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Physical Review Letters

Folded Stern-Gerlach experiment as a means for detecting nuclear magnetic resonance in individual nuclei

John A. Sidles

Published Feb 24, 1992• 138 citations
Publisher

Abstract

In a Stern-Gerlach experiment, the spatial trajectory of a particle traversing a magnetic-field gradient serves as a detector of its spin state. This paper considers the implications of adding a harmonic restoring potential to the Stern-Gerlach experiment. The behavior of such systems is shown to resemble that of a ``folded'' Stern-Gerlach experiment, in that the linear spatial trajectory of the Stern-Gerlach experiment becomes folded into the cyclic phase-space trajectory of the oscillator. Under appropriate circumstances, the coupling between the spin and the oscillator is shown to be sufficiently strong that it might find practical application as the basis of a magnetic resonance imaging device.

DOI
10.1103/physrevlett.68.1124

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This article describes systems in which the precession of a single particle spin is magnetically coupled to the excitation of an oscillator. The behavior of such systems resembles that of a ‘‘folded’’ Stern–Gerlach experiment, in which the linear spatial trajectory of the original Stern–Gerlach experiment is folded into the cyclic trajectory of an oscillator. Both quantum and semiclassical solutions to the equations of motion are derived. The results encompass any kind of oscillator which couples to a magnetic field. Examples include mechanical cantilevers with a magnetic source affixed to them, as well as inductor-capacitor resonant circuits. One potential application of oscillator-coupled magnetic resonance is the imaging of biological molecules. Some design issues relevant to molecular imaging are discussed.

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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.

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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.

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•May 30, 1995•3 citations
In a Stern-Gerlach experiment, the spatial trajectory of a particle traversing a magnetic-field gradient serves as a detector of its spin state. This paper considers the implications of adding a harmonic restoring potential to the Stern-Gerlach experiment. The behavior of such systems is shown to resemble that of a ``folded'' Stern-Gerlach experiment, in that the linear spatial trajectory of the Stern-Gerlach experiment becomes folded into the cyclic phase-space trajectory of the oscillator. Under appropriate circumstances, the coupling between the spin and the oscillator is shown to be sufficiently strong that it might find practical application as the basis of a magnetic resonance imaging device.
This article describes systems in which the precession of a single particle spin is magnetically coupled to the excitation of an oscillator. The behavior of such systems resembles that of a ‘‘folded’’ Stern–Gerlach experiment, in which the linear spatial trajectory of the original Stern–Gerlach experiment is folded into the cyclic trajectory of an oscillator. Both quantum and semiclassical solutions to the equations of motion are derived. The results encompass any kind of oscillator which couples to a magnetic field. Examples include mechanical cantilevers with a magnetic source affixed to them, as well as inductor-capacitor resonant circuits. One potential application of oscillator-coupled magnetic resonance is the imaging of biological molecules. Some design issues relevant to molecular imaging are discussed.
The Stern-Gerlach experiment has played an important role in our understanding of quantum behavior. We propose and analyze a modified version of this experiment where the magnetic field of the detector is in a quantum superposition, which may be experimentally realized using a superconducting flux qubit. We show that if incident spin-1/2 particles couple with the two-state magnetic field, a discrete target distribution results that resembles the distribution in the classical Stern-Gerlach experiment. As an application of the general result, we compute the distribution for a Gaussian waveform of the incident fermion. This analysis allows us to demonstrate theoretically: (1) the quantization of the intrinsic angular momentum of a spin-1/2 particle, and (2) a correlation between EPR pairs leading to nonlocality, without necessarily collapsing the particle’s spin wavefunction.
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.
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.
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.