Berry phase

The Berry phase or geometric phase occurs in a quantum mechanical system when, when slowly ( adiabatically ) traversing a closed path in the parameter space of the system, the system does not return to its initial state, but its wave function receives a phase factor , precisely the Berry phase. The Berry phase is not limited to quantum mechanical systems; an analogous effect can also be found in classical systems (see below ).

The Berry phase is named after Michael Berry who introduced it in 1983. However, there were other forerunners that were forgotten again. For example, S. Pancharatnam discovered the phenomenon in the context of classical physics with polarization cycles (1956), which is why it is sometimes also named after Pancharatnam.

description

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

In general, a system (described by the Hamilton operator ) must depend on at least two parameters, and the parameter space must, for example, have singularities or a nontrivial topology (illustratively: holes) in order to have a non-trivial topology (deviating from 1 for a closed path in the parameter space) ) Phase factor. Points in the parameter space where the energy levels of "neighboring" states approach the energy of the initial state (degeneracy points) are of particular importance, because only there such non-trivial phase factors can arise if the cycles enclose them. The parameter space reduced by these degeneracy points receives a non-trivial topology. Since the geometry of the ( augmented ) parameter space is of crucial importance, the Berry phase is also referred to as the geometric phase.

Examples of occurrence

Quantum mechanical Berry phase

Examples of the Berry phase are adiabatic cycles in the molecular coordinates , which can generate a phase factor in the wave function of the electrons, which can be treated in the Born-Oppenheimer approximation : the Hamilton operator and the wave function of the electrons can be parameterized by the nuclear coordinates. This was one of Berry's original examples, and one such example was discovered by Christopher Longuet-Higgins in 1958 . The case of the geometric phase in molecular physics was dealt with in particular by Alden Mead and Donald Truhar, with initial work in 1979. Spectroscopic experiments on this, which confirmed the theory of Alden and Mead, took place in the 1980s.

In his essay from 1984 Berry gives an example in which the Berry phase can be calculated explicitly relatively easily: a spin in a magnetic field that is slowly varied by the direction of the magnetic field following a closed curve . The phase is proportional in this case , where n is the spin quantum number ( ) and the opening angle from seen from the origin . For spin 1/2, this corresponded to a formula that Pancharatnam had derived for polarized light in 1956. ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle n\cdot \Omega (C)}$ ${\displaystyle -s\leq n\leq s}$ ${\displaystyle \Omega (C)}$ ${\displaystyle C}$ ${\displaystyle B=0}$

The spin 1/2 example can be extended to general quantum mechanical two-state systems, described by complex Hermitian 2 × 2 matrices. Here, too, there is a degeneracy point and a formula that describes the Berry phase by the spatial opening angle at which the cycle is viewed from the degeneracy point (phase ). ${\displaystyle {\frac {\Omega (C)}{2}}}$

An even simpler example results in the case of real symmetric 2 × 2 matrices, which in quantum mechanics correspond to time-reversal invariant systems. The case corresponds to a sentence from elementary matrix theory (Berry), whose parameter dependency is considered. In the simplest case of real symmetric 2 × 2 matrices:

${\displaystyle {\begin{pmatrix}u\,v\\v\,w\end{pmatrix}}}$

the eigenvectors are real and therefore only the prefactors +1 (phase 0 or ) and −1 (phase ) come into consideration for cycles in the parameter space for the “Berry phases” . Non-trivial Berry phases with a prefactor −1 (change of sign) only exist if the cycle encircles a degeneracy point in the parameter space. These are by the straight line , given in the parameter space (u, w, v). Only if the cycles enclose the straight line there is a change in sign in the eigenvectors. ${\displaystyle 2\pi }$ ${\displaystyle \pi }$ ${\displaystyle v=0}$ ${\displaystyle u=w}$

Another example is the Aharonov-Bohm effect . The parameter space is the usual spatial space here, but due to the magnetic field inside the closed path (there is a singularity for the field of the vector potential) it is no longer regarded as simply connected. The wave function of an electron orbiting the magnetic field receives a phase factor proportional to the magnetic flux , although at the location of the electron itself the magnetic field disappears everywhere (but not the associated vector potential ).

