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Physical Review A

Collisionless motion of neutral particles in magnetostatic traps

E.L. Surkov, Jook T. M. Walraven, Georgy V. Shlyapnikov

Published Jun 1, 1994• 27 citations
Publisher

Abstract

We present an analytical and numerical analysis of collisionless motion of neutral particles in magnetostatic traps. We treat the example of an idealized Ioffe quadrupole trap, emphasizing that the essential features of coupling between the degrees of freedom are of general relevance for static traps. In common situations the coupling between axial and radial motion predominantly occurs in a regular way. The coupling is weak, in particular, when the frequency of axial oscillations ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$ is much smaller than the radial frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\rho}}}$ and the particle energy is much lower than the Zeeman energy in the center of the trap. Then, under an adiabatic change of the potential parameters, the quantity ${\mathit{scrE}}_{\mathit{z}}$/${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$, where ${\mathit{scrE}}_{\mathit{z}}$ is a characteristic axial energy, is an approximate adiabatic invariant. If ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$ is decreasing and ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is constant, then the characteristic radial energy ${\mathit{scrE}}_{\mathrm{\ensuremath{\rho}}}$ will remain constant while ${\mathit{scrE}}_{\mathit{z}}$ decreases proportionally to ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$. This implies one-dimensional adiabatic cooling, which is an interesting option for gravitational studies of light atoms such as hydrogen and antihydrogen.

DOI
10.1103/physreva.49.4778

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We analyze several puzzling features of a recent experiment with a noninteracting gas of atoms in a quadrupole trap. After an initial momentum kick, the system reaches a stationary, quasithermal state even without collisions, due to the dephasing of individual particle trajectories. Surprisingly, the momentum distribution remains anisotropic at long times, characterized by different temperatures along the different directions. In particular, there is no transfer of the kick energy between the axial and radial trap directions. To understand these effects we discuss and solve two closely related models: a spherically symmetric trap $V(r)\ensuremath{\simeq}{r}^{\ensuremath{\alpha}}$ and a strongly confined gas along one direction (a pancake trap). We find that in the isotropic trap, the gas unexpectedly also preserves the anisotropy of the kick at long times, which we are able to explain using the conservation of angular momentum and the virial theorem. Depending on the value of $\ensuremath{\alpha}$ we find that the kick can cool or heat the orthogonal directions. The pancake trap case is quantitatively similar to the quadrupole one. We show that for the former, the temperature anisotropy and memory of the kick direction are due to the change in the two-dimensional effective potential resulting from the kick, thereby also explaining the quadrupole experimental results.

