= Phase derivative of a two-mode condensate energy
{title2=$\partial_\chi E=i\langle[H,n_R]\rangle=-i\langle[H,n_L]\rangle$}
For $|\chi\rangle=(c a_L^\dagger+e^{i\chi}s a_R^\dagger)^N|0\rangle/\sqrt{N!}$ with real $c,s$ and $c^2+s^2=1$, one has $|\chi\rangle=e^{i\chi n_R}|0_{\rm phase}\rangle$. Differentiating $E=\langle\chi|H|\chi\rangle$ gives $\partial_\chi E=i\langle[H,n_R]\rangle$. If $[H,n_L+n_R]=0$, this equals $-i\langle[H,n_L]\rangle$. The <Heisenberg equation of motion> identifies the left-to-right <particle current> as $J=\partial_\chi E/\hbar$. Reversing the definition of the relative <phase> reverses this sign.
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