Rédiger un PREreview

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.