Displacement operator

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Quantum optics operators
Ladder operators
Creation and annihilation operators
Displacement operator
Rotation operator (quantum optics)
Squeeze operator
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The displacement operator for one mode in quantum optics is the operator

,

where α is the amount of displacement in phase space, is the complex cojugate of that displacement, and and are the lowering and raising operators, respectively. The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement. Displaced states are eigenfunctions of the annihilation operator

[edit] Properties

.

Note that the residual phase, in this case , is path dependent. If the path formed by a series of displacements completes a closed loop in phase space the residual phase will be proportional to the area of that loop.

[edit] Multimode displacement

The displacement operator can also be generalized to multimode displacement.