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Communications in Theoretical Physics

Squeezing Properties of the Ion Traps

Feng Mang, Juhao Wu, Wang Kelin

Published Dec 15, 1997• 3 citations
Publisher

Abstract

With time and space transformation, we first solve the Schrödinger equation of the time-dependent harmonic oscillator (TDHO) system. The properties of the squeezing in the case of , i.e., the Paul trap and , namely a stable frequency interfered by a single-pulsing-type disturbance, are investigated by using function series expansion. Results in the former case are compared with the numerical calculations carried out before. It is found that the condition related to the most strong squeezing is the direct result of our method, not any approximation is needed. Solutions of the latter one are valuable for the study of the ion trap, where different cases are studied in detail. Finally we made a discussion for the measurement and observation of a trapped ion by using squeezing properties.

DOI
10.1088/0253-6102/28/4/385

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With time and space transformation, we first solve the Schrödinger equation of the time-dependent harmonic oscillator (TDHO) system. The properties of the squeezing in the case of , i.e., the Paul trap and , namely a stable frequency interfered by a single-pulsing-type disturbance, are investigated by using function series expansion. Results in the former case are compared with the numerical calculations carried out before. It is found that the condition related to the most strong squeezing is the direct result of our method, not any approximation is needed. Solutions of the latter one are valuable for the study of the ion trap, where different cases are studied in detail. Finally we made a discussion for the measurement and observation of a trapped ion by using squeezing properties.
"We investigate quantum dynamics for an ion confined within an oscillating quadrupole field, starting from two well known and elegant approaches. It is established that the Hamilton equations of motion, in both Schrodinger and Heisenberg representations, are equivalent to the Hill equation. One searches for a linear independent solution associated to a harmonic oscillator (HO). An adiabatic invariant, which is also a constant of motion, is introduced based on the Heisenberg representation. Thus, the state of the non-autonomous system can be determined at any subsequent moment of time. The quantum states for trapped ions are demonstrated to be Fock (number) states, while the exact solutions of the Schrodinger equation for a trapped ion are exactly the quasienergy states. Semiclassical dynamics is also investigated for many-body systems of trapped ions, where the wavefunction associated to the Schrodinger equation is prepared as a Gauss package multiplied by a Hermite polynomial. We also discuss time evolution for the system under investigation and supply the propagator."
We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general three-dimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the ('Floquet–Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small 'peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.
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.