Atom-surface interaction induced by quenched monopolar charge disorder

TL;DR

This work analyzes how quenched monopolar charge disorder inside a dielectric and on its surface shifts atomic energy levels and can produce an attractive force that competes with the nonresonant Casimir-Polder interaction. It develops a Gaussian-zero-mean disorder model and computes the disorder-induced Stark-like energy shift via second-order perturbation theory, comparing it to the Casimir-Polder shift in single-slab and two-slab planar geometries. For a single slab, surface disorder yields a downward shift decaying as $z_0^{-2}$ while bulk disorder decays as $z_0^{-1}$; in two-slab gaps, the shifts involve image-charge sums with Lerch transcendent and hypergeometric functions, reducing to the single-slab result as the separation grows. The results reveal crossovers where disorder forces dominate CP forces at large separations and show that, in a symmetric two-slab gap, the net disorder-induced force vanishes at midgap, shifting toward the slab with smaller disorder variance when variances differ. The framework enables extraction of disorder variances from force measurements and points to extensions to correlated disorder, Rydberg atoms, and other geometries.

Abstract

We study the modification to the energy level shifts of an atom induced by the quenched monopolar charge disorder inside the bulk of neighboring dielectric slabs as well as their surfaces. By assuming that the charge disorder follows Gaussian statistics with a zero mean, we find that the disorder generally results in a downward shift of the energy levels, which corresponds to an attractive force that can compete with and overcome the nonresonant Casimir-Polder force for sufficiently large atom-surface separations $z_0$. For an atom near a single semi-infinite slab with bulk (surface) charge disorder, the shift decays as $z_0^{-1}$ ($z_0^{-2}$). For both surface and bulk disorder, the shift is proportional to the variance of the charge disorder density. In addition, we investigate the behavior of the charge disorder-induced energy level shift for an atom confined to a vacuum gap between two coplanar and semi-infinite slabs of the same dielectric material, finding that the position of net zero disorder-induced force occurs closer to the surface of the slab with the smaller charge disorder variance.

Figures (7)

• Figure 1: The atom next to a single semi-infinite slab: the atom is at ${\mathbf{r}} = z_0 \, {\bf e}_z$, with the origin of the coordinate system being the point on the surface directly underneath the atom. The red dot at position ${\mathbf{r}}_{\parallel}$ represents a given local surface charge disorder. The green arrow represents the direction of the unit vector ${\bf n}({\mathbf{r}}_{\parallel})$, which points from the local charge disorder at ${\mathbf{r}}_{\parallel}$ to the atom. Not shown is the rest of the charge disorder, which can be in the bulk and/or the surface.
• Figure 2: An atom at position ${\mathbf{r}}_0 = z_0 \, {\bf e}_z$ between two coplanar, semi-infinite slabs with dielectric permittivities $\varepsilon_1$ and $\varepsilon_2$. The red dot at position $z' \, {\bf e}_z$ represents a given local charge disorder $\delta q$ in slab 2, whilst the blue dot at position $z" \, {\bf e}_z$ represents a given local charge disorder $\delta Q$ in slab 1. Not shown is the rest of the charge disorder, which is homogeneously distributed throughout the bulk of each slab.
• Figure 3: (a) Interpretation of the terms in the series for $\varphi_{I}$ (Eq. (\ref{['varphii']})): $\delta \tilde{q}_{I} \equiv 2\delta q_{I}/(\varepsilon_2+1)$ is the image charge of the impurity charge $\delta q_{I}$ positioned at $z=z_{I}$ ($z_{I} < 0$) "seen" by the atom at $z = z_0$ if only slab 2 (with dielectric permittivity $\varepsilon_2$) is present. Adding slab 1 (with dielectric permittivity $\varepsilon_1$) gives rise to an image charge of $\delta \tilde{q}_{I}$, i.e., $\delta \tilde{q}_{I'} = - \Delta_1 \delta \tilde{q}_{I}$ at $z=2d-z_{I}$. In turn, $\delta \tilde{q}_{I'}$ gives rise to another image charge $\delta \tilde{q}_{I"} = \Delta_2\Delta_1 \delta \tilde{q}_{I}$ at $z=z_{I}-2d$, and this image charge gives rise to a further image charge $\delta \tilde{q}_{I"'} = -\Delta_2\Delta_1^2 \delta \tilde{q}_{I}$ at $z=4d-z_{I}$, and so on. (b) Interpretation of the terms in the series for $\psi_{I}$ (Eq. (\ref{['psii']})): $\delta \tilde{Q}_{I} \equiv 2\delta Q_{I}/(\varepsilon_1+1)$ is the image charge of the impurity charge $\delta Q_{I}$ positioned at $z=z_{I}$ ($z_{I} > 0$) "seen" by the atom at $z = z_0$ if only slab 1 (with dielectric permittivity $\varepsilon_1$) is present. Adding slab 2 (with dielectric permittivity $\varepsilon_2$) gives rise to an image charge of $\delta \tilde{Q}_{I}$, i.e., $\delta \tilde{Q}_{I'} = - \Delta_2 \delta \tilde{Q}_{I}$ at $z=-z_{I}$. In turn, $\delta \tilde{Q}_{I'}$ gives rise to another image charge $\delta \tilde{Q}_{I"} = \Delta_1\Delta_2 \delta \tilde{Q}_{I}$ at $z=z_{I}+2d$, and this image charge gives rise to a further image charge $\delta \tilde{Q}_{I"'} = -\Delta_1\Delta_2^2 \delta \tilde{Q}_{I}$ at $z=-z_{I}-2d$, and so on.
• Figure 4: Log-log plots of the magnitude of the nonresonant Casimir-Polder energy shift (black), surface charge disorder-induced energy shift (blue dashed) and bulk charge disorder-induced energy shift (red dot-dashed) as functions of the atom-surface separation $z_0$, for a helium atom in the $n=2$ triplet state next to a vitreous ${\rm{SiO_2}}$ slab surface, with polarizability and dielectric functions given by Eqs. (\ref{['alpha-He*']}) and (\ref{['varepsilon-SiO2']}). We set $\sigma_B^2 = 1.168\times 10^{-5} \, {\rm esu^2/cm^3}$ and $\sigma_S^2 = 3.146 \times 10^{-10} \, {\rm esu^2/cm^2}$ for the bulk and surface charge disorder variances per unit area.
• Figure 5: Log-log plots of the magnitude of the nonresonant Casimir-Polder force (black), surface charge disorder-induced force (blue dashed) and bulk charge disorder-induced force (red dot-dashed) as functions of the atom-surface separation $z_0$, for a helium atom in the $n=2$ triplet state next to a vitreous ${\rm{SiO_2}}$ slab surface, with polarizability and dielectric functions given by Eqs. (\ref{['alpha-He*']}) and (\ref{['varepsilon-SiO2']}). We set $\sigma_B^2 = 1.168\times 10^{-5} \, {\rm esu^2/cm^3}$ and $\sigma_S^2 = 3.146 \times 10^{-10} \, {\rm esu^2/cm^2}$ for the bulk and surface charge disorder variances per unit area.