Bose-Einstein condensation (BEC) in the grand-canonical ensemble admits a formulation in terms of a phase-density decomposition of the condensate mode operator ${\widehat{\psi }}_{\mathbf{0}}$. In the presence of macroscopic condensate number fluctuations this representation presents nontrivial implications. In particular, we show that, for the ideal gas, under the assumption of a well-defined phase and a fluctuating condensate density, the square modulus of the anomalous average $\langle {\widehat{\psi }}_{\mathbf{0}}\rangle$ can provide only a fraction of the whole condensate density ${\rho }_{\mathbf{0}}$, and for the grand-canonical statistics of the ideal Bose gas one obtains the value $|\langle {\widehat{\psi }}_{\mathbf{0}}\rangle {|}^2=(\pi /4){\rho }_{\mathbf{0}}$. The remaining part is supplemented by the (macroscopic) fluctuations of ${\widehat{\psi }}_{\mathbf{0}}$, which become a distinctive feature of the BEC in this setting. This provides a transparent physical picture of a condensate of photons with a well-defined phase but large number fluctuations, as observed in dye-filled-microcavity photon experiments. We also propose a way to access the square modulus of the anomalous average to test theoretical predictions.

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• Bose-Einstein condensates