Abstract
Mechanical degrees of freedom are natural candidates for continuousvariable quantum information processing and bosonic quantum simulations. However, these applications require the engineering of squeezing and nonlinearities in the quantum regime. Here we demonstrate squeezing below the zeropoint fluctuations of a gigahertzfrequency mechanical resonator coupled to a superconducting qubit. This is achieved by parametrically driving the qubit, which results in an effective twophonon drive. In addition, we show that the resonator mode inherits a nonlinearity from the offresonant coupling with the qubit, which can be tuned by controlling the detuning. We, thus, realize a mechanical squeezed Kerr oscillator, in which we demonstrate the preparation of nonGaussian quantum states of motion with Wigner function negativities and high quantum Fisher information. This shows that our results can also have applications in quantum metrology and sensing.
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Main
From the oscillation of a trapped particle to the vibration of a solidstate structure, mechanical modes are ubiquitous degrees of freedom that can exhibit soughtafter properties such as high quality factors and large coupling rates to spins and electromagnetic fields. When operated in the quantum regime, mechanical modes are powerful building blocks for quantum technologies, with applications in information processing^{1,2,3}, bosonic simulations^{4,5}, memories^{6} and microwavetooptical frequency conversion^{7,8}. Moreover, their nonzero mass makes them particularly suited for sensing forces^{9,10,11}, as well as for fundamental physics investigations, ranging from tests of the superposition principle^{12} to the detection of dark matter^{13} and quantum gravity effects^{14}. To fully unlock these applications, however, it is necessary to have available a sophisticated toolbox for the preparation and manipulation of quantum states of motion, which is a nontrivial task.
In this context, mechanical resonators have recently attracted a lot of attention as new elements for hybrid quantum systems^{15,16}. In particular, gigahertzfrequency resonators can be interfaced to superconducting qubits and thus controlled with the toolbox of circuit quantum acoustodynamics (cQAD). For example, resonant interaction with a qubit was used to demonstrate the preparation of mechanical Fock states^{17,18,19} and Schrödinger cat states^{20}. Crucially, compared with their electromagnetic counterparts, mechanical resonators have small physical footprints and a high density of accessible longlived modes, making them ideal candidates for hardwareefficient quantum processors^{2} and quantum random access memories^{6}.
The realization of continuousvariable quantum computing and bosonic simulations relies on the availability of a universal gate set composed of phase shift, displacement, beamsplitter, singlemode squeezing and Kerr nonlinearity^{21,22,23,24}. The first two are relatively simple to realize through free evolution and coherent driving. Beamsplitter operations have been recently demonstrated in cQAD between surface^{3} and bulk^{25} acoustic waves. For mechanical systems, quantum noise squeezing was pioneered in trapped ions^{26}, and later demonstrated in drum oscillators using the tools of electromechanics^{27,28,29,30,31}. In cQAD, a recent experiment demonstrated the twomode squeezing of gigahertzfrequency surface acoustic waves through the modulation of one of the Bragg reflectors^{32}. Nonlinear evolutions in the quantum regime are difficult to realize with standard opto or electromechanical coupling, since this coupling is linear for small displacements. One possibility is to offresonantly couple a mechanical oscillator to a twolevel system, which gives rise to an effective nonlinearity for the phonon through a hybridization of modes. Recently, this was demonstrated in an experiment where the vibrational modes of a carbon nanotube were coupled to a quantum dot^{33}. Despite all this progress, however, the demonstration of a full gate set for universal continuousvariable quantum information processing in a single cQAD device is still lacking.
In this work, we present squeezing below the zeropoint fluctuations of a gigahertzfrequency phonon mode of a highovertone bulk acousticwave resonator (HBAR) with tunable nonlinearity. The phonon mode is coupled to a superconducting qubit, which we use as a mixing element for implementing the effective squeezing drive: by applying two microwave tones to the qubit, we activate a parametric process that creates pairs of phonons in the resonator. Moreover, this coupling gives rise to an effective Kerr nonlinearity for the phonon mode, which we tune by changing the qubit–phonon detuning. To characterize our system, we study the dependence of the squeezing rate as well as the Kerr nonlinearity on different system parameters. Having demonstrated control over both these quantities, we combine them to realize a mechanical version of a squeezed Kerr oscillator, a paradigmatic model in quantum optics. By using the qubit to perform direct Wigner function measurements, we show that operating this system in different regimes results in the preparation of nonGaussian states of motion with Wigner negativities and high quantum Fisher information (QFI).
