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Journal of Applied Mathematics and Physics

Non-Perturbative Treatment of Quantum Mathieu Oscillator

Mohammed Janati Idrissi, Abdelaziz Fedoul, Salaheddine Sayouri

Published Jan 1, 2020• 0 citations
Publisher

Abstract

We study the evolution in time of the quantum Mathieu oscillator (QMO), according to the motion of a charged particle in a radio frequency Paul trap. We adopt non-perturbative treatment based on the quantized Floquet formalism together with the resonating averages method (RAM). We prove that we can develop solutions of the time-dependent Schrödinger equation of such a system, in terms of the simple harmonic oscillator wave functions. Numerical simulations of the analytical results are performed to show the coherence and the squeezed proprieties of the wave-packet of this system.

DOI
10.4236/jamp.2020.84054

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We study the evolution in time of the quantum Mathieu oscillator (QMO), according to the motion of a charged particle in a radio frequency Paul trap. We adopt non-perturbative treatment based on the quantized Floquet formalism together with the resonating averages method (RAM). We prove that we can develop solutions of the time-dependent Schrödinger equation of such a system, in terms of the simple harmonic oscillator wave functions. Numerical simulations of the analytical results are performed to show the coherence and the squeezed proprieties of the wave-packet of this system.
The quantum motion of a charged particle in a quadrupole radio-frequency trap is solved exactly in terms of the classical trajectories. Thus the quantum stability regions are exactly given in terms of the stability regions for the associated Mathieu equations, and quantum trapping is demonstrated inside these stability area. These exact results enable us to test a commonly believed «static effective potential approximation» for a particle in a rapidly oscillating time-periodic potential: our results exhibit serious limitations to this approximation. As a subproduct of our approach, we solve the eigenvalue problem for the Floquet operator of our system. This exactly solvable system should be a good starting point for the study of quantum instabilities under small perturbations Le mouvement quantique d'une particule chargee dans un piege quadripolaire de radio-frequence est exactement resoluble en termes des solutions du mouvement classique. Il en resulte que les regions de stabilite quantique sont exactement les regions de stabilite pour les equations de Mathieu correspondantes, et on demontre le piegeage quantique a l'interieur de ces regions. Ces resultats exacts permettent de tester «l'approximation de potentiel effectif statique» communement admise pour une particule dans un potentiel periodique dans le temps rapidement oscillant, et mettent en evidence de serieuses limitations a cette approximation. Comme sous produit a cette approche, on resout l'equation aux valeurs propres pour l'operateur de Floquet de ce systeme. Ce probleme exactement resoluble devrait etre un bon point de depart pour l'etude de la stabilite quantique sous de petites perturbations
We apply the Floquet theorem and the resonating averages method to solve the Schrödinger equation of the harmonic oscillator whose frequency varies periodically with time. We determine its Floquet states and its corresponding wave functions. The formalism provides an efficiency tool for analyzing the various transition probabilities as well as the uncertainty relation of the system. The effect of a periodic external force on the Floquet states and the two-photon algebra realization are shortly discussed.
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The authors review the theory of the harmonic oscillator with time-dependent frequency by means of an approach based on an operator constant of the motion. With the help of this operator constant we define the ground state, the excited states and a coherent state of the oscillator and discuss the time dependence of these states through their Wigner functions. The authors derive the Wigner function of an arbitrary state at time t evolving in the time-dependent harmonic potential. Moreover, they calculate the correlation coefficient between position and momentum, which appears in the Schrodinger uncertainty relation. The authors illustrate their results for the example of a charged particle in the Paul trap.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.