The Berry phase can be observed , for example, in interference experiments . An experimental proof of the Berry phase in optical experiments with linearly polarized light with glass fibers wound around a cylinder (helically) succeeded Akira Tomita and Raymond Chiao in 1986 . The experiment can be described in the context of Berry's spin case discussed above and measures the solid angle during the rotation of the “spin direction”. Similar experiments with neutrons that were sent through a helically variable magnetic field were also carried out in the 1980s by T. Bitter (Heidelberg) and D. Dubbers ( Institut Laue-Langevin ). In 1987, Robert Tycko ( Bell Laboratories ) carried out an experiment to detect the Berry phase, in which the rotation of nuclear spins, which were bound to crystal axes, took place by rotating the entire crystal about axes other than its axes of symmetry.

Occurrence in classical mechanics

In classical mechanics, the Foucault pendulum provides an example of a geometric phase. The pendulum does not return to its starting point with one full revolution of the earth in 24 hours (closed path in the parameter space), but a phase shift occurs depending on the geographical latitude. The geometric phase in classical mechanics is also known as the Hannay angle (after John Hannay , a colleague of Berry in Bristol).

Mathematical definition

Mathematically, the Berry phase is an expression of a holonomy . A simple example of a holonomy is the parallel transport of a vector on the spherical surface in a triangle made of great circles: If you start from the north pole, go to the equator, follow this 90 degrees and then return to the north pole, this causes a 90 degree rotation of the parallel transported Vector (an analogy to the Berry phase). This rotation depends solely on the geometry (curvature) of the underlying space (the sphere).

Formally, the geometric phase is given by:

${\displaystyle \gamma _{n}[C]=i\oint _{C}\!\langle n({\vec {R}})|\left(\nabla _{\vec {R}}\,n({\vec {R}})\right)\,\rangle \cdot d{\vec {R}}\,}$

integrating via a closed path C in the parameter space (variable , which is generally vectorial ). The Bra-Ket notation is used for the states, is an eigenstate of the Hamilton operator of the system. is the Nabla operator of the derivative according to the parameters. Because of the normalization of the states is imaginary and thus real. ${\displaystyle {\vec {R}}}$ ${\displaystyle |n({\vec {R}})\rangle }$ ${\displaystyle \nabla _{\vec {R}}}$ ${\displaystyle |n({\vec {R}})\rangle }$ ${\displaystyle \langle n({\vec {R}})|\nabla _{\vec {R}}\,n({\vec {R}})\rangle }$ ${\displaystyle \gamma [C]}$

After running through the cycle in the parameter space, the wave function transforms to:

${\displaystyle |n_{\text{nach Pfad C}}\rangle =|n_{\text{Start}}\rangle \cdot e^{\mathrm {i} \cdot \gamma _{n}[C]}}$

In the three-dimensional case, the formula for the geometric phase can be brought into a form that is more favorable for the application by applying Stokes' theorem (conversion into a surface integral):

${\displaystyle \gamma _{n}[C]=-\,Im\iint _{\mathcal {C}}(\nabla _{\vec {R}}\,\times \,\langle n({\vec {R}})|\left(\nabla _{\vec {R}}\,n({\vec {R}})\right)\,\rangle )\cdot d{\vec {F}}=-\,\iint _{\mathcal {C}}{\vec {V}}_{n}({\vec {R}})\cdot d{\vec {F}}}$

${\displaystyle {\vec {V}}_{n}({\vec {R}})=Im\sum _{m\neq n}{\frac {\langle n({\vec {R}})|\nabla _{\vec {R}}\,{\hat {H}}|m({\vec {R}})\,\rangle \,\times \langle m({\vec {R}})|\nabla _{\vec {R}}\,{\hat {H}}|n({\vec {R}})\,\rangle }{{(E_{m}({\vec {R}})-E_{n}({\vec {R}}))}^{2}}}}$

