Localization of a quantum particle in a classical one-component plasma: fluctuation-induced random potential, localization length and mutual decoherence

Abstract

We develop a microscopic theory of disorder-induced attenuation and mutual coherence degradation for a quantum particle in a classical one-component plasma. The random potential originates from equilibrium thermal fluctuations of the ionic charge density within the random phase approximation. Its correlator retains an unscreened $1/r$ tail, leading to a Coulomb logarithm in the eikonal localization scale $\ell (k)$. In the weak-disorder regime $\ell (k)\propto k^2/\ln (\hat{I}ºL)$, while in the strong-disorder limit $\ell \propto (\ln (\hat{I}ºL){)}^{-1/3}$. Building on the same disorder model, we evaluate the mutual coherence function (Cooperon) of an electron beam and derive a closed analytical expression for the phase structure function $D_{\ddot{I}}•(\ddot{I})$. At large transverse separations the coherence decays as a power law ${\hat{I}}^3(\ddot{I})\sim \ddot{I}{}^{-\hat{I}\cdot }$, with an exponent determined by the disorder strength. The transverse coherence length $\ddot{I}{}_c$ satisfies a scaling relation $\ddot{I}{}_c\sim \hat{I}{»}_D\sqrt{\ell /L}$, linking the eikonal attenuation scale with the loss of quantum coherence. Numerical estimates for aqueous electrolytes under transmission electron microscopy conditions are given. A relativistic extension confirms that the same scaling holds for relativistic beams, with the eikonal coupling given by $A_{mrel}=1/(\hslash v)$ and approaching the finite high-energy limit $1/(\hslash c)$.

Submitted to Physical Review E