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Physical Review Letters

Theory of a Josephson Oscillator

Michael J. Stephen

Published Dec 9, 1968• 73 citations
Publisher

Abstract

A quantum mechanical theory of superconducting tunnel junctions including noise is developed. The theory is applied to the calculation of the frequency pulling, linewidth of the radiation, and voltage power spectrum in the ac Josephson effect.
DOI
10.1103/physrevlett.21.1629

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Deep theoretical understanding of the electrical response of Josephson junctions is indispensable regarding both recent discoveries of new kinds of superconductivity and technological advances such as superconducting quantum computers. Here, we study the microscopic theory of the dc current-biased I-V characteristics of Josephson tunnel junctions. We derive an analytical formula of the I-V characteristics of generic junctions. We identify subharmonics of the I-V characteristics and their underlying mechanism as the feedback effect of intrinsic ac currents generated by voltage pulses in the past. We apply our theory to analytically solve the Werthamer equation and describe various dc current-biased I-V characteristics as a function of softening of the superconducting gap. Strikingly, we identify voltage staircases of the I-V characteristics in a genuine Josephson junction without ac current bias or qubit dynamics. Our general analytical formalism opens new avenues for a microscopic understanding of I-V characteristics of Josephson junctions that have been limited to phenomenological models so far.
The full quantum theory of mixing has been widely developed by Tucker. From theoretical analysis it allows one to predict the optimal performances of a quasiparticle superconducting insulator-superconducting junction mixer from knowing the junction current-voltage characteristics. This theory, however, considers only quasiparticle processes and neglects pair effects. At frequencies close to the superconductor energy gap, the Josephson currents become unavoidable and are reputed to be the cause of strong instabilities and noise. We present here a theoretical calculation of the Josephson noise and losses, which is consistent with experimental results. The width of the Josephson noise peaks is discussed.
We consider the problem of a Josephson junction interacting with a quantized electromagnetic field contained in a microwave cavity. Inasmuch as the cavity has a quality factor QQQ, and an external current JJJ flows in and out of the junction, this is a problem in nonequilibrium statistical mechanics. We use a technique that eliminates adiabatically the radiation field directly from the total-density-matrix equation of motion. The resulting equation for the reduced density matrix is found to have a simple physical interpretation. The spectrum of the voltage fluctuations and the linewidth of the emitted radiation are then calculated for the case when the resonance cavity is at absolute zero.
A quantum-mechanical analysis of the Josephson phase escape in the presence of both dc and ac bias currents is presented. We find that the potential barrier for the escape of the phase is effectively suppressed as the resonant condition occurs, i.e., when the frequency $\ensuremath{\omega}$ of the ac bias matches the Josephson-junction energy-level separation. This effect manifests itself by a pronounced drop in the dependence of the switching current Is{I}_{s}Is​ on the power W of the applied microwave radiation and by a peculiar double-peak structure in the switching current distribution P(Is).{P(I}_{s}).P(Is​). The developed theory is in a good accord with an experiment which we also report in this paper. The obtained features can be used to characterize certain aspects of the quantum-mechanical behavior of the Josephson phase, such as the energy level quantization, the Rabi frequency of coherent oscillations, and the effect of damping.