The device used in this work is a cQAD system where a transmon qubit is flipchip bonded to an HBAR, an improved version of the devices used in previous works^{19,34}. The qubit has a frequency ω_{q} = 2π × 5.042 GHz, which can be tuned via a Starkshift drive^{34}. At this frequency, the qubit has an energy relaxation time T_{1} = 17(0.4) μs, Ramsey decoherence time \({T}_{2}^{* }\) = 24(0.7) µs and anharmonicity α = 2π × 185 MHz. The HBAR is coupled to the qubit through a piezoelectric transducer made of aluminium nitride that mediates a Jaynes–Cummings interaction with a coupling strength g = 2π × 292 kHz. The phonon mode we consider in this work has a frequency ω_{a} = 2π × 5.023 GHz, an energy relaxation time T_{1} = 132(4) μs and a Ramsey decoherence time \({T}_{2}^{* }\) = 210(9) µs.
Our system can be described by the Hamiltonian
where q and a are the bosonic annihilation operators for the qubit and phonon mode, respectively. The term \({H}_{{{{\rm{qd}}}}}/\hslash =\left({\varOmega }_{1}{{\rm{e}}}^{{\rm{i}}{\omega }_{1}t}+\right.\) \(\left.{\varOmega }_{2}{{\rm{e}}}^{{\rm{i}}{\omega }_{2}t}\right){q}^{{\dagger} }+\,{{\rm{h.c.}}}\,\) describes the two offresonant microwave drives at frequencies ω_{1,2} and amplitude Ω_{1,2} applied to the qubit (Fig. 1a). We define the detunings Δ_{1,2} = ω_{1,2} − ω_{q} and the dimensionless drive strengths ξ_{1,2} = Ω_{1,2}/Δ_{1,2}. In addition, we use a third, faroffresonant drive at approximately 8.4 GHz to control the qubit frequency via the a.c. Stark shift.
When the resonance condition ω_{1} + ω_{2} = 2ω_{a} is fulfilled, the qubit nonlinearity mediates a fourwavemixing process that results in a twophonon drive (a^{†2} + a^{2}). In addition, the phonon mode also inherits a nonlinearity a^{†2}a^{2} from the coupling to the qubit. The emergence of these squeezing and Kerr terms can be unveiled through a series of unitary transformations (Supplementary Information), and results in an effective Hamiltonian for the phonon mode that reads
Here \(\varDelta =({\omega }_{1}+{\omega }_{2}2{\omega }_{a}^{{\prime} })/2\), where \({\omega }_{a}^{{\prime} }\approx {\omega }_{a}+\frac{{g}^{2}}{{\varDelta }_{a}}\) is the frequency of the phonon mode including a normal mode shift due to the presence of the qubit. \({\varDelta }_{a}={\omega }_{a}{\omega }_{\rm{q}}^{\rm{ss}}\) is the detuning between the phonon mode and the a.c. Starkshifted qubit. The squeezing rate ϵ (Supplementary Information) is given by refs. ^{35,36}
where Σ_{21} = Δ_{1} + Δ_{2}. Finally, K is the Kerr nonlinearity: \(K\approx {g}^{4}/{\varDelta }_{a}^{3}\) for α ≫ Δ_{a} ≫ g (Supplementary Information).
The Hamiltonian in equation (2) is a paradigmatic model in quantum optics, exhibiting a plethora of interesting phenomena such as chaotic dynamics^{37}, quantum phase transitions^{38}, tunnelling^{39} and parametric amplification^{40}. Moreover, this model admits macroscopic superpositions as quantum ground states, which can be exploited for errorprotected qubit encoding^{41,42,43}. The latter application made squeezed Kerr oscillators particularly attractive for quantum information processing, which motivated their recent experimental implementation for electromagnetic modes in circuit quantum electrodynamics platforms^{44,45,46,47}. Here we present the implementation for a mechanical mode, and use it to prepare squeezed and nonGaussian quantum states of motion of a massive system.
