Squeezed states of light: going beyond quantum limits
Quantum optics theory unveiled the concept of squeezed states of light. These states were not previously considered and offered a new perspective by suggesting the possibility of surpassing the standard quantum limit in measurement precision, a long-established boundary.
To understand squeezed states, it’s helpful to first recall quasi-classical states. A quasi-classical state | \boldsymbol \alpha_\lambda\rangle is defined as an eigenstate of the annihilation operator \mathbf{a}_\lambda:
\mathbf a_\lambda | \boldsymbol \alpha_\lambda\rangle = \alpha_\lambda | \boldsymbol \alpha_\lambda\rangle, \quad \alpha_\lambda \in \mathbb C
Squeezed states are introduced by generalizing this definition. A single-mode squeezed state | \boldsymbol \alpha_\lambda, R_\lambda \rangle is defined as an eigenstate of a generalized annihilation operator \mathbf{A}_{R_\lambda}, which is a linear combination of the annihilation \mathbf{a}_\lambda and creation \mathbf{a}^\dag_\lambda operators. The coefficients in this combination involve hyperbolic functions, specifically \cosh \left(R_\lambda\right) and \sinh \left(R_\lambda\right), where R_\lambda is a real number.
The generalized annihilation operator \mathbf{A}_{R_\lambda} and its adjoint \mathbf{A}_{R_\lambda}^\dag are given by:
\begin{aligned} & \mathbf{A}_{R_\lambda} =\mathbf a_\lambda \cosh \left(R_\lambda\right) + \mathbf a_\lambda^\dag\sinh \left(R_\lambda\right) \\ & \mathbf{A}_{R_\lambda}^\dag =\mathbf a_\lambda^\dag \cosh \left(R_\lambda\right) + \mathbf a_\lambda\sinh \left(R_\lambda\right) \end{aligned}
and the squeezed states are eigenstates of \mathbf{A}_{R_\lambda}:
\mathbf{A}_{R_\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \alpha_{\lambda R} | \boldsymbol \alpha_\lambda, R_\lambda \rangle, \quad \alpha_\lambda \in \mathbb C, \; R_\lambda \in \mathbf R
To confirm that \mathbf{A}_{R_\lambda} and \mathbf{A}_{R_\lambda}^\dag indeed behave as annihilation and creation operators, we can examine their commutator:
\begin{aligned} \left[\mathbf{A}_{R_\lambda}, \mathbf{A}_{R_\lambda}^\dag\right] = & \left[\mathbf a_\lambda \cosh \left(R_\lambda\right) + \mathbf a_\lambda^\dag\sinh \left(R_\lambda\right), \mathbf a_\lambda^\dag \cosh \left(R_\lambda\right) + \mathbf a_\lambda\sinh \left(R_\lambda\right)\right] \\ = & \left(\cosh^2 \left(R_\lambda\right) - \sinh^2 \left(R_\lambda\right)\right) \left[\mathbf a_\lambda, \mathbf a_\lambda^\dag\right] \\ = & 1 \end{aligned}
This result, \left[\mathbf{A}_{R_\lambda}, \mathbf{A}_{R_\lambda}^\dag\right] = 1, is analogous to the canonical commutation relation for standard annihilation and creation operators, justifying their designation.
It’s also worth noting that the original annihilation and creation operators \mathbf{a}_\lambda and \mathbf{a}_\lambda^\dag can be expressed in terms of the generalized operators \mathbf{A}_{R_\lambda} and \mathbf{A}_{R_\lambda}^\dag:
\begin{aligned} \mathbf a_\lambda & = \mathbf{A}_{R_\lambda} \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}^\dag\sinh \left(R_\lambda\right) \\ \mathbf a_\lambda^\dag & = \mathbf{A}_{R_\lambda}^\dag \cosh \left(R_\lambda\right) - \mathbf{A}_{R_\lambda}\sinh \left(R_\lambda\right) \end{aligned}
Squeezed states are offering paths to enhance measurement precision beyond conventional quantum limits.
For more insights into this topic, you can find the details here.