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The mystery of geometric phase: Why do quantum systems acquire hidden phases?
In the field of physics, geometric phase is a phase difference that a quantum system acquires when undergoing a cyclic adiabatic process. This phenomenon not only covers the core theory of quantum mechanics, but also reveals many amazing physical phenomena. Since S. Pancharatnam independently discovered this phenomenon in classical optics in 1956, it has been developed and deepened, and further promoted by Michael Berry in 1984. The geometric phase (also known as Pancharatnam–Berry phase, Pancharatnam phase or Berry phase) has been It has become an important physical phenomenon.
The existence of geometric phase stems from the geometric properties of the parameter space of the Hamiltonian. When a system undergoes an induced parameter change process and eventually returns to its original state, if such a process is cyclical, an additional phase difference will be obtained. This phenomenon is not limited to quantum systems, but also has important application and theoretical value in classical optics.
The key to the occurrence of geometric phase is that the parameters change very slowly (adiabatically), which allows the system to remain in its energy eigenstate at every instant.
When geometric phases occur, the dependence of the state of the system is usually singular. This means that under certain parameter combinations, the state of the system may be undefined. In order to measure the geometric phase, it is usually necessary to perform an interference experiment. The Foucault pendulum in classical mechanics is a classic example in this regard.
Berry phase in quantum mechanics
In a quantum system, if it is in the nth eigenstate, the adiabatic evolution of the Hamiltonian will keep the system in the nth eigenstate and acquire a phase factor. This phase is obtained not only from the progression of the state over time, but also from the changes in the eigenstates that change as the Hamiltonian changes.
For a cyclically varying Hamiltonian, the Berry phase cannot be cancelled because it is an invariant and observable property of the system.
The existence of the Berry phase is closely related to the parameter change of the Hamiltonian, which can be calculated by integrating along a closed path. Such a process requires a phase term to describe the overall change. This causes the system to cycle through the parameter space and obtain the corresponding geometric phase.
Application Examples of Geometric Phase
Foucault pendulum
The Foucault pendulum is a very easy example to understand of geometric phase. As the pendulum moves with the Earth's rotation, the plane of its circular motion has a pre-rotation. For some particular path, the total number of rotations is a measure of the solid angles that the pendulum encompasses after traversing any closed path.
In other words, this pre-rotation is not due to the influence of inertial forces, but is caused by the rotation of the path along which the pendulum travels.
At the latitude of Paris, the pre-rotation period of the Foucault pendulum is about 32 hours, which means that at the end of a day's rotation, the plane of the pendulum has changed significantly. This phenomenon profoundly points out the close connection between geometric phase and physical system.
Polarized light in optical fibers
A second example is linearly polarized light entering a single-mode fiber. During this process, the momentum of light is always tangent to the path of the optical fiber, so the change in polarization state during the entry and exit of light can also be described by the geometric phase. The polarization direction of light when it enters the optical fiber will be out of phase with the polarization direction when it leaves.
The amount of this phase change is also measured by the solid angle enclosed by the light as it travels through the fiber.
Through these examples, we can see that geometric phase is not just a mathematical oddity, it also provides profound insights into the understanding of physical phenomena and has application potential.
Just imagine, what other physical phenomena in this world can allow us to discover more hidden mysteries through the perspective of geometric phase?
