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Quantum mechanical uncertainties and exact transition probabilities for time dependent quadratic Hamiltonian with special results for Paul ion trap

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

Published May 12, 2013• 0 citations
OpenAlex

Abstract

Large quantum uncertainties for the Paul trap are obtained near the stability border. These results are important as a trapped ion can be stored only as long as the fluctuations are smaller than the trap dimensions. Exact transition probabilities for the forced Paul trap and harmonic oscillator are also obtained based on a simplified version of the propagator for time-dependent harmonic oscillator. The results have important implications, e.g., the methods of mass spectrometry can be applied only in the region of small fluctuations, which can be calculated exactly by our method. The classical trajectory of a general forced harmonic oscillator is also calculated which may exhibit chaotic behavior.

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Large quantum uncertainties for the Paul trap are obtained near the stability border. These results are important as a trapped ion can be stored only as long as the fluctuations are smaller than the trap dimensions. Exact transition probabilities for the forced Paul trap and harmonic oscillator are also obtained based on a simplified version of the propagator for time-dependent harmonic oscillator. The results have important implications, e.g., the methods of mass spectrometry can be applied only in the region of small fluctuations, which can be calculated exactly by our method. The classical trajectory of a general forced harmonic oscillator is also calculated which may exhibit chaotic behavior.
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.
The main objective of this study is to investigate the quantum mechanical lifetime of a trapped ion in a Paul trap. The motivation for this research is that there are several applications of trapped ion devices such as mass spectrometers, atomic clocks, and quantum computers. Active research is presently ongoing in the subject of an ion trapping by electric and magnetic fields. There are many classical studies of the characteristics of trapped ions, however there have been relatively few done quantum mechanically. It might be expected that purely quantum effects could play an important role in describing the trapped ions. So, those are the main factors behind the research on this topic. Time-dependent perturbation theory was used in this study which is to find the quantum mechanical lifetime of trapped ion in a Paul trap because time-dependent perturbation theory allows describing the complex system with the help of a simple system that can be calculated easily. It is shown that the lifetime of a trapped ion depends on its transition frequency and the transition between two allowed quantum states of the ion. The transition frequency of an ion has to be very close to the applied frequency of alternating current (AC). The results show that the lifetime of a trapped ion can be adjusted as desired for experimentation.
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.
Following the paper by Combescure [Ann. Phys. (N.Y.) 204, 113 (1990)], we apply the quantum singular time-dependent oscillator model to describe the relative one-dimensional motion of two ions in a trap. We argue that the model can be justified for low-energy excited states with the quantum numbers $n\ensuremath{\ll}{n}_{\mathrm{max}}\ensuremath{\sim}100$, provided the dimensionless constant characterizing the strength of the repulsive potential is large enough ${g}_{*}\ensuremath{\sim}{10}^{5}$. Time-dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center-of-mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential since the large value of g∗{g}_{*}g∗​ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.
We investigate the quantum motion of two ions stored in a Paul trap and interacting with a time-periodic laser field. In the pseudopotential approximation and large detuning condition, we find that the relative motion is independent of the laser field, but the exact centre-of-mass motion is closely related to the laser field. By adjusting the laser intensity and frequency, we can well control the quantum motion of the centre-of-mass. We illustrate some physical properties described by the centre-of-mass states, such as the squeezed coherent property, the widths and heights of the wavepackets of probability density, the classical-quantum correspondence, the resonance ladders of expectation energy and the transition probabilities between time-dependent quantum levels.