First, let us investigate the squeezing dynamics for a large Δ_{a} = 2π × 1.5 MHz, such that \(K\propto {\varDelta }_{a}^{3}\) is substantially smaller than \(\epsilon \propto {\varDelta }_{a}^{1}\). When the parametric drives are applied, the qubit frequency is modified by the a.c. Stark shift, which then results in a change to the normal mode shift of the phonon mode (Fig. 1a). To ensure that the resonance condition for squeezing is satisfied, we correct for these shifts by performing the following calibration experiment (Fig. 1b): we apply the parametric drives for a time t_{S} = 20.0 μs, including 0.5 μs Gaussian edges, with ∣Δ_{1,2}∣ = 15 MHz and drive frequency ω_{2} = 2ω_{a} − ω_{1} + δ. We then reset the qubit to its ground state by swapping the population acquired during the offresonant driving to an ancillary phonon mode. Finally, we bring the qubit into resonance with the phonon mode we want to squeeze for time \(\uppi /(2\sqrt{2}g)\), thereby swapping part of the phonon population to the qubit, after which we measure the qubit state using a standard dispersive readout. Repeating this experiment for different δ yields the data shown in Fig. 1c, showing a peak at around δ ≈ 2π × 140 kHz that provides an indication of when the twophonon drive becomes resonant.
To fully characterize the state resulting from the twophonon drive and verify the coherence of this process, we set a desired δ and t_{S}, and perform a Wigner function measurement of the phonon mode^{34}. The results for δ = 2π × 80 kHz and t_{S} = 0, 6 and 12 μs are shown in Fig. 1d. For t_{S} = 0 μs, we obtain a measurement of the ground state, whereas for larger times, we observe a reduction in the quantum noise along one quadrature, that is, squeezing, as well as an increase in the perpendicular quadrature. Note that for the longest evolution time, we also see a distortion of the state, which is due to the residual phonon mode nonlinearity. As comprehensively shown later, this distortion limits the squeezing, and it depends on δ. Choosing δ = 2π × 80 kHz allowed us to observe the strongest squeezing, because of a partial compensation of nonlinearity K by detuning Δ ≈ g^{2}/Δ_{a} − δ/2 (equation (2)).
Given phasespace quadratures X and P, we define the variance along direction θ as V(θ) = Var[Xcosθ + Psinθ]. For an ideal ground state V_{GS} = V(θ) = 1/2; therefore, observing a smaller value implies quantum squeezing. Therefore, we define V_{min} = min_{θ}V(θ) and V_{max} = max_{θ}V(θ) as the squeezed and antisqueezed quadratures. Fitting the Wigner function at t_{S} = 6 μs with a twodimensional Gaussian, we obtain V_{min} = 0.252(6), which corresponds to a noise reduction of 3.0(1) dB below V_{GS}. From this, and V_{max} = 1.45(4), we estimate a thermal population of \(\sqrt{{V}_{\min }{V}_{\max }}1/2=0.10(1)\), corresponding to a state purity of 83(1)% (Supplementary Information). For the chosen experimental parameters, numerical simulations indicate that t_{S} ≈ 6 μs is the optimal time for the strongest squeezing, which is mostly limited by residual nonlinearity (Fig. 2a). In fact, longer evolution times result in nonGaussian states, such as the one shown for t_{S} = 12 μs, for which the above analysis is not reliable.
Alternatively, the maximum likelihood estimate allows us to reconstruct the corresponding density matrices from the measurements shown in Fig. 1d ref. ^{48}, from which we calculate the associated covariance matrices. For t_{S} = 6 μs, we obtain V_{min} = 0.236(1), whereas for t_{S} = 12 μs, we obtain V_{min} = 0.268(3), which shows less squeezing due to nonlinear evolution. We then plot the diagonal elements of the density matrices (Fig. 1e), corresponding to the Fockstate populations of the phonon states. We observe high populations of even states, characteristic of a twophonon drive. The black bars indicate a fit of the reconstructed populations to the closest ideal squeezed state.
