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arXiv:cond-mat/9903307 · IEEE Transactions on Applied Superconductivity

Modelling the nonlinear high-frequency response of a short Josephson junction under two-frequency irradiation

Anton V. Velichko, Adrian Porch

Published Jun 1, 1999• 2 citations

Abstract

The nonlinear response of a short Josephson Junction (JJ), being irradiated simultaneously with two high-frequency signals, has been studied in the framework of the nonlinear Resistively-Shunted Junction (RSJ) Model. One of the signals, hereafter referred to as "probe signal", has a small amplitude I/sub pr/<I/sub c/ (I/sub c/ is the critical current of the JJ) and frequency f/sub pr/, and is used to monitor the response of the junction to the other high-power signal with amplitude I/sub pm/ and frequency f/sub pm/, hereafter referred to as "pump signal". Varying the frequency ratio f(pm)/f/sub pr/ from 0.5 to 100, and the current amplitude of the probe signal from 0.01 to 0.9 of I/sub c/, we found that the dependence of the junction impedance at the frequency f/sub pr/, Z/sub s//sup fpr/, versus I/sub pm/ preserves its general features, independent of f/sub pm//f/sub pr/ and I/sub pr//I/sub c/ values. At the same time, some particular features, like negative values of Re(Z/sub s//sup fpr/) and "fine" structure of the steps in Z/sub s//sup fpr/ (I/sub pm/ are observed for f/sub pm//f/sub pr/<1 and for particular values of I/sub pr//I/sub c/. In general, the behavior of Z/sub s//sup pr/(I/sub pm/) is rather different from that predicted by the nonlinear RSJ model for a short JJ in the regime of single-frequency irradiation, when one and the same signal plays the roles of the pump and the probe signals simultaneously. Possible applications of the model are briefly discussed.

DOI: 10.1109/77.784889

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The nonlinear response of a short Josephson Junction (JJ), being irradiated simultaneously with two high-frequency signals, has been studied in the framework of the nonlinear Resistively-Shunted Junction (RSJ) Model. One of the signals, hereafter referred to as "probe signal", has a small amplitude I/sub pr/<I/sub c/ (I/sub c/ is the critical current of the JJ) and frequency f/sub pr/, and is used to monitor the response of the junction to the other high-power signal with amplitude I/sub pm/ and frequency f/sub pm/, hereafter referred to as "pump signal". Varying the frequency ratio f(pm)/f/sub pr/ from 0.5 to 100, and the current amplitude of the probe signal from 0.01 to 0.9 of I/sub c/, we found that the dependence of the junction impedance at the frequency f/sub pr/, Z/sub s//sup fpr/, versus I/sub pm/ preserves its general features, independent of f/sub pm//f/sub pr/ and I/sub pr//I/sub c/ values. At the same time, some particular features, like negative values of Re(Z/sub s//sup fpr/) and "fine" structure of the steps in Z/sub s//sup fpr/ (I/sub pm/ are observed for f/sub pm//f/sub pr/<1 and for particular values of I/sub pr//I/sub c/. In general, the behavior of Z/sub s//sup pr/(I/sub pm/) is rather different from that predicted by the nonlinear RSJ model for a short JJ in the regime of single-frequency irradiation, when one and the same signal plays the roles of the pump and the probe signals simultaneously. Possible applications of the model are briefly discussed.

For a current-driven Josephson junction shunted by an Ohmic resistance, the dc voltage response and impedance to external high-frequency currents is calculated with a second-order perturbation method based on the unperturbed solution for the time evolution of the voltage as given by Aslamazov and Larkin. The response is proportional to signal power and has three characteristic-frequency regions depending on whether the internal self-generated Josephson frequency, ω0, is larger than, equal to, or smaller than the signal frequency, ω. For ω0< or >ω the response could be expressed by easily observable parameters of the dc voltage-current characteristics. This permits comparison of the predictions with experimental results obtained on point-contact junctions whose dc characteristics were only approximately represented by the simple model chosen for analysis. For ω0≪ω the response is found to be proportional to (i) the slope of the dc characteristic, (ii) the inverse square of the applied frequency, (iii) the inverse battery current in terms of the critical current, and (iv) the voltage amplitude of the inherent Josephson oscillations. For ω0≫ω ``classical'' detection proportional to curvature is obtained. For ω0∼ω resonance detection occurs, which depends to a considerable degree on fluctuations of the average contact voltage. These predictions are in reasonable agreement with experiments in which point-contact junctions were illuminated with klystron radiation at 10 and 90 GHz. Evidence is presented that the voltage fluctuations limiting the resonance response are the result of noise generated in the junction. Comparison of the noise-equivalent power expected from the simple model with measured results suggests that parasitic elements, such as shunt capacity, are not negligible at 90 GHz.

