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Physical Review A

Resonant cooling in position-dependent magnetic fields

Jayme De Luca, Reginaldo de Jesus Napolitano, Vanderlei Salvador Bagnato

Published Mar 1, 1997• 4 citations
Publisher

Abstract

We consider the dynamics of a neutral atom of magnetic moment \ensuremath{\mu} in a spatially dependent magnetic field. Interaction with a resonant radio frequency introduces a peculiar nonlinear dynamics for the slow motion of the center of mass. We derive asymptotically exact equations for the slow dynamics, which are compared to the numerical solution of the full nonlinear problem. We also calculate the frequency of small oscillations for resonant trapped motions.

DOI
10.1103/physreva.55.r1597

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American Journal of Physics•Apr 1, 2000•10 citations
We consider the dynamics of a neutral atom of magnetic moment \ensuremath{\mu} in a spatially dependent magnetic field. Interaction with a resonant radio frequency introduces a peculiar nonlinear dynamics for the slow motion of the center of mass. We derive asymptotically exact equations for the slow dynamics, which are compared to the numerical solution of the full nonlinear problem. We also calculate the frequency of small oscillations for resonant trapped motions.
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.
An integrable dynamical problem of a classical neutral particle interacting with an inhomogeneous magnetic field via a magnetic dipole moment is considered; this has been studied quantum mechanically (1978). The equation for the centre-of-mass velocity is of second order resembling that of an anharmonic oscillator with velocity replacing position in the equation. The velocity undergoes curious oscillations, where the spin and centre-of-mass degrees of freedom exchange energy, although there is also, in general, a uniform component to the velocity. An application to the motion of a classical 'neutron' in a magnetic domain wall is discussed, and comparison with previous quantum treatments made.
The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schr\"odinger-Pauli wave equation, and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic field of a wave carrying orbital angular momentum and a static magnetic field. The running wave traps the atom only in the transverse direction, while the standing wave traps the atom also in the direction of the beam.
The dynamics of atom trapping in a radio-frequency-dressed-state potential formed by static and radio frequency (rf) fields has been studied using the Direct Simulation Monte Carlo (DSMC) technique. Using the simulations, a case of formation of a toroidal trap for cold 87Rb^{87}Rb87Rb atoms is investigated for atoms trapped in a static quadrupole magnetic trap and exposed to an rf-field with temporally increasing amplitude and decreasing frequency. We first calculate the adiabatic rf-dressed potential for an atom interacting with these fields and then apply DSMC algorithm to simulate the motion of the atom in this time dependent adiabatic potential. In the simulations the Landau-Zener (LZ) transition probability is calculated to know if the atom is in the trappable or untrappable dressed state. The results show that, initially at lower rf-field strength, the rf-field ejects atoms from the trap and leads to evaporative cooling of the atom cloud. However at higher rf-field strength, the atoms make LZ transition to the trappable dressed state which results in a toroidal shape to the atom cloud. The results of numerical simulations show a qualitative agreement with the experimental observations.
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.
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|.