Quantum squeezing in a nonlinear mechanical oscillator

Mechanical modes offer high quality factors, compact footprints, and strong coupling to spins and electromagnetic fields, making them promising resources for quantum technologies such as information processing, bosonic simulations, quantum memories, transduction, precision sensing, and tests of fundamental physics. Realizing universal continuous-variable quantum computing requires a gate set including phase shift, displacement, beamsplitter, single-mode squeezing, and Kerr nonlinearity. While some of these are straightforward, achieving squeezing and strong nonlinearities in the quantum regime is challenging for mechanical systems due to typically linear couplings for small displacements. Hybrid circuit quantum acoustodynamics (cQAD) platforms enable interfacing gigahertz-frequency mechanical resonators with superconducting qubits, providing a toolbox for state preparation and control. The research question addressed here is whether one can realize and control both quantum squeezing below zero-point motion and a tunable Kerr nonlinearity in a single gigahertz mechanical mode, and use these resources to prepare non-Gaussian states useful for quantum information and metrology. The study demonstrates such control in a high-overtone bulk acoustic-wave resonator (HBAR) coupled to a transmon qubit, achieving a mechanical squeezed Kerr oscillator and preparing states with Wigner negativities and enhanced quantum Fisher information.

Prior works in hybrid systems have shown resonant qubit-phonon interactions enabling mechanical Fock and Schrödinger cat states, and beamsplitter-type operations in cQAD. Mechanical squeezing was pioneered with trapped ions and later achieved in electromechanical drum resonators. Two-mode squeezing of surface acoustic waves was demonstrated via Bragg reflector modulation. Realizing strong nonlinearities in mechanical systems is difficult with standard linear opto/electromechanical couplings; an alternative is off-resonant coupling to a two-level system to induce effective Kerr-type phonon nonlinearities, as shown for vibrational modes of a carbon nanotube coupled to a quantum dot. Despite this progress, a full universal continuous-variable gate set within a single cQAD device had not been demonstrated. Squeezed Kerr oscillators have been implemented for electromagnetic modes in circuit QED and exhibit rich physics including phase transitions, tunnelling, and parametric amplification, and are attractive for bosonic encoding and error protection. This work implements the squeezed Kerr oscillator for a mechanical mode.

Platform: A cQAD device where a transmon qubit is flip-chip bonded to an HBAR. The transmon frequency is 5.042 GHz (tunable via an a.c. Stark shift drive near 8.4 GHz), with T1 = 17(0.4) μs, T2(Ramsey) = 24(0.7) μs, and anharmonicity ~185 MHz. The HBAR phonon mode used has frequency 5.023 GHz, T1 = 132(4) μs, T2(Ramsey) = 210(9) μs, and couples to the qubit via a piezoelectric AlN transducer with Jaynes–Cummings coupling g ≈ 2π × 292 kHz.

Squeezing mechanism: Two off-resonant microwave tones are applied to the qubit at frequencies ω1 and ω2. When ω1 + ω2 = 2ω (ω is the dressed phonon frequency), the qubit nonlinearity mediates a four-wave mixing process that generates an effective two-phonon drive on the mechanical mode, realizing single-mode squeezing. Off-resonant coupling also induces an effective Kerr nonlinearity on the phonon. After eliminating the qubit, the effective phonon Hamiltonian comprises detuning, two-phonon (squeezing), and Kerr terms. The squeezing rate scales with the product of the two drive amplitudes and the qubit anharmonicity and detunings; the Kerr nonlinearity scales approximately as g^2/Δq, with Δq the qubit-phonon detuning.

Calibration and pulse sequences: The parametric tones distort the qubit frequency via Stark shifts, modifying the normal-mode shift of the phonon. A calibration scans a small frequency correction δ to maintain the squeezing resonance. Sequence: apply two parametric drives for ts (e.g., 20 μs with 0.5 μs Gaussian edges), reset the qubit by swapping its acquired population to an ancillary mechanical mode, then swap phonon population to the qubit by bringing it into resonance for a time π/(2√2 g) and measure qubit state dispersively. Scanning δ identifies resonance (peak near δ ≈ 2π × 140 kHz; optimal squeezing observed at δ ≈ 2π × 80 kHz due to partial compensation of residual nonlinearity by detuning).

