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Study of the Aharonov-Anandan Phase by NMR Interferometry

Suter, Dieter, Karl Todd Mueller, Alexander Pines

Published Mar 1, 1988• 135 citations
Publisher

Abstract

Aharonov and Anandan have recently reformulated and generalized Berry’s phase by showing that a quantum system which evolves through a circuit C in projective Hilbert space acquires a geometrical phase β(C) related to the topology of the space and the geometry of the circuit. We present NMR interferometry experiments in a three-level system which demonstrate the Aharonov-Anandan phase and its topological invariance for different circuits.

DOI
10.17877/de290r-3226

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Aharonov and Anandan have recently reformulated and generalized Berry’s phase by showing that a quantum system which evolves through a circuit C in projective Hilbert space acquires a geometrical phase β(C) related to the topology of the space and the geometry of the circuit. We present NMR interferometry experiments in a three-level system which demonstrate the Aharonov-Anandan phase and its topological invariance for different circuits.
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.
Berry (1984) discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase. Aharonov and Anandan (1987), on the other hand, showed that an eigenstate transporting round a closed circuit in the projective Hilbert space suffices to generate the topological phase. The authors employ the evolution operator method to study the propagation of an eigenstate. They show that Berry's phase can be represented as a closed path integral in the Hilbert space and is independent of the choice for the base ( mod n(t))). The treatments of Berry, and Aharonov and Anandan, are shown to correspond to two different choices of the base. Therefore their two approaches are unified; they have acquired a more general viewpoint on the origin of Berry's phase.
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.
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.