Millisecond coherence times in gigahertz-frequency mechanical oscillators

MainLong-lived phonons are a compelling resource, as they permit numerous quantum operations within their coherence time, which enables high-performance quantum sensors1,2,3,4, transducers5,6,7,8 and memories9,10,11. The efficient control of long-lived phonons using optomechanical12,13,14, electromechanical15,16 and superconducting qubit systems17,18,19 has generated renewed interest in phononic device physics and technologies for quantum applications20,21. Although various mechanical oscillators have produced such long-lived phonons over a range of frequencies21,22, high-frequency (gigahertz) phonons are often desirable, as they have improved immunity to unwanted noise, permit ground-state operation at cryogenic temperatures and are more readily controlled using quantum optics and circuit quantum electrodynamics techniques21,23. In theory, crystalline media are ideal for hosting such long-lived phonons, as they have vanishing internal dissipation at cryogenic temperatures24,25,26,27. However, it has proven difficult to extend the coherence times of such gigahertz-frequency crystalline oscillators to millisecond timescales.Silicon-based nanomechanical phononic crystal resonators have shown long phonon lifetimes (>1 s) at gigahertz frequencies; however, strong coupling between phonons and the two-level system limits their coherence times to ~100 μs (refs. 10,11,14). Tight phonon confinement and strong boundary reflections within these systems make them vulnerable to complex surface interactions that may contribute to excess noise and dephasing14. Alternatively, micro-fabricated high-overtone bulk acoustic-wave resonators (μHBARs) of the type seen in Fig. 1a support gigahertz-frequency phonon modes with orders of magnitude lower surface participation13,28. In principle, lower surface participation could translate to lower dephasing rates and longer coherence times. However, in practice, μHBARs have yielded modest coherence times ( 18 K, the phonon linewidth follows the predicted T6.5 temperature dependence (dashed grey) due to bulk phonon–phonon scattering. For T 30 ms and finesse levels of F > 100,000 are probably achievable, which would benefit a variety of applications.μHBAR devices of the type demonstrated here could find direct application in both cavity optomechanical and circuit quantum electrodynamics systems. Cavity optomechanical techniques have recently been used to achieve laser-cooling of such quartz μHBARs to their ground state (Fig. 1e), paving the way for quantum optomechanical control of such long-lived phonon modes36. These techniques also enable the realization of high optomechanical coupling rates (~14 MHz), which are essential for high-speed operations43,44. Building on these results, the optomechanical control of such highly coherent μHBARs opens doors to applications in quantum transduction7, networking10 and computing45.The piezoelectric coupling23,35 of such μHBARs to superconducting qubits (Fig. 1d) may also enable advanced forms of quantum state synthesis and tomography19,23. At the lower phonon frequencies (≤5 GHz) typically used in superconducting qubit studies, μHBARs could achieve much longer coherence times. Longer oscillation periods at these lower frequencies combined with the \(1/{\lambda }_{ph}^{2}\) scaling of scattering losses35 translate to a sixfold increase in phonon coherence times, corresponding to τcoh > 40 ms for the current μHBARs. Moreover, the remediation of both μHBAR surfaces could translate to much longer coherence times (τcoh > 150 ms), making such systems a compelling resource for quantum random access memories9. Further studies at millikelvin temperatures will be necessary to quantify the relative impact of two-level systems on phonon coherence for qubit-based applications. However, the greatly reduced surface participation offered by such μHBARs, combined with the surface remediation methods presented here, are powerful tools for managing and reducing the interactions between the two-level system and phonons, which are strongly correlated with structural disorders46,47 and impurities48 at surfaces.Looking ahead, attaining mastery of the phonon–surface interactions becomes even more critical as one shrinks the phononic device size, as increasing surface participation makes them more sensitive to surface imperfections. Hence, optimized surface treatments that reduce surface-induced decoherence will probably translate to even more notable improvements for micro- and nanomechanical systems. As the methods for studying phonon decoherence and surface–phonon interactions demonstrated here can be applied to a wide range of crystals13,49, they could enable the exploration of diverse material platforms for hybrid quantum systems. Looking beyond the field of quantum acoustics, surface-induced decoherence has been reported to be limiting for a variety of different solid-state quantum systems, including superconducting qubits50 and spin defects51. Thus, the subsurface damage and disorder that we have identified as the dominant source of the phonon decoherence also probably play a critical role in these systems. Taking advantage of the highly coherent phonons, we developed a spectroscopic method for sensing the surface-induced losses to the parts per million level. By combining this method with other material characterization tools (such as AFM and GIXRD) and an optimized polishing process, we have formulated in this work a powerful strategy for surface diagnosis and remediation that can be applied to a wide range of different solid-state platforms.MethodsThe measurements described in ’Pump–probe spectroscopy of μHBAR phonons’ and ’Sources of dissipation and techniques to enhance phonon coherence’ were obtained using the two-colour pump–probe spectroscopy technique.

