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Comprehensive Review of Gauge and Reparameterization Invariant Geometric Phase for Open Quantum Systems Under the Most General Conditions

Abhirup Chatterjee, Sobhan Kumar Sounda

Published Nov 17, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.20944/preprints202511.1232.v1

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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.
We introduce an operational framework to analyze nonadiabatic Abelian and non-Abelian, cyclic and noncyclic, geometric phases in open quantum systems. In order to remove the adiabaticity condition, we generalize the theory of dynamical invariants to the context of open systems evolving under arbitrary convolutionless master equations. Geometric phases are then defined through the Jordan canonical form of the dynamical invariant associated with the superoperator that governs the master equation. As a by-product, we provide a sufficient condition for the robustness of the phase against a given decohering process. We illustrate our results by considering a two-level system in a Markovian interaction with the environment, where we show that the nonadiabatic geometric phase acquired by the system can be constructed in such a way that it is robust against both dephasing and spontaneous emission.
The geometric phase is defined for any arbitrary quantum evolution using a ``reference section'' of the bundle covering the curve in the projective Hilbert space. A canonical one-form is defined whose line integral gives the desired geometric phase. It is manifestly gauge, phase, and reparametrization invariant for all quantum evolutions. A simple proof of the vanishing nature of the geometric phase along the geodesic is given. Also, an elementary proof of the nonadditive nature of the geometric phase is given. In the limit of cyclic evolution of a pure quantum state, this phase reduces to the Aharonov and Anandan phase, precisely. It is observed that in addition to the geometric phase, other geometric structures exist, such as the ``length'' and ``distance'' during any arbitrary quantum evolution. The relations among all of these geometric quantities are pointed out. Finally, two simple examples are studied to illustrate the ideas introduced in this paper.
The geometric phase in quantum mechanics was originally elucidated in the context of the adiabatic theorem. This and other limitations were later removed. We show how this phase can be presented as the simplest functional of smooth open curves of unit vectors in Hilbert space, invariant under the two groups of local phase changes and monotonic reparametrisations. This viewpoint clarifies the role of the early Pancharatnam criterion for two Hilbert space vectors to be `in phase', in the later geometric phase context. Noncyclic evolution is handled in a very direct manner. Connections to the Bargmann invariants of quantum mechanics are immediate and can be exploited. A class of geometric phases arises in connection with unitary representations of Lie groups. In this case a surprisingly complete analysis of the structure of the geometric phase is possible, motivated by the Wigner-Eckart theorem. The interplay between geometric phase ideas and Lie group unitary representations is extraordinarily rich; the differential geometric structures of Lie groups and coset spaces play very important roles and allow classification of all possible geometric phases associated with a given Lie group. Examples involving SU(2), SU(3) and the metaplectic group Mp(2) are presented.
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.
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1∕21∕21∕2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.