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Fudan xuebao. Ziran Kexue ban

Geometric Phase in Two-state System and in Coherent State of Forced Oscillator

Suqing Chen

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The formula for deriving the geometric phase in two-state system is dicussed, emphasizing on the property of Bargmann invariant. A comparison with the coherent state of forced oscillator is also made.

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The concept of Geometric Phase in Quantum Mechanics is generally formulated en- tirely in terms of geometric structure of complex Hilbert Space. This tutorial article gives a general mathematical overview of the Geometric phase introduced by Berry with the conditions over the system’s evolution embedded through adiabaticity and cyclicity, which were subsequently relaxed sequentially by Aharonov-Anandan, Samuel-Bhandari, and later by Mukunda and Simon by using the idea of Bargmann Invariants. The arti- cle presents a thorough and illustrative overview regarding the mathematical derivation behind the upliftment of the conditions, which results in the generalized definition of the Geometric phase for quantum systems. In addition to that, the article also gives an overall idea about the general analytical aspects of finding the Geometric phase for three level open quantum systems undergoing decoherence and dephasing due to interaction with the surroundings in the weak system reservoir coupling limit described by quantum master equations.
Properties of geometric phase under Galilean transformation are investigated within a general framework. We found that the geometric phase in the system commonly met with in the laboratory is Galilean invariant. However, this is not always true for general non-relativistic quantum systems. The physical origins of the frame dependence of geometric phase is discussed in detail. We emphasize that the change of geometric phase caused by Galilean boosts does not destroy the results of measurement for two-state interference. It is consistent with the general constraint of Galilean invariance on a non-relativistic quantum system.
We present a theory of the geometric phase based logically on the Bargmann invariant of quantum mechanics, and null phase curves in ray space, as the fundamental ingredients. Null phase curves are themselves defined entirely in terms of the (third order) Bargmann invariant, and it is shown that these are the curves natural to geometric phase theory, rather than geodesics used in earlier treatments. The natural symplectic structure in ray space is seen to play a crucial role in the definition of the geometric phase. Logical consistency of the formulation is explicitly shown, and the principal properties of geometric phases are deduced as systematic consequences.