We study, both classically and quantum mechanically, the problem of a neutral particle with a spin angular momentum S, mass m, and magnetic moment , moving in one dimension in an inhomogeneous magnetic field given by BϭB 0 ẑϩB Ќ Ј xŷ. This problem serves for us as a toy model to study the trapping of neutral particles. We identify Kϵͱ͓S 2 (B Ќ Ј ) 2 /mB 0 3 ͔, which is the ratio between the precessional frequency of the particle and its vibrational frequency, as the relevant parameter of the problem. Classically, we find that when is antiparallel to B, the particle is trapped provided that KϽ0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin Sϭប/2 particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find that the lifetime T esc of the particle in its trapped ground state is T esc ϭ(T vib /2) ϫ(1/ͱ8K)exp(2/K), where T vib ϭ2ͱmB 0 /(B Ќ Ј ) 2 is the classical period of the particle when placed in the adiabatic potential Vϭ͉B͉.

In this paper we analyze the dynamic stability of the hovering magnetic top from first principles without using any preliminary assumptions. We write down the equations of motion for all six degrees of freedom and solve them analytically around the equilibrium solution. Using this solution we then find conditions which the height of the hovering top above the base, its total mass, and its spinning speed have to satisfy for stable hovering.The calculation presented in this paper can be used as a guide to the analysis and synthesis of magnetic traps for neutral particles.

In this paper we analyze the spinning motion of the hovering magnetic top. We have observed that its motion looks different from that of a classical top. A classical top rotates about its own axis which precesses around a vertical fixed external axis. The hovering magnetic top, on the other hand, has its axis slightly tilted and moves rigidly * Also with the Department of Physics, University of California, San Diego, La Jolla, 92093 CA, USA 1 as a whole about the vertical axis. We call this motion synchronous, because in a stroboscopic experiment we see that a point at the rim of the top moves synchronously with the top axis.We show that the synchronous motion may be attributed to a small deviation of the magnetic moment from the symmetry axis of the top. We show that as a consequence, the minimum angular velocity required for stability is given by 4µHI 1 /I 2 3 for I 3 > I 1 and by µH/(I 3 − I 1 ) for I 3 < I 1 . Here, I 3 and I 1 are the principal and secondary moments of inertia, µ is the magnetic moment, and H is the magnetic field. For comparison, the minimum angular for a classical top is given by 4µHI 1 /I 2 3 both for I 3 < I 1 and for I 3 > I 1 . We also give experimental results that were taken with a top whose moment of inertia I 1 can be changed. These results show very good agreement with our calculations.

We calculate exactly the modes of motion of the Time-averaged Orbiting Potential (TOP) trap with its four degrees of freedom, namely the three translations and the spin, taken into account. We find that, when gravity is neglected, there are two parameters in the problem namely, the angular velocity of the rotating field and its strength. We present the stability diagram in these parameters. We find the mode frequencies calculated from the time-averaged potential model used by the inventors of the TOP is an excellent approximation to our exact results. However, for other parameters, this may not be the case.

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