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Journal of Magnetic Resonance

Geometric phase detection via NMR interferometry

S Fricke, Jeffrey A. Reimer

Published Jun 1, 2026• 1 citation

Abstract

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.

DOI: 10.1016/j.jmr.2026.108099

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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 Aharonov-Anandan phase is a contribution to the phase acquired by the cyclic evolution of a quantum state, which depends only on the geometric properties of its trajectory. We report the study and the exploitation of the Aharonov-Anandan phase by nuclear magnetic resonance interferometry techniques in homonuclear spin-1/2 pairs in the near-equivalence limit. We introduce a new method for engineering effective zero-quantum Hamiltonians with an arbitrary phase in the transverse plane. We use this method to generate a variety of cyclic zero-quantum paths, enabling direct study of the geometric Aharonov-Anandan phase to probe the rotational characteristics of the zero-quantum subspace. We show that the geometric Aharonov-Anandan phase may be used for efficient double-quantum excitation in strongly coupled spin pairs. We find that geometric double-quantum excitation outperforms the standard method by a factor of 2 in experiments performed on a typical case involving near-equivalent spin pairs.

Recently, it has been proposed to perform quantum computation through the Berry phase (adiabatic cyclic geometric phase) shift with NMR. This geometric quantum gate will hopefully be fault tolerant to certain types of error because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through the Aharonov-Anandan phase (non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. Moreover, in the scheme we have chosen the zero dynamic phase time evolution loop in obtaining the geometric phase shift; we need not perform any extra operation to cancel the dynamic phase.

Geometric phases encode the holonomy acquired under cyclic control of a quantum Hamiltonian: for nondegenerate eigenstates the effect is Abelian (a scalar phase), while in degenerate manifolds the Wilczek-Zee generalization yields a matrix-valued holonomy that can mix states and depends on path ordering. We present an overview of measuring Abelian and non-Abelian geometric phases in nitrogen-vacancy (NV) centers in diamond and propose a non-rotating measurement protocol enabled by the intrinsic

{ }^{14} ~\mathrm{N}

hyperfine manifold. As a first step, we propose to measure the Abelian Berry phase on a single hyperfine-resolved two-level transition using microwaveengineered closed loops with dynamical-phase cancellation by loop reversal. We then propose a non-Abelian protocol that engineers a two-dimensional degenerate dark subspace in an NV-

{ }^{14} ~\mathrm{N}

manifold via RF dressing and three phase-coherent microwave couplings (a tripod Hamiltonian). Two closed control loops implemented by winding independent microwave phases generate non-commuting SU(2) holonomies, which can be verified using final-only population readout and minimal tomography. We also present timing trade-off curves, full unitary Schrödinger evolution of the four-level tripod, and projected dark-subspace diagnostics that quantify both leakage and non-commuting holonomy.

Beyond the adiabatic limit, the Aharonov-Anandan phase is a generalized description of Berry's phase. In this regime, systems with time-independent Hamiltonians may also acquire observable geometric phases. Here we report on a measurement of the Aharonov-Anandan phase in photonics. Different from previous optical experiments on geometric phases, the implementation is based on light modes confined in evanescently coupled waveguides rather than polarization-like systems, thereby physical models in more than two-dimensional Hilbert spaces are achievable. In a tailored photonic lattice, we realize time-independent quantum-driven harmonic oscillators initially prepared in the vacuum state and achieve a measurement of the Aharonov-Anandan phase via integrated interferometry.