Scattering of Squeezed Light by a Dielectric Slab

TL;DR

This work develops a fully quantum framework based on Green-function quantization to study how squeezed light scatters from a dissipative, dispersive dielectric slab. It derives input–output relations and computes both single-mode and continuum squeezing, showing that dispersion, absorption, and internal reflections irreversibly degrade squeezing by mixing it with vacuum or thermal noise, except in the ideal lossless, perfectly matched limit where resonant transmission can preserve squeezing. The analysis reveals phase-sensitive noise filtering, spectral reshaping, and temporal-delays (including Fabry–Pérot–like resonances) that reshape the squeezing spectrum and reduce the effective spectral squeezing parameter $\rho'_{\Gamma}(\omega)$. These results provide a rigorous baseline for using squeezed light in realistic optical media, informing design choices for quantum communication and high-precision sensing where material loss and dispersion cannot be neglected.

Abstract

We develop a quantum theory for the scattering of squeezed coherent light by a dissipative dielectric slab. Using the Green-function quantization approach, we derive the transformation of the field quadratures and show how dispersion, absorption, and multiple reflections distort the incident squeezing. We find that the slab can selectively attenuate or amplify quadrature noise depending on the slab parameters and provide expressions for the output power spectra.

Figures (7)

• Figure 1: Geometry of the dielectric slab and notation for the field operators. The slab has thickness $2l$ and is positioned between $x=-l$ and $x=+l$. Different domains are labeled 1-3. The dielectric function $\epsilon(x,\omega)$ changes value at the interfaces.
• Figure 2: Variance of the transmitted squeezed quadrature ${\Delta \hat{X}_{R 3}^{\prime}}^2$ as a function of dielectric slab thickness for a squeezed coherent state with parameter$\rho = 0.8$ (initial variance = 0.045, equivalent to 6.9 dB of squeezing below the standard quantum limit). The calculation uses representative parameters for a dielectric material with refractive index $\eta = 1.5$ and extinction coefficient $\kappa=0.005$ at the squeezed light wavelength of $\left (\lambda=1064 nm\right )$ nm. The oscillatory behavior with period $\lambda/(4\eta) = 0.177~\mu m$ results from quantum interference effects within the slab, with the variance approaching an asymptotic value as thickness increases. The inset shows the variance of the orthogonal quadrature ${\Delta \hat{X}_{L 3}^{\prime}}^2$. The horizontal dashed gray line shows the original squeezed variance of the input pulse, and the dashed black line shows the standard quantum limit (SQL) or coherent state variance of $1/4$
• Figure 3: Variance of the reflected squeezed quadrature ${\Delta \hat{X}_{L1}^{\prime}}^2$ as a function of slab thickness for a squeezed coherent state with parameter $\rho = 0.8$, $\eta = 1.5$ and $\kappa = 0.0075$ at the squeezed light wavelength of $\lambda = 1064 nm$ . The purple line shows the theoretical ultimate variance limit for an infinitely thick slab. The inset shows the variance of the orthogonal quadrature ${\Delta \hat{Y}_{L1}^{\prime}}^2$ for the reflected pulse, which exhibits anti-squeezing behavior with the variance approaching an ultimate limit of $0.290$ as thickness increases, above the standard quantum limit.
• Figure 4: Relative frequency shift, $\Delta\omega_{\Gamma}/ \omega_c$, for transmitted (main) and reflected (inset) squeezed pulses as a function of slab refractive index,$\eta_c$, using representative parameters($\lambda_c=633$$nm$, $\mathcal{L}_I / l =80$, and $\rho_{I}=1.5$) for two values of extinction coefficient, $\kappa_c$(solid for $\kappa_c$=0.002, and dotted for $\kappa_c$=0.02). The oscillatory structure arises from Fabry-Perot resonances within the slab. Increased loss ('$\kappa_c$=0.02', dotted) dampens these resonances compared to the lower-loss case ('$\kappa_c$=0.002', solid). The shift vanishes for a matched, lossless slab ('$\eta_c$→1', '$\kappa_c$→0'), All oscillating pattern shifts downward as $\kappa$ increases, as shown in the detailed inset(lower left).
• Figure 5: Change in relative mean-square spectral length, $\mathcal{L}_{I}^2 / \mathcal{L}_{\Gamma}^2$, as a function of $\eta_c$ with same representative values of Fig. \ref{['fig:Relative_Delta_omega_vs_eta']}, for transmitted pulse(main plot), and reflected one(inset plot) for two values of extinction coefficient, $\kappa_c$(solid for $\kappa_c$=0.002, and dotted for $\kappa_c$=0.02). A value greater than one indicates spectral broadening due to group velocity dispersion, while a value less than one indicates spectral narrowing. The reflected pulse exhibits more extreme distortion, as its spectrum is more sensitive to the Fabry-Perot resonances of the slab.