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Physical Review A

Algebraic time-ordering techniques and harmonic oscillator with time-dependent frequency

G. Dattoli, S. Solimeno, A. Torre

Published Oct 1, 1986• 41 citations
Publisher

Abstract

The Wei-Norman algebraic techniques for time-ordering problems are applied to the study of the evolution of quantum states ruled by a harmonic-oscillator Hamiltonian with a time-dependent frequency. The slowly varying frequency case is studied; the adiabatic theorem is rederived together with higher-order corrections. The analogy with the propagation of a paraxial beam through a self-focusing fiber is also pointed out.

DOI
10.1103/physreva.34.2646

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The Wei-Norman algebraic techniques for time-ordering problems are applied to the study of the evolution of quantum states ruled by a harmonic-oscillator Hamiltonian with a time-dependent frequency. The slowly varying frequency case is studied; the adiabatic theorem is rederived together with higher-order corrections. The analogy with the propagation of a paraxial beam through a self-focusing fiber is also pointed out.
Abstract In this work, we give a quantitative answer to the question: how sudden or how adiabatic is a frequency change in a quantum harmonic oscillator (HO)? We do that by studying the time evolution of a HO which is initially in its fundamental state and whose time-dependent frequency is controlled by a parameter (denoted by ϵ ) that can continuously tune from a totally slow process to a completely abrupt one. We extend a solution based on algebraic methods introduced recently in the literature that is very suited for numerical implementations, from the basis that diagonalizes the initial Hamiltonian to the one that diagonalizes the instantaneous Hamiltonian. Our results are in agreement with the adiabatic theorem and the comparison of the descriptions using the different bases together with the proper interpretation of this theorem allows us to clarify a common inaccuracy present in the literature. More importantly, we obtain a simple expression that relates squeezing to the transition rate and the initial and final frequencies, from which we calculate the adiabatic limit of the transition. Analysis of these results reveals a significant difference in squeezing production between enhancing or diminishing the frequency of a HO in a non-sudden way.
The unitary operator V ( t ) that diagonalizes the time-dependent quadratic Hamiltonian (TDQH) into a time-dependent harmonic oscillator (TDHO) is obtained using a Lie algebra. The method involves a factorization of the TDQH into a TDHO through a unitary Bogoliubov transformation in terms of creation and annihilation operators with time-dependent coefficients. It is shown that this operator can be easily achieved by means of the factorization, together with the commonly known Wei–Norman theorem. We discuss the conditions under which this unitary operator converges to the evolution operator U ( t ) of the Schrödinger equation for the TDQH, giving then a straightforward calculation of the evolution operator with respect to the procedures published in the literature.
Abstract The solution of the time‐dependent Schrödinger's equation for a perturbed harmonic oscillator is obtained using a solvable Lie algebra. We choose a harmonic oscillator interacting with a one‐mode field, where the perturbation happens to be periodic in time. This leads to one of the simplest Floquet problems. Using the Wei–Norman theorem, the Floquet wave function is obtained as well as the semiclassical Floquet shift in the energy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
The temporal evolution of the state vector relative to a harmonic oscillator with time-dependent frequency is examined. The Schrödinger equation is solved by choosing the instantaneous eigenstates of the Hamiltonian as the basis, thus getting an infinite set of coupled linear differential equations. This formulation is particularly suitable for studying the cases in which the Hamiltonian undergoes a very slow or a sudden variation from an initial constant value into a final one. A rigorous proof of adiabatic invariance to all orders in the slowness parameter is given for the transition probabilities. An application to the evolution of an initially coherent state is made.