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Physical Review

Quantized Phase Effect and Josephson Tunneling

Michael Martin Nieto

Published Mar 10, 1968• 27 citations
Publisher

Abstract

By using properly defined sine and cosine phase variables, a model is proposed which yields a rigorous quantum-mechanical derivation of Josephson tunneling with exact current eigenstates. The model predicts that for dc tunneling the coherent fluids on both sides of the junction must be in the same state, and the "phase is quantized." This prediction can in principle be investigated experimentally, and may afford a fundamental test of quantum mechanics.

DOI
10.1103/physrev.167.416

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By using properly defined sine and cosine phase variables, a model is proposed which yields a rigorous quantum-mechanical derivation of Josephson tunneling with exact current eigenstates. The model predicts that for dc tunneling the coherent fluids on both sides of the junction must be in the same state, and the "phase is quantized." This prediction can in principle be investigated experimentally, and may afford a fundamental test of quantum mechanics.
Experiments to investigate the quantum behavior of a macroscopic degree of freedom, namely the phase difference across a Josephson tunnel junction, are described. The experiments involve measurements of the escape rate of the junction from its zero voltage state. Low temperature measurements of the escape rate for junctions that are either nearly undamped or moderately damped agree very closely with predictions for macroscopic quantum tunneling, with no adjustable parameters. Microwave spectroscopy reveals quantized energy levels in the potential well of the junction in excellent agreement with quantum-mechanical calculations. The system can be regarded as a "macroscopic nucleus with wires."
A macroscopic variable, the phase difference across a Josephson junction, is shown to obey quantum mechanics by observing the phenomena of energy-level quantization and quantum tunneling.The relevant parameters of the junction were determined .!.!! .ill!:!, in the thermal regime from the dependence of the escape rate on bias current and from resonant activation in the presence of microwaves.The first observation of quantized energy levels for this system is reported.The positions of these energy levels are in quantitative agreement with the quantum mechanical predictions based on the measured junction parameters.A model is presented that predicts correctly the observed resonance shapes.The measured value of the escape rate out of the zero-voltage state for an underdamped (Q -30) junction was found to be independent of temperature at low temperatures.This low temperature escape rate was in excellent agreement with the prediction for macroscopic quantum tunneling, with no adjustable parameters.Jolw rJt..
In this paper, we apply quantum Hamilton mechanics to describe the dynamical motion of an electronic Cooper pair tunneling through a Josephson junction. Quantum Hamilton equations provide us a set of canonical equations q̇ = f(q, p) and ṗ = g(q, p) to model the tunneling dynamics of a Cooper pair. Instead of using the conventional probabilistic description, we solve complex quantum trajectory q(t) = qR+ qI·i from the Hamilton equations to demonstrate the tunneling dynamics on a geometrical phase plane. In order to control the dynamics of a cooper pair, we add a gate voltage parameter ngto the quantum Hamiltonian. By adjusting the magnitude of ng, we successfully control the tunneling dynamic of the Cooper pair such that the predicted current-voltage relation is in excellent agreement with the experimental measurements.
The formalism that we have previously developed for the phase difference between two quantized electromagnetic fields, which is intimately connected with the measurement process, is explored further, both theoretically and experimentally. We calculate the higher moments of the measured cosine and sine operators for certain two-mode Fock states \ensuremath{\Vert}n1{\mathit{n}}_{1}n1​,n2{\mathit{n}}_{2}n2​〉, and show how the measurement itself exerts a bias on the outcome. We find that the corresponding phase difference becomes uniform over the interval 0 to 2\ensuremath{\pi} only in the limit n1{\mathit{n}}_{1}n1​,n2{\mathit{n}}_{2}n2​\ensuremath{\rightarrow}\ensuremath{\infty}. On the other hand, the phase difference associated with the product of a coherent state \ensuremath{\Vert}v〉 with a Fock state can be random for large \ensuremath{\Vert}v\ensuremath{\Vert}, because of the availability of an infinite number of photons. Several of our theoretical predictions are compared with predictions based on the Susskind-Glogower and the Pegg-Barnett operators, and they are also tested by experiment. We find that the experimental results confirm our theory in every case, and this includes tests of the higher moments of the measured cosine operator.
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.