Wannier-bloch Oscillators

We consider a Wannier-Stark problem for small eld f in the one-ladder case. We prove that a generical rst band state is a metastable state (Wannier-Bloch oscillator) with the lifetime determined by the imaginary part of the Wannier-Stark ladder. The innnite resonances of the ladder cause Bloch oscillations as a global beating eeect. For an adiabatic time = ft large enough, but much smaller than the resonance lifetime, we have a new version of the acceleration theorem and well speciied Bloch oscillators. In the x representation and in the adiabatic scale: x ! x(f) = =f + y the state vanishes externally to a pulsating region of jj deened by jj < + () where + (n) = 0 and + (n + 1=2) is the maximum value equal to the rst band width. For and such that jj is in this region and for y in a xed domain, the state approaches a nite combination of oscillating Bloch states.