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Proceedings of the IEEE

General energy relations for Josephson junctions

P. Russer

Published Jan 1, 1971• 38 citations
Publisher

Abstract

The Josephson junction acts as a nonlinear lossless inductor and can be applied for mixing and parametric amplification in the microwave region. Since an alternating current can be generated in the junction by an applied dc voltage, a dc power flow must be taken into consideration. Energy relations for the Josephson junction are derived which are similar to the Manley-Rowe equations but with an additional term for the dc power. These equations show the possibility of realizing dc-pumped parametric amplifiers.

DOI
10.1109/proc.1971.8133

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The Josephson junction acts as a nonlinear lossless inductor and can be applied for mixing and parametric amplification in the microwave region. Since an alternating current can be generated in the junction by an applied dc voltage, a dc power flow must be taken into consideration. Energy relations for the Josephson junction are derived which are similar to the Manley-Rowe equations but with an additional term for the dc power. These equations show the possibility of realizing dc-pumped parametric amplifiers.
A Josephson element is a nonlinear lossless element that is characterized by an internal frequency, ω0;, the autonomous frequency, proportional to the dc voltage across the element. Detection, mixing, generation, harmonic generation and mixing, and parametric amplification are possible using such an element. We present here general relations for the power flow at the frequencies of interest in such an element. These general power flow relations depend upon whether the autonomous frequency is phase-locked, either harmonically or subharmonically, to a frequency in the system or is unlocked. The results here generalize those given previously by Russer for the harmonically locked case.
Abstract. In this work, we derive the dynamics of the lossy DC pumped non-degenerate Josephson parametric amplifier (DCPJPA). The main element in a DCPJPA is the superconducting Josephson junction. The DC bias generates the AC Josephson current varying the nonlinear inductance of the junction. By this way the Josephson junction acts as the pump oscillator as well as the time varying reactance of the parametric amplifier. In quantum-limited amplification, losses and noise have an increased impact on the characteristics of an amplifier. We outline the classical model of the lossy DCPJPA and derive the available noise power spectral densities. A classical treatment is not capable of including properties like spontaneous emission which is mandatory in case of amplification at the quantum limit. Thus, we derive a quantum mechanical model of the lossy DCPJPA. Thermal losses are modeled by the quantum Langevin approach, by coupling the quantized system to a photon heat bath in thermodynamic equilibrium. The mode occupation in the bath follows the Bose-Einstein statistics. Based on the second quantization formalism, we derive the Heisenberg equations of motion of both resonator modes. We assume the dynamics of the system to follow the Markovian approximation, i.e. the system only depends on its actual state and is memory-free. We explicitly compute the time evolution of the contributions to the signal mode energy and give numeric examples based on different damping and coupling constants. Our analytic results show, that this model is capable of including thermal noise into the description of the DC pumped non-degenerate Josephson parametric amplifier.
In this contribution, we investigate different types of Josephson junction arrays that are embedded in coplanar microwave transmission lines, acting as traveling-wave parametric amplifiers. For the realization of traveling-wave parametric amplifiers, different Josephson junction array geometries have been proposed in the literature. Here, we investigate the amplifier’s Josephson nonlinearity along with the gain performance, following a general method to derive the nonlinear wave-mixing coefficients from the current-phase relation of different junction array topologies. These nonlinear coefficients are then incorporated into a corresponding Hamiltonian, which is used in order to solve the Heisenberg equations of motion of the photon number operator, determining the gain of the amplifier.
Due to the inherent nonlinearity in the Josephson junction (JJ) element, there is great potential to develop nonlinear parametric devices, such as amplifiers and mixers. In this paper, we investigate the time domain small-signal and large-signal analysis on the nonlinear flux-current characteristics of an array of lumped JJs. The procedure is similar to that applied on the nonlinear current-voltage characteristics of a semiconductor p-n junction diode, and results are presented in closed-form equations. The gain improvement due to parametric interaction of the pump with the signal is also discussed at the end of the paper.
Conventional negative resistance effects can be generated with a Josephson junction quite similar to that obtained with Varactor diodes, with the additional advantage that the internal, self-generated oscillations serve as a pump, i. e., only a dc source is needed instead of a high frequency pump source. However, another negative resistance effect typical for a Josephson junction is possible as has been predicted recently for the current controlled, highly damped junction model. The effect has recently been demonstrated experimentally at 9 GHz. Characteristically, the pump (Josephson) frequency is smaller than the frequency of the amplified signal and the pump mechanism is effective over a broad range of frequencies in contrast to the conventional parametric negative resistance effect where the pump frequency exceeds that of the signal and idler and where the frequency of operation is limited to a narrow range. The broad-band feature renders the amplifier operation uncritical to tuning and bias voltage fluctuations. The fact that the junction is the element of lowest impedance in the circuit alleviates to a degree the matching problem generally encountered with Josephson contacts. An assessment of the competitiveness of this effect in microwave technology is given.