Steady-State Entanglement Generation via Casimir-Polder Interactions

TL;DR

The paper addresses generating robust two-atom entanglement near a planar surface by harnessing Casimir-Polder fluctuations. It develops a macroscopic QED framework using the Green's tensor to derive surface-modified decay rates $\Gamma_{ij}$, dipole couplings $\Omega_{ij}$, and CP shifts inside a Born-Markov master equation, enabling analysis of the relative decay $D(\mathbf{r}_1,\mathbf{r}_2)=\Gamma-\Gamma_{12}$. A key result is that steady-state entanglement with concurrence $C(\rho_{\mathrm{steady}})=0.5$ is achieved when $D(\mathbf{r}_1,\mathbf{r}_2)=0$, with parallel dipoles ($xx$) showing near-zero decay near suitable surfaces; numerical examples reveal substantial entanglement near perfect conductors and Nb superconductors ($C_{xx}\approx 0.37$–$0.41$ at $\tilde{x}=1$, $\tilde{z}=0.2$) and more modest but enhanced entanglement near a gold surface at an optimal distance. The work demonstrates a constructive use of fluctuation-mediated interactions for nanoscale entanglement generation, with implications for near-surface quantum sensing and extensions to curved geometries.

Abstract

We investigate the generation of steady-state entanglement between two atoms resulting from the fluctuation-mediated Casimir-Polder (CP) interactions near a surface. Starting with an initially separable state of the atoms, we analyze the atom-atom entanglement dynamics for atoms placed at distances in the range of $\sim25$ nm away from a planar medium, examining the effect of medium properties and geometrical configuration of the atomic dipoles. We show that perfectly conducting and superconducting surfaces yield an optimal steady-state concurrence value of approximately 0.5. Furthermore, although the generated entanglement decreases with medium losses for a metal surface, we identify an optimal distance from the metal surface that assists in entanglement generation by the surface. While fluctuation-mediated interactions are typically considered detrimental to the coherence of quantum systems at nanoscales, our results demonstrate a mechanism for leveraging such interactions for entanglement generation.

Figures (9)

• Figure 1: Schematic of the model featuring two two-level atoms near a planar medium of permittivity $\epsilon (\omega)$. The atomic dipoles ${\mathbf d}_1$ and ${\mathbf d}_2$ are oriented along $\theta$ with respect to the $x$-axis.
• Figure 2: (a) The relative decay in free space (green dashed and black dotted curves) and the total relative decay near the perfect conductor at a distance of $\tilde{z} = 0.2$, as functions of $\tilde{x}$ for the $zz$ (dash-dotted blue curve) and $xx$ (solid red curve) configurations. The inset shows the density matrix's time-dependent concurrence for the system with $zz$ (dash-dotted blue curve) and $xx$ (solid red curve) configurations near the perfect conductor surface. (b) and (c), The schematic depicts the dipole-image model for the perpendicular dipoles in $zz$ configuration and the parallel dipoles in $xx$ configuration, along with their corresponding images in the medium.
• Figure 3: Contour plot depicting $D(\tilde{x},\tilde{z})$ in the vicinity of (a) superconducting, and (b) a gold surface where Zero relative decay is required for steady-state entanglement generation. (c) The evolution of concurrence between atoms over time for the $xx$ configuration with $\Tilde{x} = 1$ in free space (black dotted), and at a distance $\Tilde{z} = 0.2$ from the perfect conductor (solid red), the superconducting (blue dash-dotted), and the gold surface (yellow dashed). However, atom-atom entanglement can persist for a long time even if the relative decay does not vanish completely. The selected position ($\Tilde{x}=1,\Tilde{z}=0.2$) is indicated by the white $\otimes$ sign in contour plots.
• Figure 4: (a) The single emitter decay $\Gamma(\tilde{z})$ as a function of $\tilde{z}$ and (b) The relative decay $D(\tilde{x},\tilde{z})$ as a function of $\tilde{x}$ where $\tilde{z} = 0.01$ for $xx$ (the solid red curve), and $zz$ (dashed blue curve) configurations near a perfect conductor.
• Figure 5: The contour plot of the relative decay as a function of $\tilde{x}$, and $\tilde{z}$ near a perfect conducting surface for $xx$ configuration.