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An important idea in the development of SQUIDs is the idea of flux quantisation. Consider a superconducting ring with a magnetic field orientated as shown below in figure 3. The arrow-headed path indicates our integration contour. Since the `supercurrent` is contained within the outer sheaths this contour passes through a current-free region.

**Figure 3**

The wave function of the Cooper pair can be expressed as:

Where n is the pair density and the phase. Since the condensate is very coherent the phase factor can be taken to be constant over the material. Now we use the equation below, which is derived from quantum mechanical considerations:

where m^{*} is the mass of the Cooper pair.

Using this and the fact that the pairs have no momentum (since no current flows), we arrive at the expression;

The last term arises due to the fact that the Cooper pair wave function, integrated over a closed loop only differs in phase factors of 2n, where n is any integer (including zero).

However, the term on the left-hand side is just an expression for the magnetic flux passing through the ring. Therefore, we rearrange the equation to find

as seen previously. As well as providing confirmation of the BCS theory it also is used in the theory behind SQUIDs where the phase difference of Cooper pairs depends on this magnetic flux.

The Josephson junction is composed of two superconductors separated by a thin insulating layer. The junction is connected in a circuit with an external voltage applied across a resistor, the current, I, and the voltage drop, V, across the junction can be measured.

Cooper pairs have attributes of Bosons; many Cooper pairs
can occupy the same state. If the temperature is almost 0K, the majority of
Cooper pairs will be in the ground state and hence the wavefunction
_{1} is common to all the Cooper pairs in superconductor 1, and
_{2} is common to all the Cooper pairs in superconductor 2.

If H is the Hamiltonian operator then applying Schroedinger’s Equation to the system gives:

Where K is a tunneling constant, characteristic of the junction. If K where zero then there would be no tunneling and the above equations would just describe the lowest energy state in each isolated superconductor. The K term however allows the wavefunctions to tunnel through the barrier from one superconductor into the other.

We assume that _{1} is an approximate solution
with respect to H, so H is replaced by the energy of the state for each
superconductor.

Now if both superconductors were identical, the energies
U_{1}and U_{2} would be the same and they could be cancelled.
However an external voltage V is applied to the junction which causes a
difference in energy of qV between the two superconductors (q = 2e, e = charge
of electron). Taking the zero of the energy scale as halfway in between
gives:

(1) |

To proceed further we need to consider the physical meaning
of the wavefunction in this system. |(r)|^{2} gives the
probability of finding an electron pair at a particular point, r, in the
superconductor. But since there are very many electron pairs all with the same
wavefunction, |(r)|^{2} is proportional to the density of
electron pairs at that point r. However the electron pair density is in fact
the same for all r because the charge density is uniform. To see this we just
need to consider that if more electrons were concentrated at one point than
another they would repel each other and the charge density would be evened out.
So the wavefunction squared is proportional to the charge density, , in
the superconductor. This means that the wavefunctions _{1} and
_{2} can be written

; | (2) |

_{1} and _{2} are the respective
electron pair densities of superconductors 1 and 2. _{1} and
_{2} are the phases of the wavefunctions on either side of the
junction.

When (2) is substituted into (1), rearranging and equating the real and imaginary parts gives the following 4 equations which can be used to describe the Josephson effects.

(3) |

(4) |

(5) |

(6) |

Where = _{2} - _{1}

The density of the tunneling current passing through the
junction is equal to d_{1}/dt or -d_{2}/dt. This
gives the current density as

The value of can be found from (5) and (6)

.

Integrating with respect to t gives

(7) |

Where _{0} is (t) at time t=0 and is
dependent on the original phase difference between the electron pairs on either
side of the junction.

For a DC voltage V, (7) gives:

(8) |

There is an external voltage U_{ext} applied to the circuit but because the Josephson junction is connected in series with a resistor it does not necessarily follow that there will be a voltage drop across the junction. Indeed if the external voltage is low enough then there will be no voltage drop across the junction.

If there is no voltage drop across the junction (V=0 in (8)) then the current density is given by

(9) |

The current density varies from **J**_{0} to
-**J**_{0}depending on the value of _{0}. This current
would soon be stopped by the build up of charge on one side of the junction were
it not for the current from the external battery, which collects and replaces
charge on each side of the junction. As the external voltage is increased the
phase difference _{0} between the two sides of the junction is
modified and the current density increases until it reaches **J**_{0}
at which point the phase difference can no longer keep up and a voltage drop
occurs across the junction.

If there is a DC voltage drop V across the junction, then (8) gives

This is an alternating current with angular frequency
(q/)V. This is large for normal voltages and for a typical voltage drop
of about 1mV it gives a Josephson current oscillating at 3x10^{12} Hz,
i.e. oscillations in the infrared region of the spectrum.

A SQUID uses two Josephson junctions in parallel and relies on the interference of the currents from each junction. Figure 4 below shows the schematic diagram of a SQUID.

**Figure 4: Basic SQUID**

The total current J_{tot} will be the sum of the
currents through ‘a’ and ‘b’. The currents in each junction are J_{a}
and J_{b} and the respective phases are _{a} and
_{b}. Analysing the phase difference between X and Y, we note
that this difference is the same whether you go through junction ‘a’ or
junction ‘b’.

Along the top path, the phase difference is:

.

And along the bottom path, it is:

.

These must be equal and when subtracted give the difference
in phases across each individual junction. This is just the line integral of
**A** around the whole loop:

.

The integral is equal to the magnetic flux through the loop and this gives:

.

Hence, the phase changes as the magnetic flux is varied. We
write the total current as the sum of J_{a} and J_{b}. To
simplify the formula, we set:

, .

From (9) we obtain the result:

Simple trigonometry gives:

.

We can quickly obtain the maximum current from the device for any given since the maximum value of the sine term is 1:

.

The maximum current occurs when

.

This is akin to flux quantization and the current takes on its maximum values when the above is true. This results in the following relation of the Josephson current to the applied magnetic field.

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