Electric field average and variance in squeezed states
We start by considering the electric field operator \mathbf{E}_{\lambda}(\mathbf r,t) for a specific mode \lambda:
\mathbf{E}_{\lambda}(\mathbf r,t) = i \mathbf e_\lambda \mathscr{E}_{\lambda}^{(1)} \left( \mathbf{a}_{\lambda} e^{i(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t)} - \mathbf{a}_{\lambda}^{\dagger} e^{-i(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t)} \right)
Here, \mathbf{a}_{\lambda} and \mathbf{a}_{\lambda}^{\dagger} are the annihilation and creation operators, \mathbf e_\lambda is the polarization vector, \mathscr{E}_{\lambda}^{(1)} is a constant, \mathbf{k}_{\lambda} is the wave vector, and \omega_{\lambda} is the frequency.
To understand the behavior of the electric field in a squeezed state | \boldsymbol \alpha_\lambda, R_\lambda \rangle, I first computed its average value. By utilizing the properties of squeezed states and coherent states, I found that the average electric field in a squeezed state can be expressed in terms of a modified coherent amplitude \alpha_\lambda^\prime:
\alpha_\lambda^\prime = \cosh \left(R_\lambda\right) \alpha_{\lambda R} - \sinh \left(R_\lambda\right) \bar\alpha_{\lambda R}
This result suggests that the average electric field in a squeezed state behaves similarly to that in a coherent state |\boldsymbol \alpha_\lambda^\prime\rangle, but with a transformed amplitude \alpha_\lambda^\prime. In the specific case where the initial coherent amplitude \alpha_\lambda is real, this transformation simplifies even further to \alpha_\lambda^\prime = \alpha_{\lambda R} e^{-R_\lambda}.
Next, I turned my attention to the variance of the electric field, a quantity that reveals the fluctuations around the average value. The variance (\Delta E_{\lambda}(\mathbf r,t))^2 is defined as:
\left(\Delta E_{\lambda}(\mathbf r,t)\right)^2 = \langle \left(\mathbf{E}_{\lambda}(\mathbf r,t) \right)^2 \rangle - \left(\langle \mathbf{E}_{\lambda}(\mathbf r,t) \rangle \right)^2
After a more involved calculation, I arrived at the following expression for the variance in a squeezed state:
\left(\Delta E_{\lambda}(\mathbf r,t)\right)^2 = \left[\mathscr{E}_{\lambda}^{(1)} \right]^2 \left( e^{2 R_\lambda} \cos^2\left(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t \right) + e^{-2 R_\lambda} \sin^2\left(\mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t \right) \right)
This result shows that, unlike in quasi-classical coherent states where the variance is constant, the variance of the electric field in a squeezed state is not constant. It depends on both position \mathbf r and time t through the term \mathbf{k}_{\lambda} \cdot \mathbf{r} - \omega_{\lambda} t. This space-time dependence reflects the non-classical nature of the squeezed states. The squeezing parameter R_\lambda directly controls the magnitude of these fluctuations, demonstrating how squeezing modifies the quantum statistics of the electromagnetic field.
For more insights into this topic, you can find the details here.