In terms of regular [creation and annihilation operators](creation%20and%20annihilation%20operators.md) that act in [Fock Space](Fock%20Space.md) and the [[squeezing operator]] we may define a pair of _pseudo-creation and annihilation operators_ by sandwiching the creation an annihilation operators as follows: $\hat{A}^\dagger(\xi)=\hat{S}(\xi)\hat{a}^\dagger\hat{S}^\dagger(\xi)$ and $\hat{A}(\xi)=\hat{S}(\xi)\hat{a}\hat{S}^\dagger(\xi)$ Note that in a lot of literature we also encounter $\hat{S}^\dagger(\xi)\hat{a}^\dagger\hat{S}(\xi)$ and $\hat{S}^\dagger(\xi)\hat{a}\hat{S}(\xi).$ Both forms are interchangeable however they are not equivalent since the difference between them is in the sign of the squeezing parameter ([Property 1](squeezed%20state%20pseudo-creation%20and%20annihilation%20operators.md#Properties)). Thus it is important to remember that $(\hat{S}\big(\xi)\hat{a}\hat{S}^\dagger(\xi)\big)^\dagger \neq \hat{S}^\dagger(\xi)\hat{a}^\dagger\hat{S}(\xi).$ # Properties 1. $\hat{S}(\xi)\hat{a}^\dagger\hat{S}^\dagger(\xi)=\hat{S}^\dagger(-\xi)\hat{a}^\dagger\hat{S}(-\xi)$ and $\hat{S}(\xi)\hat{a}\hat{S}^\dagger(\xi)=\hat{S}^\dagger(-\xi)\hat{a}\hat{S}(-\xi).$ # Hyperbolic form Using the [baker-Campbell-Hausdorff lemma](Baker-Campbell-Hausdorff%20lemma.md) and the definition of the [squeezing operator](Squeezing%20Operator.md) we may [derive](squeezed%20state%20pseudo-creation%20and%20annihilation%20operators.md#Derivation%20of%20hyperbolic%20form) the following expressions: $\hat{A}^\dagger(\xi)=\hat{a}^\dagger\cosh{s}-\hat{a}e^{i\theta}\sinh{s}$ $\hat{A}(\xi)=\hat{a}\cosh{s}-\hat{a}^\dagger e^{-i\theta}\sinh{s}$ # Derivation of hyperbolic form #QuantumMechanics/QuantumOptics #QuantumMechanics/QuantumDynamics