Squeeze Operator

Operator

For one bosonic mode, write $\zeta=re^{i\phi}$. A common single-mode squeeze operator is

$$S(\zeta) = \exp \left[ \frac12 \left( \zeta^*a^2-\zeta a^{\dagger2} \right) \right].$$

With this convention,

$$S(\zeta)^\dagger aS(\zeta) = a\cosh r - e^{i\phi}a^\dagger\sinh r.$$

The squeezed vacuum is

$$\lvert\zeta\rangle = S(\zeta)\lvert0\rangle.$$

Assumptions

• The mode is bosonic with $[a,a^\dagger]=1$.
• The sign convention for $S(\zeta)$ is declared.
• Quadrature definitions determine which variance is squeezed.
• The ideal squeezed state is normalizable for finite $r$; singular EPR-like limits require separate care.

Common Mistakes

• Comparing squeeze formulas without checking the sign convention in the exponent.
• Forgetting that squeezing one quadrature increases noise in the conjugate quadrature.
• Confusing a single-mode squeezed state with a two-mode squeezed entangled state.
• Treating the limit $r\to\infty$ as an ordinary finite-energy state.

Canonical Links

• Squeezed States: First Encounter
• Squeezed States as Entangled Modes
• Gaussian States Preview
• Displacement Operator

References

• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.
• C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge University Press, 2005.