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Physics Letters A

Symplectic structure of the Aharonov-Anandan geometric phase

Luis J. Boya, José F. Cariñena, José M. Gracia-Bondı́a

Published Dec 1, 1991• 26 citations
Publisher
DOI
10.1016/0375-9601(91)90539-k

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We show the existence of a nonadiabatic geometric phase, i.e., an Aharonov-Anandan (AA) phase, in the Aharonov-Casher (AC) topological interference effect in one-dimensional mesoscopic rings. We find the AC phase is the phase accumulated by the spin wave function during a cyclic evolution, and show it is the sum of a geometric AA phase and a dynamical phase. In the adiabatic limit, the AA phase becomes the spin-orbit Berry phase introduced by Aronov and Lyanda-Geller. By solving exactly the model of a quasi-one-dimensional ring formed by the 2DEG on a semiconductor heterostructure, we discuss the observability of the AA phase in the AC effect.

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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.

arXiv:1510.08601•Optics Letters•Apr 14, 2016•14 citations
We show the existence of a nonadiabatic geometric phase, i.e., an Aharonov-Anandan (AA) phase, in the Aharonov-Casher (AC) topological interference effect in one-dimensional mesoscopic rings. We find the AC phase is the phase accumulated by the spin wave function during a cyclic evolution, and show it is the sum of a geometric AA phase and a dynamical phase. In the adiabatic limit, the AA phase becomes the spin-orbit Berry phase introduced by Aronov and Lyanda-Geller. By solving exactly the model of a quasi-one-dimensional ring formed by the 2DEG on a semiconductor heterostructure, we discuss the observability of the AA phase in the AC effect.
We show that cyclic quantum evolution can be realized and the Aharonov-Anandan (AA) geometric phase can be determined for any spin-j system driven by periodic fields. Two methods are extended for the study of this problem: the generalized spin-coherent-state technique and the Floquet quasienergy approach. Using the former approach, we have developed a generalized Bloch-sphere model and presented a SU(2) Lie-group formulation of the AA geometric phase in the spin-coherent state. We show that the AA phase is equal to j times the solid angle enclosed by the trajectory traced out by the tip of a generalized Bloch vector. General analytic formulas are obtained for the Bloch vector trajectory and the AA geometric phase in terms of external physical parameters. In addition to these findings, we have also approached the same problem from an alternative but complementary point of view without recourse to the concept of coherent-state terminology. Here we first determine the Floquet quasienergy eigenvalues and eigenvectors for the spin-j system driven by periodic fields. This in turn allows the construction of the time-evolution propagator, the total wave function, and the AA geometric phase in a more general fashion.
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.