To characterize the usefulness of the prepared squeezed states for applications such as quantum metrology, we measure their lifetime. For this, we prepare a squeezed state and wait for a variable time t_{w} before performing the Wigner function measurement (Fig. 1b). The (anti)squeezing for different t_{w} values is shown in Fig. 1f. As expected from the free evolution of a squeezed state in the presence of relaxation, we observe that its variances gradually return to those of the ground state (Supplementary Information). Fitting the squeezed variance measurements with \((1{\rm{e}}^{{\gamma }_{\rm{d}}t}(12{V}_{\min }))/2\), where V_{min} is the variance at t_{w} = 0, gives a decay time \({\gamma }_{\rm{d}}^{1}\) = 78(11) µs. Although the decay time of V_{max} of 125(12) μs is compatible with phonon T_{1}, we attribute the lower decay time \({\gamma }_{\rm{d}}^{1}\) of the minimum variance to the fact that the squeezed quadrature is more sensitive to dephasing than the antisqueezed quadrature.
We now investigate the dependence of the squeezing rate ϵ on the experimental parameters. We extract ϵ by measuring the evolution of V_{min} as a function of the squeezing time t_{S}. Concretely, ϵ is inferred from a fit of the squeezing measurements with the function V_{min}(t) = (γ + 4ϵe^{–t(γ+4ϵ)})/2(γ + 4ϵ), which describes the squeezing dynamics in the presence of decay at rate γ (Supplementary Information). Note that ϵ can be estimated from the evolution at short times t_{S} ≪ 1/K, where the state is still Gaussian. For this reason, we obtain V_{min} from a Gaussian fit, after having checked that for short times, it is consistent with the one obtained from state reconstruction. An example is shown in Fig. 2a, for δ = 2π × 80 kHz, Δ_{a} = 2π × 1.5 MHz, ξ_{1} = 0.28 and ξ_{2} = 0.26. Fitting of V_{min}(t) gives us an effective decay time γ^{−1} = 12.8(11) μs and squeezing rate ϵ = 2π × 7.6(3) kHz. Note that this effective decay time is shorter than bare phonon T_{1}, as these measurements are subject to Purcell decay via the qubit and to additional dephasing resulting from finite qubit population and parametric driving.
Repeating the measurement shown in Fig. 2a for different drive powers ξ_{1}ξ_{2} and qubit–phonon detunings Δ_{a}, we obtain the rates shown in Fig. 2b,c. The precise value of these controllable parameters is extracted from independent measurements in the following way. We set the desired drive strengths ξ_{1,2}, which we previously calibrated via their Stark shift on the qubit^{25}. The detuning Δ_{a} is then set by changing the qubit frequency via the independent Starkshift drive. In Fig. 2b,c, we compare the measured squeezing rate ϵ to the one predicted by equation (3). We see a disagreement that we attribute to the effects of higher order in ξ_{1}ξ_{2}, which are not included in equation (3). For this reason, we add a comparison with the rates expected from Floquet theory^{35} and from a timedomain simulation of our system Hamiltonian (equation (1)), which shows good agreement with the measurements. In the current experiment, the squeezing rate is limited by the finite power of the parametric drives.
After having characterized the squeezing rate, we proceed with investigating the nonlinearity K of the phonon mode. In Fig. 3a, we show a spectroscopic measurement taken by applying a 400 μs probe tone detuned by Δ_{p} = ω_{p} − ω_{a} from the phonon mode and then measuring the qubit population. When the probe tone is resonant with the phonon mode, it drives the phonon mode into a steady state, from which population leaks to the qubit, which results in our spectroscopic readout signal. This lets us infer the population of the phonon mode (Supplementary Information), which we denote on the second y axis (Fig. 3a). We observe an asymmetric resonance peak characteristic of a nonlinear Duffing oscillator. Fitting these data with the equation of motion of a classical driven Duffing oscillator^{49}, we extract the nonlinearity of the mode (Supplementary Information). Repeating this analysis for different probe amplitudes and detunings Δ_{a} gives us the data shown in Fig. 3b. These are in good agreement with the predictions obtained from the numerical diagonalization of equation (1) without drives, as well as with the analytical expression \(K\approx {g}^{4}/{\varDelta }_{a}^{3}\) obtained from the fourthorder perturbation theory (Supplementary Information). This shows that we can tune the phonon nonlinearity by approximately one order of magnitude.
