Ultrastrong Parametric Coupling between a Superconducting Cavity and a Mechanical Resonator

We present a new optomechanical device where the motion of a micromechanical membrane couples to a microwave resonance of a three-dimensional superconducting cavity. With this architecture, we realize ultrastrong parametric coupling, where the coupling not only exceeds the dissipation in the system but also rivals the mechanical frequency itself. In this regime, the optomechanical interaction induces a frequency splitting between the hybridized normal modes that reaches 88% of the bare mechanical frequency, limited by the fundamental parametric instability. The coupling also exceeds the mechanical thermal decoherence rate, enabling new applications in ultrafast quantum state transfer and entanglement generation.

Figure 1

Parameter space diagram showing four regimes of parametric optomechanical coupling $g$ as a function of cavity dissipation $\kappa$ and mechanical frequency $\mathrm{\Omega }$. For a parametric drive at $\mathrm{\Delta }=-\mathrm{\Omega }$, strong coupling coincides with the normal-mode splitting condition $\kappa <4g$. Ultrastrong coupling arises when $g$ further increases to a significant fraction of $\mathrm{\Omega }$ until reaching the limit for stability at $2g=\sqrt{{\mathrm{\Omega }}^2+{\kappa }^2/4}$. Labeled points denote previous optomechanical experiments in the strong-coupling regime: (A) optical Fabry-Pérot cavity [5], (B) lumped element microwave circuit [7], (C) toroidal optical microcavity [8], and (D) microwave loop-gap cavity [9]. The star indicates the highest coupling achieved in this Letter.

Figure 2

Device schematic. (a) A cavity, milled from two aluminum blocks, supports a microwave resonance near 12 GHz. Microwave signals couple in and out of the cavity through two asymmetrically coupled ports. (b) An aluminum mechanically compliant capacitor patterned on a sapphire substrate is galvanically connected to the cavity walls through superconducting wire bonds, loading the fundamental cavity resonance frequency down to 6.5 GHz. (c) A micrograph image shows the capacitor, which has a fundamental mechanical resonance at 9.7 MHz and a vacuum gap of approximately 30 nm.

Figure 3

Measured and calibrated cavity transmission from port 1 to port 2 for varied drive strengths $n_d$ from weak coupling [(a),(b)] through strong coupling [(c),(d)] and up to ultrastrong coupling (e). The data (blue) are fit to theory (black) containing the first five mechanical modes. The vertical red line indicates the frequency of the applied microwave drive, adjusted to maintain $\mathrm{\Delta }=-\mathrm{\Omega }$. The structure below the drive frequency at the highest powers directly shows the importance of counterrotating terms in the ultrastrong-coupling regime.

Figure 4

(a) Measured system rates as a function of drive strength. In the weak-coupling regime, the optical damping first exceeds the mechanical dissipation ($C=1$) and then exceeds the thermal decoherence ($C_q=1$). Once $2g$ reaches $\kappa /2$, the eigenmodes split, indicating the onset of strong coupling. Eventually, ultrastrong-coupling corrections become important before the system approaches a parametric instability at $2g=\mathrm{\Omega }$. (b) Mechanical eigenfrequencies as a function of the optomechanical coupling. In the ultrastrong-coupling regime, the splitting ${\mathrm{\Omega }}_s={\mathrm{\Omega }}_{+}-{\mathrm{\Omega }}_{-}$ exceeds the linear strong-coupling approximation ${\mathrm{\Omega }}_s\simeq 2g$ (black), reaching a maximal value ${\mathrm{\Omega }}_s\approx 0.88\mathrm{\Omega }$.

Figure 5

Mechanical susceptibility inferred from microwave measurements at our highest drive power (blue) with fit line (black) compared with the bare susceptibility (gray). The susceptibility in the ultrastrong-coupling regime shows asymmetry at the two eigenfrequencies as well as a marked increase in its value at zero frequency, in agreement with the full theory.