Quantum squeezing in a nonlinear mechanical oscillator

Mechanical modes provide high-Q, strongly coupled platforms for quantum technologies, enabling information processing, bosonic simulations, quantum memories, and transduction. Their finite mass also makes them attractive for precision force sensing and fundamental tests. A universal continuous-variable (CV) gate set requires phase shift, displacement, beamsplitter, single-mode squeezing, and Kerr nonlinearity; while displacements and phase shifts are straightforward, generating strong squeezing and nonlinearities in the quantum regime is challenging in mechanical systems due to typically linear opto/electromechanical couplings at small displacements. Hybrid approaches in circuit quantum acoustodynamics (cQAD) couple gigahertz mechanical resonators to superconducting qubits, enabling deterministic state preparation (Fock and cat states) and beamsplitter operations. Off-resonant coupling of a mechanical mode to a two-level system can endow the mechanics with an effective Kerr nonlinearity. Despite progress, a single cQAD device demonstrating both sub–zero-point squeezing and tunable Kerr nonlinearity, essential for universal CV processing, has been lacking. This work demonstrates quantum squeezing below zero-point fluctuations in a high-overtone bulk acoustic-wave resonator (HBAR) mode coupled to a transmon qubit, with a qubit-mediated, tunable mechanical Kerr nonlinearity. By combining these, the authors realize a mechanical squeezed Kerr oscillator and prepare non-Gaussian motional states with Wigner negativity and high quantum Fisher information (QFI), relevant for metrology.

Prior efforts in hybrid mechanical platforms and cQAD have shown: preparation of mechanical Fock and Schrödinger cat states via resonant qubit–phonon interactions; beamsplitter operations between acoustic modes; quantum noise squeezing in trapped ions and electromechanical drum resonators; two-mode squeezing of surface acoustic waves via modulated reflectors; and effective mechanical nonlinearities via off-resonant coupling to two-level systems (e.g., nanotube–quantum dot systems). The universal CV gate set is well-established in theory, with squeezing and Kerr central to non-Gaussian state engineering and error-protected encoding (e.g., Kerr-cat qubits) extensively explored in circuit QED for electromagnetic modes. However, a unified demonstration of sub–zero-point single-mode squeezing and tunable Kerr nonlinearity in a gigahertz mechanical mode within cQAD had not been realized.

Device and parameters: A transmon qubit is flip-chip bonded to an HBAR. Qubit frequency ωq/2π = 5.042 GHz (tunable via a far-off-resonant a.c. Stark drive at ≈8.4 GHz), energy relaxation T1 = 17(0.4) µs, Ramsey T2 = 24(0.7) µs, anharmonicity α/2π = 185 MHz. The targeted HBAR phonon mode has ω/2π = 5.023 GHz, T1 = 132(4) µs, Ramsey T2 = 210(9) µs. Qubit–phonon Jaynes–Cummings coupling g/2π = 292 kHz via an AlN piezoelectric transducer. Hamiltonian and drives: The cQAD Hamiltonian includes the qubit and phonon modes with a Jaynes–Cummings interaction and two off-resonant qubit drives at frequencies ω1 and ω2 with amplitudes Ω1, Ω2. The drives induce an a.c. Stark shift of the qubit (and hence a normal-mode shift δ of the phonon), and, when ω1 + ω2 = 2ω is satisfied, a four-wave-mixing process mediates an effective two-phonon drive on the phonon. After a series of unitary transformations, the effective phonon Hamiltonian is H/ħ = −Δ a†a − ε(a†2 + a2) − K a†a2 (effective squeezing and Kerr terms), with Δ = (ω1 + ω2 − 2ω)/2, ω ≈ ωa + δ, and Δq = ω − ωq the detuning between phonon and Stark-shifted qubit. The squeezing rate is approximately ε = 2 ε1 ε2 / Δq (with drive-dependent parameters; higher-order corrections are discussed), and the inherited Kerr nonlinearity K is tunable via detuning (approximate forms given in the Supplementary Information). Calibration for resonant squeezing: Two parametric drives are applied for ts = 20.0 µs with 0.5 µs Gaussian edges. After driving, the qubit is reset by swapping any acquired population to an ancillary phonon mode. The qubit is then brought into resonance with the target phonon for π/(2√2 g) to swap part of the phonon population to the qubit, followed by dispersive readout. Scanning the parametric drive correction δ identifies the resonance of the two-phonon drive via a peak in the qubit excited-state probability near δ ≈ 2π × 140 kHz. State tomography: With δ set (e.g., 2π × 80 kHz) and varying squeezing times ts = 0, 6, 12 µs, the phonon Wigner function is measured using the qubit as an ancilla, enabling direct Wigner tomography. Gaussian fits yield variances Vmin and Vmax; maximum likelihood reconstruction provides density matrices, covariance matrices, and Fock-state populations. Squeezing rate extraction: Vmin(ts) is measured for short times (ts < 1/κ, where the state remains near-Gaussian) and fit to Vmin(t) = (γ + 4 ε e^{−γ t} + 4) / [2(γ + 4 ε)] to infer ε in the presence of decay at rate γ. Drive strengths are calibrated via qubit Stark shifts; qubit–phonon detuning Δ is set by the Stark-shift drive. Measurements are compared to analytical predictions, Floquet theory, and time-domain simulations of the full Hamiltonian. Kerr nonlinearity characterization: A 400 µs probe tone detuned from the phonon is applied while monitoring qubit population. When resonant, phonon steady-state population leaks to the qubit, yielding an asymmetric Duffing-like resonance. Fitting to a classical driven Duffing oscillator model extracts the mode nonlinearity. Repeating across probe amplitudes and qubit–phonon detunings maps the tunable nonlinearity and validates it against exact diagonalization and perturbative expressions. Non-Gaussian state preparation and QFI: With ε1 ε2 = 0.07 and Δ0/2π = 0.53 MHz fixed, varying δ tunes Δ/κ to explore different regimes of the squeezed Kerr oscillator (single-, double-, triple-well effective potentials in a semiclassical picture). Wigner functions are measured for various ts and detunings; simulations of the effective model (including a reduced phonon lifetime of ~40 µs due to Purcell decay) reproduce observations. Maximum likelihood reconstructions are used to compute the quantum Fisher information (QFI) FQ[ρ, A(θ)] for displacement estimation, maximizing over θ to obtain Fmax and benchmarking against the coherent-state limit FQ = 2, as well as against Fock(1) and cat states from prior work.

