Squeeze operator

In quantum physics, the squeeze operator for a single mode of the electromagnetic field is[1] where the operators inside the exponential are the ladder operators.

It is a unitary operator and therefore obeys

is the identity operator.

Its action on the annihilation and creation operators produces The squeeze operator is ubiquitous in quantum optics and can operate on any state.

For example, when acting upon the vacuum, the squeezing operator produces the squeezed vacuum state.

The squeezing operator can also act on coherent states and produce squeezed coherent states.

The squeezing operator does not commute with the displacement operator: nor does it commute with the ladder operators, so one must pay close attention to how the operators are used.

There is, however, a simple braiding relation,

γ = α cosh ⁡ r +

sinh ⁡ r

[2] Application of both operators above on the vacuum produces displaced squeezed state: Or Squeezed coherent state: As mentioned above, the action of the squeeze operator

on the annihilation operator

can be written as

sinh ⁡ (

{\displaystyle S^{\dagger }(z)aS(z)=\cosh(|z|)a-{\frac {z}{|z|}}\sinh(|z|)a^{\dagger }.}

To derive this equality, let us define the (skew-Hermitian) operator

The left hand side of the equality is thus

We can now make use of the general equality

which holds true for any pair of operators

To compute

thus reduces to the problem of computing the repeated commutators between

As can be readily verified, we have

Using these equalities, we obtain

{\displaystyle [\underbrace {A,[A,\dots ,[A} _{k},a]\dots ]]={\begin{cases}|z|^{k}a,&{\text{ for }}k{\text{ even}},\\-z|z|^{k-1}a^{\dagger },&{\text{ for }}k{\text{ odd}}.\end{cases}}}

so that finally we get

= a cosh ⁡

i θ

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