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Physica Scripta

Exact formulated approach to transition probabilities of linearly driven parametric oscillators

Wensen Liu

Published Mar 1, 2009• 0 citations
Publisher

Abstract

The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.

DOI
10.1088/0031-8949/79/03/035001

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The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.
Abstract We analyze transition probabilities of harmonic oscillator system with spatial LQC (Linear-Quadratic-Cubic) perturbation in time-dependent. This system initially was in the ground state with no perturbation at t < 0, then at t ≥ 0, the system is perturbed by spacial LQC perturbation in time-dependent until t → ∞. We use the time-dependent perturbation theory to analyze this problem. In the initial state, before there is no perturbation, we define the ground state with the base ket of harmonic oscillator without perturbation. Next, when the perturbation is applied to the system, we compute the transition amplitude base on the system state presented above and then we get total wave function that depends on time. By getting this wave function, we can compute transition probability for the system. As a result, there are three transition probabilities, namely the transitions from the ground state to the first, second, and third excited state. There is no transition to others.
The transition probabilities for a one-dimensional harmonic oscillator with a time-dependent interaction potential are obtained. Two different bilinear approximations to the interaction potential are obtained by expanding it in Taylor series around two reference coordinate values, namely, the equilibrium position q0=0 and the expectation value of the coordinate q0=〈q〉, keeping only first- and second-order terms. The effect of the choice of q0 on the accuracy of the results is investigated for several values of the interaction potential parameters. The model posed by the semiclassical approach to the collision between a structureless particle and a harmonic oscillator via an exponential interaction potential is chosen as an illustrative example. The expansion around 〈q〉 is found to be related to an iterative procedure, based on Lie algebraic methods, recently proposed to treat time-dependent quantum-mechanical problems.
We present a Lie algebraic approach to a Hamiltonian class covering driven, parametric quantum harmonic oscillators where the parameter set-mass, frequency, driving strength, and parametric pumping-is time-dependent. Our unitary-transformation-based approach provides a solution to our general quadratic time-dependent quantum harmonic model. As an example, we show an analytic solution to the periodically driven quantum harmonic oscillator without the rotating wave approximation; it works for any given detuning and coupling strength regime. For the sake of validation, we provide an analytic solution to the historical Caldirola-Kanai quantum harmonic oscillator and show that there exists a unitary transformation within our framework that takes a generalized version of it onto the Paul trap Hamiltonian. In addition, we show how our approach provides the dynamics of generalized models whose Schrödinger equation becomes numerically unstable in the laboratory frame.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
A variety of dynamics in this world can be approximately treated by a model: a driven and damped parametric oscillator. An intensive investigate of this time-dependent model from both classical and quantum mechanical points of view provides a dynamical insight to resolve the classical and quantum dynamics in order to understand the more complicated time-dependent evolutions not only happens in the macroscopic classical scale for the synchronized behaviors but also in the coherent microscopic scale for a quantum state control. By using a Lie transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamical behaviors of the driven and damped parametric oscillator to give a universal physical mechanism for the classical parametric control on the quantum state evolution in a dissipation system.
We use an algebraic method to model the molecular collision dynamics of a collinear triatomic system. Beginning with a forced oscillator, we develop a mathematical framework upon which inelastic and reactive collisions are modeled. The model is considered algebraic because it takes advantage of the properties of a Lie algebra in the derivation of a time-evolution operator. The time-evolution operator is shown to generate both phase-space and quantum dynamics of a forced oscillator simultaneously. The model is considered semi-classical because only the molecule's internal degrees-of-freedom are quantized. The relative translation between the colliding atom and molecule in an exchange reaction (AB+C ->A+BC) contains no bound states and any possible tunneling is neglected so the relative translation is treated classically. The purpose of this dissertation is to develop a working model for the quantum dynamics of a collinear reactive collision. After a reliable model is developed we apply statistical mechanics principles by averaging collisions with molecules in a thermal bath. The initial Boltzmann distribution is of the oscillator energies. The relative velocities of the colliding particles is considered a thermal average. Results are shown of quantum transition probabilities around the transition state that are highly dynamic due to the coupling between the translational and transverse coordinate.