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Physical Review Letters

Study of the Aharonov-Anandan quantum phase by NMR interferometry

Dieter Suter, Karl Todd Mueller, Alexander Pines

Published Mar 28, 1988• 241 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 \ensuremath{\beta}(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
10.1103/physrevlett.60.1218

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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 \ensuremath{\beta}(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.
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.
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.