械振荡器成为可能。然而，运动与光或电之间的有效线性相互作用阻止了对更广泛的运动量子态的访问，例如猫态或能量压缩态。运动与光的强二次耦合可以绕过这个限制^4,5,6。虽然已经有二次耦合光机械系统的实验演示^5,7,8，这些还没有进入非经典的运动状态。在这里，我们通过将运动二次耦合到库珀对盒量子比特的能级来创建非经典状态。通过改变振荡器和量子比特状态的微波频率驱动，我们将振荡器耗散稳定在平均声子数为 43 和亚泊松数波动约为 3 的状态。在这种能量压缩状态下，我们观察到二次耦合的一个显着特征：由量子位跃迁引起的机械振荡器的反冲，非常类似于分子中的振动跃迁^9,10。