We consider a current biased, resistively shunted josephson junction irradiated at two frequencies. The perturbation technique introduced by Aslamasov and Larkin is used in the calculations. Both signals are treated as perturbations. The second order calculation yields the size of the mixing steps atV_{\pm}=h(\omega_{1}\pm\omega_{2})/2e. As in the case of a single frequency we show that subharmonic mixing steps are absent. The amplitude of the voltage oscillation at the difference and sum frequencies is shown to be non-zero at all voltages. We calculate the microwave resistance for one frequency ω2to third order in the perturbation. There are negative resistance regions nearV\pm(as well as nearV_{2}= h\omega_{2}/2e). Near V, the negative resistance region appears for bias voltage V just aboveV_{-}, while near V the region appears for V just belowV_{+}. This means that when an incident frequency mixes with a cavity mode the mixing step atV_{-}will be inverted compared to the cavity step itself.

We have studied the responses of high temperature superconducting (HTS) Josephson junctions to monochromatic and polychromatic electromagnetic irradiations from 50 GHz to 4.25 THz. The highest operating frequency and the maximum value of normalized response are shown to be proportional to the I/sub C/R/sub N/ product of the junction, where I/sub C/ and R/sub N/ are the critical current and normal resistance of the junction, respectively. Using a junction with I/sub C/R/sub N/ of 330 /spl mu/V, direct detections of monochromatic irradiations from 76 GHz to 3.1 THz have been obtained, indicating that fast detectors can be realized to cover around 50 : 1 frequency band by operating only one junction at one temperature. Also reported are the superposition of polychromatic signals, as well as some possible applications of the system.

Experiments are reported in which two millimeter-wave signals incident on point-contact Josephson junctions produced changes in the junction dc voltage versus current characteristic and an intermediate frequency output whose amplitude depended sensitively on both junction bias and applied power. Equations are derived, based on Josephson's phenomenological equations, for the Josephson current in a junction exposed to two applied rf signals. When the applied signals differ appreciably in frequency, additional constant-voltage steps in the $V\ensuremath{-}I$ curve are predicted which are spaced at the difference frequency. These steps have been observed in experiments employing sources at 64 and 72 Gc/sec. Results of mixing experiments utilizing two sources nearly equal in frequency are reported at 23 and at 72 Gc/sec. In this case the two waves beat together and are equivalent in their effect to a single signal amplitude modulated at the difference frequency. Also explained on the same basis are experiments in which the third harmonic of a signal at 24 Gc/sec mixed with a signal at 72 Gc/sec. These results demonstrate the existence of the Josephson mixing mechanism as opposed to classical nonlinear mixing, and they show that it is operative at microwave and millimeter-wave frequencies over a wide range of power.

A theory is presented for the nonequilibrium voltage states of an irradiated Josephson junction shunted by an external resistor but with no external current or voltage biasing. This device, referred to as a free-running Josephson junction, is modeled in a small--radiation-amplitude, deterministic regime extending the previous work of Shenoy and Agarwal. The time-averaged induced voltage is treated as a dynamical variable, the external radiation is modeled as a current source, and the induced junction-radiation vector potential, with and without a mode structure, is treated to first order in the driving currents. A dynamical equation for the time-averaged induced voltage yields a (nonequilibrium) steady-state relation between the time-averaged induced voltage and the incident radiation amplitude valid for a wide range of voltages, including zero. Regions of bistability occur in the voltage--versus--incident-amplitude curves, some of which are dependent on the external resistor. The zero-voltage state breaks down, as the external radiation amplitude is increased, at a critical value of the incident-radiation amplitude inversely proportional to the external resistance.