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Journal of Physics A Mathematical and General

Energy gaps of a trapped ion interacting with a laser field

Wenhua Hai, Mang Feng, Xiwen Zhu, Lei Shi, Kelin Gao, Ximing Fang

Published Nov 11, 1999• 21 citations
Publisher

Abstract

We consider the secular motion of an ion stored in a Paul trap and interacting with a standing laser field. It is shown that the Schrödinger equation of the system is analogous to the linearized equation of a classically chaotic system. From the mathematical simulation between the two systems we reveal several new properties in the fully quantum treatment. The results lead to some parameter and energy gaps in which the motional ground state does not exist.

DOI
10.1088/0305-4470/32/47/306

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We consider the secular motion of an ion stored in a Paul trap and interacting with a standing laser field. It is shown that the Schrödinger equation of the system is analogous to the linearized equation of a classically chaotic system. From the mathematical simulation between the two systems we reveal several new properties in the fully quantum treatment. The results lead to some parameter and energy gaps in which the motional ground state does not exist.
A classically chaotic system consisting of a Paul trapped ion and a sequences of standing laser pulses is treated quantum-mechanically. Under the circumstance of time-dependence, we derive the transition probability from the ion's motional state to ', and find, in the first-order approximation, the classically chaotic character disappears. Theoretical analysis and numerical calculations show that by regulating the phase parameter we can control the transition probability. When reaches some specific values, the transition from the state to ' is forbidden and, for some laser periods, resonance occurs, which leads to the corresponding transitions between different motional states. The time-evolution of an initial motional state just over one period is also studied in detail.
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.
We study quantum motion around a classical heteroclinic point of a single trapped ion interacting with a strong laser standing wave. We construct a set of exact coherent states of the quantum system and from the exact solutions reveal that quantum signatures of chaos can be induced by the adiabatic interaction between the trapped ion and the laser standing wave, where the quantum expectation values of position and momentum correspond to the classically chaotic orbit. The chaotic region on the phase space is illustrated. The energy crossing and quantum resonance in time evolution and the exponentially increased Heisenberg uncertainty are found. The results suggest a theoretical scheme for controlling the unstable regular and chaotic motions.
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.