text: Early quantum models included a soliton pair creation model by Maki and a proposal by John Bardeen that condensed CDW electrons tunnel coherently through a tiny pinning gap, fixed at ±k unlike the Peierls gap. Makis theory lacked a sharp threshold field and Bardeen only gave a phenomenological interpretation of the threshold field. However, a 1985 paper by Krive and Rozhavsky pointed out that nucleated solitons and antisolitons of charge ±q generate an internal electric field E* proportional to q/ε. The electrostatic energy (1/2)ε[E ± E*] prevents soliton tunneling for applied fields E less than a threshold E = E*/2 without violating energy conservation. Although this Coulomb blockade threshold can be much smaller than the classical depinning field, it shows the same scaling with impurity concentration since the CDWs polarizability and dielectric response ε vary inversely with pinning strength.
  Building on this picture, as well as a 2000 article on time-correlated soliton tunneling, a more recent quantum model proposes Josephson-like coupling (see Josephson effect) between complex order parameters associated with nucleated droplets of charged soliton dislocations on many parallel chains. Following Richard Feynman in The Feynman Lectures on Physics, Vol. III, Ch. 21, their time-evolution is described using the Schrödinger equation as an emergent classical equation. The narrow-band noise and related phenomena result from the periodic buildup of electrostatic charging energy and thus do not depend on the detailed shape of the washboard pinning potential. Both a soliton pair-creation threshold and a higher classical depinning field emerge from the model, which views the CDW as a sticky quantum fluid or deformable quantum solid with dislocations, a concept discussed by Philip Warren Anderson.