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Physical review. B, Condensed matter

Coupled quasiparticle-boson systems: The semiclassical approximation and discrete nonlinear Schrödinger equation

M. I. Salkola, A. R. Bishop, V. M. Kenkre, S. Raghavan

Published Aug 1, 1995• 46 citations
Publisher

Abstract

The validity of the semiclassical approximation is studied for a system comprising one quasiparticle coupled to a boson degree of freedom. Using a two-site Holstein model as an example, it is shown that the semiclassical approximation becomes exact in a nontrivial adiabatic limit. Furthermore, in the model's polaron regime, there exists a hierarchy of time scales that rationalizes the quantum dynamics of the Holstein model. For the single-mode case considered, the discrete nonlinear Schr\"odinger equation is found to be valid only in a highly limited antiadiabatic regime.

DOI
10.1103/physrevb.52.r3824

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The validity of the semiclassical approximation is studied for a system comprising one quasiparticle coupled to a boson degree of freedom. Using a two-site Holstein model as an example, it is shown that the semiclassical approximation becomes exact in a nontrivial adiabatic limit. Furthermore, in the model's polaron regime, there exists a hierarchy of time scales that rationalizes the quantum dynamics of the Holstein model. For the single-mode case considered, the discrete nonlinear Schr"odinger equation is found to be valid only in a highly limited antiadiabatic regime.
The validity of the discrete nonlinear Schr\"odinger equation and related semi-classical transport equations is studied with the help of an exact calculation of the spin-boson system in the experimentally relevant context of the fluorescence depolarization of stick dimers. The consequences of an initial excitation corresponding to low-energy dynamics are investigated. A number of features are found including the sensitive role of the polarization angle of incident light in determining the regimes of validity. Our earlier finding that the semiclassical approximation improves in validity in a well defined ``massive oscillator'' limit involving small phonon frequency is borne out. In the opposite limit of large phonon frequency, the semiclassical approximation and the discrete nonlinear Schr\"odinger equation are found to coincide with each other but to differ significantly from the exact evolution. \textcopyright{} 1996 The American Physical Society.
The semiclassical and quantum dynamics of two ultrastrongly coupled nonlinear resonators cannot be explained using the discrete nonlinear Schrödinger equation or the Bose-Hubbard model, respectively. Instead, a model beyond the rotating wave approximation must be studied. In the semiclassical limit this model is not integrable and becomes chaotic for a finite window of parameters. For the quantum dimer we find corresponding regions of stability and chaos. The more striking consequence for both semiclassical and quantum chaos is that the tunneling time between the sites becomes unpredictable. These results, including the transition to chaos, can be tested in experiments with superconducting microwave resonators.
The Hamiltonian of a two-level system coupled to a boson mode is transformed to Bargmann's Hilbert space of analytical functions. The quantum dynamics is calculated with a suitable choice of a complete set of orthonormal functions in Bargmann space. A semiclassical approximation which is formulated as a dynamical Hamiltonian system is compared with the quantum dynamics. For the parameters under consideration, semiclassical dynamics is not at all appropriate to approximate the quantum dynamics. These parameters are suited to view the system as a two-site model of small-polaron dynamics.
Self-trapping has been traditionally studied on the assumption that quasiparticles interact with harmonic phonons and that this interaction is linear in the displacement of the phonon. To complement recent semiclassical studies of anharmonicity and nonlinearity in this context, we present below a fully quantum-mechanical analysis of a two-site system, where the oscillator is described by a tunably anharmonic potential, with a square well with infinite walls and the harmonic potential as its extreme limits, and wherein the interaction is nonlinear in the oscillator displacement. We find that even highly anharmonic polarons behave similar to their harmonic counterparts in that self-trapping is preserved for long times in the limit of strong coupling, and that the polaronic tunneling time scale depends exponentially on the polaron binding energy. Further, in agreement, with earlier results related to harmonic polarons, the semiclassical approximation agrees with the full quantum result in the massive oscillator limit of small oscillator frequency and strong quasiparticle-oscillator coupling.
We adapt a variational procedure to calculate ground-state properties of the Holstein model in the adiabatic limit. At strong coupling, this adaption leads to rapid convergence of results. The intermediate coupling regime is further handled with an adaptive algorithm. We also use semiclassically derived results for the adiabatic end point, along with weak coupling perturbation theory. These establish weak and strong coupling (or large and small polaron, respectively) regimes in two dimensions or higher. As is well known, these are connected smoothly, but the crossover becomes increasingly abrupt as the phonon frequency decreases.
We calculate the real space and time formation of a polaron quasiparticle from a bare electron. The time-dependent Schr\"odinger equation for the Holstein model of electron-phonon coupling is numerically integrated in a large Hilbert space to obtain the time evolution of the electron and phonon densities and the electron-phonon (el-ph) correlation functions. The quantum dynamical nature of the phonons is preserved. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. A potential barrier between the quasifree and heavy polaron states exists in dimensions $D\ensuremath{\geqslant}2$, consistent with earlier adiabatic theory. We compare to recent experiments.