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Josephson effect

The Josephson effect is named after British physicist Brian David Josephson who predicted its existence in 1962. We see the Josephson Effect through a flow of electric current as electron pairs, called Cooper pairs, between two superconducting materials that are separated by an extremely thin insulator. This arrangement is called a Josephson junction. Their properties are exploited in SQUIDs which allow the measurement of tiny magnetic fields. It is also speculated that Josephson junctions may allow the realisation of qubits, the key elements of an eventual quantum computer.

The key here is the fact that the two superconductors act to preserve their long-range order across the insulating barrier. Rapid alternating currents occur within the insulator when a steady voltage is applied across the superconductors. The current flow is known as the Josephson current and the quantum tunneling of the insulator by the Cooper pairs is the Josephson effect. (related physics on quantum tunneling: see also Casimir Effect.)

The phase of an electron wave function in one superconductor is relative to the fixed phase relationship with the other superconductor. This relationship between the two superconducting fields is called phase coherence and is the primary enabler of the Josephson Effect.

The basic equations coverning the dynamics of the Josephson effect are

$U(t) = \frac{h}{2 e} \frac{1}{2 \pi} \frac{\partial \phi}{\partial t}, \ \ \ I(t) = I_c \sin (\phi (t))$

where U(t) and I(t) are the voltage and current across the Josephson junction, $φ(t)$ is the phase difference between the wave functions in the two superconductors comprising the junction, and $I c$ is a constant, the critical current of the junction. The physical constant, $\frac{h}{2 e}$ is the magnetic flux quantum, the inverse of which is the Josephson constant. Three main effects are easily derived from these relations:

1. The DC Josephson effect: With no applied voltage across the insulator, the phase is constant and there may flow a direct current due to tunneling. This DC Josephson current is proportional the sine of the phase difference across the insulator and may take values between $- I c$ and $I c$ .

2. The AC Josephson effect: With a fixed voltage, $U D C$ across the junctions, the phase will vary linear with time and the current will be an AC current with amplitude $I c$ and frequency $\frac{2 e}{h} U_{DC}$ . This means, a Josephson junction can act as a perfect voltage-to-frequency converter.

3. The inverse AC Josephson effect: If the phase takes the form $φ(t) = φ 0 + n ω t + a sin(ω t)$ , the voltage and current will be

$U(t) = \frac{\hbar}{2 e} \omega ( n + a \cos( \omega t) ), \ \ \ I(t) = I_c \sum_{m = -\infty}^{\infty} J_n (a) \sin (\phi_0 + (n + m) \omega t)$

The DC components will then be

$U_{DC} = n \frac{\hbar}{2 e} \omega, \ \ \ I(t) = I_c J_{-n} (a) \sin \phi_0$

Hence, for distinct DC voltages, the junction may carry a DC current and the junction acts like a perfect frequency-to-voltage converter. This effect is used for the representation and maintenance of the SI voltage, where an alternating phase is induced by radiation of microwaves with a well-known frequency onto the Josephson junction.

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Josephson effect