Extended Data Fig. 1a shows the reflection-mode apparatus used to perform non-invasive Brillouin-based measurements of the fabricated μHBARs. The pump (blue) and probe (red) waves, which have distinct wavelengths, were used to simultaneously transduce and detect phonons within the μHBAR, which enabled background-free measurement of the transduced elastic-wave motion. The μHBAR array was mounted on a dielectric mirror such that both the pump and probe waves produced standing-wave field patterns that closely matching those of the standing-wave phonon modes within the μHBAR (Extended Data Fig. 1b). This permitted efficient Brillouin (or acousto-optic) coupling to the μHBAR phonon modes while eliminating the need for active interferometric stabilization required by previous methods13,28. The pump and probe waves were focused to a spot size closely matching the fundamental Gaussian modes of the μHBAR, which permitted efficient coupling to phonon modes with frequencies near the Brillouin frequency of z-cut quartz (ΩB/2π ≈ 12.66 GHz) using 1,550-nm light (Supplementary Information Section IX).The frequencies of the pump and probe waves were chosen to fall within the Brillouin phase-matching bandwidth to enable efficient transduction and detection of elastic-wave motion using the coherent anti-Stokes Brillouin scattering process, as diagrammed in Extended Data Fig. 1c. Phonons were generated within the μHBAR using an intensity-modulated pump wave (blue) that created two optical tones with a frequency separation (Ω) near the Brillouin frequency (ΩB), as seen in the upper arm of Extended Data Fig. 1a. Stimulated Stokes scattering resulted in the transfer of energy between these optical pump tones, which excited phonons within the μHBAR. The elastic-wave motion associated with these phonons was simultaneously detected using a continuous-wave probe laser (red). This elastic-wave motion imprinted Stokes and anti-Stokes sidebands on the probe wave through phase-matched Brillouin scattering (lower arm of Extended Data Fig. 1a). Heterodyne detection of the anti-Stokes sideband was then used to measure the phase and amplitude of the coherently driven phonons.To prevent optical crosstalk, orthogonally polarized pump (s-polarized) and probe (p-polarized) waves were used to excite and detect the phonons within the μHBAR resonator. The pump wave (s-polarized) was synthesized by modulating 1,549.068-nm laser light using an intensity modulator (IM1) that was driven by a vector network analyser (VNA). This modulator was operated at the null-bias point to produce two first-order sideband tones with frequency separation Ω when the intensity modulator was driven at frequency Ω/2. This led to excitation of the μHBAR phonon modes when the frequency separation (Ω) was tuned through the Brillouin frequency (ΩB). This pump wave was then amplified with an erbium-doped optical fibre amplifier to boost its optical intensity before entering the polarization beam splitter (PBS) and exciting phonons within the μHBAR. The 1,549.120-nm probe wave (p-polarized) was amplified using a second erbium-doped optical fibre amplifier before passing through the PBS to interact with the μHBAR. The phase and amplitude of the excited phonons were measured through heterodyne detection of an anti-Stokes sideband that was imprinted on the probe wave through Brillouin scattering.Coherent measurement of the elastic-wave motion was performed by detecting the heterodyne beat note produced by the interference of the anti-Stokes sideband with an optical local oscillator (LO). The optical LO tone was imprinted on the probe wave using a separate intensity modulator (IM2) driven at frequency Ω1. Although intensity modulation produces both +Ω1 and −Ω1 sidebands, only +Ω1 was used as the optical LO. The reflected probe was then band-pass-filtered such that only the anti-Stokes sideband and the +Ω1 LO tone were transmitted, which enabled coherent detection of the elastic-wave motion at frequency Ω − Ω1 using a high-speed photodetector. As the detected radio-frequency (RF) signal power was proportional to the anti-Stokes optical power, the RF power was linearly proportional to the phonon population inside the μHBAR device.During the frequency-domain spectral measurements, the VNA was used to sweep the drive frequency (Ω) through the μHBAR resonance while simultaneously measuring the phase and amplitude of the anti-Stokes beat note at frequency Ω − Ω1. A slow sweep speed was used to ensure a steady-state measurement of the resonant response. During the ring-down measurements, while the VNA was sweeping the drive frequency through the cavity resonance, the intensity modulation of the pump wave was abruptly switched off when the drive frequency reached the peak resonance of the target phonon mode (Ωo). The lock-in amplifier then demodulated the anti-Stokes beat note at Ωo − Ω1 and recorded its phase and amplitude during the free-induction decay. This abrupt turn-off was enabled by adding a fast RF switch between the VNA output and IM1, which was controlled by an arbitrary waveform generator. The abrupt increase in amplitude of the anti-Stokes beat note during resonant excitation of the μHBAR was used to trigger the arbitrary waveform generator, which then caused the RF switch to open, abruptly turn off the RF drive to IM1 and permit a lock-in measurement of the free-induction decay of the μHBAR phonon mode.