Loss Mechanisms in High-coherence Multimode Mechanical Resonators Coupled to Superconducting Circuits

Abstract

Circuit quantum acoustodynamics (cQAD) devices have a wide range of applications in quantum science, all of which depend crucially on the quantum coherence of the mechanical subsystem. In this context, high-overtone bulk acoustic-wave resonators (HBARs) are particularly promising, since they have shown very high quality factors with negligible dephasing. However, the introduction of piezoelectric films, which are necessary for coupling to a superconducting circuit, can lead to additional loss channels, such as surface scattering and two-level systems (TLS). Here, we study the acoustic dissipation of HBAR resonators in cQAD systems and find that the defect density of the piezoelectric material and its interface with the bulk are limiting factors for the coherence. We measure acoustic modes with phonon lifetimes up to 400 $μ$s and lifetime-limited coherence times approaching one millisecond in the quantum regime. When coupled to a superconducting qubit, this leads to a hybrid system with a large quantum coherence cooperativity of $C_{T_2}=1.1\times10^5$. These results represent a new milestone for the performance of cQAD devices and offer concrete paths forward for further improvements.

Figures (24)

• Figure 1: Device geometry. (a) Schematic illustration of a longitudinal overtone mode in an HBAR, where the dome-shaped piezoelectric layer mediates the coupling between the mechanical motion and the antenna's out-of-plane electric field $\vec{E}$. The diagram also indicates the bulk thickness $t_B$, piezoelectric layer thickness $t_P$, and mode waist $w_0$. (b) Photograph of the HBAR resonator flip-chip bonded to a flux-tunable superconducting qubit. (c) Circuit model of the system in (b). (d) Photograph of the HBAR resonator flip-chip bonded to a coplanar waveguide microwave antenna. (e) Driven microwave circuit modelling the system in (d). The scale-bars in (b) and (d) correspond to $2~$mm.
• Figure 2: Spectroscopy mode characterization. (a) Typical broadband reflection spectrum measured through the microwave antenna, with a narrower span (b) highlighting the spectral features of each mode. The inset shows the complex reflection response of mode $n=332$, with the corresponding fit shown as a solid line. (c) Schematic illustrating the material stack used to model the resonator. The defect layer is only included to model the HVPE-deposited HBARs (Samples B.1 and C) (d) Frequency spacing between successive modes as a function of mode frequency. The solid line is a fit to the multi-material model. (e) Energy participation ratio for each layer in the model. (f-h) Internal quality factor of each resonant mode as a function of mode frequency. The error bars, corresponding to one standard deviation of the $Q_{\text{i}}$ fit uncertainty, are smaller than the marker size. The quality factor predicted by the energy participation model including surface scattering and absorption is shown as a solid red line. The dashed gray line shows only the surface scattering contribution. Stars indicate modes used for the power and temperature measurements below.
• Figure 3: TLS loss characterization. The color scheme identifies corresponding phonon modes across panels. The left panels show measurements of Sample A, and the right panels of Sample B.1. (a) Internal quality factors $Q_{\text{i}}$ as a function of the average phonon population for selected mechanical overtones in Samples A and B.1. The solid lines are fits to the resonant TLS loss model in Eq. \ref{['eq:TLSpower']}. (b) Fractional frequency shift of the phonon modes with temperature. The solid lines are fits to Eq. \ref{['eq:TLStemp']}. (c) Inverse of the TLS loss tangent ($Q_{\text{TLS}}$) for all phonon modes calculated from the fractional frequency shifts of the modes between $10~$mK and $4~$K, shown as dots. The color-coded stars and circles are the $Q_{\text{TLS}}$ extracted from the fits in (a) and (b) respectively. The solid red line is a fit to the energy participation model in Eq. \ref{['eq:TLSparticipation']} used to extract the TLS loss tangent of AlN in each sample. In all the panels, error bars represent one standard deviation in the variable fit uncertainty (or propagated uncertainty), except the horizontal error bars in panel (b), which indicate the temperature variation within $\pm1~$minute of data acquisition.
• Figure 4: Phonon coherence times in the quantum regime. (a-b) Pulse sequences to measure phonon $T_1$ and $T_2^*$ through the qubit, and corresponding phonon population as a function of wait time $\tau$ in both measurements. The red lines are fits to a decaying exponential (a) or exponentially decaying sinusoid (b). (c) Phonon mode relaxation time ($T_1$) values in Sample A, obtained either by swapping a single excitation with the qubit and observing its decay (pink), or calculated from the internal quality factors extracted by microwave spectroscopy at high powers (blue). (d) Violin plots comparing the $T_1$ distributions of both measurement methods. (e) Phonon decay $T_1$, and coherence times $T_2^*$ and $T_2^e$ for six phonon modes in Sample B.2, with $T_2^*$ times approaching the lifetime limit $2T_1$. The band around $2T_1$ represents the uncertainty. (f) Correlation between inverse internal quality factors measured with the qubit and with microwave spectroscopy in sample A, with a linear fit. In all the panels, error bars represent one standard deviation in the variable fit uncertainty, or the propagated uncertainty.
• Figure 5: Dephasing limit and strong coupling in cQAD. (a) Decay ($T_1$) and coherence times ($T_2^*$) of mechanical modes reported in works that achieve strong coupling between a mechanical resonator in the quantum ground state and a superconducting qubit bozkurt_mechanical_2025undershute_decoherence_2025yang_mechanical_2024lee_strong_2023wollack_quantum_2022von_lupke_parity_2022chu_creation_2018satzinger_quantum_2018chu_quantum_2017oconnell_quantum_2010. The modes characterized in this work for sample B.2 are displayed as blue stars. The dashed black line represents the lifetime limit of coherence with negligible dephasing. (b) Overview of the $T_1$ and $T_2^*$ cooperativities in the same works and additional references bolgar_quantum_2018moores_cavity_2018arrangoiz-arriola_resolving_2019bienfait_phonon-mediated_2019kervinen_sideband_2020crump_coupling_2023 demonstrating resonant strong coupling with only $T_1$ data provided (black markers).