• Demonstration of single-mode mechanical squeezing below zero-point fluctuations in a gigahertz HBAR mode. For δ = 2π × 80 kHz and ts = 6 µs, Gaussian fit yields Vmin = 0.252(6) (3.0(1) dB below 1/2) and Vmax = 1.45(4). Maximum likelihood reconstruction gives Vmin = 0.236(1) at ts = 6 µs and Vmin = 0.268(3) at ts = 12 µs (reduced by nonlinear distortion at longer times).
• Estimated thermal population from Vmin Vmax − 1/2 ≈ 0.10(1), corresponding to state purity ≈ 83(1)%.
• Fock-basis populations show dominant even-number occupation, characteristic of a two-phonon drive.
• Lifetime of squeezing: the squeezed variance relaxes with a fitted decay constant reported as τ^{-1} = 78(11) µs; the antisqueezed variance decays with ~125(12) µs, comparable to phonon T1. The faster decay of the squeezed quadrature indicates higher sensitivity to dephasing.
• Squeezing rate example: fitting Vmin(t) for δ = 2 × 80 kHz, Δ = 2π × 1.5 MHz, ε1 = 0.28, ε2 = 0.26 yields γ^{−1} = 12.8(1.1) µs and ε/2π = 7.6(3) kHz. Measured ε versus drive powers and detunings agrees with Floquet and time-domain simulations; deviations from the lowest-order analytical formula are attributed to higher-order effects. In the present setup, ε is limited by available parametric drive power.
• Tunable mechanical Kerr nonlinearity: spectroscopy reveals an asymmetric Duffing response. Extracted nonlinearity versus qubit–phonon detuning and probe amplitude matches exact diagonalization and perturbative predictions, demonstrating approximately an order-of-magnitude tunability via detuning.
• Realization of a mechanical squeezed Kerr oscillator enabling non-Gaussian states with Wigner negativity. From data and simulations, effective parameters around K/2π ≈ 14(1) kHz and ε/2π ≈ 11(1) kHz (for selected settings) reproduce observed dynamics.
• Metrological usefulness quantified by QFI: certain parameter regimes (e.g., point (iii) after 6 µs) yield Fmax exceeding both the coherent-state limit (2) and values from previously measured states (including Fock(1) and a prior large cat state), indicating enhanced displacement sensitivity.

By parametrically driving a qubit dispersively coupled to a high-Q gigahertz mechanical mode, the experiment implements an effective two-phonon (squeezing) drive and imparts a tunable Kerr nonlinearity to the phonon. This addresses the central challenge for mechanical continuous-variable quantum control: engineering both strong squeezing and intrinsic nonlinearity in the quantum regime. The sub–zero-point noise reduction verifies coherent mechanical squeezing, while control of the Kerr term via qubit–phonon detuning enables access to rich squeezed-Kerr dynamics, including the generation of non-Gaussian states with Wigner negativity. The extracted squeezing rates, dependence on drive powers and detuning, and Duffing spectroscopy establish quantitative control over the effective Hamiltonian parameters. The observed high QFI demonstrates practical metrological advantages for displacement sensing with massive mechanical systems. Together with previously demonstrated beamsplitter operations in the same cQAD platform, these capabilities complete the gate toolbox required for universal continuous-variable processing and bosonic simulations with HBAR modes, opening avenues for scalable, hardware-efficient processors and sensors.

The work demonstrates quantum squeezing below the zero-point level in a gigahertz HBAR mechanical mode, alongside a tunable qubit-mediated Kerr nonlinearity, realizing a mechanical squeezed Kerr oscillator. The platform enables preparation of non-Gaussian states with Wigner negativity and enhanced quantum Fisher information for displacement sensing. With beamsplitter operations already available in this architecture, the results complete the universal CV gate set for HBAR-based quantum information processing and bosonic simulations. Future directions include optimizing parametric processes with control algorithms to increase speed and fidelity, leveraging multiple HBAR modes for hardware-efficient quantum chemistry simulations and nonlinear boson sampling, and combining squeezing with cat-state preparation to improve robustness against phonon loss.

• Residual mechanical nonlinearity distorts the state during longer squeezing durations, limiting achievable squeezing and introducing non-Gaussian features.
• The squeezing rate is constrained by the available parametric drive power and higher-order effects not captured by lowest-order analytical models.
• Effective decay of squeezed states is accelerated by Purcell loss via the qubit and dephasing due to finite qubit population and parametric driving, yielding shorter lifetimes than the bare phonon T1.
• Calibration complexities (a.c. Stark shifts and normal-mode shifts) require careful detuning corrections; imperfect compensation can reduce squeezing efficiency.
• Reported decay constants (e.g., τ^{-1}) and parameter notations may involve experimental uncertainties and model-dependent interpretations; Wigner reconstructions are sensitive to Hilbert-space truncation and calibration errors.