Squeeze operator

In quantum physics, the squeeze operator for a single mode of the electromagnetic field is^[1]

{\hat {S}}(z)=\exp \left({1 \over 2}(z^{*}{\hat {a}}^{2}-z{\hat {a}}^{\dagger 2})\right),\qquad z=re^{i\theta }

where the operators inside the exponential are the ladder operators. It is a unitary operator and therefore obeys $S(\zeta )S^{\dagger }(\zeta )=S^{\dagger }(\zeta )S(\zeta )={\hat {1}}$ , where $\hat 1$ is the identity operator.

Its action on the annihilation and creation operators produces

{\hat {S}}^{\dagger }(z){\hat {a}}{\hat {S}}(z)={\hat {a}}\cosh r-e^{i\theta }{\hat {a}}^{\dagger }\sinh r\qquad {\text{and}}\qquad {\hat {S}}^{\dagger }(z){\hat {a}}^{\dagger }{\hat {S}}(z)={\hat {a}}^{\dagger }\cosh r-e^{-i\theta }{\hat {a}}\sinh r

The squeeze operator is ubiquitous in quantum optics and can operate on any state. For example, when acting upon the vacuum, the squeezing operator produces the squeezed vacuum state.

The squeezing operator can also act on coherent states and produce squeezed coherent states. The squeezing operator does not commute with the displacement operator:

{\hat {S}}(z){\hat {D}}(\alpha )\neq {\hat {D}}(\alpha ){\hat {S}}(z),

nor does it commute with the ladder operators, so one must pay close attention to how the operators are used. There is, however, a simple braiding relation, ${\hat {S}}(z){\hat {D}}(\alpha )={\hat {D}}(\gamma ){\hat {S}}(z)$ .^[2]

Application of both operators above on the vacuum produces squeezed states:

{\hat {D}}(\alpha ){\hat {S}}(r)|0\rangle =|\alpha ,r\rangle

References

↑ Gerry, C.C. & Knight, P.L. (2005). Introductory quantum optics. Cambridge University Press. p. 182. ISBN 978-0-521-52735-4.
↑ M. M. Niter and D. Truax (1995) quant-ph/9506025 , eqn (15), doi:10.1002/prop.2190450204

Operators in physics

General

Space and time	Parity Time evolution D'Alembert operator

Particles	C-symmetry

Operators for operators	Ladder operator Anti-symmetric operator

Quantum

Fundamental	Position Momentum Rotation

Energy	Total energy Kinetic energy Hamiltonian

Angular momentum	Spin Orbital Total

Electromagnetism	Transition dipole moment

Optics	Displacement Squeeze Quantum correlator Hanbury Brown and Twiss effect

Particle physics	Creation and annihilation Casimir invariant

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Squeeze operator

See also

References