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Physical Review A

Iterative Bogoliubov transformations and anharmonic oscillators

Rocío Jáuregui, J. Récamier

Published Sep 1, 1992• 27 citations
Publisher

Abstract

An algebraic method for studying quantum anharmonic oscillators is proposed. It consists in the iterative application of generalized (squeezing) Bogoliubov transformations to the usual creation and annihilation operators. The convergence of this procedure is analyzed for quartic, cubic plus quartic, and symmetric-double-well Hamiltonians. For these systems, we show the usefulness of the resulting harmonic-oscillator number states in calculating the energy spectra.

DOI
10.1103/physreva.46.2240

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An algebraic method for studying quantum anharmonic oscillators is proposed. It consists in the iterative application of generalized (squeezing) Bogoliubov transformations to the usual creation and annihilation operators. The convergence of this procedure is analyzed for quartic, cubic plus quartic, and symmetric-double-well Hamiltonians. For these systems, we show the usefulness of the resulting harmonic-oscillator number states in calculating the energy spectra.
We present a new method to study the eigenvalues and eigenfunctions of anharmonic oscillators. It involves a new class of Bogoliubov transformations and leads to the introduction of k-photon coherent states. We consider the Hamiltonians for the simple harmonic and anharmonic oscillators as the two generators of a Lie algebra, whose other generators may be found exactly, or up to any desired order of the perturbation parameter involved. An element of this Lie group, turning out to be the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one, thus facilitating the calculation of the eigenvalues and eigenfunctions of the former. The transformation of the ordinary annihilation and creation operators leads to generalized ones, corresponding to generalized oscillation modes, and also helps us out to introduce multi-photon coherent states. We specifically consider four-photon coherent states in detail and study time dependent position and momentum uncertainties in these states.
We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.
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.
The time evolution of a quantum single-quartic anharmonic oscillator is considered. The study is carried on in operational form by use of the raising and lowering operators of the oscillator. The equation of motion is solved by application of a new integration method based on iteration techniques, and the rigorous solutions that describe the time development of the displacement and momentum operators of the oscillator are obtained. These operators are presented as a Laplace transform and a subsequent inverse Laplace transform of suitable functionals. Finally, the results are employed to describe the time evolution of a quasiclassical anharmonic oscillator.
The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.
Using Lie algebra methods, we find a new class of Bogoliubov transformations which generalize the notion of squeezed states. The Hamiltonians for the simple harmonic and anharmonic oscillators, turn out to be the generators of a Lie group, whose other generators may be found exactly, or up to any desired order of the perturbation parameter. An element of this Lie group, which is realized as the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one. The transformation of the ordinary annihilation and creation operators under this unitary transformation leads to the introduction of multi-photon coherent states. We specifically consider four-photon coherent states in detail and study the time dependent position and momentum uncertainties in these states.