We may perform an analogous consideration for a plane electron wave exp[ikr]. In this case it is useful to discuss the scattering in k-space. This is done in ref /2/, /3/. Again one considers multiple scattering and the interference generates a plane wave, propagating in the opposite direction -k. We call this back- scattered wave the "coherent echo" of the original plane wave. In two dimensions this echo decays as 1/t. In Fig.3 we plotted the momentum of the original wave (which decays within the short time ) and the coherent echo as a function of time (for details see ref /3/). The coherent echo disappears when the two partial electron waves lose their coherence, for example by inelastic electron-phonon scattering. This happens for t>, where is the inelastic lifetime of the conduction electrons.

Fig.3: The momentum of a plane electron wave, scattered by lattice defects. The momentum decays after the scattering time . However, at later times a coherent echo forms, propagating in the opposite direction.

This physical interpretation is based on Kubo diagram which was originally discovered by Langer and Neal /4/ in the late 60's. Then in 1979 and 1980 a number of papers by Abrahams et al. /5/, Anderson et al. /6/, Gorkov et al. /7/, Altshuler et al. /8/ Hikami et al /9/ etc derived the theory of this quantum interference.

The formation of the coherent echo is not restricted to electrons. It requires only an original plane wave and many scattering centers. Our interpretation of the formal theory stimulated scattering experiments with laser light (see for example /10/) and lead to a new field of research, weak localization of light. The enhanced coherent back-scattering in real space can also be considered as a coherent echo.

The next step in our consideration is to realize that the fraction of the returning electrons, i.e., the total coherent echo, can be counted by a resistance measurement. This can be seen most easily in the plane wave picture. A classical electron with momentum k yields a contribution to the current of ek/m over the period , i.e., its total contribution is proportional to k. In the presence of multiple scattering our coherent echo yields an opposite contribution to the current which is equal to the integrated echo. The relative size of the correction is

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