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Molecular Physics

Geometric phase in NMR interferometry experiment

Nikolai R. Skrynnikov, B. C. Sanctuary

Published Dec 20, 1994• 4 citations
Publisher

Abstract

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.

DOI
10.1080/00268979400101831

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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.
Inspired by the optical experimental scheme [Phys. Rev. Lett. 86, 369 (2001)], we propose and implement a NMR experiment to observe an Abelian geometric phase shift arising from the cyclic evolution of U(2)-invariant states. Such a phase shift geometrically refers to the Bargmann invariant in a four-dimensional SU(3)/U(2) parameter space, while the usual geometric phases arise in the evolution of U(1)-invariant states and is related to solid angles in a two-dimensional $\text{SU}(2)/\text{U}(1)\ensuremath{\approx}{S}^{2}$ parameter space (or Poincar\'e sphere). We present the experiment in a four-level system and investigate its topological invariance for different circuits. This may provide a realistic candidate to design complex geometric quantum circuits for future quantum computers.
We introduce a benchtop NMR method for interferometric detection of geometric phase (also known as Berry/Aharonov-Anandan phase) via spin coherence holonomy in bulk ensembles using phase-wound echo trains. Consecutive π pulses with cyclic phase incrementation drive closed spinor trajectories on the Bloch sphere, while a parity-based analysis isolates the geometric contribution and cancels dynamical offsets by comparing positive and negative winding experiments with a zero-winding reference. The extracted phase increases linearly with echo number, reverses sign under winding reversal, and is independent of echo time in the adiabatic regime, consistent with spinor holonomy arising from parallel transport. Experiments performed in strongly inhomogeneous, low-field conditions demonstrate robustness to B₀ gradients and diffusion. This approach establishes a practical foundation for probing geometric phase effects in NMR, with extensions to non-adiabatic transport, heterogeneous systems, and quantum sensing applications.
The total phase acquired during an evolution of a quantal system has two components: the usual dynamical phase, $\ensuremath{-}1/\ensuremath{\Elzxh}\ensuremath{\int}H(t)dt,$ given by the integrated expectation value of the Hamiltonian, and a geometric phase ${\ensuremath{\Phi}}_{G}(C)$ which depends only on the geometry of the curve traced in ray space. We have performed an interference experiment using polarized neutrons which clearly demarcates the separate contributions of the dynamical and geometric phases to the total. In the experiment, the two phases arise from two distinct physical operations, a translation and a rotation of a spin flipper within the interferometer, respectively. This work also constitutes the first direct observation of the Pauli anticommutation; a purely geometric phase shift of \ensuremath{\pi} radians appears after a reversal of the current in one of the spin flippers. The experiment was carried out at the 10 MW University of Missouri Research Reactor using a skew-symmetric perfect-silicon-crystal neutron interferometer. A detailed description of the experiment and its interpretation is given in this paper.
Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires one to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mixed quantum states. We show how different they are from the well-understood, noiseless, pure-state case.
A different way to realize nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic resonance, where the dynamical and geometric phases are implemented separately without the usual operational process. Therefore the phase accumulated in the geometric gate is a pure geometric phase for any input state. In comparison with the conventional geometric gates by rotating operations, our approach simplifies experimental implementations making them robust to certain experimental errors. In contrast to the unconventional geometric gates, our approach distinguishes the total and geometric phases and offers a wide choice of the relations between the dynamical and geometric phases.
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.