Phase qubit

In quantum computing, and more specifically in superconducting quantum computing, the phase qubit is a superconducting device based on the superconductor–insulator–superconductor (SIS) Josephson junction,[1] designed to operate as a quantum bit, or qubit.

The major distinction among the three is the ratio of Josephson energy vs charging energy[3] (the necessary energy for one Cooper pair to charge the total capacitance in the circuit): A phase qubit is a current-biased Josephson junction, operated in the zero voltage state with a non-zero current bias.

A Josephson junction is a tunnel junction,[6] made of two pieces of superconducting metal separated by a very thin insulating barrier, about 1 nm in thickness.

[7] The difference in the complex phases of the two superconducting wavefunctions is the most important dynamic variable for the Josephson junction, and is called the phase difference

The Josephson equation[1] relates the superconducting current (usually called the supercurrent)

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

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 set of three parallel circuit elements is biased by an external current source

[9] Solving the circuit equations yields a single dynamic equation for the phase, The terms on the left side are identical to those of a particle with coordinate (location)

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.

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 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.

The current through the junction is related to this phase value by If we consider small variations

(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

The nonlinear inductor represents the response of the Josephson junction to changes in bias current.

This corresponds to the oscillation frequency of the phase particle in the bottom of one of the minima of the washboard potential.

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.

At very low temperatures, much less than 1 K (achievable using a cryogenic system known as a dilution refrigerator), with a sufficiently high resistance and small capacitance Josephson junction, quantum energy levels [11] become detectable in the local minima of the washboard potential.

Clear resonances at certain frequencies were observed, which corresponded well with the quantum transition energies obtained by solving the Schrödinger equation[12] for the local minimum in the washboard potential.

Classically only a single resonance is expected, centered at the plasma frequency

Quantum mechanically, the potential minimum in the washboard potential can accommodate several quantized energy levels, with the lowest (ground to first excited state) transition at an energy

, but the higher energy transitions (first to second excited state, second to third excited state) shifted somewhat below this due to the non-harmonic nature of the trapping potential minimum, whose resonance frequency falls as the energy increases in the minimum.

Observing multiple, discrete levels in this fashion is extremely strong evidence that the superconducting device is behaving quantum mechanically, rather than classically.

The phase qubit uses the lowest two energy levels in the local minimum; the ground state

The slope in the washboard potential is set by the bias current

This changes the energy difference between the ground and first excited states.