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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 , where 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.
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,
[2]
Application of both operators above on the vacuum produces displaced squeezed state:
As mentioned above, the action of the squeeze operator on the annihilation operator can be written as
To derive this equality, let us define the (skew-Hermitian) operator , so that .
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 and . To compute thus reduces to the problem of computing the repeated commutators between and .
As can be readily verified, we have
^ M. M. Nieto and D. Truax (1995), Nieto, Michael Martin; Truax, D. Rodney (1997). "Holstein‐Primakoff/Bogoliubov Transformations and the Multiboson System". Fortschritte der Physik/Progress of Physics. 45 (2): 145–156. arXiv:quant-ph/9506025. doi:10.1002/prop.2190450204. S2CID14213781. Eqn (15). Note that in this reference, the definition of the squeeze operator (eqn. 12) differs by a minus sign inside the exponential, therefore the expression of is modified accordingly ().