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Proceedings of the National Academy of Sciences

Continuous Stern-Gerlach effect: Principle and idealized apparatus

HANS G. DEHMELT

Published Apr 1, 1986• 115 citations
Publisher

Abstract

A nondestructive type of Stern-Gerlach effect for an individual electron is described that has been thoroughly demonstrated in experiments at the University of Washington. This "continuous Stern-Gerlach effect" makes use of an inhomogeneous magnetic field provided by a weak auxiliary magnetic bottle and is nondestructive in contrast to all previous versions of the effect. As in the classic Stern-Gerlach effect, changes in the spin state are detected via changes in classical particle trajectories; however, what is observed now is not a deflection of the orbit but rather a change of its frequency in the storage well. A simplified model of the continuous Stern-Gerlach effect at zero temperature is introduced to develop the relation between minimum measurement time required for determination of spin state, driven oscillation amplitude, and zero-point fluctuations in the storage well. The problem of the reduction of the wavefunction by the interaction of the electron with the apparatus is addressed following W. Pauli.

DOI
10.1073/pnas.83.8.2291

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A nondestructive type of Stern-Gerlach effect for an individual electron is described that has been thoroughly demonstrated in experiments at the University of Washington. This "continuous Stern-Gerlach effect" makes use of an inhomogeneous magnetic field provided by a weak auxiliary magnetic bottle and is nondestructive in contrast to all previous versions of the effect. As in the classic Stern-Gerlach effect, changes in the spin state are detected via changes in classical particle trajectories; however, what is observed now is not a deflection of the orbit but rather a change of its frequency in the storage well. A simplified model of the continuous Stern-Gerlach effect at zero temperature is introduced to develop the relation between minimum measurement time required for determination of spin state, driven oscillation amplitude, and zero-point fluctuations in the storage well. The problem of the reduction of the wavefunction by the interaction of the electron with the apparatus is addressed following W. Pauli.
A nondestructive type of Stern-Gerlach ef- fect for an individual electron is described that has been thoroughly demonstrated in experiments at the University of Washington. This Stern-Gerlach makes use of an inhomogeneous magnetic field provided by a weak auxiliary magnetic bottle and is nondestructive in contrast to all previous versions of the effect. As in the classic Stern-Gerlach effect, changes in the spin state are detected via changes in classical particle trajectories; however, what is observed now is not a deflection of the orbit but rather a change of its frequency in the storage well. A simplified model of the continuous Stern-Gerlach effect at zero temperature is introduced to develop the relation between minimum measurement time required for determination of spin state, driven oscillation amplitude, and zero-point fluctuations in the storage well. The problem of the reduction of the wavefunction by the interaction of the electron with the apparatus is addressed following W. Pauli.
This paper resumes the discussion of the continuous Stern-Gerlach effect, a method to continuously and nondestructively determine the spin state of the same individual electron, quasipermanently confined in a Penning trap in ultrahigh vacuum at liquid helium temperatures. Here the focus is on limitations to spin-state detection due to thermal and zero-point noise and on the quantum-mechanical measurement process. Illustrations of the continuous spin measurement process in the presence of selected perturbations are provided. The alteration of the spin state brought about by the frequency measurement process is exhibited. To resolve Zeno's paradox in a specific example, a slow resonant spin flip is discussed, when interrupted by frequent spin-state measurements. The continuous Stern-Gerlach effect is shown to be a near-ideal example for the quantum mechanical measurement process, for which all steps may be followed in quantitative detail.
The classical Stern-Gerlach experiment is analyzed with an emphasis on the spin dynamics. The central question asked is whether there occurs a relaxation of the spin angular momentum during the time the particle passes through the Stern-Gerlach magnet. We examine in particular the transverse relaxation, involving angular momentum exchange between the spin of the particles and the spins of the magnet. A method is presented describing relaxation effects at an individual particle level. This leads to a stochastic equation of motion for the spins. This is coupled to a classical equation of motion for the particle translation. The experimental situation is then modeled through simulations of individual trajectories using two sets of parameter choices and three different sets of initial conditions. The two main conclusions are: (A) if the coupling between the magnet and the spin is solely described by the Zeeman interaction with the average magnetic field the simulations show a clear disagreement with the experimental observation of Stern and Gerlach. (B) If one, on the other hand, also allows for a T(2) relaxation time shorter than the passage time one can obtain a practically quantitative agreement with the experimental observations. These conclusions are at variance with the standard textbook explanation of the Stern-Gerlach experiment.
The conflict between Bohr's assertion that the magnetic moment of the electron cannot be measured with experiments based on the concept of classical trajectories, and the measurement of the magnetic moment of electrons in a modified Penning trap by Dehmelt et al. has led us to reevaluate other implications of Bohr's assertion. We show that, contrary to the analysis of Bohr and Pauli, the assumption of classical trajectories in a Stern-Gerlach--like device can result in a high degree of spin separation for an electron beam. This effect may persist within a fully quantum-mechanical analysis. The magnetic fields considered are such that a tabletop Stern-Gerlach electron spin filter is feasible.
Observation of two separated beam spots at a detection screen downstream of a Stern–Gerlach magnet does not, in fact, demonstrate that the wavefunction of a neutral spin one-half particle has remained in a spin superposition while traveling through that magnetic field. The wavefunction may have been reduced to just one spin-direction eigenfunction, as D. Bohm suggested, by immediate momentum and energy transfer with the magnet, rather than by subsequent, which-way determination at the screen. The same two beam spots at the detector screen will result. Einsteinian relativity, and the understanding of Schrödinger evolution applicability through a static potential, forbid continuation of a spin superposition through the Stern–Gerlach field. A calculation for single wavepacket development there conforms to observations from the Stern–Gerlach experiment. And several experiments corroborate immediate reduction to a single spin eigenfunction in the magnetic field. Additionally, Ramsey’s separated, oscillating fields observations, and related experiments, do not rebut this understanding.
A detailed analysis of the classic Stern–Gerlach experiment is presented. An analytical simple solution is presented for the quantum description of the translational and spin dynamics of a silver atom in a magnetic field with a gradient along a single z-direction. This description is then used to obtain an approximate quantum description of the more realistic case with a magnetic field gradient also in a second y-direction. An explicit relation is derived for how an initial off center deviation in the y-direction affects the final result observed at the detector. This shows that the “mouth shape” pattern at the detector observed in the original Stern–Gerlach experiment is a generic consequence of the gradient in the y-direction. This is followed by a discussion of the spin dynamics during the entry of the silver atom into the magnet. An analytical relation is derived for a simplified case of a field only along the z-direction. A central question for the conceptual understanding of the Stern–Gerlach experiment has been how an initially unpolarized spin ends up in a polarized state at the detector. It is argued that this can be understood with the use of the adiabatic approximation. When the atoms first experience the magnetic field outside the magnet, there is in general a change in the spin state, which transforms from a degenerate eigenstate in the absence of a field into one of two possible non-degenerate states in the field. If the direction of the field changes during the passage through the device, there is a corresponding adiabatic change of the spin state. It is shown that an application of the adiabatic approximation in this way is consistent with the previously derived exact relations.