Resultados respecto al objetivo general y objetivos específicos

It is convenient to consider a thick superconducting ring incorporating a weak link constriction as an inductor Λ enclosing a weak link capacitor C, as shown in Figure 7.2(a). In this model it is assumed that charge can move through the weak link in integer units of q=2e and flux can move across the weak link in integer units of Φ0, both processes being quantum mechanical in nature. For this weak link ring treated as a single macroscopic quantum

object, the canonically conjugate variables are the magnetic flux Φ threading through the ring and effectively the charge at the weak link11. Thus:

Figure 7.2 (a) Idealised representation of a weak link ring, (b) Leaky inductor/capacitor circuit model of a weak link ring.

[8]

which implies the uncertainty relation:

[9]

The role of these conjugate variables can best be explained by example. Let us again consider the thick superconducting ring. Here, the flux is extremely well defined and localised at discrete values of nΦ0. To see what has happened to the charge we can imagine making a cut across the ring of infinitely

narrow section (Figure 7.2(b)) such that C→∞. It follows that charge on either side of this cut must be ill-defined since quantum-mechanical pair transfer processes (0, ± q, ± 2q…± Nq) can occur across the cut up to extremely high order in N. This is equivalent to saying that macroscopic screening supercurrents can flow in the ring. Thus, from a quantum-mechanical viewpoint, as implied by [9], precise quantisation of flux in units of

Φ0=h/2e will only be observed in circumstances where charge can be completely delocalised around the ring; that is, for the case of the thick

superconducting ring.

Rigid quantisation of flux cannot be maintained when a weak link constriction is incorporated in the ring since, by definition, this constriction is made small enough in section to allow quantum-mechanical coupling between different flux states of the ring. For a relatively large cross section constriction we can arrange for nearest neighbour coupling [nΦ0→(n±1)Φ0] to be dominant. If we assume that the matrix element for this nearest neighbour coupling is

where Ω/4π is the quantum-mechanical frequency for the transfer of Φ0 bundles of flux across the weak link, then, for small Ω, we can write a

matrix equation for all such nearest-neighbour couplings of the form:

[10]

which yields the Φx-dependent ground-state energy of the weak link ring.

Figure 7.3 (a) Ground-state energy band E versus Φx in the flux mode shown for (b) Screening current

In Figure 7.3(a) we show the ground-state energy, periodic in Φ0, as a function of Φx, computed using a small value of Ω As can be

seen, the introduction of a weak link into the ring creates a splitting at the crossing points of the energy parabolas in Figure 7.1. It can also be seen that for small Ω coupling between adjacent flux states is only important close to the maxima in E versus Φx, at which points an amplitude superposition (e.g.

) exists between these macroscopically different quantum-mechanical states of the ring. In any one of the almost parabolic sections between these maxima the flux state of the weak link ring is rather well defined and localised about a particular value of nΦ0. As implied by [9], this

requires that the weak link constriction is large enough in cross-section to accommodate a macroscopic screening current; that is, quantum-mechanical pair-charge transfer processes ±Nq can take place through the weak link up to very high order in N. This is demonstrated very clearly in Figure 7.3(b), where the screening super-current is plotted as a function of the external applied flux Φx, as calculated from the ground-state

energy E(Φx) of Figure 7.3(a). The screening current response Is(Φx), shown in Figure 7.3(b), is necessarily the sum of harmonics12 such that:

[11]

In terms of the uncertainty relation [9], if the flux threading the ring inductor is relatively well defined about discrete values of nΦ0, the charge on the weak

link capacitor must be a superposition of a large number of integer pair-charge states. This is clearly not in the limit of weak link behaviour, as defined by Josephson8_{, where only nearest-neighbour superpositions are considered. In the ‘flux mode’ limit (small}

couplings) we can therefore treat the charge Q as continuous (i.e. no longer discrete) and write a wave function in terms of an angular displacement

θ=2πQ/q as:

[12]

which, from [1], yields for the ground-state energy a Schrödinger equation of the form:

[13]

We note, first, that [13] is not the Schrödinger equation, based on the Josephson definition of weak link behaviour, which has been used previously to calculate the properties of weak link rings operating in the flux mode regime10, and, second, that [13] contains just one cosine in the potential.

lator modes of the weak link ring. If the effective capacitance of the weak link is C then these oscillator modes are created at a frequency:

[14]

Thus, neglecting the zero-point energy, we can introduce a new energy (from [6]):

[15]

If, in the flux-mode limit, flux can move in or out of the ring in units of Φ0 while at the same time the photon number can change from m to m′, with

coupling Cmm′, the matrix equation for all possible nearest-neighbour flux-state couplings now becomes:

[16]

where:

[17]

λ is a dimensionless parameter, and the combinatorial factors arise because photons are identical bosons.

The energies calculated from [16] form a set of bands in Φx space, with periodicity Φ0 and band number κ. It can be seen that n and m are no longer

good quantum numbers and are replaced as such by the band number κ. The ground state and first two excited state energy bands (Eκ(Φx) versus Φx,

κ=1, 2 and 3) are shown in Figure 7.4 for a choice of small Ω and ωc and λ=0.5. The effect of the ring-

oscillator mode photons is to create ever more complex band patterns as the band number κ is increased. Again, the expectation value of the macroscopic screening super-current flowing around the ring is just:

[18]

and the ring magnetic susceptibility for a particular band is given by: