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Communications in Theoretical Physics

Laser Controlling Wavepacket Trains of a Paul Trapped Ion

Cai Li-Hua, Hai Wen-Hua, Wu Yun-Wen

Published Jun 1, 2006• 4 citations
Publisher

Abstract

We have studied the quantum and classical motions of a single Paul trapped ion interacting with a time-periodic laser field. By using the test-function method, we construct n exact solutions of quantum dynamics that describe the generalized squeezed coherent states with the expectation orbits being the corresponding classical ones. The space-time evolutions of the exact probability densities show some wavepacket trains. It is demonstrated analytically that by adjusting the laser intensity and frequency, we can control the center motions of the wavepacket trains. We also discuss the other physical properties such as the expectation value of energy, the widths and heights of the wavepackets, and the resonance loss of stability.

DOI
10.1088/0253-6102/45/6/025

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We have studied the quantum and classical motions of a single Paul trapped ion interacting with a time-periodic laser field. By using the test-function method, we construct n exact solutions of quantum dynamics that describe the generalized squeezed coherent states with the expectation orbits being the corresponding classical ones. The space-time evolutions of the exact probability densities show some wavepacket trains. It is demonstrated analytically that by adjusting the laser intensity and frequency, we can control the center motions of the wavepacket trains. We also discuss the other physical properties such as the expectation value of energy, the widths and heights of the wavepackets, and the resonance loss of stability.
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 investigate the secular motion of a single Paul-trapped ion in the Lamb-Dicke regime, which interacts with a sequence of standing laser pulses. By using the ansatz method, we get an exact quantum solution of the system. Based on the wave-packet trains described by the exact solution, we find that: i) The center, height and width of the wave-packet trains depend on the strength of laser pulses, the deformation and spread of the wave-packet trains can be controlled by adjusting the strength of laser. ii) Energy expectation values of the ion show jumps at the instantaneous switching on of the laser pulses. In the time intervals when the laser pulses are switched off some narrow energy bands are generated. iii) When the strength of laser pulses reaches a critical value, the system changes its stability.
By using Husimi’s test-function method, we construct n exact solutions of a single Paul trapped ion modelled by a quantum harmonic oscillator with a time-dependent ‘spring constant’. Any n th solution is a combination of the Fock state and coherent state, which describes a wavepacket train consisting of n + 1 packets. Its centre oscillates like the classical harmonic oscillator with variable frequency, while the width and height of each packet change simultaneously. When the deformation is small, it behaves like a propagated and non-deformed train, and the large deformation is identified with breathing of the wavepacket train.
We consider a harmonically trapped and strongly driven electron in a one-dimensional (1D) quantum wire. A set of exact complete solutions of the time-dependent Schrödinger equation is constructed, which shows the properties of generalized squeezed coherent states. The corresponding probability densities behave like some breathing and oscillating wave-packet-trains. The resonance ladders of the expectation energy are found and the transition probabilities between two stationary states are exactly calculated. The results reveal that the quantum motions of the considered system can be controlled by adjusting the laser driving field.
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.
Due to an infinite series solution of the classical Mathieu equation, using single-radio-frequency (rf) driving makes it difficult to manipulate motional states of a Paul-trapped ion. Here, we apply double-rf driving consisting of two external fields to generate the squeezed coherent states of simple forms. Stability parameter regions of the system are found in which both the first and second rf fields may be very strong but only the latter can be weak. Based on exact solutions, we investigate the time evolutions of expectation energies and controllable transitions between not only two stationary states, but also two nonstationary states. Such a quantum-control scenario may be tested experimentally.