Number-Phase Fluctuations in Isolated Superconductors

We improve the Bardeenn Cooper Schrieffer wave function with a fixed particle number so as to incorporate many-body correlations beyond the mean-field treatment. It is shown that the correlations lower the ground-state energy far more than Cooper-pair condensation in the weak-coupling region. Moreover, they naturally bring a superposition over the number of condensed particles. Thus, Cooper-pair condensation is special among the various bound-state formations of quantum mechanics in that number fluctuations are necessarily present in the condensate through the dynamical exchange of particles with the non-condensate reservoir. On the basis of this result, we propose Delta N-con . Delta phi greater than or similar to 1 as the uncertainty relation relevant to the number-phase fluctuations in superconductors and superfluids, where the number of condensed particles N-con is used instead of the total particle number N. The formula implies that a macroscopic phase phi can be established even in number-fixed superconductors and superfluids since Delta N-con >> 1.