Redundant information encoding in QED during decoherence

Broadly understood decoherence processes in quantum electrodynamics, induced by neglecting either the radiation [L. Landau, Z. Phys. 45, 430 (1927)] or the charged matter [N. Bohr and L. Rosenfeld, K. Danske Vidensk. Selsk, Math.-Fys. Medd. XII, 8 (1933)], have been studied from the dawn of the theory. However, what happens in between, when a part of the radiation may be observed, as is the case in many real-life situations, has not been analyzed yet. We present such an analysis for a nonrelativistic, pointlike charge and thermal radiation. In the dipole approximation, we solve the dynamics and show that there is a regime where, despite the noise, the observed field carries away almost perfect and hugely redundant information about the charge momentum. We analyze a partial charge-field state and show that it approaches a so-called spectrum broadcast structure.

Figure 1

The considered physical model. A nonrelativistic charged particle interacts with an electromagnetic field, treated as the environment. The particle is described by a wave packet, narrow compared to the shortest relevant radiation wavelength, and moves with initial velocity ${\mathbf{v}}_0$, chosen along the $z$ axis. The sphere represents the “celestial sphere” of the mode directions $\mathbf{k}/k$. A part of this sphere, given by a solid angle ${\mathrm{\Omega }}_{\text{unob}}$, is not monitored and the corresponding modes are traced out. The rest is divided into small portions (only one portion shown) $\mathrm{\Delta }{\mathrm{\Omega }}_0$, centered each around some average direction ${\mathbf{k}}_0/k_0$, which represent, e.g., detection regions of approximately pointlike detectors.

Figure 2

A sample plot of the decoherence damping factor ${\mathrm{\Gamma }}_{\mathbf{p},{\mathbf{p}}^{\prime }}\left(t\right)$ (upper trace) and the state fidelity $-ln{B}_{\mathbf{p},{\mathbf{p}}^{\prime }}^{\text{mac}}\left(t\right)$ (lower trace) as a function of time, measured in the inverse cutoff units ${\overline{\mathrm{\Omega }}}^{-1}$ and plotted on a logarithmic scale. The parameters for the plot are the following: $\overline{\mathrm{\Omega }}={10}^{15} {\mathrm{s}}^{-1}, \alpha ={10}^5, \delta p_0/\left(m_{0}c\right)=5\times {10}^{-2}, \hslash \overline{\mathrm{\Omega }}/\left(k_{B}T\right)=4\times {10}^2$, the unobserved portion ${\mathrm{\Omega }}_{\text{unob}}$ is given by $0\le \theta \le \pi /4$, and the observed macrofraction size is $\mathrm{\Delta }{\mathrm{\Omega }}_0/\left(4\pi \right)=0.05$. This gives ${\tau }_{\text{dip}}=20{\overline{\mathrm{\Omega }}}^{-1}$ and ${\tau }_F\approx 7.95{\overline{\mathrm{\Omega }}}^{-1}$.