A definition a bogoliubov transformation is defined as $$b=ua+va^\dagger~,~ b^\dagger=u^*a^\dagger+v^*a$$ But, using squeeze operator $$S=\exp{\left[\frac{1}{2}(z (a^\dagger)^2-z^*a^2)\right]}$$ we can claim that $$b=SaS^\dagger $$ is also a bogoliubov transform. Using S, how do we find the value of $u$ and $v$ corresponding to the first set of expressions? I have tried applying BCH formula to $b=SaS^\dagger$ but didn't get anything helpful.
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