Displacement operator

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In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics,

,

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, which is an eigenstate of the annihilation (lowering) operator.

Properties[edit]

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[edit]

The Kermack-McCrae identity gives two alternative ways to express the displacement operator:

Multimode displacement[edit]

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

.

See also[edit]

References[edit]

  1. ^ Christopher Gerry and Peter Knight: Introductory Quantum Optics. Cambridge (England): Cambridge UP, 2005.