Coherent Bose-Einstein condensation with fluctuating density
Bose-Einstein condensation in the grand canonical ensemble admits a formulation in terms of a phase-density decomposition of the condensate mode operator $\hat{\psi}_{\bf 0}$. 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 full hierarchy of correlation functions is determined by the statistics of the density. Within this framework, the modulus squared of the anomalous average $\langle \hat{\psi}_{\bf 0}\rangle$ can provide only a fraction of the whole condensate density $\rho_{\bf 0}$ and for the grand canonical statistics of the ideal Bose gas one obtains the value $|\langle {\hat \psi}_{\bf 0}\rangle|^2 =(\pi/4) \rho_{\bf 0}$. The remaining part is supplemented by the (macroscopic) fluctuations of $\hat{\psi}_{\bf 0}$, 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.