Number-Phase Fluctuations in Isolated Superconductors

TL;DR

This paper refines the BCS wave function to include many-body correlations, revealing that these correlations significantly lower the ground-state energy and introduce number-phase fluctuations, leading to a new uncertainty relation in superconductors.

Contribution

It introduces an improved number-fixed BCS wave function that accounts for correlations and proposes a new number-phase uncertainty relation relevant to superconductors.

Findings

Correlations lower the ground-state energy more than pairing in weak coupling.

Number fluctuations are inherent in the condensate due to particle exchange.

A new uncertainty relation $ ext{ extDelta} N_{ m con} ext{ extDelta} extphi extgreater 1$ is proposed.

Abstract

We improve the Bardeen-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 $\varDelta N_{\rm con}\cdot \varDelta \phi\gtrsim 1$ as the uncertainty relation relevant to the number-phase fluctuations in superconductors and superfluids, where the number of condensed particles $N_{\rm con}$ 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 $\varDelta N_{\rm con}\gg 1$.