Another way to understand the behavior of a Josephson junction in the zero-voltage state is to consider the SIS tunnel junction as a nonlinear inductor. When the phase is trapped in one of the minima, the phase value is limited to a small range about the phase value at the potential minimum, which we will call . The current through the junction is related to this phase value by
- .
If we consider small variations in the phase about the minimum (small enough to maintain the junction in the zero voltage state), then the current will vary by
These variations in the phase give rise to a voltage through the ac Josephson relation,
This last relation is the defining equation for an inductor with inductance
This inductance depends on the value of phase at the minimum in the washboard potential, so the inductance value can be controlled by changing the bias current . For zero bias current, the inductance reaches its minimum value,
As the bias current increases, the inductance increases. When the bias current is very close (but less than) the critical current, the value of the phase is very close to, as seen by the dc Josephson relation, above. This means that the inductance value becomes very large, diverging as reaches the critical current .
The nonlinear inductor represents the response of the Josephson junction to changes in bias current. When the parallel capacitance from the device geometry is included, in parallel with the inductor, this forms a nonlinear resonator, with resonance frequency
- ,
which is known as the plasma frequency of the junction. This corresponds to the oscillation frequency of the phase particle in the bottom of one of the minima of the washboard potential.
For bias currents very near the critical current, the phase value in the washboard minimum is
- ,
and the plasma frequency is then
clearly showing that the plasma frequency approaches zero as the bias current approaches the critical current.
The simple tunability of the current-biased Josephson junction in its zero voltage state is one of the key advantages the phase qubit has over some other qubit implementations, although it also limits the performance of this device, as fluctuations in current generate fluctuations in the plasma frequency, which causes dephasing of the quantum states.
Read more about this topic: Phase Qubit