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arXiv:0705.2257

Some geometric features of Berry's phase

Alejandro José Cabrera

Published May 15, 2007• 0 citations
arXivHTMLPDF

Abstract

In this letter, we elaborate on the identification and construction of the differential geometric elements underlying Berry's phase. Berry bundles are built generally from the physical data of the quantum system under study. We apply this construction to typical and recently investigated systems presenting Berry's phase to explore their geometric features.

DOI
10.48550/arxiv.0705.2257

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In this letter, we elaborate on the identification and construction of the differential geometric elements underlying Berry's phase. Berry bundles are built generally from the physical data of the quantum system under study. We apply this construction to typical and recently investigated systems presenting Berry's phase to explore their geometric features.
The discovery of the geometric phase is one of the most interesting and intriguing findings of the last few decades. It led to a deeper understanding of the concept of phase in quantum mechanics and motivated a surge of interest in fundamental quantum mechanical issues, disclosing unexpected applications in very diverse fields of physics. Although the key ideas underlying the existence of a purely geometrical phase had already been proposed in 1956 by Pancharatnam, it was Michael Berry who revived this issue 30 years later. The clarity of Berry's seminal paper, in 1984, was extraordinary. Research on the topic flourished at such a pace that it became difficult for non-experts to follow the many different theoretical ideas and experimental proposals which ensued. Diverse concepts in independent areas of mathematics, physics and chemistry were being applied, for what was (and can still be considered) a nascent arena for theory, experiments and technology. Although collections of papers by different authors appeared in the literature, sometimes with ample introductions, surprisingly, to the best of my knowledge, no specific and exhaustive book has ever been written on this subject. The Geometric Phase in Quantum Systems is the first thorough book on geometric phases and fills an important gap in the physical literature. Other books on the subject will undoubtedly follow. But it will take a fairly long time before other authors can cover that same variety of concepts in such a comprehensive manner. The book is enjoyable. The choice of topics presented is well balanced and appropriate. The appendices are well written, understandable and exhaustive - three rare qualities. I also find it praiseworthy that the authors decided to explicitly carry out most of the calculations, avoiding, as much as possible, the use of the joke `after a straightforward calculation, one finds...' This was one of the sentences I used to dislike most during my undergraduate studies. A student is inexperienced in such matters and needs to look at details. This book is addressed to graduate physics and chemistry students and was written thinking of students. However, I would recommend it also to young and mature physicists, even those who are already `into' the subject. It is a comprehensive work, jointly written by five researchers. After a simple introduction to the subject, the book gradually provides deeper concepts, more advanced theory and finally an interesting introduction and explanation of recent experiments. For its multidisciplinary features, this work could not have been written by one single author. The collaborative effort is undoubtedly one of its most interesting qualities. I would definitely recommend it to anyone who wants to learn more on the geometric phase, a topic that is both beautiful and intruiguing.
The geometric phase is the comparison the of a physical quantity due to the changes on a closed cruve in the system state space. These phases are experimentally measurable because of, that the space of a time dependent quantum system may be a curveture, therefore the Holonomy arises from this curveture. The Berry phase has three key features, which makes it very important. First, the Berry phase is gauge invariance. Especial functions are defined by aequal linear equation, hence it has a multiplicative guage freedom in a general phase factor independent of the parameter. Berry phase (forget from on phase factor ) with this phase factor remains unchanged provided that the special function is on a single loop value. This makes the Berry phase to be a physical quantity that is experimentally measureable by inter ference phenomena. Secondly, Berry phase is a geometric concept. that’s mean it can be written as a linear integral of loop in parameter space, which does not depend on the rate of change on the loop. This property makes it mesureable the Berry phase in terms of local geometric quantites in the parameter space. In fact, Berry showed that this phase can be written as a find integral (Berry curve) on the surface. Berry curve plays an essential role in effective dynamics with variables with degrees of slow release. Third, the Berry phase is similar to that of guage field and differential geometry theory. This makes beauty, intuition and the same concept in physics.