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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 restriction4,5,6. Although there have been experimental demonstrations of quadratically coupled optomechanical systems5,7,8, 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 molecules9,10.

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Fig. 1: Quadratically coupled electromechanics.
Fig. 2: Determining the phonon distribution from a qubit spectrum.
Fig. 3: Dissipative energy squeezing.
Fig. 4: Dissipatively stabilized sub-Poissonian state.

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Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The codes that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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We thank D. Palken, L. Sletten, R. Delaney, F. Beaudoin and L. Tian for fruitful discussions. We gratefully acknowledge R. Simmonds and F. Lecocq for their help with the fabrication of the device. We thank M. Malnou and D. Palken for providing us with a Josephson parametric amplifier. We acknowledge funding from National Science Foundation (NSF) under grant no. PHY-1734006. J.J.V. acknowledges financial support from the European Union’s H2020 programme under Marie Skłodowska-Curie grant no. 841618

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Authors and Affiliations



X.M., J.J.V. and K.W.L. conceived and designed the experiment, X.M. and J.J.V. fabricated the device and performed the experiment, X.M., S.K., J.J.V. and K.W.L. derived the theory and analysed the data and X.M., K.W.L. and J.D.T. prepared the manuscript. All authors provided suggestions for the experiment, discussed the results and contributed to the manuscript.

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Correspondence to X. Ma.

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Peer review information Nature Physics thanks Adrian Bachtold, Mario Gely, Gary Steele and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Discussion, Figs. 1–14 and Table 1.

Source data

Source Data Fig. 2

Numerical data used to generate Fig. 2.

Source Data Fig. 3

Numerical data used to generate Fig. 3.

Source Data Fig. 4

Numerical data used to generate Fig. 4.

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Ma, X., Viennot, J.J., Kotler, S. et al. Non-classical energy squeezing of a macroscopic mechanical oscillator. Nat. Phys. 17, 322–326 (2021).

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