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Birkhäuser Boston eBooks

Upper Bounds on Bogolubov’s Inner Product: Quantum Systems of Anharmonic Oscillators

M. Corgini

Published Jan 1, 2000• 1 citation
Publisher

Abstract

An upper bound on the so-called Bogolubov Inner Product for creation and annihilation bose operators is obtained in the case of certain quantum systems of anharmonic oscillators as a straightforward application of domination of semigroups. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-1372-7_3

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An upper bound on the so-called Bogolubov Inner Product for creation and annihilation bose operators is obtained in the case of certain quantum systems of anharmonic oscillators as a straightforward application of domination of semigroups. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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We construct a quantum Markov semigroup on the von Neumann algebra B ( L 2 (ℝ + ;ℂ)) of all bounded operators on a Hilbert space L 2 (ℝ + ;ℂ) with the following property: the biggest *-subalgebra of B ( L 2 (ℝ + ;ℂ)) contained in the domain of the infinitesimal generator is not σ-weakly dense. Our semigroup is an extension of a classical Markov semigroup on L ∞ (ℝ + ;ℂ) with the same property. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
For a quantum system with energy E, there is a limitation in quantum computation which is identified by the minimum time needed for the state to evolve to an orthogonal state. In this paper, we will compute the minimum time of orthogonalization (i.e. quantum speed limit) for a simple anharmonic oscillator and find an upper bound on the rate of computations. We will also investigate the growth rate of complexity for the anharmonic oscillator by treating the anharmonic terms perturbatively. More precisely, we will compute the maximum rate of change of complexity and show that for even order perturbations, the rate of complexity increases while for the odd order terms it has a decreasing behavior.
We discuss a recently proposed method [R. J\'auregui and J. R\'ecamier, Phys. Rev. A 46, 2240 (1992)] based on the application of iterative Bogoliubov transformations to anharmonic oscillators and show that, if the algorithm converges, one easily obtains the final result directly in one step. We prove that the Bogoliubov transformation can be written in terms of scaling and translation parameters and present exact results for the coefficients of the Bogoliubov transformation for some selected examples.