Localization of a quantum particle in a classical one-component plasma: fluctuation-induced random potential, localization length and mutual decoherence
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 $\ell(k) \propto k^2 / \ln(κL)$, while in the strong-disorder limit $\ell \propto (\ln(κL))^{-1/3}$. 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 $D_ϕ(ρ)$. At large transverse separations the coherence decays as a power law $γ(ρ)\sim ρ^{-η}$, with an exponent determined by the disorder strength. The transverse coherence length $ρ_c$ satisfies a scaling relation $ρ_c \sim λ_D \sqrt{\ell/L}$, 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 $A_{ m rel}=1/(\hbar v)$ and approaching the finite high-energy limit $1/(\hbar c)$.