The quantised Josephson junction equation that underpins the behaviour of charge qubits and other tunnel devices is usually derived through cannonical quantisation of the classical macroscopic Josephson relations. However, this approach may neglect effects due to the fact that the charge qubit consists of a superconducting island of finite size connected to a large superconductor. We show that the well known quantised Josephson equation can be derived directly and simply from a microscopic many-body Hamiltonian. By choosing the appropriate strong coupling limit we produce a highly simplified Hamiltonian that nevertheless allows us to go beyond the mean field limit and predict further finite-size terms in addition to the basic equation.

We consider an array of l b Cooper Pair Boxes, each of which is coupled to a superconducting reservoir by a capacitive tunnel junction. We discuss two effects that probe not just the quantum nature of the islands, but also of the superconducting reservoir coupled to them. These are analogues to the well-known quantum optical effects 'superradiance,' and 'revival.' When revival is extended to multiple systems, we find that 'entanglement revival' can also be observed. In order to study the above effects, we utilise a highly simplified model for these systems in which all the singleelectron energy eigenvalues are set to be the same (the strong coupling limit), as are the charging energies of the Cooper Pair Boxes, allowing the whole system to be represented by two coupled quantum spins, one finite, which represents the array of boxes, and one representing the reservoir, which we consider in the limit of infinite size. Although this simplification is drastic, the model retains the main features necessary to capture the phenomena of interest. Given the progress in superconducting box experiments over recent years, it is possible that experiments to investigate both of these interesting quantum coherent phenomena could be performed in the forseeable future. § Current address:

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