The Quantum Mechanics of Particles in Time-Dependent Quadrupole Fields

Quadrupole Fields, Roy J. Glauber

We solve the quantum mechanical problem of the motion of charged particle in a quadrupole field that varies arbitrarily with time. The time-dependent wave functions are shown to be in a simple one-to-one correspondence with the wave functions of an ordinary static-field harmonic oscillator. We define and generate a set of coherent states for the charged particle that provide Gaussian localization about the classical particle trajectories. When the field variation is periodic in time, these minimum uncertainty states are shown to pulsate periodically in size and in phase. Under the action of external fields, coherent states retain their coherence. We discuss the way in which the Kapitza pseudopotential approach approximates the present exact results.

pAthanasios Petridisp

M. Tohyama

The monopole and quadrupole oscillations of a Bose-Einstein condensate (BEC) gas trapped in a spherically symmetric potential are calculated using the time-dependent Hartree approximation which enables us to use an orthonormal single-particle basis for the description of both the condensate and noncondensate components. It is found that the monopole mode has a much smaller damping rate than the quadrupole mode. The obtained temperature dependence of the frequency shifts and the damping rates of the collective oscillations are found to be consistent with experimental data obtained for BEC gases in anisotropic trapping potentials. A comparison between the time-dependent Hartree approximation and the time-dependent Hartree-Fock approximation is also made and it is found that there is a rather small difference.

Sergei Nagaitsev, V. Danilov

Quadrupole ion traps can be transformed into nonlinear traps with integrable motion by adding special electrostatic potentials. This can be done with both stationary potentials (electrostatic plus a uniform magnetic field) and with time-dependent electric potentials. These potentials are chosen such that the single particle Hamilton-Jacobi equations of motion are separable in some coordinate systems. The electrostatic potentials have several free adjustable parameters allowing for a quadrupole trap to be transformed into, for example, a double-well or a toroidal-well system. The particle motion remains regular, non-chaotic, integrable in quadratures, and stable for a wide range of parameters. We present two examples of how to realize such a system in case of a time-independent (the Penning trap) as well as a time-dependent (the Paul trap) configuration.

Zoltán Papp