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We present an analytical and numerical analysis of collisionless motion of neutral particles in magnetostatic traps. We treat the example of an idealized Ioffe quadrupole trap, emphasizing that the essential features of coupling between the degrees of freedom are of general relevance for static traps. In common situations the coupling between axial and radial motion predominantly occurs in a regular way. The coupling is weak, in particular, when the frequency of axial oscillations \ensuremathΩz{\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}\ensuremathΩz​ is much smaller than the radial frequency \ensuremathΩ\ensuremathρ{\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\rho}}}\ensuremathΩ\ensuremathρ​ and the particle energy is much lower than the Zeeman energy in the center of the trap. Then, under an adiabatic change of the potential parameters, the quantity scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​/\ensuremathΩz{\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}\ensuremathΩz​, where scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​ is a characteristic axial energy, is an approximate adiabatic invariant. If \ensuremathΩz{\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}\ensuremathΩz​ is decreasing and \ensuremathΩp{\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}\ensuremathΩp​ is constant, then the characteristic radial energy scrE\ensuremathρ{\mathit{scrE}}_{\mathrm{\ensuremath{\rho}}}scrE\ensuremathρ​ will remain constant while scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​ decreases proportionally to \ensuremathΩz{\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}\ensuremathΩz​. This implies one-dimensional adiabatic cooling, which is an interesting option for gravitational studies of light atoms such as hydrogen and antihydrogen.
We analyze the collisionless motion of atoms in magnetic traps in relation to evaporative cooling. For the example of a long Ioffe quadrupole trap we investigate both the regular and stochastic regimes of motion. We emphasize a strong influence of the regime of collisionless motion on the process of evaporative cooling. For evaporation of atoms across an axial potential barrier the stochastic motion of atoms at energies above the barrier enables three-dimensional evaporation, i.e., particles acquiring in elastic collisions a total enery higher than the barrier height E0{\mathit{E}}_{0}E0​ escape from the trap. The regular motion leads to axial evaporation: only atoms which due to collisions acquire an axial energy higher than E0{\mathit{E}}_{0}E0​ leave the trap. The rate of axial evaporation is smaller by a factor \ensuremath{\sim}\ensuremath{\eta}=E0{\mathit{E}}_{0}E0​/T\ensuremath{\gg}1, where T is the gas temperature. This has important consequences for evaporative cooling, which we discuss in relation to trapped atomic hydrogen. \textcopyright{}1996 The American Physical Society.
We consider quantized motion of neutral atoms cooled below the recoil limit in a quadrupole magnetostatic trap. Because of Majorana transitions to untrapped levels near the point of zero field at the trap center, all quantum levels have a nonzero decay rate. The Schrödinger equation associated with the potential gμB · S (S is the total atomic spin) takes the form of coupled equations in r when the spinor components are expanded in spherical harmonics. We integrate the multichannel problem numerically to obtain asymptotic phase shifts, resonance energies, and widths. For S = 1/2, the lowest levels have widths somewhat less than their spacing. Thus the trap quantum-level structure might possibly be observable if the atoms are sufficiently cold, namely, in the 0.1-μK regime for most atoms and attainable trap field gradients. The width decreases rapidly with increasing MJ, the angular momentum about the symmetry axis. Spectroscopic linewidths of a few hertz are possible if there is enough population in the lowest levels with a few MJ quanta. The decay rate of the lowest levels, however, is probably too rapid for studying Bose–Einstein condensation in such a 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 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.
In view of the recent successful confinement of decelerated sodium atoms in a magnetostatic trap, it is of interest to evaluate possible trap-field configurations. Neutral atoms in a Zeeman sublevel whose energy increases with field can be confined by a field whose magnitude \ensuremath{\Vert}B\ensuremath{\Vert} increases with distance from the center. Because this same basic requirement applies also to traps for neutrons and for plasmas (in the guiding-center approximation), trap configurations developed previously for these purposes are of interest for neutral atoms. However, the desired properties differ considerably because of very different objectives and different behavior of very cold atoms as compared with hot plasmas. We characterize basic trap configurations using both the exact expressions for the field, and a multipole polynomial expansion that facilitates studies of symmetry properties and classical or quantum orbits. Polynomial terms for the field components are derived and coefficients obtained by comparison with Taylor-series expansions and by global fit. Contours of \ensuremath{\Vert}B\ensuremath{\Vert} for various trap configurations are also presented. Under certain restrictive conditions, \ensuremath{\Vert}B\ensuremath{\Vert}, and hence the effective potential, can be made isotropic to second order.
We analyze several puzzling features of a recent experiment with a noninteracting gas of atoms in a quadrupole trap. After an initial momentum kick, the system reaches a stationary, quasithermal state even without collisions, due to the dephasing of individual particle trajectories. Surprisingly, the momentum distribution remains anisotropic at long times, characterized by different temperatures along the different directions. In particular, there is no transfer of the kick energy between the axial and radial trap directions. To understand these effects we discuss and solve two closely related models: a spherically symmetric trap $V(r)\ensuremath{\simeq}{r}^{\ensuremath{\alpha}}$ and a strongly confined gas along one direction (a pancake trap). We find that in the isotropic trap, the gas unexpectedly also preserves the anisotropy of the kick at long times, which we are able to explain using the conservation of angular momentum and the virial theorem. Depending on the value of $\ensuremath{\alpha}$ we find that the kick can cool or heat the orthogonal directions. The pancake trap case is quantitatively similar to the quadrupole one. We show that for the former, the temperature anisotropy and memory of the kick direction are due to the change in the two-dimensional effective potential resulting from the kick, thereby also explaining the quadrupole experimental results.