Microwave radiometry of a quantum-critical, hybrid Josephson array

TL;DR

The study demonstrates microwave radiometry as a calibrated, non-invasive method to measure radiation from a tunable two-dimensional Josephson junction array across superconducting, anomalous-metallic, and insulating regimes. By converting emitted microwave power into an effective sample temperature $T_s$ and correlating it with impedance via $|S_{11}|$, the authors reveal that the anomalous metal heats more readily than the quantum-critical or insulating states, indicating a breakdown of thermal equilibrium with the cryostat. Near the SIT, finite-bias noise shows universal $T_s \propto \sqrt{I}$ scaling consistent with theoretical predictions for non-equilibrium quantum critical behavior, and data from two devices collapse onto a single curve when plotted against current (or current density), suggesting a universal, non-equilibrium description near criticality. The work establishes microwave radiometry as a powerful probe of non-equilibrium physics in Josephson arrays and related near-critical systems, offering a path to explore universal aspects of quantum critical dynamics and thermalization in low-dimensional superconducting devices.

Abstract

Arrays of Josephson junctions can be tuned through anomalous metallic, quantum-critical, and insulating regimes. We introduce a new experimental probe, capturing microwave radiation across all three regimes, using a two-dimensional array of superconductor-semiconductor hybrid Josephson junctions as a model system. Our approach allows in-situ calibration of the sample's circuit parameters and provides isolation from measurement back-action effects. We measure the radiation temperature of the anomalous metal, and find that it is hotter than both the quantum-critical and insulating regimes. We further show that the anomalous-metallic regime is more susceptible to additional heating than other regimes, explaining its emergence in otherwise thermalized systems. Turning to the quantum-critical regime, we discover nonlinear scaling of radiative noise with applied bias, consistent with theoretical predictions of universal non-equilibrium behavior at quantum critical points.

Figures (19)

• Figure 1: a, Schematic of the device showing the Al/InAs Josephson junction array capacitively coupled to a microwave readout chain, with a top-gate to tune carrier density in the InAs two dimensional electron gas. A circulator and filtering (not pictured) prevents back-action noise from disrupting the anomalous metallic state, and allows it to be probed with a weak microwave tone. The incident noise power spectral density $P$ can also be measured. Standard four-probe transport measurement is omitted from the schematic for clarity. For details of the measurement setup, see Supplementary Figs. \ref{['fig:fig_fridge']}-\ref{['fig:fig_fridge_dc']}. Scanning electron micrograph of the Al islands is shown in the inset. b, Measured resistance $R$ versus cryostat temperature $T_\mathrm{cryo}$ at different top-gate voltages (indicated by colors). c, Measured radiation $P$ as a function of $T_\mathrm{cryo}$ for selected top-gate voltages. The most excess radiation is observed at a top-gate voltage of $-8.484~\mathrm{V}$, shown in panels b and c in salmon color. Solid lines are a model for an effective saturation radiation, $P_\mathrm{sat}$, determined via best fit to each curve. The values of $P_\mathrm{sat}/k_B$ for each top-gate voltage are $-7.88~\mathrm{V}$: $49~\mathrm{mK}$; $-8.38~\mathrm{V}$: $76~\mathrm{mK}$; $-8.484~\mathrm{V}$: $89~\mathrm{mK}$; $-8.88~\mathrm{V}$: $37~\mathrm{mK}$.
• Figure 2: a, Measured resistance $R$ as a function of top-gate voltage $V_\mathrm{g}$ for different cryostat temperatures (indicated by colors). Resistance curves cross at the critical point with resistance $\approx 60~\mathrm{k \Omega}$ (indicated). b, Excess noise $\Delta P$ versus $V_\mathrm{g}$ for different cryostat temperatures. $\Delta P$ is defined as the measured noise power spectral density $P$ minus the value measured deep in the superconducting phase for each trace; it represents the emitted radiation in excess of thermal equilibrium. Large excess noise is observed at low temperature (darkest blue trace). Curves at $0.1~\mathrm{K}$ and $1~\mathrm{K}$ are shifted down for clarity by $20$ and $40~\mathrm{mK}$, respectively. c, Reflection coefficient $|S_{11}|^2$ (left axis) and $\Delta P$ at $10~\mathrm{mK}$ from panel b (right axis) versus $V_\mathrm{g}$. Excess noise is strongly negatively correlated with $|S_{11}|^2$.
• Figure 3: a, Parametric plot of noise power spectral density $P$ vs microwave reflection $|S_{11}|^2$, both measured as a function of gate voltage. Power of microwave drive tone (at $547~\mathrm{MHz}$) at room temperature is indicated in legend; there is nominally $91~\mathrm{dB}$ of attenuation between room temperature and the sample. A linear fit is performed to each curve to extract $T_\mathrm{s}$ and $\alpha (T_\mathrm{s} - T_\mathrm{cryo})$ according to Eq. (\ref{['eq:beamsplitter']}). b, Slopes vs intercepts extracted from the fits shown in panel a. A subsequent linear fit (black) yields the net loss $\alpha = 64.5~\mathrm{dB}$, and the cryostat temperature $T_\mathrm{cryo} = 50~\mathrm{mK}$. The fit from the independent open circuit calibration ('oc', gray) of $63.9~\mathrm{dB}$ is also shown. Uncertainty is less than the marker size. c, Measured resistance as a function of cryostat temperature is shown for the trace at $-8.38~\mathrm{V}$ from Fig. \ref{['fig:fig1']}b, displaying anomalous metallic resistance saturation. $T_\mathrm{s}$ as determined from Fig. \ref{['eq:beamsplitter']} using $\alpha$, $T_\mathrm{cryo}$ from the fits is marked on the resistance curve. Resistance saturation begins approximately at this temperature.
• Figure 4: a, Sample resistance as a function of cryostat temperature at different gate voltages for various drive tone powers at $200~\mathrm{MHz}$. Shown are superconducting/weakly anonomalous-metallic (blue, $V_g=-8.292~\mathrm{V}$), quantum-critical (red, $V_g=-8.552~\mathrm{V}$), and insulating (green, $V_g=-8.632~\mathrm{V}$ and purple, $V_g=-8.664~\mathrm{V}$). Resistance saturation is more pronounced on the superconducting side of the transition compared to the insulating side. b, Sample temperature $T_\mathrm{s}$ versus drive power $P_\mathrm{drive}$ for different gate voltages. Colors indicate gate voltage and shades indicate powers, matching a. Heating by $P_\mathrm{drive}$ is more pronounced on the superconducting side of the transition compared to the insulating side.
• Figure 5: a, Circuit schematic of lossy transmission-line model describing our device. b, c Real and imaginary parts of $S_{11}$ above the superconducting transition temperature, both measured data and results of fitting to our model. At the most negative gate voltages a disagreement of 5 percent is indicated on both plots. d, e Real and imaginary parts of $S_{11}$ at base temperature of the cryostat, both measured data and results of fitting to our model. f, Fit inductance using our circuit model (dataset CM) and from extracted critical currents (dataset CC). g Measured zero-bias resistance corresponding to the datasets used to fit inductance. Dataset CM refers to the data used in the circuit model, while dataset CC refers to the data used to extract critical currents. In f and g the data has been shifted along the x-axis by uniformly subtracting an amount $V^*$ corresponding to the location of the SIT in transport measurements for each dataset. Bands on fits in d, e, f represent propagated uncertainties, determined by varying normal-state parameters found in a, b over a 5% range, refitting at each parameter value, and taking the standard deviation of the results.