Linear and Nonlinear Properties of a Compact High-Kinetic-Inductance $\mathrm{WSi}$ Multimode Resonator

The kinetic inductance (KI) of superconducting devices can be exploited for reducing the footprint of linear elements as well as for introducing nonlinearity to the circuit. We characterize the linear and nonlinear properties of a multimode resonator fabricated from amorphous tungsten silicide ($\mathrm{WSi}$), with a fundamental frequency of $f_1=172$ MHz. We show how the multimode structure of the device can be used to extract the different quality factors and to aid the nonlinear characterization. In the linear regime, the footprint is reduced by a factor of approximately $2.9$ with standard lateral dimensions, with no significant degradation of the internal quality factor compared to a similar $\mathrm{Al}$ device. In the nonlinear regime, we observe self-positive frequency shifts at low powers, which can be attributed to saturation of tunneling two-level systems. The cross-mode nonlinearities are described well by a Kerr model with a self-Kerr coefficient on the order of $|K_{11}|/2\pi \approx 1.5\times {10}^{-7}$ Hz per photon. These properties, together with a reproducible fabrication process, make $\mathrm{WSi}$ a promising candidate for creating linear and nonlinear circuit quantum electrodynamics elements.

Figure 1

The multimode resonator device. (a),(b) A sketch of the device: (b) shows the coupling to the feed line via an interdigitated capacitor—white, metal trace and ground; green, exposed $\mathrm{Si}$ substrate. (c) A SEM image of the surface of a sputtered 30-nm $\mathrm{WSi}$ film: grains of order 10 nm are visible.

Figure 2

Reduction of the footprint of the multimode resonator. (a) Multimode spectra of two geometrically identical $\mathrm{Al}$ and $\mathrm{WSi}$ multimode resonators. The lines are linear fits for the frequencies of the modes. The $\mathrm{WSi}$ device obtains a frequency spacing that is smaller by a factor of approximately $2.9$ from the $\mathrm{Al}$ device using the same length. (b) The raw transmission spectrum of the $\mathrm{WSi}$ device, showing the first 23 modes. The profile is affected by the frequency response of the amplifiers. The lower peaks at 300 and 2844 MHz are probably an electronic feed-through and a slot mode. (c) The magnitude of typical transmission data around resonance. The red line is the magnitude of the fit to the complex data [32, 33].

Figure 3

The quality factors of the devices for different modes. (a) The coupling quality factors for the $\mathrm{Al}$ (blue) and $\mathrm{WSi}$ (magenta) devices (log-log plot). The dashed lines are fits for the expected $1/n$ dependence, given in Eq. (3). (b) The extracted internal factors for the $\mathrm{Al}$ (blue) and $\mathrm{WSi}$ (magenta) devices. $Q_i$ is lower for the $\mathrm{WSi}$ device probably because a larger fraction of the field energy is stored in the oxides on the surfaces. The dashed line is a fit to a model of thermal saturation of loss-inducing tunneling two-level systems, given in Eq. (4). The lower internal $Q$ at low modes for the $\mathrm{WSi}$ device is unexpected and might be a result of coupling to unwanted slot modes [see Fig. 2]. The inset in (b) shows the loaded (total) quality factor, which is dominated by internal loss (coupling) for the $\mathrm{WSi}$ ($\mathrm{Al}$) device. The lines are guides to the eye.

Figure 4

Self-nonlinearity frequency shifts. The differential frequency shift for the first four modes (denoted by different marker shapes as detailed in the legend) of the $\mathrm{WSi}$ device as a function of the stored energy (in units of first-mode photons) for two different cool-downs. The empty (full) markers are measurements from the first (second) cool-down. The lines are guides to the eye. The inset shows the (vertically shifted) frequency shifts of the first seven modes, measured at the first cool-down. The positive frequency shifts hinder the evaluation of the bare self-Kerr coefficients.

Figure 5

Cross-Kerr frequency shifts. (a) The differential frequency shift for different modes of the $\mathrm{WSi}$ device as a function of the stored energy in various pumped modes (in units of first-mode photons). In the legend, ($n$,$m$) denotes (probed mode $n$, pumped mode $m$). The inset shows linear fits to the frequency shifts of the second mode. (b) The extracted cross-Kerr coefficients $K_{nm}$ when pumping mode $m$ as a function of the probed mode $n$. The dashed lines are fits to $K_{nm}\propto K_{1m}n$. The inset shows the fitted $K_{1m}$ divided by the pumped mode number $m$ as a function of $m$, which should equal $K_{11}$. The dashed line is the average of $K_{1m}/m$.