Quantum squeezing in a nonlinear mechanical oscillator (2024)

From the oscillation of a trapped particle to the vibration of a solid-state structure, mechanical modes are ubiquitous degrees of freedom that can exhibit sought-after 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, memories⁶ and microwave-to-optical frequency conversion^7,8. Moreover, their non-zero mass makes them particularly suited for sensing forces^9,10,11, as well as for fundamental physics investigations, ranging from tests of the superposition principle¹² to the detection of dark matter¹³ and quantum gravity effects¹⁴. 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 non-trivial task.

In this context, mechanical resonators have recently attracted a lot of attention as new elements for hybrid quantum systems^15,16. In particular, gigahertz-frequency 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 states²⁰. Crucially, compared with their electromagnetic counterparts, mechanical resonators have small physical footprints and a high density of accessible long-lived modes, making them ideal candidates for hardware-efficient quantum processors² and quantum random access memories⁶.

The realization of continuous-variable quantum computing and bosonic simulations relies on the availability of a universal gate set composed of phase shift, displacement, beamsplitter, single-mode 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 surface³ and bulk²⁵ acoustic waves. For mechanical systems, quantum noise squeezing was pioneered in trapped ions²⁶, and later demonstrated in drum oscillators using the tools of electromechanics^{27,28,29,30,31}. In cQAD, a recent experiment demonstrated the two-mode squeezing of gigahertz-frequency surface acoustic waves through the modulation of one of the Bragg reflectors³². 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 off-resonantly couple a mechanical oscillator to a two-level 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 dot³³. Despite all this progress, however, the demonstration of a full gate set for universal continuous-variable quantum information processing in a single cQAD device is still lacking.

In this work, we present squeezing below the zero-point fluctuations of a gigahertz-frequency phonon mode of a high-overtone bulk acoustic-wave 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 non-Gaussian 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 flip-chip 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 Stark-shift drive³⁴. At this frequency, the qubit has an energy relaxation time T₁ = 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 T₁ = 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 off-resonant 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, far-off-resonant drive at approximately 8.4 GHz to control the qubit frequency via the a.c. Stark shift.

When the resonance condition ω₁ + ω₂ = 2ω_a is fulfilled, the qubit nonlinearity mediates a four-wave-mixing process that results in a two-phonon drive (a^†2 + a²). In addition, the phonon mode also inherits a nonlinearity a^†2a² 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. Stark-shifted qubit. The squeezing rate ϵ (Supplementary Information) is given by refs. ^35,36

where Σ₂₁ = Δ₁ + Δ₂. 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 dynamics³⁷, quantum phase transitions³⁸, tunnelling³⁹ and parametric amplification⁴⁰. Moreover, this model admits macroscopic superpositions as quantum ground states, which can be exploited for error-protected 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 non-Gaussian 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 ω₂ = 2ω_a − ω₁ + δ. We then reset the qubit to its ground state by swapping the population acquired during the off-resonant 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 two-phonon drive becomes resonant.

To fully characterize the state resulting from the two-phonon 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 mode³⁴. 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 ≈ g²/Δ_a − δ/2 (equation (2)).

Given phase-space 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_θV(θ) and V_max = max_θV(θ) as the squeezed and antisqueezed quadratures. Fitting the Wigner function at t_S = 6 μs with a two-dimensional 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 non-Gaussian 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 Fock-state populations of the phonon states. We observe high populations of even states, characteristic of a two-phonon 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 \((1-{\rm{e}}^{-{\gamma }_{\rm{d}}t}(1-2{V}_{\min }))/2\), 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 T₁, 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, ξ₁ = 0.28 and ξ₂ = 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 T₁, 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 ξ₁ξ₂ 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 qubit²⁵. The detuning Δ_a is then set by changing the qubit frequency via the independent Stark-shift 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 ξ₁ξ₂, which are not included in equation (3). For this reason, we add a comparison with the rates expected from Floquet theory³⁵ and from a time-domain 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 oscillator⁴⁹, 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 fourth-order 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 non-Gaussian 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 two-dimensional 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 triple-node 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 ξ₁ξ₂ = 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 non-classicality^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 non-classicality. 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 work²⁰. 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 zero-point fluctuations of a gigahertz-frequency phonon mode of an HBAR device with tunable nonlinearity. This allows us to prepare non-Gaussian states of motion characterized by a high QFI, which can find immediate application in quantum sensing with mechanical degrees of freedom⁵⁶. Our results, in combination with the beamsplitter operation demonstrated in the same platform elsewhere²⁵, complete the toolbox for universal continuous-variable 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 hardware-efficient quantum chemistry simulations^5,36, as well as for nonlinear boson sampling⁵⁷. 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 states²⁰ for increasing their robustness to phonon losses^58,59.