Phase qubit

 ( Quantum Information Science ) 

$I\; =\; I\_0\; \backslash sin\; \backslash delta$ (Josephson current-phase relationship)

Here $I\_0$ is the critical current of the tunnel junction, determined by the area and thickness of the tunnel barrier in the junction, and by the properties of the superconductors on either side of the barrier. For a junction with identical superconductors on either side of the barrier, the critical current is related to the superconducting gap $\backslash Delta$ and the normal state resistance $R\_n$ of the tunnel junction by the Ambegaokar–Baratoff formula

$I\_0\; =\; \backslash frac\{\backslash pi\; \backslash Delta\}\{2\; e\; R\_n\}$ (Ambegaokar–Baratoff formula)

The Gor'kov phase evolution equation gives the rate of change of the phase (the "velocity" of the phase) as a linear function of the voltage $V$ as

$V\; =\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash frac\{d\; \backslash delta\}\{d\; t\}$ (Gor'kov-Josephson phase evolution equation)

This equation is a generalization of the Schrödinger equation for the phase of the BCS theory. The generalization was carried out by Gor'kov in 1958.

$\backslash frac\{\backslash hbar\; C\}\{2\; e\}\; \backslash ,\; \backslash frac\{d^2\; \backslash delta\}\{dt^2\}\; +\; \backslash frac\{\backslash hbar\}\{2\; e\; R\}\; \backslash frac\{d\; \backslash delta\}\{dt\}\; =\; I\; -\; I\_0\; \backslash sin\; \backslash delta$.

The terms on the left side are identical to those of a particle with coordinate (location) $\backslash delta$, with mass proportional to the capacitance $C$, and with friction inversely proportional to the resistance $R$. 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 $U(\backslash delta)$ given by

$U(\backslash delta)\; =\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash left\; (\; -I\_0\; \backslash cos\; \backslash delta\; -\; I\; \backslash ,\; \backslash delta\; \backslash right\; )$.

This is the "washboard potential", so-called because it has an overall linear dependence $-I\; \backslash ,\; \backslash delta$, modulated by the washboard modulation $-I\_0\; \backslash ,\; \backslash cos\; \backslash delta$.

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 $I\_0$ 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 $I$ above the critical current, i.e. for $\backslash left|\; I\; \backslash right|\; >\; I\_0$, and for large resistances $R$ 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.

$I\; =\; I\_0\; \backslash sin\; \backslash delta\_0$.

If we consider small variations $\backslash Delta\; \backslash delta$ in the phase about the minimum $\backslash delta\_0$ (small enough to maintain the junction in the zero voltage state), then the current will vary by

$\backslash Delta\; I\; =\; \backslash left\; (I\_0\; \backslash cos\; \backslash delta\_0\backslash right)\; \backslash Delta\; \backslash delta$.

These variations in the phase give rise to a voltage through the ac Josephson relation,

$\backslash Delta\; V\; =\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash frac\{d\; \backslash Delta\; \backslash delta\}\{dt\}\; =\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash frac\{1\}\{I\_0\; \backslash cos\; \backslash delta\_0\}\; \backslash frac\{d\; \backslash Delta\; I\}\{dt\}\; =\; L\; \backslash frac\{d\; \backslash Delta\; I\}\{dt\}$

This last relation is the defining equation for an inductor with inductance

$L\; =\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash frac\{1\}\{I\_0\; \backslash cos\; \backslash delta\_0\}$.

This inductance depends on the value of phase $\backslash delta\_0$ at the minimum in the washboard potential, so the inductance value can be controlled by changing the bias current $I$. For zero bias current, the inductance reaches its minimum value,

$L\_\{\backslash rm\; min\}\; =\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash frac\{1\}\{I\_0\}\; =\; \backslash frac\{\backslash hbar\; R\_n\}\{\backslash pi\; \backslash Delta\}$.

As the bias current increases, the inductance increases. When the bias current is very close (but less than) the critical current $I\_0$, the value of the phase $\backslash delta\_0$ is very close to $\backslash pi/2$, as seen by the dc Josephson relation, above. This means that the inductance value $L$ becomes very large, diverging as $I$ reaches the critical current $I\_0$.

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 $LC$ resonator, with resonance frequency

$\backslash omega\_p\; =\; \backslash frac\{1\}\{\backslash sqrt\{L\; C\}\}\; =\; \backslash sqrt\{\backslash frac\{2\; e\; I\_0\; \backslash cos\; \backslash delta\_0\}\{\backslash hbar\; C\}\}$,

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

$\backslash cos\backslash delta\_0\; =\; \backslash sqrt\{1-(I/I\_0)^2\}$,

and the plasma frequency is then

$\backslash omega\_p\; \backslash approx\; \backslash sqrt\{\backslash frac\{2\; e\; I\_0\}\{\backslash hbar\; C\}\}\; \backslash left\; ^\{1/4\}$,

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.