Phase Qubit - The McCumber-Stewart Model

The McCumber-Stewart Model

The ac and dc Josephson relations control the behavior of the Josephson junction itself. The geometry of the Josephson junction, two plates of superconducting metal separated by a thin tunnel barrier, is that of a parallel plate capacitor, so in addition to the Josephson element the device includes a parallel capacitance . The external circuit is usually simply modeled as a resistor in parallel with the Josephson element. The set of three parallel circuit elements is biased by an external current source, thus the current-biased Josephson junction. Solving the circuit equations yields a single dynamic equation for the phase,

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The terms on the left side are identical to those of a particle with coordinate (location), with mass proportional to the capacitance, and with friction inversely proportional to the resistance . The particle moves in a conservative force field given by the term on the right, which corresponds to the particle interacting with a potential energy given by

This is the washboard potential, so-called because it has an overall linear dependence, modulated by the washboard modulation .

The zero voltage state describes one of the two distinct dynamic behaviors displayed by the phase particle, and corresponds to when the particle is trapped in one of the local minima in the washboard potential. These minima exist for bias currents, i.e. for currents below the critical current. With the phase particle trapped in a minimum, it has zero average velocity and therefore zero average voltage. A Josephson junction will allow currents up to to pass through without any voltage; this corresponds to the superconducting branch of the Josephson junction's current-voltage characteristic.

The voltage state is the other dynamic behavior displayed by a Josephson junction, and corresponds to the phase particle free-running down the slope of the potential, with a non-zero average velocity and therefore non-zero voltage. This behavior always occurs for currents above the critical current, i.e. for, and for large resistances also occurs for currents somewhat below the critical current. This state corresponds to the voltage branch of the Josephson junction current-voltage characteristic. For large resistance junctions the zero-voltage and voltage branches overlap for some range of currents below the critical current, so the device behavior is hysteretic.