Magnetic trapping of neutral particles: Classical and quantum-mechanical study of a Ioffe-Pritchard type trap

Recently, we developed a method for calculating the lifetime of a particle inside a magnetic trap with respect to spin flips, as a first step in our efforts to understand the quantum mechanics of magnetic traps. The one-dimensional toy model that was used in this study was physically unrealistic because the magnetic field was not curl free. Here, we study, both classically and quantum mechanically, the problem of a neutral particle with spin S, mass m, and magnetic moment mu, moving in three dimensions in an inhomogeneous magnetic field corresponding to traps of the Ioffe-Pritchard "clover-leaf" and "baseball" type. Defining by omega(p), omega(z), and omega(r) the precessional, the axial, and the lateral vibrational frequencies, respectively, of the particle in the adiabatic potential V-eff =mu\B\, we find classically the region in the (omega(r)/omega(p))-(omega(z)/omega(p)) plane where the particle is trapped. Quantum mechanically, we study the problem of a spin-one particle in the same field. Treating omega(r)/omega(p) and omega(z)/omega(p) as small parameters for the perturbation from the adiabatic Hamiltonian, we derive a closed-form expression for the transition rate 1/T-esc of the particle from its trapped ground state. In the extreme cases, the expression for 1/T-esc reduces to 4 pi omega(r) exp(-2 omega(p)/omega(r)) for omega(p)much greater than omega(r)much greater than omega(z), to 8 root 2 pi root omega(p)omega(i)exp(-2 omega(p)/omega(i)) for omega(p)much greater than omega(r)=omega(z)equivalent to omega(i) and to root pi/2 omega(r)(omega(z)/omega(p))(3/2)exp(-2 omega(p)/omega(z)) for omega(p)much greater than omega(z)much greater than omega(r). (C) 2000 American Institute of Physics. [S0021-8979(00)10005-2].