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..
We examine resonant activation from the zero-voltage state of a current-biased Josephson junction in the limit of low damping. In this limit, the quantum dynamics of a Josephson junction can be described by a master equation. Our finite temperature analysis includes transitions between any two levels in the potential well and accounts for escape near the top of the well. Our results show that energy level quantization in lightly damped junctions should be observable even for temperatures in the thermal regime, where only classical behavior is observed for junctions with moderate damping. Finally we discuss implications for ongoing experiments in quantum computation.
We employ a functional integral technique to calculate the macroscopic quantum tunneling rate at zero temperature of a current-biased Josephson junction weakly coupled to a resonator. We allow for the effects of environmental dissipation on both the junction and the resonator via baths of harmonic oscillators, and we consider both cases of weak and strong junction damping. The resonator can be, in principle, any quantum harmonic oscillator that couples to the junction bilinearly in either the coordinate or the velocity. The Josephson phase difference $\ensuremath{\phi}$ is the tunneling variable in this system, and the low-lying bound states in the junction's potential energy well are weakly metastable due to tunneling through a finite-width energy barrier. There has been some interest in using such biased junctions as qubits with the resonator providing multiqubit coupling. The main result of this work is that coupling to the resonator has a suppressive effect on the junction's tunneling: the stronger the coupling strength between the junction and resonator, the greater the reduction of the tunneling rate. Including damping to the junction also suppresses tunneling, a well-known result, but damping of the resonator actually reduces the magnitude of suppression; i.e., damping the resonator partially counteracts the suppression provided directly by the junction-resonator coupling. Details of the junction-resonator coupling yield interesting variations on this theme that may be useful for qubit design. For example, the coupling
Uint between a current-biased junction and an AlN dilatational resonator should depend on the angular frequency of the resonator ${\ensuremath{\omega}}_{R}$ in a power-law fashion [A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)], specifically ${U}_{\mathit{int}}\ensuremath{\propto}{({\ensuremath{\omega}}_{R})}^{n}$, where
n=1∕2. We find that for any value of
n in the range
0<n<1 the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for
n=1∕2, the tunneling rate is maximally suppressed for ${\ensuremath{\omega}}_{R}∕{\ensuremath{\omega}}_{J}\ensuremath{\approx}1$, where ${\ensuremath{\omega}}_{J}$ is the bias current-dependent plasma frequency of the junction.
We report experiments on energy level quantization in Nb/AlOx/Nb Josephson junctions while measurements on SQUIDs are in progress. Data on the presence of energy level quantization are shown at temperatures above the classical-quantum regime crossover temperature. This has been possible by extending the measurements of the rate of the escape from the zero-voltage state at higher sweep bias rate in order to induce non-stationary conditions in the energy potential describing the junction dynamics. Data are compatible with extremely low "effective" dissipation in our system. Theoretical predictions on quantum effects related to the presence of non-stationary conditions in the potential associated to a rf SQUID are reported in view of new possible experiments.