Displacement operator

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Quantum optics operators
Ladder operators
Creation and annihilation operators
Displacement operator
Rotation operator
Squeeze operator
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The displacement operator in quantum optics is the operator

$D(\alpha)=\exp \left ( \alpha \hat{a} - \bar{\alpha} \hat{a}^\dagger \right )$ ,

where $α$ is the amount of displacement in phase space, $\bar{\alpha}$ is the complex cojugate of that displacement, and $\hat{a}$ and $\hat{a}^\dagger$ 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.

[edit] Properties

$D(\alpha)D(\beta)=\exp \left(\frac{\alpha \bar{\beta}-\beta \bar{\alpha}}{2} \right)D(\alpha + \beta)$ .

Note that the residual phase, in this case $\textrm{Re} \left(\frac{\alpha \bar{\beta}-\beta \bar{\alpha}}{2}\right)$ , 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.