Phase derivative of a two-mode condensate energy (${\partial }_{\chi }E=i⟨[H,n_R]⟩=-i⟨[H,n_L]⟩$)

For $|\chi ⟩=(c{a}_L^{†}+e^{i\chi }s{a}_R^{†}{)}^N|0⟩/\sqrt{N!}$ with real $c,s$ and $c^2+s^2=1$, one has $|\chi ⟩=e^{i\chi n_R}|0_{\mathrm{phase}}⟩$. Differentiating $E=⟨\chi |H|\chi ⟩$ gives ${\partial }_{\chi }E=i⟨[H,n_R]⟩$. If $[H,n_L+n_R]=0$, this equals $-i⟨[H,n_L]⟩$. The Heisenberg equation of motion identifies the left-to-right particle current as $J={\partial }_{\chi }E/\hslash$. Reversing the definition of the relative phase reverses this sign.