In this formula only the expected values ​​of the derivative of the Hamilton operator appear. In addition, the dominant contribution of the degenerate states becomes clear. In analogy to electrodynamics, the formula can be interpreted in such a way that it corresponds to a “magnetic field” (mathematically: curvature of a form of connection) and its vector potential (mathematically: form of connection). ${\displaystyle \nabla _{\vec {R}}\,{\hat {H}}}$ ${\displaystyle |n\rangle }$ ${\displaystyle {\vec {V}}_{n}({\vec {R}})}$ ${\displaystyle \langle n({\vec {R}})|\left(\nabla _{\vec {R}}\,n({\vec {R}})\right)\,\rangle }$

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

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

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

literature

• Alfred Shapere, Frank Wilczek (Ed.): Geometric Phases in Physics . World Scientific, 1989 (reprint volume in introductions by Berry: The quantum phase- five years after and Roman Jackiw : Three elaborations on Berry's connection, curvature and phase , as a reprint from International J. Mod. Phys. , Volume 3, 1988, Pp. 285–297)
• Jeeva Anandan, Joy Christian, Kazimir Wanelik: Resource Letter GPP-1: Geometric Phases in Physics . In: American Journal of Physics . tape 65 , no. 3 , 1997, p. 180 , doi : 10.1119 / 1.18570 , arxiv : quant-ph / 9702011 .
• C. Alden Mead : The geometric phase in molecular systems . In: Reviews of Modern Physics . tape 64 , no. 1 , January 1, 1992, p. 51-85 , doi : 10.1103 / RevModPhys.64.51 .
• Michael V. Berry: The geometric phase . In: Sci. Amer. tape 259 , no. 6 , December 1988, pp. 46-52 ( phy.bris.ac.uk [PDF]).
• Roman Jackiw: Berry's phase: topological ideas from atomic, molecular and optical physics . In: Comments on Atomic and Molecular Physics . tape 21 , 1988, pp. 71–82 ( cern.ch [PDF]).
• Arno Bohm , Quian Niu, Ali Mostafazadeh, Hiroyasu Koizumi, Josef Zwanziger: The geometric phase in quantum systems - foundations, mathematical concepts and applications in molecular and condensed matter physics . Springer Verlag, 2003 (deals with the quantum Hall effect, among other things)
• Dariusz Chruscinski, Andrzej Jamiolkowski: Geometric phases in classical and quantum mechanics , Birkhäuser 2004

Web links

• Berry's geometric phase: a review. (No longer available online.) University of Milan, March 18, 2013, archived from the original on March 27, 2015 ; accessed on June 18, 2016 . 
• Daniel Rohrlich: Berry's Phase . In: Greenberger u. a .: Compendium of Quantum Physics , Preprint 2007, arxiv : 0708.3749

