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arXiv:cond-mat/0604273 · Physical Review A

Topological constraints on magnetostatic traps

R. Gerritsma, R. J. C. Spreeuw

Published Oct 6, 2006• 19 citations
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Abstract

We theoretically investigate properties of magnetostatic traps for cold atoms that are subject to externally applied uniform fields. We show that Ioffe-Pritchard traps and other stationary points of $B$ are confined to a two-dimensional curved surface, or manifold $\mathcal{M}$, defined by $\mathrm{det}(\ensuremath{\partial}{B}_{i}∕\ensuremath{\partial}{x}_{j})=0$. We describe how stationary points can be moved over the manifold by applying external uniform fields. The manifold also plays an important role in the behavior of points of zero field. Field zeroes occur in two distinct types, in separate regions of space divided by the manifold. Pairs of zeroes of opposite type can be created or annihilated on the manifold. Finally, we give examples of the manifold for cases of practical interest.

DOI
10.1103/physreva.74.043405

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We theoretically investigate properties of magnetostatic traps for cold atoms that are subject to externally applied uniform fields. We show that Ioffe-Pritchard traps and other stationary points of BBB are confined to a two-dimensional curved surface, or manifold M\mathcal{M}M, defined by det(\ensuremath∂Bi∕\ensuremath∂xj)=0\mathrm{det}(\ensuremath{\partial}{B}_{i}∕\ensuremath{\partial}{x}_{j})=0det(\ensuremath∂Bi​∕\ensuremath∂xj​)=0. We describe how stationary points can be moved over the manifold by applying external uniform fields. The manifold also plays an important role in the behavior of points of zero field. Field zeroes occur in two distinct types, in separate regions of space divided by the manifold. Pairs of zeroes of opposite type can be created or annihilated on the manifold. Finally, we give examples of the manifold for cases of practical interest.
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.
Knowledge of the fundamental limitations on a magnetic trap for neutral particles is of paramount interest to designers as it allows for the rapid assessment of the feasibility of specific trap requirements or the quality of a given design. In this paper, performance limitations are defined for convexity of magnetic trapping potential and bias field using a local approximation in the trapping center. As an example, the fundamental bounds are computed for current supporting regions in the form of a spherical shell, a cylindrical region, and a box. A Pareto-optimal set considering both objectives is found and compared with known designs of the baseball trap and Ioffe-Pritchard trap. The comparison reveals a significant gap in the performance of classical trap designs from fundamental limitations. This indicates a possibility of improved trap designs and modern techniques of shape synthesis are applied in order to prove their existence. The topologically optimized traps perform almost two times better as compared to conventional designs. Last, but not least, the developed framework might serve as a prototype for the formulation of fundamental limitations on plasma confinement in a wider sense.
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 scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​/${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$, where scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​ 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 scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​ 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.