Having demonstrated control over the parameters of equation (2), we now show that different regimes of this Hamiltonian can be used to prepare nonGaussian states. The dynamics of the system is fully determined by the two dimensionless parameters Δ/K and ϵ/K, where the former is controlled by the drive frequencies and the latter, by the drive amplitudes. This twodimensional parameter space is divided into three different regions by two phase transitions at Δ = ±2ϵ (refs. ^{50,51,52}). In a semiclassical description, these regions are associated with an effective single, double and triplenode potential in the frame rotating at half the driving frequency^{52,53} (Fig. 4a).
To prepare interesting mechanical states, we start with the phonon mode in the ground state, and then let it evolve according to the Hamiltonian in equation (2). We choose ξ_{1}ξ_{2} = 0.07 and Δ_{a} = 2π × 0.53 MHz. This fixes ϵ/K, but leaves us control over Δ/K by changing the parametric drive correction δ. The Wigner functions measured at different values of t_{S} (Fig. 1b) are shown in Fig. 4b. We note parameter regimes that result in states significantly deviating from the Gaussian states. Moreover, in some cases, regions with negative Wigner function appear, which are a direct indication of nonclassicality^{54,55}. The states we observe can be qualitatively understood from an evolution of the ground state in the semiclassical potential associated with the chosen parameter regime. From exact diagonalization, we obtain K = 2π × 14(1) kHz, whereas from a Gaussian fit of the states at t_{S} = 3 μs, we estimate ϵ = 2π × 11(1) kHz. Figure 4c shows the numerical simulations of equation (2) with these parameters, which show good agreement with our measurements. For these simulations, we use a lower phonon lifetime of 40 μs, estimated by taking into account the Purcell decay via the qubit, and include a global rotation to take into account that the measurements shown in Fig. 4b are performed in the rotating frame of the qubit.
To have a pragmatic characterization of the states that we can prepare, we quantify their usefulness for a metrological protocol by means of the QFI. For a perturbation generated by the operator \(\hat{A}\), the QFI associated with state ρ is defined as \({F}_{{\rm{Q}}}[\,\rho ,A]=2{\sum }_{k,l}\frac{{({\lambda }_{k}{\lambda }_{l})}^{2}}{({\lambda }_{k}+{\lambda }_{l})} \left\langle k\right\vert A\left\vert l\right\rangle { }^{2}\), where λ_{k} and k〉 are the eigenvalues and eigenvectors of ρ, respectively, and the summation goes over all k, l such that λ_{k} + λ_{l} > 0. Taking the parameter estimation task to be the measurement of a displacement amplitude, we get A(θ) = Xsinθ + Pcosθ, where θ specifies the displacement direction. From this, we define \({F}_{{\rm{Q}}}^{\,\max }={\max }_{\theta }{F}_{{\rm{Q}}}[\,\rho ,A(\theta )]\), which gives the maximum sensitivity attainable by the state. Note that a coherent state has F_{Q}[α〉, A(θ)] = 2, meaning that if we consider coherent states as classical resources, then any \({F}_{{\rm{Q}}}^{\,\max } > 2\) implies nonclassicality. We numerically estimate \({F}_{{\rm{Q}}}^{\,\max }\) by first reconstructing the density matrix of the state and then maximizing F_{Q}[ρ, A(θ)] over θ. The results are shown in Fig. 4d, together with the values we obtain for a Fock 1〉 state and for the Schrödinger cat states from another work^{20}. The Fisher information is computed for the time evolution of states at four different detunings. We observe that the states corresponding to point (iii) in Fig. 4a exhibit significantly larger values after 6 μs than previously measured states, whereas the states in the squeezed regime do not surpass them.
