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Review of Scientific Instruments

The theory of oscillator-coupled magnetic resonance with potential applications to molecular imaging

John A. Sidles, Joseph L. Garbini, Gary P. Drobny

Published Aug 1, 1992• 81 citations
Publisher

Abstract

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.

DOI
10.1063/1.1143837

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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.
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.
We consider the process of a single-spin measurement using magnetic resonance force microscopy (MRFM) with a cyclic adiabatic inversion (CAI). This technique is also important for different applications, including a measurement of a qubit state in quantum computation. The measurement takes place through the interaction of a single spin with a cantilever modelled by a quantum oscillator in a coherent state in a quasi-classical range of parameters. The entire system is treated rigorously within the framework of the Schrödinger equation. For a many-spin system our equations accurately describe conventional MRFM experiments involving CAI of the spin system. Our computer simulations of the quantum spin–cantilever dynamics show that the probability distribution for the cantilever position develops two asymmetric peaks with the total relative probabilities mainly dependent on the initial angle between the directions of the average spin and the effective magnetic field, in the rotating frame. We show that each of the peaks is correlated with the direction of the average spin (being along or opposite to the direction of the effective magnetic field). This generates two possible outcomes of a single-spin measurement, similar to the Stern–Gerlach effect. We demonstrate that the generation of the second peak can be significantly suppressed by turning on adiabatically the amplitude of the rf magnetic field. We also show that MRFM CAI can be used both for detecting a signal from a single spin, and for measuring the single-spin state by measuring the phase of the cantilever driving oscillations.
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 motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a semiclassical framework. The concept of coherent internal states is used in the formulation of the semiclassical approximation from the full quantum-mechanical expression. The classical trajectories are defined only for certain spin states, that satisfy the conditions for being coherent internal states. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.