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Control of the Geometric Phase and Nonequivalence between Geometric-Phase Definitions in the Adiabatic Limit
Xiaosong Zhu1,2,3, Peixiang Lu2,3, Manfred Lein1
1Leibniz University Hannover, Institute of Theoretical Physics, 30167 Hannover, Germany.
Physical Review Letters
|February 4, 2022
Summary
Geometric phases like the Berry and Aharonov-Anandan (AA) phases accumulate during quantum state evolution. This study reveals their low-frequency limits may differ, especially near energy degeneracies, impacting quantum state manipulation.
Area of Science:
- Quantum mechanics
- Geometric phase theory
- Floquet theory
Background:
- Geometric phases, including Berry and Aharonov-Anandan (AA) phases, arise when quantum states return to their initial state after time evolution.
- These phases are crucial for understanding adiabatic and nonadiabatic processes in quantum systems.
Purpose of the Study:
- To evaluate Berry and AA geometric phases using Floquet theory for systems in time-dependent external fields.
- To investigate the behavior of these phases, particularly when paths involve eigenenergy degeneracies.
- To explore the relationship between adiabatic and nonadiabatic geometric phases in the low-frequency limit.
Main Methods:
- Application of Floquet theory to analyze time evolution of quantum states in time-dependent external fields.
- Focus on quantum state evolution paths that traverse points of eigenenergy degeneracy.
- Analysis of the low-frequency limits of Berry and AA phases.
Main Results:
- Contrary to expectations, the low-frequency limits of Berry and AA phases do not always coincide.
- Degeneracies can cause slow convergence to adiabaticity, leading to non-zero contributions to the AA phase.
- Adiabatic steering through degeneracies allows control over geometric phases, including a π shift in the Berry phase.
- Revisiting a specific degeneracy crossing example, it was found that both phase definitions can yield the same result at specific driving frequencies.
Conclusions:
- The coincidence of Berry and AA phases is not guaranteed, especially near degeneracies, challenging previous assumptions.
- Degeneracies offer opportunities for controlling geometric phases and manipulating quantum states.
- Symmetry in time evolution around degeneracy points can lead to zero dynamical phase, simplifying quantum state control.
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