In conclusion, we have demonstrated squeezing below the zeropoint fluctuations of a gigahertzfrequency phonon mode of an HBAR device with tunable nonlinearity. This allows us to prepare nonGaussian states of motion characterized by a high QFI, which can find immediate application in quantum sensing with mechanical degrees of freedom^{56}. Our results, in combination with the beamsplitter operation demonstrated in the same platform elsewhere^{25}, complete the toolbox for universal continuousvariable quantum information processing and bosonic quantum simulation in HBARs. This opens up the possibility to use the large number of modes available in these devices for hardwareefficient quantum chemistry simulations^{5,36}, as well as for nonlinear boson sampling^{57}. Refining parametric processes with optimal control algorithms could also improve the operation speed and fidelity. In addition, the possibility of using our protocol to squeeze arbitrary states can be used in combination with the preparation of cat states^{20} for increasing their robustness to phonon losses^{58,59}.
Data availability
Source data are provided with this paper. These data are also available via Zenodo at https://doi.org/10.5281/zenodo.10838493 (ref. ^{60}). 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
Analysis scripts for this paper are available via Zenodo at https://doi.org/10.5281/zenodo.10838493 (ref. ^{60}).
References
Gustafsson, M. V. et al. Propagating phonons coupled to an artificial atom. Science 346, 207–211 (2014).
Chamberland, C. et al. Building a faulttolerant quantum computer using concatenated cat codes. PRX Quantum 3, 010329 (2022).
Qiao, H. et al. Splitting phonons: building a platform for linear mechanical quantum computing. Science 380, 1030–1033 (2023).
Toyoda, K., Hiji, R., Noguchi, A. & Urabe, S. Hong–Ou–Mandel interference of two phonons in trapped ions. Nature 527, 74–77 (2015).
Chen, W. et al. Scalable and programmable phononic network with trapped ions. Nat. Phys. 19, 877–883 (2023).
Hann, C. T. et al. Hardwareefficient quantum random access memory with hybrid quantum acoustic systems. Phys. Rev. Lett. 123, 250501 (2019).
Mirhosseini, M., Sipahigil, A., Kalaee, M. & Painter, O. Superconducting qubit to optical photon transduction. Nature 588, 599–603 (2020).
Delaney, R. D. et al. Superconductingqubit readout via lowbackaction electrooptic transduction. Nature 606, 489–493 (2022).
Biercuk, M. J., Uys, H., Britton, J. W., VanDevender, A. P. & Bollinger, J. J. Ultrasensitive detection of force and displacement using trapped ions. Nat. Nanotechnol. 5, 646–650 (2010).
Schreppler, S. et al. Optically measuring force near the standard quantum limit. Science 344, 1486–1489 (2014).
Ivanov, P. A., Vitanov, N. V. & Singer, K. Highprecision force sensing using a single trapped ion. Sci. Rep. 6, 28078 (2016).
Schrinski, B. et al. Macroscopic quantum test with bulk acoustic wave resonators. Phys. Rev. Lett. 130, 133604 (2023).
Carney, D. et al. Mechanical quantum sensing in the search for dark matter. Quantum Sci. Technol. 6, 024002 (2021).
Bonaldi, M. et al. Probing quantum gravity effects with quantum mechanical oscillators. Eur. Phys. J. D 74, 178 (2020).
Clerk, A. A., Lehnert, K. W., Bertet, P., Petta, J. R. & Nakamura, Y. Hybrid quantum systems with circuit quantum electrodynamics. Nat. Phys. 16, 257–267 (2020).
Chu, Y. & Gröblacher, S. A perspective on hybrid quantum opto and electromechanical systems. Appl. Phys. Lett. 117, 150503 (2020).
Satzinger, K. J. et al. Quantum control of surface acousticwave phonons. Nature 563, 661–665 (2018).
ArrangoizArriola, P. et al. Coupling a superconducting quantum circuit to a phononic crystal defect cavity. Phys. Rev. X 8, 031007 (2018).
Chu, Y. et al. Creation and control of multiphonon Fock states in a bulk acousticwave resonator. Nature 563, 666–670 (2018).
Bild, M. et al. Schrödinger cat states of a 16microgram mechanical oscillator. Science 380, 274–278 (2023).
