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Physical Review A

Explicit demonstration of spinor character for a spin-½ nucleus via NMR interferometry

Michael E. Stoll, Alexander J. Vega, R. W. Vaughan

Published Oct 1, 1977• 62 citations
Publisher

Abstract

The results of a nuclear-magnetic-resonance experiment are presented which directly demonstrate the spinor character of a spin-\textonehalf{} nucleus, $^{13}\mathrm{C}$. The interferometric spectroscopic technique used and its potential applications are discussed.

DOI
10.1103/physreva.16.1521

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The results of a nuclear-magnetic-resonance experiment are presented which directly demonstrate the spinor character of a spin-\textonehalf{} nucleus, 13C^{13}\mathrm{C}13C. The interferometric spectroscopic technique used and its potential applications are discussed.
The components of spinors corresponding to half integral spin value transform under rotation of the spatial coordinate system with a matrix of even dimensionality. This leads to a state function phase factor with period 4π rad in the rotation angle. A spectroscopic interference experiment to observe such phase factors is described in analogy with double-beam interferometry. The experiment is performed with the techniques of high resolution nuclear magnetic resonance spectroscopy and the rotational symmetry of composite spin [Formula: see text], 1, and [Formula: see text] particles is explored by calculation and observation. Side effects relating to inhomogeneity dephasing and relaxation are evident in the experimental recordings, but they are discussed only briefly. A relation is established between the spectroscopic interference experiment and earlier "spin tickling" experiments.
Nuclear magnetic resonance (NMR) is a direct macroscopic manifestation of the quantum mechanics of the intrinsic angular momentum of atomic nuclei. It is best known for its extraordinary range of applications, which include molecular structure determination, medical imaging, and measurements of flow and diffusion rates. Most recently, liquid-state NMR spectroscopy has been found to provide a powerful experimental tool for the development and evaluation of the coherent control techniques needed for quantum information processing. This burgeoning new interdisciplinary field has the potential to achieve cryptographic, communications, and computational feats far beyond what is possible with known classical physics. Indeed, NMR has made the demonstration of many of these feats sufficiently simple to be carried out by high school summer interns working in our laboratory (see the last two authors). In this paper the basic principles of quantum information processing by NMR spectroscopy are described, along with several illustrative experiments suitable for incorporation into the undergraduate physics curriculum. These experiments are spin–spin interferometry, an implementation of the quantum Fourier transform, and the quantum simulation of a harmonic oscillator.
Nuclear Magnetic Resonance (NMR) forms a natural test-bed to perform quantum information processing (QIP) and has so far proven to be one of the most successful quantum information processors. The nuclear spins in a molecule treated as quantum bits or qubits which are the basic building blocks of a quantum computer. The development of NMR over half a century puts it in a platform where we can utilize its excellent control techniques over an ensemble of spin systems and perform quantum computation in a highly controlled way. Apart from a successful quantum information processor, NMR is also a highly powerful quantum platform where many of the potentially challenging quantum mechanical experiments can be performed.
Experimental detection of geometric phase effects is based on interferometry techniques. The observation of a nonadiabatic geometric phase in nuclear magnetic resonance has been reported by D. Suter, K. T. Mueller, and A. Pines (1988, Phys. Rev. Lett., 60, 1218). A rigorous interpretation of their results is presented here. The experiment exploits the concept of fictitious spin-1/2 subsystems embedded in a spin system of higher order. A relevant expression for the geometric phase is derived here using the Euler rotation representation for the SU(2) group. In order to correlate the experimental data to the geometric phase, a decomposition of the density matrix in a basis of pure states is used and the isomorphism between the cyclic evolution of pure states and mixed states is constructed. Rigorous treatment of the dynamic phase, along with technical aspects of the experiment, shows that the observed echo shift is related directly to the geometric phase, in agreement with Suter et al. The observed geometric phase is classified as a frame-related phase.
Modern experimental techniques can generate magnetic fields of the form H(t) = H0 z-hat + H1 [x-hat cos(ωt) + y-hat sin(ωt)], at frequencies within an order of magnitude of the nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) frequencies, ωn0 and ωe0, respectively, when acting on atoms or molecules. We derive simple closed-form expressions for the exact nuclear- and electronic-spin wave functions that enable controlled transitions between entangled states, allowing an atom or molecule to function as a quantum computer. These solutions also enable precise NMR or EPR measurements of nuclear moments in atoms and molecules. We present examples relevant to measurements of the nuclear moments of 14N, 7Li, and 133Cs. Because existing hyperfine measurements of the lowest three nuclear moments of 133Cs are mutually inconsistent, the proposed NMR/EPR experiments provide a route to measuring all seven of its nuclear moments with high precision.