We study, both classically and quantum-mechanically, the problem of a neutral particle with spin S, mass m and magnetic moment mue, moving in two-dimensions in an inhomogeneous magnetic field given by B_x=B'*x; B_y=-B'*y; B_z=B; We identify K, the ratio between the precessional frequency of the particle and its vibration frequency, as the relevant parameter of the problem. Classicaly, we find that when the magnetic moment is antiparallel to B, the particle is trapped provided that K
We study, both classically and quantum-mechanically, the problem of a neutral particle with spin S, mass m and magnetic moment mue, moving in two-dimensions in an inhomogeneous magnetic field given by B_x=B'*x; B_y=-B'*y; B_z=B; We identify K, the ratio between the precessional frequency of the particle and its vibration frequency, as the relevant parameter of the problem. Classicaly, we find that when the magnetic moment is antiparallel to B, the particle is trapped provided that K
We study, both classically and quantum mechanically, the problem of a neutral particle with a spin angular momentum S, mass m, and magnetic moment μ, moving in one dimension in an inhomogeneous magnetic field given by B=B0ẑ+B⊥′xŷ. This problem serves for us as a toy model to study the trapping of neutral particles. We identify K≡[S2(B⊥′)2/μmB03], 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 μ is antiparallel to B, the particle is trapped provided that K<0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin S=ℏ/2 particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find that the lifetime Tesc of the particle in its trapped ground state is Tesc=(Tvib/2π)(1/8πK)exp(2/K), where Tvib=2πmB0/μ(B⊥′)2 is the classical period of the particle when placed in the adiabatic potential V=μ|B|.
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 1D 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 3D 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, we find classically the region in the $(ømega_{r}% (omega_r -- omega_z) 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. We find that in the extreme cases, the expression for 1/T_{esc} is dominated by the largest of the two frequencies omega_r and omega_z.
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 μ, moving in three dimensions in an inhomogeneous magnetic field corresponding to traps of the Ioffe–Pritchard “clover-leaf” and “baseball” type. Defining by ωp, ωz, and ωr the precessional, the axial, and the lateral vibrational frequencies, respectively, of the particle in the adiabatic potential Veff =μ|B|, we find classically the region in the (ωr/ωp)−(ωz/ωp) plane where the particle is trapped. Quantum mechanically, we study the problem of a spin-one particle in the same field. Treating ωr/ωp and ωz/ωp as small parameters for the perturbation from the adiabatic Hamiltonian, we derive a closed-form expression for the transition rate 1/Tesc of the particle from its trapped ground state. In the extreme cases, the expression for 1/Tesc reduces to 4πωr exp(−2ωp/ωr) for ωp≫ωr≫ωz, to 82πωpωi exp(−2ωp/ωi) for ωp≫ωr=ωz≡ωi and to π/2 ωr(ωz/ωp)3/2exp(−2ωp/ωz) for ωp≫ωz≫ωr.
We have been studying the use of inhomogeneous magnetic fields for the trapping of neutral atoms via the interaction with their magnetic moments. Apart from gravitational forces (which may turn out to be of great importance in atom traps), only electromagnetic forces are of real interest for trapping neutral atoms, and these forces must necessarily arise from the interaction of multipole moments and inhomogeneous fields. In this paper we restrict ourselves to static traps and magnetic dipoles, even though optical, quasi‐static, and Stark effect traps have been discussed. We present some discussion of the field configurations that might be used, of the motion of ‘‘warm’’ trapped atoms (about 1 K), and of the states of ‘‘cold’’ trapped atoms (about 0.1 μK). The basic scale of the interaction is set by the ratio of the Bohr magneton to the Boltzmann constant, about 1.5 Tesla/Kelvin.
Magnetic trapping of laser-cooled neutral atoms has been demonstrated at NBS and is progressing in other laboratories. Confinement of any object requires exchanging kinetic for potential energy, and for neutral atoms, this necessarily derives from shifts of internal energy levels. This is implemented through the force experienced by the atomic magnetic moment in a carefully designed, highly inhomogeneous field. Since magnetic fields of convenient strength can shift atomic energy levels by only a few gigahertz (temperature ≌ 0.1 K), neutral atom traps are very shallow and must be loaded with very cold atoms. We present some motivations for using magnetic traps and then discuss the constraints and optimal configurations of various arrangements. For example, no trap can be isotropic. Next we discuss both the classical and quantum mechanical motions of trapped atoms. These motions are important for at least two reasons: First, several schemes under study for further cooling depend on knowing the position and velocity of the atoms, as well as the vector field at each point along the orbit. Second, magnetic traps depend on the atomic moment remaining aligned with the field as the atom orbits in the trap, and this precludes rapid motion through a low-field region.