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where α is the amount of displacement in optical phase space, α* is the complex conjugate of that displacement, and â and â† are the lowering and raising operators, respectively. The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude α. It may also act on the vacuum state by displacing it into a coherent state. Specifically, where is a coherent state. Displaced states are eigenfunctions of the annihilation (lowering) operator.
The displacement operator is a unitary operator, and therefore obeys , where I is the identity matrix. Since , the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude (). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.
The product of two displacement operators is another displacement operator, apart from a phase factor, has the total displacement as the sum of the two individual displacements. This can be seen by utilizing the Baker-Campbell-Hausdorff formula.
which shows us that:
When acting on an eigenket, the phase factor appears in each term of the resulting state, which makes it physically irrelevant.
Two alternative ways to express the displacement operator are:
The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as
where is the wave vector and its magnitude is related to the frequency according to . Using this definition, we can write the multimode displacement operator as
and define the multimode coherent state as
- Gerry, Christopher, and Peter Knight: Introductory Quantum Optics. Cambridge (England): Cambridge UP, 2005.
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