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Physical review. B, Condensed matter

Small–radiation-amplitude dynamical voltage model of an irradiated, externally unbiased Josephson tunnel junction

Robert T. McAdory

Published May 1, 1988• 0 citations

Abstract

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.

DOI: 10.1103/physrevb.37.7368

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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.

We report an experimental and theoretical study of low-voltage resistive states observed in small tunnel Josephson junctions subject to microwave radiation. The observed features result from Shapiro steps in the current–voltage characteristics and appear when both thermal fluctuations and high frequency dissipation are strong. Without microwave radiation, Josephson junctions have a phase diffusion supercurrent branch characterized by a finite small resistance and hysteretic switching to a higher voltage range under these conditions. When microwave radiation is applied, three different types of resistive states are observed in the current-voltage characteristics. First, a phase diffusion branch steadily evolves and its maximum voltage Vm increases with the microwave power. Another interesting observed feature is a zero-crossing resistive state characterized by a negative resistance. Finally, we find that the low-voltage resistive state can split into numerous hysteretic fine branches resembling incoherent Shapiro-like steps. The appearance of a particular resistive state depends on the interrelations among the Josephson energy EJ, the energy kBT of thermal fluctuations, and the microwave frequency ω. A theoretical analysis based on incoherent multi-photon absorption by a junction biased in the Josephson phase diffusion regime is in good agreement with the experimental observations.

Relaxation oscillations at subharmonics of the Josephson frequency may occur in a resistively shunted superconducting tunnel junction, where the shunt loop has a large inductance. We have studied the variation in the relaxation period with the shunt inductance and resistance, the Josephson critical current, and the bias current. Steps in the current-voltage curve were noted at bias points where the subharmonic number changed. Radiation powers of the order of a nW were coupled out of small, single, tunnel junctions at 10 GHz. Linewidths scaled with the frequency square as compared with those of the Josephson mode oscillations. Wide signals centered around harmonics of the relaxation frequency were due to amplified noise. Injection locking to a fraction of the frequency of a relatively strong signal occurred, while weaker signals were amplified. Gains up to 15 dB were achieved within a fairly large bandwidth, but the noise temperature was high. The importance of parametrically up-converted noise from low idler frequencies is emphasized.

We consider quantum fluctuations in a current biased overdamped Josephson junction in the regime when the bias current is larger than the critical current. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. Quantum effects are important when the resistor temperature is low compared to the Josephson frequency, even in the case when noise is measured at frequencies lower than the temperature. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and calculate the widths of the Josephson radiation lines.

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.

Quantum fluctuations in a current-biased Josephson junction, described in terms of the RCSJ model, are considered. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and to calculate the widths of the Josephson radiation lines.