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Molecular Physics

Application of a two-step approach to the adiabatic approximation of coupled oscillators

Chen‐Shiung Hsue

Published Nov 1, 1984• 1 citation
Publisher

Abstract

A recently proposed two-step approach is applied to the adiabatic approximation of a coupled oscillator system. Near perfect agreement with exact quantum calculation has been obtained. Both the simplicity and the quality of this approach compare favourably with all the other methods in the coupled two-mode oscillator model.
DOI
10.1080/00268978400102791

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The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.
Adiabatic approximations are a powerful tool for simplifying nonlinear quantum dynamics, and are applicable whenever a system exhibits a hierarchy of time scales. Current interest in small nonlinear quantum systems, such as few-mode Bose-Hubbard models, warrants further development of adiabatic methods in the particular context of these models. Here we extend our recent work on a simple four-mode Bose-Hubbard model with two distinct dynamical time scales, in which we showed that among the perturbations around excited stationary states of the system is a slow collective excitation that is not present in the Bogoliubov spectrum. We characterized this mode as a resonant energy exchange with its frequency shifted by nonlinear effects, and referred to it as a second Josephson oscillation, in analogy with the second sound mode of liquid helium II. We now generalize our previous theory beyond the mean-field regime, and construct a general Bogoliubov free quasiparticle theory that explicitly respects the system's adiabatic invariant as well the exact conservation of particles. We compare this theory to the numerically exact quantum energy spectrum with up to 40 particles, and find good agreement over a significant range of parameter space.
The dynamics of a qubit in a structured environment is investigated theoretically. One point of view of the model is the spin-boson model with a Lorentz shaped spectral density. An alternative view is a qubit coupled to harmonic oscillator (HO), which in turn coupled to a Ohmic environment. Two different methods are applied and compared for this problem. One is a perturbation method based on a unitary transformation. Since the transformed hamiltonian is of rotating wave approximation (RWA) form, we call it the transformed rotating wave approximation (TRWA) method. And the other one is the numerically exact method of the quasi-adiabatic propagator path-integral (QUAPI) method. TRWA method can be applied from the first point of view. And the QUAPI method can applied from both points of views. We find that from the 1st point of view QUAPI only works well for large ΓΓΓ. Since the memory time is too long for the practical evaluation of QUAPI when ΓΓΓ is small. We call this treatment as QUAPI1. And from the 2nd point of view, QUAPI works well for small ΓΓΓ, since the non-adiabatic effect become more important as ΓΓΓ increases, one need smaller time-step and more steps to obtain accurate result which also quickly runs out the computational resources. This treatment is called QUAPI2. We find that the TRWA method works well for the whole parameter range of ΓΓΓ and show good agreement with QUAPI1 and QUAPI2. On the other hand, we find that the decoherence of the qubit can be reduced with increasing coupling between HO and bath. This result may be relevant to the design of quantum computer.