State characterization: The phonon Wigner function is measured using the qubit to perform parity measurements after phase-space displacements. Wigner functions are acquired after evolution times ts = 0, 6, 12 μs under the squeezing drive. Variances along phase-space quadratures are extracted via 2D Gaussian fits to estimate squeezed (Vmin) and anti-squeezed (Vmax) quadratures; maximum-likelihood reconstruction yields density matrices and Fock-state populations, confirming even-parity dominance from a two-phonon drive. Squeezed state decay is measured by inserting a variable wait tw after state preparation and tracking Vmin and Vmax versus tw.

Squeezing rate extraction: Vmin(ts) for short times (ts less than the onset of significant non-Gaussianity) is fit to a decay-including model Vmin(t) = (γ + 4 ε e^{−γ t}) / [2 (γ + 4 ε)] to extract the squeezing rate ε and effective decay γ. Measurements are repeated versus drive powers and qubit-phonon detuning. Predictions from a simple analytical expression are compared to Floquet and time-dependent simulations of the full system Hamiltonian for validation.

Kerr nonlinearity measurement: A 400 μs weak probe tone near the mechanical resonance is applied while monitoring qubit population, which reflects the steady-state phonon population due to leakage via the qubit. The resulting asymmetric resonance lineshape (Duffing response) is fit using the classical driven Duffing oscillator equation of motion to extract the effective Kerr nonlinearity. This procedure is repeated for multiple probe amplitudes and qubit-phonon detunings, and the results are compared with exact diagonalization of the undriven Hamiltonian and perturbative estimates K ≈ g^2/Δq.

Non-Gaussian state preparation and QFI: Parameter regimes are set by the dimensionless ratios Δ/K and ε/K (Δ set by drive frequencies; ε by drive amplitudes). The parameter space exhibits regions separated by phase transitions at Δ = ±2 ε. Starting from the ground state, evolution under chosen parameters (for example, ε1 ε2 = 0.07, Δ0 ≈ 2π × 0.53 MHz) yields non-Gaussian states with Wigner negativities. Simulations of the effective model using K ≈ 2π × 14(1) kHz, ε ≈ 2π × 11(1) kHz, and an effective phonon lifetime ~40 μs (including Purcell decay) reproduce measured Wigner functions. The quantum Fisher information (QFI) for displacement estimation is computed from reconstructed density matrices and maximized over displacement direction; values are compared to coherent-state and other experimental benchmarks.

• Achieved single-mode mechanical squeezing below the zero-point level in a gigahertz HBAR mode coupled to a superconducting transmon, using a two-phonon drive mediated by parametric driving of the qubit.
• Wigner tomography shows reduced noise along one quadrature and increased noise along the orthogonal quadrature; maximum squeezing observed around an evolution time of ~6 μs.
• Quantitatively, from a 2D Gaussian fit at ts = 6 μs: Vmin = 0.252(6), corresponding to 3.0(1) dB noise reduction below the ground-state variance V0 = 0.5; Vmax = 1.45(4). The inferred thermal population is Vmin Vmax − 1/2 = 0.10(1), corresponding to state purity ≈ 83(1)%. Maximum-likelihood reconstruction yields Vmin = 0.236(1) at 6 μs and Vmin = 0.268(3) at 12 μs, indicating nonlinear distortion at longer times.
• Squeezed state lifetime: the squeezed quadrature variance relaxes with a time constant ~78(11) μs; the anti-squeezed quadrature decays with ~125(12) μs, consistent with phonon T1 ≈ 132 μs. The squeezed quadrature shows enhanced sensitivity to dephasing.
• Squeezing rate extraction: fitting Vmin(t) yields an effective decay γ−1 = 12.8(11) μs during the driven evolution and a squeezing rate ε ≈ 2π × 7.6(3) kHz (for δ = 2 × 80 kHz, Δ ≈ 2π × 1.5 MHz, and drive strengths ε1 ≈ 0.28, ε2 ≈ 0.26). Dependence on drive power and detuning matches Floquet and time-domain simulations; deviations from a simple analytical formula are attributed to higher-order effects. The achievable ε is limited by available parametric drive power.
• Tunable Kerr nonlinearity: spectroscopy reveals an asymmetric Duffing-like lineshape; fitting yields a Kerr nonlinearity consistent with numerical diagonalization and perturbative K ≈ g^2/Δq. The phonon nonlinearity is tunable by approximately an order of magnitude via the qubit-phonon detuning.
• Mechanical squeezed Kerr oscillator realized: operating in different Δ/K and ε/K regimes produces non-Gaussian states of motion with Wigner negativities. Parameters inferred from data and simulations include K ≈ 2π × 14(1) kHz and ε ≈ 2π × 11(1) kHz in the non-Gaussian regime explored.
• Metrological utility: the quantum Fisher information for displacement estimation exceeds the coherent-state limit in several regimes; for a particular detuning (region (iii) in the phase diagram), QFI after ~6 μs exceeds values from previously measured mechanical states, while purely squeezed-regime states do not surpass those benchmarks.

