Inverse Bogoliubov vacuum condition

For $a_{\mathrm{out}}=\alpha a_{\mathrm{in}}+\beta b_{\mathrm{in}}^{†}$ with real canonical coefficients, the inverse is $a_{\mathrm{in}}=\alpha a_{\mathrm{out}}-\beta b_{\mathrm{out}}^{†}$. The in-vacuum annihilation condition therefore has coefficient $v=-\beta$ when written as $(u{a}_{\mathrm{out}}+v{b}_{\mathrm{out}}^{†})|0_{\mathrm{in}}⟩=0$. Forward and inverse maps must not be confused.