Geometric phase

The Berry phase or geometric phase occurs at a quantum mechanical system if the slow ( adiabatic ) passing through a closed path in the parameter space of the system, the system does not return to its initial state, but its wave function is replaced by a phase factor, just the Berry phase. The Berry phase is not limited to quantum mechanical systems, a similar effect is also found in classical systems ( see below).

The Berry phase is named after Michael Berry, who introduced it in 1983. However, there were already other precursors, which fell back into oblivion. For example, S. Pancharatnam discovered the phenomenon in the framework of classical physics in polarization cycles ( 1956), and sometimes it is therefore also named additionally after Pancharatnam.

Description

For adiabatic changes follows from the adiabatic theorem of quantum mechanics, that the system returns to its initial state, however, as Michael Berry discovered occur a dependent on the geometry of the parameter space phase factor in the wave function.

In general, a system (described by the Hamiltonian ) must depend on at least two parameters, and the parameter space for example, have singularities or a non-trivial topology ( clear: holes ) in order for a closed path in the parameter space a non-trivial ( different from 1 ) phase factor to obtain. Of particular importance are points in the parameter space in which the energy levels of " neighboring" states the energy of the initial state approach ( degenerate points), because only there can such a non- trivial phase factors arise when the cycles enclose them. The reduced to this degeneracy parameter space points receives a non-trivial topology. Since the geometry of the ( augmented ) parameter space is of crucial importance, the Berry phase is also called the geometric phase.

Examples of the occurrence of

Quantum mechanical Berry phase

Examples of the Berry phase are adiabatic cycles in the molecule coordinates that can generate a phase factor in the wave function of the electrons can be treated in the Born- Oppenheimer approximation: the Hamiltonian and the wave function of the electron can be parameterized by the nuclear coordinates. That was one of the original examples of Berry and one such example was discovered in 1958 by Christopher Longuet- Higgins. The case of the geometric phase in molecular physics in particular was treated by Alden Mead and Donald Truhar, with initial work in 1979. Spectroscopic experiments to confirm the theory of Alden and Mead, took place in the 1980s.

In his essay of 1984 Berry gives an example in which the Berry phase can be relatively easily calculated explicitly: a spin in a magnetic field that is slowly varied by changing the direction of the magnetic field of a closed curve. Phase is proportional in this case, where n is the spin quantum number (), and the opening angle of from the origin is. For spin 1/2 which corresponds to a formula that Pancharatnam 1956 had derived for polarized light.

The spin 1/2 example can be extended to general quantum mechanical two-state systems described by complex Hermitian 2 × 2 matrices. Again, there is a degeneration point and a formula that describes the Berry phase by the three-dimensional aperture angle at which the cycle is considered from the point of degeneration ( phase).

An even simpler example arises in the case of real symmetric 2 × 2 matrices, which correspond in quantum mechanics zeitumkehrinvarianten systems. The case corresponds to a set of elementary matrix theory ( Berry ), whose dependence parameter is considered. In the simplest case, real symmetrical 2 × 2 matrices:

Are the eigenvectors are real and thus it come in cycles in the parameter space for the " Berry phase " only the prefactors 1 (Phase 0 or ) and -1 ( phase) into consideration. Non-trivial Berry phases with prefactor -1 ( change of sign ), it is only if the cycle is a degeneracy point is circled in the parameter space. These are given by the straight line in the parameter space (u, w, V). Only if the cycles of the line surround, there are sign changes in the eigenvectors.

Another example is the Aharonov -Bohm effect. The parameter space is here the usual spatial domain, but ( the vector potential of the field there is a singularity ) not more than simply construed connected due to the magnetic field in the interior of the closed path. The wave function of the magnetic field umzirkelnden electron obtains a phase factor proportional to the magnetic flux, although the position of the electron itself, the magnetic field vanishes everywhere (but not the associated vector potential ).

Berry phase, for example, experiments in the interference observed. An experimental proof of the Berry phase in optical experiments with linearly polarized light with a cylinder ( helical) winding glass fibers until 1986, Akira Tomita and Raymond Chiao. The experiment is within the scope of Berry above diskutiertem spin writable case, and measures the solid angle at the rotation of the " spin- direction ". Analogous experiments with neutrons that were sent by a helically -varying magnetic field, were also carried out in the 1980s by T. Bitter ( Heidelberg) and D. Dubbers ( Institut Laue -Langevin ). Robert Tycko ( Bell Laboratories ) resulted in 1987 an experiment to demonstrate the Berry phase, in which the rotation of the nuclear spins, which were bonded to the crystal axes, by rotating the crystal to the total of its various axes of symmetry axes occurred.

Occur in classical mechanics

In classical mechanics, the Foucault pendulum provides an example of a geometric phase. The pendulum does not return in a complete revolution of the Earth in 24 hours ( closed path in the parameter space ) to its starting point, but it occurs is dependent on the latitude phase shift. The geometric phase in classical mechanics is also called the Hannay angle ( by John Hannay, a colleague of Berry in Bristol).

Mathematical definition

Mathematically, the Berry phase expression of a holonomy. A simple example of a holonomy is the parallel transport of a vector on the sphere in a triangle of great circles: one starts from the North Pole, goes to the equator, following this 90 degrees and then returns to the North Pole, causes a 90- degree rotation of the parallel transported vector ( an analogy to the Berry phase ). This rotation depends only on the geometry ( curvature ) of the underlying space ( the sphere ) from.

Formally, the geometric phase is given by:

Wherein a closed path in the parameter space C (variable, which is generally vectorially ) is integrated. There the Bra- Ket notation is used for the states is an eigenstate of the Hamiltonian of the system. is the nabla operator of differentiation with respect to the parameters. Because of the normalization of the states is imaginary and therefore real.

The wave function transforms into after passing through the cycle in the parameter space:

In the three-dimensional case, the formula may be brought for the geometric phase in a more favorable for the application form by applying the rate of Stokes (conversion into a surface integral ):

In this formula only appear on the expectation values ​​of the derivative of the Hamiltonian. In addition, the dominant contribution to the degenerate states is clear. The formula can be interpreted in analogy to electrodynamics so that a " magnetic field" corresponds to ( mathematically: curvature of a connection form) and its vector potential ( mathematically related form).

For higher dimensions of the parameter space of the differential form calculus must be used.

Berry phase has been generalized in various ways, for example in the case of degenerate conditions, which are mixed by a unitary matrix ( which occurs here, instead of a simple phase factor ). This case is known as non- Abelian Berry phase and Wilczek -Zee phase (after Frank Wilczek and Anthony Zee ).

The Berry phase has been used to describe many different physical phenomena under a unified point of view, including anomalies in quantum field theory.