Advanced microwave SQUID multiplexer model incorporating readout power effects and Josephson junction inhomogeneities

TL;DR

The paper addresses the limited parameter range of existing μMUX models by introducing a numerical framework that solves rf‑SQUID dynamics for $β_\mathrm{L}<1$ with arbitrary current–phase relations. It demonstrates substantially improved agreement with experimental data at higher readout powers and across a broader design space than previous analytical models. A key advance is the inclusion of a mesoscopic barrier inhomogeneity model, which reveals that junction barrier variations, while qualitatively similar to increasing $β_\mathrm{L}$, imprint distinct signatures on the resonance response and must be accounted for accurate characterization. The work provides a practical tool for designing and optimizing next‑generation cryogenic detector readouts and is compatible with non‑tunneling junction devices, offering enhanced predictive power for μMUX performance.

Abstract

We present an advanced model for describing the readout power dependence of the resonance characteristics of a microwave SQUID multiplexer. Our model proves valid for SQUID screening parameters up to $β_\mathrm{L}<1$, hence covering the full range of practically relevant design parameters. We demonstrate that our model significantly improves agreement with experimental data compared to the existing models, thereby enabling optimization beyond the previously accessible parameter space. Moreover, our model supports non-sinusoidal current-phase relations of the rf-SQUID's Josephson junction, allowing, for the first time, for the modeling of devices based on Josephson tunnel junctions with inhomogeneous tunnel barriers. We show that the effects of such inhomogeneities are qualitatively similar to, yet distinct from, those of the screening parameter, making their inclusion essential for accurate characterization. Incorporating these effects yields great improved agreement with measurements, even at readout power conditions well beyond typical operating parameters.

Figures (7)

• Figure 1: Equivalent circuit diagram of a single $\upmu$MUX channel. The channel comprises an rf-SQUID containing a Josephson junction with critical current $I_\mathrm{c}$ and a closed superconducting loop with inductance $L_\mathrm{S}$. The SQUID is inductively coupled to the input coil $L_\mathrm{in}$ and the inductor $L_\mathrm{T}$, which loads a superconducting quarter-wave microwave resonator. The resonator is coupled to a transmission line via a coupling capacitor $C_\mathrm{c}$. A modulation coil $L_\mathrm{mod}$ is coupled to the SQUID for flux ramp modulation. Owing to the mutual couplings, the load inductor can be described by an effective inductance $L_\mathrm{T,eff}(\Phi_\mathrm{tot})$ that depends on the total magnetic flux $\Phi_\mathrm{tot}$ through the SQUID.
• Figure 2: Measured dependence of the resonance frequency $f_\mathrm{r}(I_\mathrm{mod})$ on the modulation coil current $I_\mathrm{mod}$ for a representative $\upmu$MUX channel. The corresponding resonance curves were recorded using a vector network analyzer with low microwave probe tone power, corresponding to the limit $\varphi_\mathrm{rf} \rightarrow 0$. Panel (a) shows a fit to the data using the analytical model described by Wegner et. al.Weg22. In (b) the data was fitted using the model presented in this work, assuming an ideal (i.e., perfectly homogeneous) Josephson tunnel junction barrier. Panel (c) shows a fit obtained with our new model assuming an inhomogeneous tunneling barrier thickness with Gaussian probability density distribution characterized by a mean barrier thickness $\bar{d}=2\,\mathrm{nm}$ and standard deviation $\sigma=0.48\,\mathrm{nm}$). Panels (d), (e), and (f) show magnified views of the fits in the panels above, where $\langle f_\mathrm{r} \rangle = 4.386\,\mathrm{GHz}$ denotes the resonance frequency averaged over $I_\mathrm{mod}$. The best agreement is achieved using the model that includes a finite tunnel barrier inhomogeneity.
• Figure 3: Flowchart outlining our numerical simulation framework. Red, rounded boxes indicate user-defined input parameters, while green, rectangular boxes represent calculations of respective quantities. An optional step (Step 0) allows deriving the current phase relation of a Josephson tunnel junction with inhomogeneous barrier thickness. Step 1 determines the current-flux relation of the rf-SQUID as a function of the externally applied magnetic flux. Step 2 calculates the resonance frequency shift using the current-flux relation obtained in step 1.
• Figure 4: rf-SQUID supercurrent and its time derivative, calculated using our advanced $\upmu$MUX model, are shown. Panel (a) shows the SQUID's current–flux relation (green solid line). In addition, the calculated SQUID response to a microwave signal with amplitude $\Phi_\mathrm{rf} = \Phi_0/8$, applied at a flux offset $\Phi_\mathrm{ext} = \Phi_0/4$, is schematically depicted (red solid lines). Panel (b) and (c) show the resulting time-dependence $I_\mathrm{S}(t)$ of the supercurrent and its time derivative. The time derivative contains multiple frequency components, which are extracted using a discrete Fourier transform. The corresponding cosine and sine coefficients $a_n$ and $b_n$ are shown in panel (d). When later calculating the effective load inductance $L_\mathrm{T,eff}$, only the first Fourier coefficient $a_1$ is relevant, as all other components either do not populate the resonator or vanish.
• Figure 5: Comparison between the analytical $\upmu$MUX model presented by Wegner et al.Weg22 and the present numerical model, assuming an rf-SQUID with hysteresis parameter $\beta_\mathrm{L} = 0.9$. Panel (a) displays the dependence of the resonance frequency $f_\mathrm{r}$ on the applied magnetic flux in the limit of negligible power of the microwave probe tone ($\Phi_\mathrm{rf} \rightarrow 0$). $\langle f_\mathrm{r} \rangle = 5\,\mathrm{GHz}$ denotes the resonance frequency averaged over $\Phi_\mathrm{ext}$. Panel (b) shows the resonance frequency as a function of microwave probe tone power $\Phi_\mathrm{rf}$. The solid line corresponds to $\Phi_\mathrm{ext} = n\Phi_0$, the dashed line to $\Phi_\mathrm{ext} = (n + 1/2)\Phi_0$, and the dotted line to $\Phi_\mathrm{ext} = (n \pm 1/4)\Phi_0$. The dashed light gray lines mark the minimum and maximum resonance frequency predicted by the power-independent model. The numerical model shows the expected behavior and matches the exact power-independent solution in the low-power limit, while the analytical model exhibits ripples caused by the limitations of its Taylor expansion.