# Displacement operator

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The **displacement operator** for one mode in quantum optics is the shift operator

where is the amount of displacement in optical phase space, is the complex conjugate of that displacement, and and 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 . 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.

## Properties

The displacement operator is a unitary operator, and therefore obeys , where is the identity operator. 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.^{[1]}

## Alternative expressions

Two alternative ways to express the displacement operator are:

## Multimode displacement

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

## References

- ↑ Gerry, Christopher, and Peter Knight:
*Introductory Quantum Optics*. Cambridge (England): Cambridge UP, 2005.