Lloyd, S. & Braunstein, S. L. Quantum computation over continuous variables. Phys. Rev. Lett. 82, 1784 (1999).
Mari, A. & Eisert, J. Positive Wigner functions render classical simulation of quantum computation efficient. Phys. Rev. Lett. 109, 230503 (2012).
Braunstein, S. L. & van Loock, P. Quantum information with continuous variables. Rev. Mod. Phys. 77, 513 (2005).
Weedbrook, C. et al. Gaussian quantum information. Rev. Mod. Phys. 84, 621 (2012).
von Lüpke, U., Rodrigues, I. C., Yang, Y., Fadel, M. & Chu, Y. Engineering multimode interactions in circuit quantum acoustodynamics. Nat. Phys. 20, 564–570 (2024).
Meekhof, D. M., Monroe, C., King, B. E., Itano, W. M. & Wineland, D. J. Generation of nonclassical motional states of a trapped atom. Phys. Rev. Lett. 76, 1796 (1996).
Wollman, E. E. et al. Quantum squeezing of motion in a mechanical resonator. Science 349, 952–955 (2015).
Pirkkalainen, J.M., Damskägg, E., Brandt, M., Massel, F. & Sillanpää, M. A. Squeezing of quantum noise of motion in a micromechanical resonator. Phys. Rev. Lett. 115, 243601 (2015).
Lecocq, F., Clark, J. B., Simmonds, R. W., Aumentado, J. & Teufel, J. D. Quantum nondemolition measurement of a nonclassical state of a massive object. Phys. Rev. X 5, 041037 (2015).
Delaney, R. D., Reed, A. P., Andrews, R. W. & Lehnert, K. W. Measurement of motion beyond the quantum limit by transient amplification. Phys. Rev. Lett. 123, 183603 (2019).
Youssefi, A., Kono, S., Chegnizadeh, M. & Kippenberg, T. J. A squeezed mechanical oscillator with millisecond quantum decoherence. Nat. Phys. 19, 1697–1702 (2023).
Andersson, G. et al. Squeezing and multimode entanglement of surface acoustic wave phonons. PRX Quantum 3, 010312 (2022).
Samanta, C. et al. Nonlinear nanomechanical resonators approaching the quantum ground state. Nat. Phys. 19, 1340–1344 (2023).
von Lüpke, U. et al. Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics. Nat. Phys. 18, 794–799 (2022).
Zhang, Y. et al. Engineering bilinear mode coupling in circuit QED: theory and experiment. Phys. Rev. A 99, 012314 (2019).
Wang, C. S. et al. Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor. Phys. Rev. X 10, 021060 (2020).
Milburn, G. J. & Holmes, C. A. Quantum coherence and classical chaos in a pulsed parametric oscillator with a Kerr nonlinearity. Phys. Rev. A 44, 4704 (1991).
ChávezCarlos, J. et al. Spectral kissing and its dynamical consequences in the squeezedriven Kerr oscillator. npj Quantum Inf. 9, 76 (2023).
Wielinga, B. & Milburn, G. J. Quantum tunneling in a Kerr medium with parametric pumping. Phys. Rev. A 48, 2494 (1993).
Boutin, S. et al. Effect of higherorder nonlinearities on amplification and squeezing in Josephson parametric amplifiers. Phys. Rev. Appl. 8, 054030 (2017).
Cochrane, P. T., Milburn, G. J. & Munro, W. J. Macroscopically distinct quantumsuperposition states as a bosonic code for amplitude damping. Phys. Rev. A 59, 2631 (1999).
Goto, H. Bifurcationbased adiabatic quantum computation with a nonlinear oscillator network. Sci. Rep. 6, 21686 (2016).
Puri, S., Boutin, S. & Blais, A. Engineering the quantum states of light in a Kerrnonlinear resonator by twophoton driving. npj Quantum Inf. 3, 18 (2017).
Leghtas, Z. et al. Confining the state of light to a quantum manifold by engineered twophoton loss. Science 347, 853–857 (2015).
Grimm, A. et al. Stabilization and operation of a Kerrcat qubit. Nature 584, 205–209 (2020).
