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Physical Review A

Uniquely defined geometric phase of an open system

Nikola Burić, Milan Radonjić

Published Jul 8, 2009• 26 citations
Publisher

Abstract

Various types of unravelings of Lindblad master equation have been used to define the geometric phase for an open quantum system. Approaches of this type were criticized for lacking in unitary symmetry of the Lindblad equation [A. Bassi and E. Ippoliti, Phys. Rev. A 73, 062104 (2006)]. We utilize quantum state diffusion (QSD) approach to demonstrate that a geometric phase invariant on the symmetries of the Lindblad equation can be defined. It is then shown that such a definition of the geometric phase could be either invariant on the decomposition of the initial mixed state or gauge invariant, but not both. This alternative is inherent to the definitions based on quantum trajectories. The QSD geometric phase is computed for a qubit in different types of environments.

DOI
10.1103/physreva.80.014101

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Various types of unravelings of Lindblad master equation have been used to define the geometric phase for an open quantum system. Approaches of this type were criticized for lacking in unitary symmetry of the Lindblad equation [A. Bassi and E. Ippoliti, Phys. Rev. A 73, 062104 (2006)]. We utilize quantum state diffusion (QSD) approach to demonstrate that a geometric phase invariant on the symmetries of the Lindblad equation can be defined. It is then shown that such a definition of the geometric phase could be either invariant on the decomposition of the initial mixed state or gauge invariant, but not both. This alternative is inherent to the definitions based on quantum trajectories. The QSD geometric phase is computed for a qubit in different types of environments.
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.
A geometrical description of open quantum systems is presented. This is based on a connection on a vector bundle over a Grassmann manifold, as well as the Anandan geometric phase for pure states [J. Anandan, Phys. Lett. A 133, 171 (1988)]. The geometric phase proposed here also naturally includes the geometric phase in adiabatic open quantum systems proposed by Sarandy and Lidar [Phys. Rev. A 73, 062101 (2006)]. The present geometrical description can be applied to all the quantum systems described by time-local master equations for density operators.
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 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.