Cavity Quantum Acoustic Device in the Multimode Strong Coupling Regime

We demonstrate an acoustical analog of a circuit quantum electrodynamics system that leverages acoustic properties to enable strong multimode coupling in the dispersive regime while suppressing spontaneous emission to unconfined modes. Specifically, we fabricate and characterize a device that comprises a flux tunable transmon coupled to a $300\mu \mathrm{m}$ long surface acoustic wave resonator. For some modes, the qubit-cavity coupling reaches 6.5 MHz, exceeding the cavity loss rate (200 kHz), qubit linewidth (1.1 MHz), and the cavity free spectral range (4.8 MHz), placing the device in both the strong coupling and strong multimode regimes. With the qubit detuned from the confined modes of the cavity, we observe that the qubit linewidth strongly depends on its frequency, as expected for spontaneous emission of phonons, and we identify operating frequencies where this emission rate is suppressed.

Figure 1

Device diagram and acoustical cavity spectrum. (a) A cartoon schematic of the SAW cavity, acoustically coupled qubit, and the microwave network for control and measurement. (b) A false color SEM image of the fabricated device before the Josephson junction was patterned. Two Bragg reflectors (blue), each consisting of 400 Al strips, are spaced by $L=275\mu \mathrm{m}$ to define a SAW cavity, and a split-junction transmon qubit (red) was placed at $L_{\mathrm{eff}}/4$ from the left reflector. A cavity-IDT (pink) with 125 periods at $L/2$ is used to drive and readout the cavity modes. The center and ground conductor of a coplanar waveguide (yellow) contact either side of the IDT. Measurements consist of detecting the reflection of a microwave tone applied to the cavity IDT. A directional coupler separates incident and reflected waves, so that the reflected signal is passed through a high electron mobility transistor (HEMT) amplifier and measured. False color SEM images of (c) a split Josephson junction with a $7\times 7\mu {\mathrm{m}}^2$ loop area, (d) a split-finger IDT with the upper electrode in green and the lower in purple [29], and (e) several Al stripes within a Bragg reflector. The characteristic wavelength of the cavity is indicated, ${\lambda }_c=v_s/f_c=677\mathrm{nm}$, where the center frequency is $f_c=4.253\mathrm{GHz}$. (f) A microwave reflection measurement of the SAW cavity reveals 11 (numbered) prominent longitudinal modes within the mirror bandwidth.

Figure 2

Resonantly coupled multimode CQAD system. (a) The plot shows the cavity reflection (color scale) as the transmon is tuned by varying the applied magnetic flux. The red dashed line indicates the transmon’s flux dependent resonance. The inset shows the expected qubit frequency (red), which has a maximum at 5.08 GHz. The green section indicates the measured range of panel (a). (b) Enlargement of the first set of avoided crossings shown in panel (a). (c) A model for the acoustic spectrum based on the interaction Hamiltonian, Eq. (2).

Figure 3

Hybridization of modes in the strong multimode regime. (a) The solid lines show the squared coefficients of the eigenvector (participation) in the uncoupled mode basis as the qubit tunes with magnetic field. At $-0.252$ flux quanta, longitudinal modes ${\omega }_7$ and ${\omega }_8$ and transverse mode ${\omega }_{7t}$ nearly equally contribute to 89% of the eigenvector. The “others” trace shows the combined contribution of the remaining modes. (b) The acoustic spectrum [from Fig. 2] between modes ${\omega }_7$ and ${\omega }_8$. The dotted line shows the eigenvalue fit from the model corresponding to the superposition of modes from (a).

Figure 4

Measurements in the dispersive regime. (a) The dispersive shift of cavity mode ${\omega }_8$ versus qubit frequency (${\omega }_{\mathrm{q}}$). The purple, orange, and blue dashed lines indicate ${\omega }_8/2\pi$, ${\omega }_8/2\pi +\alpha$, and ${\omega }_8/2\pi +3\alpha /2$, respectively, where $\alpha =273\mathrm{MHz}$ is the transmon’s anharmonicity [29]. The green line shows a prediction from the standard transmon model, and the solid blue line shows a prediction from diagonalizing a generalized Jaynes-Cummings Hamiltonian with a 4-level transmon. (b) The qubit linewidth versus qubit frequency. Red points are measurements, and the blue line shows the expected spontaneous emission rate of the qubit into SAWs based on the qubit-IDT geometry, with an offset to account for the intrinsic decoherence, i.e., processes other than spontaneous emission that decohere the qubit. The green region indicates the mirror bandwidth.