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Physical Review A

Propagator for the general time-dependent harmonic oscillator with application to an ion trap

Gal Harari, Y. Ben-Aryeh, A. Mann

Published Dec 2, 2011• 27 citations
Publisher

Abstract

We present the simplest possible formula for the propagator of the general time-dependent quadratic Hamiltonian, including linear terms. The method is based on the use of a linear time-dependent invariant and requires only the solution of a linear homogeneous second-order ordinary differential equation corresponding to the classical quadratic Hamiltonian. We give an example for the case of the Paul trap.

DOI
10.1103/physreva.84.062104

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We present the simplest possible formula for the propagator of the general time-dependent quadratic Hamiltonian, including linear terms. The method is based on the use of a linear time-dependent invariant and requires only the solution of a linear homogeneous second-order ordinary differential equation corresponding to the classical quadratic Hamiltonian. We give an example for the case of the Paul trap.
In this work we present the simplest generic form of the propagator for the time-dependent quadratic Hamiltonian. We manifest the simplicity of our method by giving explicitly the propagators for a free particle in time-dependent electric field and the Paul trap. Exact transition amplitudes and uncertainties are calculated analytically for the Paul trap and harmonic oscillator (HO). The results show that near the instability regions very large quantum mechanical uncertainties are obtained as demonstrated in a special figure. The method is also applied to calculating the trajectory of a classical forced time-dependent HO.
Abstract In this paper, we find the quantum propagator for a general time-dependent quadratic Hamiltonian. The method is based on the properties of the propagator and the fact that the quantum propagator fulfills two independent partial differential equations originating from Heisenberg equations for positions and momenta. As an application of the method, we find the quantum propagator for a linear chain of interacting oscillators for both periodic and Dirichlet boundary conditions. The state and excitation propagation along the harmonic chain in the absence and presence of an external classical source is studied and discussed. The location of the first maxima of the probability amplitude P(n, τ) is a straight line in the (n, τ)-plane, indicating a constant speed of excitation propagation along the chain.
A generalization of a driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered. The generalization, which gives a general quadratic Hamiltonian system, does not change the classical equation of motion. Based on the observation by Feynman and Hibbs, the propagators (kernels) of the systems are calculated from the classical action, in terms of solutions of the classical equation of motion: two homogeneous solutions and one particular solution. The kernels are then used to find wave functions that satisfy the Schr\"odinger equation. One of the wave functions is shown to be that of a Gaussian pure state. In every case considered, we prove that the kernel does not depend on the way of choosing the classical solutions, while the wave functions depend on the choice. The generalization, which gives a rather complicated quadratic Hamiltonian, is simply interpreted as applying unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.