Geometric Phase in Quantum Mechanics is generally formulated entirely in terms of geometric structure of the Complex Hilbert Space. We will exploit this fact in case of mixed states for three level open systems undergoing depolarization using the eight di- mensional Poincare sphere and non unit vector rays in H3 within the limit of pure state approach may be found to be in agreement with the Pancharatnam Phase, Berry Phase and Aharonov-Anandan Phase. We will consider the Depolarization of the Three di- mensional fields by introducing the stokes parameters and by redefining the definition of Degree of Polarization using the SU(3) algebra to describe the quantum light depolariza- tion of Non-paraxial beams by minimal reservoir coupling models modeled by Lindblad Master equations and connect the solution of that with the geometric phase for systems undergoing depolarization.

In this article we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian. The idea of the mesoscopic lead construction and brute force chain mapping techniques has been introduced which can be used to bypass the strong system bath coupling situations.

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.

This tutorial presents an illustrative overview behind the idea of the two time mea- surement protocol scheme in connection with the method of Full Counting Statistics, a widely used tool in the context of Quantum Thermodynamics and quantum noises. This article presents the mathematical tools to derive the quantum master equation for the de- formed reduced density operator for the system in the most general way by tracing out the bath degrees of freedom for different types of interaction Hamiltonians generally used to model the open quantum system problems in the limit of weak system bath coupling. The derivation of the microscopic Q.M.E has also been described. Finally, the article provides an application of the Full counting statistics method for the well known Resonant Level Model of thermoelectric transport and its connection with the Fluctuation symmetry.

This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.

Geometric Phase in Quantum Mechanics is generally formulated entirely in terms of geometric structure of the Complex Hilbert Space. We will exploit this fact in case of mixed states for three level open systems undergoing depolarization using the eight dimensional Poincare sphere in the SU(2) Polarisation picture and non unit vector rays in H3 within the limit of pure state approach may be found to be in agreement with the Pancharatnam Phase, Berry Phase and Aharonov-Anandan Phase.