This chapter reviews the basic features of the quantum inverse scattering method. Experimental realization of Bose–Einstein condensates in dilute atomic alkali gases has stimulated a diverse range of theoretical and experimental research activity. A particularly exciting possibility is that a pair of Bose–Einstein condensates may provide a model tunable system in which to observe macroscopic quantum tunnelling. The model Hamiltonian is also realizable in Josephson coupled superconducting metallic nanoparticles, which has applications in the implementation of solid-state quantum computers. The realization of nanotechnology as a viable industry is presenting a wealth of challenging problems in theoretical physics. At the nanoscale level, the quantum fluctuations are sufficiently large enough that the approximation is invalid. The experimental work of D. C. Ralph, C. T. Black and M. Tinkham on the discrete energy spectrum in small metallic aluminium grains generated interest in understanding the nature of superconducting correlations at the nanoscale level.