Efficient flip-chip and on-chip-based modulation of flux-tunable superconducting resonators

TL;DR

This work addresses the challenge of delivering flux signals efficiently to flux-tunable superconducting resonators (FTRs) when the SQUID termination cannot be placed near the flux source. It introduces two chip-based modulation approaches—flip-chip galvanic input coils and on-chip input coils with air bridges—implemented on aluminum CPW resonators terminated by large-loop dc-SQUIDs with asymmetry to suppress branch switching. The authors quantify flux-transfer efficiency ($\eta_2$) and demonstrate flux modulation by more than $1\ \mathrm{GHz}$ with responsivities up to $\partial \omega_r/\partial \Phi \approx 2\pi\times 20\ \mathrm{GHz}/\Phi_0$, achieving $\eta_2$ up to $21\%$ (flip-chip) and $19\%$ (on-chip), while reporting $Q_i$ values around $3\times 10^4$ and $7\times 10^3$ and Kerr coefficients in the $200$–$400\ \mathrm{kHz}/n_c$ range. A proof-of-principle flux-detector demonstration yields about $1.6\%$ efficiency, highlighting practical sensitivity for flux measurements. The results establish viable low-current flux modulation strategies and identify concrete design routes ( washer-type SQUIDs, multiwinding coils, and nonlinear element options) to further enhance performance for scalable quantum sensing and computing.

Abstract

We demonstrate the efficient modulation of flux-tunable superconducting resonators (FTRs) using flip-chip or on-chip-based input coils. The FTRs we use are aluminum-based quarter-wave coplanar waveguide resonators terminated with 100um or 200um-wide square loop dc superconducting quantum interference devices (SQUIDs) employing 1um-sized Josephson junctions. We employ SQUIDs with a geometric loop inductance of up to 0.7nH to increase the flux transfer efficiency. The geometric inductance of the SQUID results in a non-zero screening parameter $β_L$, whose branch switching effect is mitigated by using asymmetric junctions. We achieve flux modulation of the FTRs by more than one GHz and flux responsivities of up to tens of GHz/$Φ_0$ with uA-scale on-chip currents. We compare flip-chip with on-chip input-coil-based flux modulation, where the former is realized through galvanically connected and closely spaced chips, while the latter is achieved through superconducting air-bridge connections. We achieve a flux-transfer efficiency from the input coil to the SQUID loop of up to 20%. Our work paves the way for efficient low current flux modulation of FTRs and sensitive measurement of flux signals.

Figures (13)

• Figure 1: Chip-based modulation of a flux-tunable resonator (FTR) using integrated input coils. (a) Equivalent lumped element circuit model: the FTR (red) is inductively coupled via a flux transformer (green) to an external magnetic source. (b) Left: design of a 1µm-wide Josephson junction. Right: false-colored SEM image of the fabricated junction where the dark electrode is patterned first; the thin appendages on either side are artifacts of the shadow evaporation technique. (c) Schematic of a flip-chip input coil (green) concentric with the SQUID (red) and galvanically connected to the bottom chip via superconducting indium bumps paradkar_2025. (d) Device based on schematic shown in (c): 200µm-wide SQUID-loop FTR on a silicon bottom chip with a NbN input coil on a sapphire top chip (false-colored). (e) Schematic of an on-chip input coil (green) patterned concentrically around the SQUID loop (red). (f) Device based on schematic shown in (e): optical image (false-colored) of a 100µm-wide SQUID-loop FTR; the input coil is routed over the CPW via an aluminum air bridge Osman_2021Osman_2024.
• Figure 2: Flux transfer efficiency $\eta_2$ between the input coil and the SQUID loop. Efficiency as a function of (a) the ratio of input coil loop width to SQUID loop width $d_i/d_s$ and (b) axial loop separation $h$. The black lines represent analytic results assuming one-dimensional square loops, while the red markers are results from FEM simulations of realistic geometries of the on-chip (star) and flip-chip (cross) configurations. The dotted vertical lines in (a) are at $d_i/d_s$ of $1.05$ and $1.25$ and in (b) at $h$ of $0µm$ and $50µm$. FEM-simulated magnetic field distribution for (c) flip-chip and (d) on-chip configurations assuming an input coil current of 100µA.
• Figure 3: Multiple $\Phi_0$ modulations of an FTR with a 200µm-wide SQUID modulated at 0.33 intra-cavity photons via (a,c) an external bias coil and (b,d) via a flip-chip-based input coil. (a,b) show the background-subtracted FTR modulation in dependence of coil current and applied flux. (c,d) show the extracted resonance frequency of the FTR (dots), the solid line is the model accounting for finite screening and junction asymmetry.
• Figure 4: Modulation of FTRs via flip-chip-based (a,c,e) and on-chip-based (b,d,f) input coil measured at 0.33 and 0.83 intra-cavity photons, respectively. (a,b) show background-subtracted plots of the FTR modulation in dependence of input coil current and input flux. (c,d) show the resonance frequency (dots), extracted via a circle-fit routine, as a function of input flux. The solid line and the dotted line are the fit result for the model according to Eq. (\ref{['eq:FTRmod']}) and its derivative, respectively. (e,f) show the intrinsic and coupling quality factors of the FTRs as a function of input flux.
• Figure 5: Kerr nonlinearity of FTRs. (a) and (b) show the lowering of the resonance frequency as a function of intra-cavity photon number due to the Kerr coefficient for the flip-chip and on-chip modulated FTRs, respectively. (c,d) Circle fit results obtained from power sweep measurements at the zero-flux-bias point of the FTRs with flip-chip-based input coil (c) and on-chip-based input coil (d). The data is fit using Eq. (\ref{['eq:Kerr']}) to obtain the Kerr coefficients. (e) and (f) show the intrinsic, coupling, and loaded quality factors ($Q_L^{-1} = Q_i^{-1} + Q_c^{-1}$), and the Kerr coefficient as a function of intra-cavity photons. The dashed lines in both plots show the power at which the flux modulation measurements, shown in Fig. \ref{['fig:Fig_4']}, were taken for the respective devices.