  One electron system minimally coupled to a quantized radiation field is
considered. It is assumed that the quantized radiation field is {\it massless},
and {\it no} infrared cutoff is imposed. The Hamiltonian, $H$, of this system
is defined as a self-adjoint operator acting on $\LR\otimes\fff\cong
L^2(\BR;\fff)$, where $\fff$ is the Boson Fock space over
$L^2(\BR\times\{1,2\})$. It is shown that the ground state, $\gr$, of $H$
belongs to $\cap_{k=1}^\infty D(1\otimes N^k)$, where $N$ denotes the number
operator of $\fff$. Moreover it is shown that, for almost every electron
position variable $x\in\BR$ and for arbitrary $k\geq 0$, $\|(1\otimes \N)\gr
(x) \|_\fff \leq D_ke^{-\delta |x|^{m+1}}$ with some constants $m\geq 0$,
$D_k>0$, and $\delta>0$ independent of $k$. In particular $\gr\in
\cap_{k=1}^\infty D (e^{\beta |x|^{m+1}}\otimes N^k)$ for $0<\beta<\delta/2$ is
obtained.
