Realizing a Circuit Analog of an Optomechanical System with Longitudinally Coupled Superconducting Resonators

We realize a superconducting circuit analog of the generic cavity-optomechanical Hamiltonian by longitudinally coupling two superconducting resonators, which are an order of magnitude different in frequency. We achieve longitudinal coupling by embedding a superconducting quantum interference device into a high frequency resonator, making its resonance frequency depend on the zero point current fluctuations of a nearby low frequency $LC$ resonator. By applying sideband drive fields we enhance the intrinsic coupling strength of about 15 kHz up to 280 kHz by controlling the amplitude of the drive field. Our results pave the way towards the exploration of optomechanical effects in a fully superconducting platform and could enable quantum optics experiments with photons in the yet unexplored radio frequency band.

Figure 1

(a) Circuit representation of the device architecture. (b) Analogous schematic of a generic optomechanical system with one of the cavity mirrors being movable and attached to a spring (red). (c) Schematic of the experimental setup. (d)–(f) Optical images of the sample at three different magnifications showing both resonators in (d), and the coupling region in (e) and (f).

Figure 2

(a) Measured transmission spectrum $S_{21}$ of the tunable high frequency resonator vs ${\mathrm{\Phi }}_{\mathrm{ext}}$. The inset shows a line cut at around zero flux bias together with a fit to the theory. A model fit to the measured resonance frequencies as a function of magnetic flux is shown as the dashed red line. The green circle indicates the chosen bias point. (b) Schematic of the two-tone spectroscopy experiment. A drive field together with the nonlinear coupling mediates coherent up-conversion from a low frequency input field ${\omega }_{\mathrm{in}}$ to the output field ${\omega }_{\mathrm{out}}$. (c) Measured squared amplitude of the up-converted field at frequency ${\omega }_{\mathrm{out}}$ as a function of ${\omega }_{\mathrm{in}}$ (blue points) normalized to its maximum value and Lorentzian fit (solid red line).

Figure 3

(a) Transmission spectroscopy data $|S_{21}{|}^2$ as a function of probe frequency ${\omega }_{\mathrm{in}}$ for varying drive frequencies ${\omega }_d$. (b) Individual transmission spectra at selected drive frequencies as indictated by the arrows in (a). The data (blue points) are fit to the input-output formula (red lines) in Eq. (3). Individual data sets are offset for clarity.

Figure 4

(a) Data and fit to Eq. (3) of the measured transmission spectra on resonance for various drive amplitudes successively increasing by a factor 1.06 from ${\alpha }_d=11.4$ to 19.2. The drive frequency ${\omega }_d$ is adjusted in each measurement to account for the power-dependent Stark shift of the high frequency resonator. Individual data sets are offset by integer values for clarity. (b)–(c) Coupling strength and Stark shift vs ${\alpha }_d$ with linear and quadratic fits, respectively. Error bars are standard errors resulting from the fit of the data in (a) to Eq. (3). Standard errors of the fitted Stark shifts are smaller than the size of the data points.