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Journal of Physics A Mathematical and General

Harmonic oscillator subject to parametric pulses: an amplitude (Milne) oscillator approach

Karl-Erik Thylwe, H. J. Korsch

Published Apr 10, 2001• 11 citations
Publisher

Abstract

A harmonic oscillator subject to a parametric pulse is examined. The aim of the paper is to present a new theory for analysing transitions caused by parametric pulses. The new theoretical notions which are introduced relate the pulse parameters in a direct way with transition matrix elements. The harmonic-oscillator transitions are expressed in terms of the asymptotic properties of a companion oscillator, the Milne (amplitude) oscillator. A traditional phase-amplitude decomposition of the harmonic-oscillator solutions results in the so-called Milne's equation for the amplitude, and the phase is determined by an exact relation to the amplitude. This approach is extended in the present analysis with new relevant concepts and parameters for pulse dynamics of classical and quantal systems. The amplitude oscillator has a particularly nice numerical behaviour. In the case of strong pulses it does not possess any of the fast oscillations induced by the pulse on the original harmonic oscillator. Furthermore, the new dynamical parameters introduced in this approach are closely related to the relevant characteristics of the pulse. The relevance to quantum mechanical problems such as reflection and transmission from a localized well and the mechanical problem of controlling vibrations is illustrated.

DOI
10.1088/0305-4470/34/16/317

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A harmonic oscillator subject to a parametric pulse is examined. The aim of the paper is to present a new theory for analysing transitions caused by parametric pulses. The new theoretical notions which are introduced relate the pulse parameters in a direct way with transition matrix elements. The harmonic-oscillator transitions are expressed in terms of the asymptotic properties of a companion oscillator, the Milne (amplitude) oscillator. A traditional phase-amplitude decomposition of the harmonic-oscillator solutions results in the so-called Milne's equation for the amplitude, and the phase is determined by an exact relation to the amplitude. This approach is extended in the present analysis with new relevant concepts and parameters for pulse dynamics of classical and quantal systems. The amplitude oscillator has a particularly nice numerical behaviour. In the case of strong pulses it does not possess any of the fast oscillations induced by the pulse on the original harmonic oscillator. Furthermore, the new dynamical parameters introduced in this approach are closely related to the relevant characteristics of the pulse. The relevance to quantum mechanical problems such as reflection and transmission from a localized well and the mechanical problem of controlling vibrations is illustrated.
An exact dynamical parametrization of pulse-induced transition amplitudes in a Rosen–Zener- or Nikitin-type two-level system is constructed. The three dynamical parameters are closely related to the shape of the interaction pulse and are convenient to calculate. The Milne equation with a complex coefficient function is essential for these calculations. Its complex solution is non-oscillatory and makes the computation of transition probabilities efficient. The paper reviews the quantum calculations for the rectangular pulse, which has well-defined duration and strength. By comparing transition matrix elements from a rectangular pulse with those of a general symmetric pulse, we introduce effective strength and effective duration for a general pulse. It is also possible to define an equivalent rectangular pulse, with respect to the transition probabilities, for each general pulse.
In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.
The quantum harmonic oscillator is a model system that is actively used in many applications of quantum and nonlinear physics, molecular chemistry and biophysics. In this paper, a formula is obtained for the probability amplitude of the transition of a quantum harmonic oscillator from an oscillatory state with the numberν\nuνto a state with the numberν+n\nu+nν+n, which is affected by an external force depending on time using a method based on the Magnus and Baker-Campbell-Hausdorff decomposition. A formula is obtained for the special case of interaction of a quantum harmonic oscillator with an ultrashort pulse field. As an example, graphs of the probabilities of the oscillator transition from the first oscillatory state to the second and third, the average energy received and the rms fluctuation of the energy received for different values of the ratio of the carrier frequency of the pulse and the frequency of the oscillator are constructed. Conclusions are drawn about the correctness of using the concept of average energy in relation to this problem.
Abstract The dynamics of time evolution of quantum oscillator excitation by electromagnetic pulses is investigated theoretically for an arbitrary field amplitude in a pulse. We consider a harmonic oscillator without damping and excitation between stationary states. The general formula for the excitation of quantum states as a function of time is derived in terms of instantaneous energy of an associated classical oscillator in the field of an electromagnetic pulse. The derived expression is used in detailed analysis of the time dependence of the quantum oscillator excitation probability beyond the range of perturbation theory for various pulse parameters including total excitation from the ground state, excitation from excited states, and excitation spectra.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.
By the time a student of physics is ready to tackle quantum mechanics he has become familiar with the classical harmonic vibrator through many examples, and knows the crucial role it played in the development of Planck's ideas. It is natural then to concentrate on the mathematical aspects of the harmonic oscillator equation in quantum mechanics, the solution of Schrödinger's equation, normalization of the wave-function, calculation of mean values and of matrix elements, leaving the physics to look after itself. At a more advanced level the harmonic vibrator provides the entry into field theories, second quantization etc., and as a general rule it tends to be viewed more as a vehicle for instruction in more interesting matters than as a physical system having its own considerable interest and importance. Here we shall seek to redress the balance and study the vibrator as a thing in itself, without losing sight of the variety of physical problems to which the results can be applied. One especially important set of applications, however, will get little attention at this stage – the vibrations of compound bodies and of extended physical systems provide such wealth of interest as to justify a volume to themselves; and it is to such matters that the whole of the third part of this work will be devoted. As soon as one begins to contemplate the harmonic vibrator one becomes aware of the exceptional nature of its behaviour, in that it conforms more closely than any other system to the classical rules.