The work addresses the challenge of engineering both squeezing and strong, tunable nonlinearities in a single mechanical mode in the quantum regime. By parametrically driving a superconducting qubit coupled to an HBAR, the experiment implements an effective two-phonon drive and an inherited Kerr nonlinearity, realizing a mechanical squeezed Kerr oscillator. The observed squeezing below the zero-point level, together with control of the Kerr strength via detuning, demonstrates key ingredients for universal continuous-variable operations in mechanical platforms. The state tomography confirms coherent squeezing dynamics at short times and non-Gaussian features at longer times due to nonlinearity, consistent with the effective model. Measured squeezing rates as functions of drive strength and detuning agree with numerical simulations beyond the lowest-order analytical approximation, highlighting the role of higher-order effects during strong driving. The preparation of states with Wigner negativities and elevated QFI establishes their non-classicality and potential for quantum-enhanced sensing. The shorter effective decay time during driven evolution stems from Purcell loss through the qubit and drive-induced dephasing; nonetheless, the squeezed states retain coherence on tens to hundreds of microseconds, compatible with mechanical T1. The ability to tune Δ/K and ε/K enables access to different dynamical phases and state families, as predicted for squeezed Kerr oscillators, expanding the mechanical quantum control toolbox.

The study demonstrates below-zero-point mechanical squeezing in a gigahertz HBAR mode with a tunable Kerr nonlinearity, implemented via parametric driving of a coupled superconducting qubit. This realizes a mechanical squeezed Kerr oscillator capable of generating non-Gaussian states with Wigner negativities and high quantum Fisher information, suitable for quantum metrology and sensing. Combined with previously demonstrated beamsplitter interactions in the same platform, these results complete the essential toolbox for universal continuous-variable quantum information processing and bosonic simulations with HBARs. The large number of long-lived modes in HBAR devices offers a path toward hardware-efficient processors and applications such as quantum chemistry simulations and nonlinear boson sampling. Future directions include optimizing parametric processes with control algorithms to increase speed and fidelity, and using state-dependent squeezing to improve the robustness of cat states against phonon loss.

• Residual phonon-mode nonlinearity distorts states and limits achievable squeezing at longer evolution times.
• Finite parametric drive power constrains the maximum squeezing rate, limiting the speed and depth of squeezing.
• During driven evolution, effective decay is faster than the bare phonon T1 due to Purcell loss via the qubit and additional dephasing from finite qubit population and strong driving, reducing state lifetimes.
• Analytical lowest-order expressions for the squeezing rate deviate from measurements at higher drive strengths; accurate predictions require Floquet or time-domain simulations.
• QFI estimates are sensitive to state-reconstruction details; the dominant error arises from Hilbert space truncation in maximum likelihood reconstruction, along with possible calibration imperfections and Wigner background fluctuations.