Frattini, N. E. et al. The squeezed Kerr oscillator: spectral kissing and phaseflip robustness. Preprint at arXiv https://arxiv.org/abs/2209.03934 (2022).
Iyama, D. et al. Observation and manipulation of quantum interference in a superconducting Kerr parametric oscillator. Nat. Commun. 15, 86 (2024).
Chou, K. S. et al. Deterministic teleportation of a quantum gate between two logical qubits. Nature 561, 368–373 (2018).
Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. in Reviews of Nonlinear Dynamics and Complexity Ch. 1 (John Wiley & Sons, 2008).
Dykman, M. I., Maloney, C. M., Smelyanskiy, V. N. & Silverstein, M. Fluctuational phaseflip transitions in parametrically driven oscillators. Phys. Rev. E 57, 5202 (1998).
Wustmann, W. & Shumeiko, V. Parametric resonance in tunable superconducting cavities. Phys. Rev. B 87, 184501 (2013).
Venkatraman, J., Cortinas, R. G., Frattini, N. E., Xiao, X. & Devoret, M. H. A driven quantum superconducting circuit with multiple tunable degeneracies. Preprint at https://arxiv.org/abs/2211.04605 (2023).
Eichler, A. & Zilberberg, O. Classical and Quantum Parametric Phenomena (Oxford Univ. Press, 2023).
Kenfack, A. & Zyczkowski, K. Negativity of the Wigner function as an indicator of nonclassicality. J. Opt. B: Quantum Semiclass. Opt. 6, 396 (2004).
Walschaers, M. NonGaussian quantum states and where to find them. PRX Quantum 2, 030204 (2021).
Guo, J., He, Q. & Fadel, M. Quantum metrology with a squeezed Kerr oscillator. Phys. Rev. A 109, 052604 (2024).
Spagnolo, N., Brod, D. J., Galvão, E. F. & Sciarrino, F. Nonlinear boson sampling. npj Quantum Inf. 9, 3 (2023).
Le Jeannic, H., Cavaillès, A., Huang, K., Filip, R. & Laurat, J. Slowing quantum decoherence by squeezing in phase space. Phys. Rev. Lett. 120, 073603 (2018).
Pan, X. et al. Protecting the quantum interference of cat states by phasespace compression. Phys. Rev. X 13, 021004 (2023).
Marti, S., Fadel, M. & von Lüpke, U. Analysis code for ‘Quantum squeezing in a nonlinear mechanical oscillator’. Zenodo https://doi.org/10.5281/zenodo.10838493 (2024).
Acknowledgements
We thank A. Eichler, F. Adinolfi and H. Doeleman for useful discussions and feedback on the paper, and M. Drimmer for contributing to the device fabrication. Device fabrication was performed at the FIRST cleanroom of ETH Zürich and the BRNC cleanroom of IBM Zürich. We acknowledge support from the Swiss National Science Foundation under grant 200021_204073. M.F. was supported by the Swiss National Science Foundation Ambizione via grant no. 208886 and The Branco Weiss Fellowship—Society in Science, administered by ETH Zürich.
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Open access funding provided by Swiss Federal Institute of Technology Zurich.
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U.v.L. and M.F. conceived the experiments. U.v.L., Y.Y., M.B. and A.O. fabricated the device. S.M., U.v.L., Y.Y. and M.F. wrote the experiment control sequences. S.M., U.v.L. and M.F. performed the measurements and analysed the data. S.M., U.v.L., O.J. and M.F. performed the numerical simulations of the experiments. U.v.L. and M.F. derived the theoretical models. Y.C. and M.F. supervised the work. S.M., U.v.L. and M.F. wrote the paper with feedback from all authors.
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Nature Physics thanks Laure Mercier de Lépinay, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Supplementary Figs. 1–6, Table 1 and discussion.
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Marti, S., von Lüpke, U., Joshi, O. et al. Quantum squeezing in a nonlinear mechanical oscillator. Nat. Phys. (2024). https://doi.org/10.1038/s41567024025456
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DOI: https://doi.org/10.1038/s41567024025456
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