Inverse Bogoliubov vacuum condition
= Inverse Bogoliubov vacuum condition
{c}
For $a_{\rm out}=\alpha a_{\rm in}+\beta b_{\rm in}^\dagger$ with real canonical coefficients, the inverse is $a_{\rm in}=\alpha a_{\rm out}-\beta b_{\rm out}^\dagger$. The <in-vacuum> annihilation condition therefore has coefficient $v=-\beta$ when written as $(u a_{\rm out}+v b_{\rm out}^\dagger)|0_{\rm in}\rangle=0$. Forward and inverse maps must not be confused.