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Jerusalem Symposia on Quantum Chemistry and Biochemistry

Tunneling and Dynamic Tunneling by an Algebraic Approach

Raphael David Levine

Published Jan 1, 1986• 1 citation
Publisher
DOI
10.1007/978-94-009-4752-8_1

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We study the quantum dynamics of the Bose-Hubbard model on a ladder formed by two rings coupled by the tunneling effect. By implementing the Bogoliubov approximation scheme, we prove that, despite the presence of the inter-ring coupling term, the Hamiltonian decouples in many independent sub-Hamiltonians ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ associated with momentum-mode pairs $\ifmmode\pm\else\textpm\fi{}k$. Each sub-Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{k}$ is then shown to be part of a specific dynamical algebra. The properties of the latter allow us to perform the diagonalization process, to find the energy spectrum and the conserved quantities of the model, and to derive the time evolution of important physical observables. We then apply this solution scheme to the simplest possible closed ladder, the double trimer. After observing that the excitations of the system are weakly populated vortices, we explore the corresponding dynamics by varying the initial conditions and the model parameters. Finally, we show that the inter-ring tunneling determines a spectral collapse when approaching the border of the dynamical-stability region.
The paper describes a numerical procedure for simulating, in the frequency domain, the large-signal quantum behaviour of superconducting tunnel-junction circuits. It is shown that the steady-state dynamics of a circuit can be decomposed by using the method of harmonic balance, and that the resulting system of coupled nonlinear algebraic equations can be solved through harmonic Newton. The presentation is generalised in a number of ways: first, the scheme is described in terms of a generic circuit comprising a tunnel junction and a Thevenin voltage source; secondly, the circuit equations are normalised to the gap voltage and the gap current of the tunnel barrier; and thirdly, an imperfect junction is modelled by convolving an ideal BCS current/voltage characteristic with a Gaussian function.