Displacement Operator

Operator

For one bosonic mode, the displacement operator is

$$D(\alpha) = \exp \left( \alpha a^\dagger-\alpha^*a \right).$$

It displaces the annihilation operator by

$$D(\alpha)^\dagger aD(\alpha) = a+\alpha.$$

The coherent state generated from the oscillator vacuum is

$$\lvert\alpha\rangle = D(\alpha)\lvert0\rangle.$$

Assumptions

• The mode satisfies $[a,a^\dagger]=1$.
• The phase-space convention for $\alpha$ is specified.
• The formula is for a single bosonic mode unless a multimode generalization is stated.
• Global phases from composition laws are physically important in interference calculations.

Common Mistakes

• Reversing $D^\dagger aD$ and $DaD^\dagger$ without changing the sign of $\alpha$.
• Confusing displacement with squeezing; displacement changes first moments, not covariance.
• Assuming $\alpha$ is directly a position displacement without the oscillator scaling factors.
• Dropping the noncommuting phase in products of displacement operators.

Canonical Links

• Coherent States
• Creation and Annihilation Operators
• Heisenberg Group
• Gaussian States Preview

References

• R. J. Glauber, “Coherent and incoherent states of the radiation field”, Physical Review 131, 2766-2788, 1963.
• C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge University Press, 2005.