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Journal of Physics C Solid State Physics

Quantum and semiclassical dynamics of a two-level system coupled to a bosonic degree of freedom

N Klenner, Jasmin Weis, M Doucha

Published Aug 30, 1986• 15 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/19/24/011

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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.
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 eigenvalue problem of a N-level system coupled to a bosonic degree of freedom is solved without using RWA. For that purpose, the bosonic degree of freedom is transformed to Bargmann's Hilbert space of analytical functions. In this representation the Schrodinger equation is a system of N coupled linear differential equations of first order. Using a discrete symmetry, these equations are simplified by a suitable transformation of the independent variable. Starting from the simplified equations, the authors develop a method to solve the eigenvalue problem of the N-level system. In addition, they present a simple approximate treatment and compare it with the exact results. The approximation turns out to be quite good up to outer level resonance and can be used to explain the differently structured regions in the energy spectra.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.
The ab initio description of evolutionary processes in extended electron-phonon systems (polaronic transport, excitonic transfer, etc) up to the present is beyond numerical accessibility, since it requires the simultaneous knowledge of all eigenfunctions and eigenvalues. Therefore, usually rough approximations are made, such as a semiclassical treatment. However, as we have shown in a recent paper, the full quantum-mechanical treatment drastically deviates from the semiclassical approximation (even in a qualitative manner). In the concept discussed here unitary product transformations are introduced, the constituents of which account for the two antagonistic tendencies inherent in every coupled electron-phonon Hamiltonian. We apply our procedure to the concrete case of the dimer-oscillator model by choosing for each of the antagonistic tendencies respectively a one parameter unitary operator, such that full analytical diagonalization is reached in the opposing limits of the Hamiltonian constituents. In the intermediate regime the two parameters of the transformation are suitably optimized. In this manner the generation of the full spectrum of eigensolutions involves two analytically fixed parameters only. The evolutionary behaviour resulting from our procedure is contrasted with the exact numerical result as well as with the one from the semiclassical approach and also with a more simple (`displacive') unitary transformation frequently used in the literature. It is shown that our calculation approaches the exact result in a satisfactory manner in all intrinsic physical parameter regimes (coupling and transfer) and overcomes the drastic shortcomings of previous calculations.
Statistical and dynamical properties of a quasiparticle coupled to polarization vibrations in a dimer model are investigated using a full quantum-mechanical approach. The propagation of the system is described in terms of spin-boson coherent states and compared to various quasiclassical descriptions of the system. The spectrum of the energy eigenvalues and the level spacing distributions of the system are calculated and turn out to be almost regular, especially in the strong coupling limit. It is demonstrated that the usual quasiclassical approach to this problem is not quite satisfactory. This is because the quantum fluctuations in the spin variable give rise to entangled states and correlations which hence should be included in a quasiclassical description. We show that a limited but satisfactory quasiclassical approximation may be given using a suitable integrable reference Hamiltonian.