Quantum Acoustics with Superconducting Qubits in the Multimode Transition-Coupling Regime

TL;DR

This work tackles the transition-coupling regime in circuit quantum acoustodynamics (cQAD) by pairing a seven-mode SAWR with a tunable transmon qubit on a flip-chip platform, exploring the regime where the coherent coupling $g$ is comparable to dissipation rates $\gamma$ and $\kappa$. The authors combine phonon-oscillation spectroscopy, dispersive AC Stark measurements, and resonant energy-exchange dynamics to map mode-dependent couplings and decoherence, describing the resonant subspace with a Jaynes–Cummings-type Hamiltonian: $H_{ ext{disp}}^{\text{RWA}}/\hbar = g_m(\sigma_{+} a_m^{\dag} + \sigma_{-} a_m)$. They observe a controllable transition from underdamped coherent oscillations to overdamped decay as a function of mode index and coupler bias, with a sinusoidal mode-coupling pattern $g_m = g_0 \sin\left(\frac{\pi}{2} m + \phi_q\right)$ and Purcell effects from nonresonant modes quantified. Importantly, the multimode cavity enables both dispersive readout and fast reset, with an optimal transition-coupling window yielding rapid reset while preserving qubit coherence, informing scalable quantum-chip designs and pointing toward optical-readout integrations for piezo-optomechanical transducers.

Abstract

Hybrid mechanical-superconducting systems for quantum information processing have attracted significant attention due to their potential applications. In such systems, the weak coupling regime, dominated by dissipation, has been extensively studied. The strong coupling regime, where coherent energy exchange exceeds losses, has also been widely explored. However, the transition-coupling regime, which lies between the above two and exhibits rich, unique physics, remains underexplored. In this study, we fabricate a tunable coupling device to investigate the coupling of a superconducting transmon qubit to a seven-mode surface acoustic wave resonator (SAWR), with a particular focus on the transition-coupling regime. Through a series of phonon oscillation experiments and studies in the dispersive regime, we systematically characterize the performance of the SAWR. We then explore the complex dynamics of energy exchange between the qubit and the mechanical modes, highlighting the interplay between dissipation and coherence. Finally, we propose a protocol for qubit readout and fast reset with a multimode mechanical cavity using one mode for readout and another mode for reset. We have demonstrated in simulation that the qubit achieves both fast reset and high coherence performance when the qubit is coupled to the reset mode in the transition-coupling regime.

Figures (11)

• Figure 1: The device. (a) Optical microscope image of the bottom sapphire chip, where the superconducting quantum circuits are fabricated. The black dashed box highlights the region containing the SAWR. (b) Enlarged view of the black dashed box in panel (a), showing the optical microscope image of the SAWR, which is fabricated on the top $\text{LiNbO}_3$ chip. The SAWR includes three interdigital transducer (IDT) ports. (c) Magnified view of the red box in panel (b), providing an optical microscope image of the three IDTs. (d) Measured transition frequency of the transmon qubit as a function of bias voltage. (e) Enlarged view of the green box in panel (c), showing the SEM image of the middle IDT, which has a period $p$ and a finger width of $d_I=p/6=$144nm. (f) Enlarged view of the blue box in panel (c), showing a scanning electron microscope (SEM) image of the left and right IDTs, which serve as input and output transducers. These IDTs are connected to ports 1 and 2 on the bottom chip via indium bumps. Each finger has a width of $d_s=$216nm, and the IDT period is $p=4d_s=$864nm.
• Figure 2: Phononic response of SAWR. (a) Schematic of the SAWR, with the main structure consisting of three IDTs and two Bragg gratings. (b) $S_{21}$ of the SAWR with fits applied to the data from the near-resonant regions of seven peaks. (c) Time-resolved measurement of the SAWR. It is performed by applying a 12ns electrical Gaussian pulse to the Port1 and acquiring the output signal from the Port2. The electrical signals are collected by the Analog-to-Digital Converter (ADC) with a time resolution of $1\,\text{ns}$.
• Figure 3: Acoustic AC stark shift measurements in the dispersive regime. The transmon is biased at the idle point and the gmon is set to $V_g = -3.0$. (a) The qubit transition frequency shifts as a function of the SAW pump frequency. The dashed line indicates the transition frequency from the first excited state to the second excited state of the qubit at the idle point, $f_{12} = 4.5979\,\text{GHz}$. (b) Energy level diagrams illustrating the conventional dispersive regime $(\chi < 0)$ and the straddling dispersive regime $(\chi > 0)$. (c) Linear relationship between the qubit transition frequency shift and the SAW pump power. (d) AC Stark shift delay measurements for different SAW pump frequencies. (e) The inset shows the experimental pulse sequence. The main plot presents the normalized data from (d) after $2\,\text{$\mu$s}$, expressed as $\delta \omega_{\text{q}}/\delta \omega_{\text{q}}^{\text{max}}$. The solid line represents a fit to the data. Colors and line styles in (c), (d), and (e) correspond to different modes.
• Figure 4: Tunable coupling and resonant regime measurements. (a) Schematic of the model, illustrating a two-level atom coupled to a multimode cavity. The transition frequency of the two-level atom is $\omega_q = 2\pi f_q$, with an intrinsic dissipation rate $\gamma$. The frequency of resonant mechanical mode is $\omega_r = 2\pi f_r$, with a dissipation rate $\kappa$. (b) Population dynamics of the $|e\rangle$ state under resonant evolution for different coupling regimes, considering the coupling strength $g$ relative to the dissipation rate and transition frequency. As $g$ increases, the system transitions from the weak coupling regime to the strong, ultrastrong, and deep strong coupling regimes, with each regime exhibiting distinct evolution characteristics. $P_{SS}$ denotes the steady-state population. (c) Dependence of the qubit transition frequency on the gmon bias. Red points indicate frequency values obtained from spectral fitting, while the black solid line represents a fit to these values. (d) Qubit relaxation time $T_1$ of the qubit at the idle point as a function of the gmon bias. Red points denote $T_1$ values extracted from decay measurements, and the black solid line represents a fit to the data. (e) Pulse sequence diagram for the resonant evolution experiment. (f) Resonant evolution data and fits for the qubit interacting with 6th mode, with orange points corresponding to $V_g = -3.0$ and blue points to $V_g = 0.75$. (g) Resonant evolution data for seven modes, with orange heatmap for $V_g = -3.0$ and blue heatmap for $V_g = 0.75$. (h) Coupling strengths extracted from the fit in (g), exhibiting a sinusoidal dependence on the mode index.
• Figure 5: Intrinsic reset and readout protocol. (a) The two modes of the SAWR are utilized as the dispersive readout mode and the fast reset mode. During the reset process, the qubit is biased into resonance with the reset mode via an external magnetic flux and then decays to the ground state via the reset mode. The coupling strengths between the qubit and the reset mode, as well as the dispersive readout mode, can be engineering through their sinusoidal dependence on the mode index. (b) Numerical simulation results of the $|e\rangle$ state population of qubit during the resonant evolution with the reset mode under different coupling strengths. The yellow line in the figure represents the reset time required to achieve 99% fidelity, while the green line corresponds to 99.9% fidelity. As the coupling strength increases, the reset efficiency improves, reaching its minimum reset time in the transition-coupling regime. (c) The dependence of the reset mode's impact on qubit decoherence as a function of coupling strength. In the weak and transition-coupling regimes, this effect is negligible, whereas in the strong coupling regime, it becomes increasingly significant. The red, yellow, and brown dashed lines in the figure indicate the values of $g_r / \kappa_r$ corresponding to $\gamma_p / \gamma$ ratios of 1%, 5%, and 10%, respectively.