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Acta Physica Sinica

Geometric quantum phase for three-level mixed state

Rao Huang-Yun, Liu Yi-Bao, Jiang Yan-Yan, Liping Guo, Wang Zisheng

Published Jan 1, 2012• 0 citations

Abstract

By expanding the density matrix of the open system in terms of Gell-mann matrix in a three-level system, we parameterize coefficients of expansion by some azimuthal angles and find an identity mapping of the density matrices onto interior points of the unit Poincaré sphere. Thus, the relations between the points on the unit Poincaré sphere and wave functions are extended to connect the interior points in the sphere with the nonunit vector rays corresponding to an open system in complex Hilbert space. Thus,the geometric phases for the open system are proposed to be observed by the nonunit vector rays,where the geometric phase of the pure state is the limiting case of our definition. The results show that this geometric phase merely with duplicate three-dimensional Hilbert projection space geometry structure related, has nothing to do with the open system concrete evolution way; and it depends on population inversion and is a slippy and single-value curve of Bloch radius. Therefore, the mixed state of open system retains indeed a memory of its motion in the form of a geometric phase factor.

DOI: 10.7498/aps.61.020302

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A mapping is established in connecting density matrices, associated with an evolution of a quantum open system, with vector ray in a complex projective Hilbert space. By using the corresponding vector ray to represent the open two-level system, we may observe the geometric phase for the open two-level system. The geometric phase of the open two-level system depends only on the smooth (open or closed) curve in the complex projective Hilbert space of ray, which is formulated entirely in terms of geometric structures on this space.

This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension

d

and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.

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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.