Quartz-superconductor quantum electromechanical system

We propose and analyze a quantum electromechanical system composed of a monolithic quartz bulk acoustic wave oscillator coupled to a superconducting transmon qubit via an intermediate $LC$ electrical circuit. Monolithic quartz oscillators offer unprecedentedly high effective masses and quality factors for the investigation of mechanical oscillators in the quantum regime. Ground-state cooling of such mechanical modes via resonant piezoelectric coupling to an $LC$ circuit, which is itself sideband cooled via coupling to a transmon qubit, is shown to be feasible. The fluorescence spectrum of the qubit, containing motional sideband contributions due to the couplings to the oscillator modes, is obtained and the imprint of the electromechanical steady state on the spectrum is determined. This allows the qubit to function both as a cooling resource for, and transducer of, the mechanical oscillator. The results described are relevant to any hybrid quantum system composed of a qubit coupled to two (coupled or uncoupled) thermal oscillator modes.

Figure 1

(a) Schematic of the system under consideration. The system is composed of a BAW quartz oscillator (orange shading), with electrodes placed across it forming the capacitance of an $LC$ tank circuit (green shading). The quartz (mechanical) oscillator is represented by an electrical equivalent circuit [13]. The $LC$ tank circuit is completed by a tunable inductance on a superconducting chip, which also contains a superconducting transmon qubit (blue shading). This is represented by two Josephson junctions in parallel with a shunt capacitor. Physically, the elements of the BAW oscillator and superconducting chip are demarcated by the dashed orange and blue boxes, respectively. The transmon qubit is flux driven (green circuit), and the qubit's fluorescence is monitored by capacitive outcoupling to a waveguide and amplification. (b) Equivalent (dissipation-free) electrical circuit for the whole quartz-superconductor quantum electromechanical system. In labeling the components, the subscripts “$m$,” “$c$,” and “$t$” denote the mechanical, electrical circuit, and transmon modes, respectively. Such an equivalent circuit enables the derivation of the Hamiltonian (1), as described in Appendix pp1.

Figure 2

Steady-state mechanical occupation, $\langle {\widehat{a}}_{\mathrm{m}}^{†}{\widehat{a}}_{\mathrm{m}}\rangle$, as a function of the qubit relaxation rate, ${\gamma }_{\downarrow }$, for ${\gamma }_{\mathrm{c}}/2\pi =\left\{50,100,200\right\}\mathrm{kHz}$, corresponding to the black, blue, and red curves (respectively). Numerical results obtained in the sideband picture are shown as solid lines, and analytical results in the adiabatic limit are shown as dashed lines. Ground-state cooling is shown to be feasible with the anticipated parameters. The results from the two cases are consistent provided that ${\gamma }_{\downarrow }\gg {\overline{g}}_{\mathrm{cq}}$, corresponding to the right-hand side of the plot. For ${\gamma }_{\downarrow }\sim {\overline{g}}_{\mathrm{cq}}$, corresponding to the left-hand side of the plot, the adiabatic approximation breaks down and leads to an overestimate of the cooling achievable. Increasing the $LC$ tank circuit quality factor (which is a relatively uncertain parameter) over the specified range improves the cooling performance of the system. Further increases of the quality factor might be possible via a redesign of the $LC$ tank circuit, with a corresponding improvement in the cooling expected.

Figure 3

Contributions to the qubit fluorescence spectrum under the driving condition ${\delta }_{\mathrm{d}}={\omega }_p$, at $\omega \sim {\omega }_p$ (i.e., the upper motional sideband frequency). The frequency $\omega$ is defined relative to the qubit drive frequency. The contributions are scaled by the total spectral density at the upper motional sideband $(\omega ={\omega }_p)$, as given by Eq. (27). The contributions are due to (a) coupling to the electrical circuit mode given by Eq. (33b) with $p=c$, (b) coupling to the mechanical oscillator mode given by Eq. (33b) with $p=m$, and (c) qubit excitation from its environment given by Eq. (28). Given the different vertical scalings on these plots, the total noise spectral density, as given by Eq. (27), is given by the contribution from the electrical circuit mode. This motional sideband enables transduction of the electromechanical system. Note that the scaling of the frequency axis also varies substantially between these plots.