Individual evidence

1. ↑ ^{a b} M. V. Berry: Quantal Phase Factors Accompanying Adiabatic Changes . In: Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences . tape 392 , no. 1802 , August 3, 1984, pp. 45-57 , doi : 10.1098 / rspa.1984.0023 .
2. ↑ Some of the forerunners Berry describes in: Michael Berry: Anticipations of the Geometric Phase . In: Physics Today . tape 43 , no. 12 , 1990, pp. 34-40 , doi : 10.1063 / 1.881219 .
3. ^ S. Pancharatnam: Generalized Theory of Interference, and Its Applications. Part I. Coherent Pencils . In: Proceedings of the Indian Academy of Sciences, Section A . tape 44 , 1956, pp. 247-262 (on- line ).
4. ↑ See Michael Berry: Anticipations of the Geometric Phase . In: Physics Today . tape 43 , no. 12 , 1990, pp. 34-40 , doi : 10.1063 / 1.881219 . Another early work was Gerhard Herzberg , HC Longuet-Higgins: Intersection of potential energy surfaces in polyatomic molecules . In: Discussions of the Faraday Society . tape 35 , January 1, 1963, p. 77-82 , doi : 10.1039 / DF9633500077 .
5. ^ C. Alden Mead, Donald G. Truhlar: On the determination of Born-Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei . In: The Journal of Chemical Physics . tape 70 , no. 5 , March 1, 1979, pp. 2284-2296 , doi : 10.1063 / 1.437734 .
6. ↑ Guy Delacrétaz, Edward R. Grant, Robert L. Whetten, Ludger Wöste, Josef W. Zwanziger: Fractional Quantization of Molecular Pseudorotation in Na ₃ . In: Physical Review Letters . tape 56 , no. 24 , June 16, 1986, pp. 2598-2601 , doi : 10.1103 / PhysRevLett.56.2598 .
7. ↑ Which especially excludes systems with magnetic fields.
8. ↑ Michael Berry: Anticipations of the Geometric Phase . In: Physics Today . tape 43 , no. 12 , 1990, pp. 34-40 , doi : 10.1063 / 1.881219 .
9. ↑ An adiabatic approximation is not required here.
10. ↑ Akira Tomita, Raymond Y. Chiao: Observation of Berry's Topological Phase by Use of an Optical Fiber . In: Physical Review Letters . tape 57 , no. 8 , August 25, 1986, pp. 937-940 , doi : 10.1103 / PhysRevLett.57.937 . Theoretical suggestions were previously made in Raymond Y. Chiao, Yong-Shi Wu: Manifestations of Berry's Topological Phase for the Photon . In: Physical Review Letters . tape 57 , no. 8 , August 25, 1986, pp. 933-936 , doi : 10.1103 / PhysRevLett.57.933 . The experiment is considered to be the first modern proof of the Berry effect. There were then discussions as to whether the experiment was a classical or a quantum effect.
11. ↑ T. Bitter, D. Dubbers: Manifestation of Berry's topological phase in neutron spin rotation . In: Physical Review Letters . tape 59 , no. 3 , July 20, 1987, pp. 251-254 , doi : 10.1103 / PhysRevLett.59.251 .
12. ^ Robert Tycko: Adiabatic Rotational Splittings and Berry's Phase in Nuclear Quadrupole Resonance . In: Physical Review Letters . tape 58 , no. 22 , June 1, 1987, pp. 2281-2284 , doi : 10.1103 / PhysRevLett.58.2281 .
13. ↑ Treated u. a. in the cited article: Michael Berry: Anticipations of the Geometric Phase . In: Physics Today . tape 43 , no. 12 , 1990, pp. 34-40 , doi : 10.1063 / 1.881219 .
14. ^ JH Hannay: Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian . In: Journal of Physics A: Mathematical and General . tape 18 , no. 2 , February 1, 1985, p. 221-230 , doi : 10.1088 / 0305-4470 / 18/2/011 .
15. ↑ Berry himself prefers the term anholonom , Michael Berry: Anticipations of the Geometric Phase . In: Physics Today . tape 43 , no. 12 , 1990, pp. 34-40 , doi : 10.1063 / 1.881219 . In his original work from 1984, Berry points out that Barry Simon referred him to the interpretation as holonomy. See also Barry Simon: Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase . In: Physical Review Letters . tape 51 , no. 24 , December 12, 1983, pp. 2167-2170 , doi : 10.1103 / PhysRevLett.51.2167 .
16. ↑ Roman Jackiw also speaks of the Berry curvature and the Berry connection
17. ^ Frank Wilczek, A. Zee: Appearance of Gauge Structure in Simple Dynamical Systems . In: Physical Review Letters . tape 52 , no. 24 , June 11, 1984, pp. 2111-2114 , doi : 10.1103 / PhysRevLett.52.2111 .
18. ↑ z. B. Philip Nelson, Luis Alvarez-Gaumé : Hamiltonian interpretation of anomalies . In: Communications in Mathematical Physics . tape 99 , no. 1 , March 1985, p. 103-114 , doi : 10.1007 / BF01466595 .