In Bose-Einstein condensate (BEC), people often say there is a well defined macroscopic phase. What exactly the macroscopic phase is? (a phase factor $\mathrm{e} ^{i\phi}$ in a many-body wavefuction?) Is the macroscopic phase the same as the coherent phase?
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There are different ways to define this phase. In mean-field (low temperature, weak interaction regime), the many-body wave function
$\psi(x_1, x_2,...)=\prod_i \Phi(x_i)$
where $\Phi(x)$ is sometimes called the macroscopic wavefunction (because all the bosons are in the same state described by $\Phi$). In the simplest case (homogeneous system), one can show that (by finding the solution of the time-independent Gross-Pitaevski equation (wiki))
$\Phi(x)=|\Phi_0| e^{i\phi}$,
where $|\Phi_0|^2$ gives the density of the condensate (and in this regime, a good approximation of the density of particles), whereas $\phi$ is a phase which is arbitrary (in the sense that all value are allowed, and that it is not given by the parameters of the problem). This is the macroscopic phase.
Adam
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