Nonadiabatic effects in a soliton ground state of an extended Jahn-Teller system in one dimension.

Abstract

We have investigated conditions for the existence and for the stability of a soliton ground state of the Fulton-Gouterman-transformed extended one-dimensional Jahn-Teller model with low electron density against quantum fluctuations of optical phonons. The Jahn-Teller band splitting occurs but the respective gap is narrowed by the self-trapping effect modified by quantum fluctuations and by many-phonon effects due to the participation of phonons in the electron transfer. The electron transfer parameter ${\mathit{T}}_{\mathit{R}}$ is reduced by the self-trapping effect which is further modified by the effects of complex squeezing parameters. These effects determine then the value of the strength of the parameter \ensuremath{\Omega}\ifmmode\bar\else\textasciimacron\fi{}=\ensuremath{\Omega}/${\mathit{T}}_{\mathit{R}}$, which is a measure of the influence of the quantum fluctuations. We have shown that the electron channels are coupled due to the participation of the phonons, which couple the original electron levels, in the electron transfer. However, the dominating contribution of these phonons is diagonal and the coupling of the channels becomes negligible if the change of the phonon wave vector by the scattering with an electron at the distance of the soliton width is negligible. Under these conditions, the problem of the Fulton-Gouterman quantum ground state of the Jahn-Teller model becomes qualitatively equivalent to the related problem of a Holstein polaron (a soliton of the nonlinear Schr\odinger equation). The soliton ground state is shown to be unstable against quantum fluctuations for weak electron-phonon coupling and for \ensuremath{\Omega}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\gtrsim}2/3${\mathrm{\ensuremath{\kappa}}}^{3}$ (\ensuremath{\kappa}=${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ is an inverse soliton width). The fluctuations compete the self-trapping polaron effect: For \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}${\mathrm{\ensuremath{\mu}}}_{\mathrm{crit}}$, \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{} an effective coupling constant, \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}=${\mathrm{\ensuremath{\alpha}}}^{2}$exp\ensuremath{\Gamma}(r,\ensuremath{\theta})/4${\mathrm{\ensuremath{\Omega}}}^{2}$T, the stability of the soliton of the nonlinear Schr\odinger equation describing the traveling lattice distortion is destroyed. The soliton was found to be stable only for a sufficiently strong effective electron-phonon coupling \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\gtrsim}${\mathrm{\ensuremath{\mu}}}_{\mathrm{crit}}$. Because the soliton effect on the phonon displacements couples with the squeezing effect through \ensuremath{\Gamma} (or \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}), the nonadiabatic effects are either amplified, if the net effect of squeezing decreases \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}, or weakened, if the net effect of squeezing increases \ensuremath{\mu}\ifmmode\bar\else\textasciimacron\fi{}. The latter case can occur if the phonon displacement and/or the squeezing parameter are complex quantities. The many-phonon effects are shown to contribute only for large quantum fluctuations \ensuremath{\Omega}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\gtrsim}2${\mathrm{\ensuremath{\kappa}}}^{2}$ beyond the validity of the above condition for the use of the perturbation theory. The soliton ground state with \ensuremath{\lambda}\ensuremath{\gtrsim}(2T/3\ensuremath{\Omega}${)}^{1/3}$ is destabilized by the quantum fluctuations. \textcopyright{} 1996 The American Physical Society.