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arXiv:quant-ph/0010068 · Physical Review Letters

Geometric Phase of Three-Level Systems in Interferometry

Barry C. Sanders, Hubert de Guise, Stephen D. Bartlett, Weiping Zhang

Published Jan 15, 2001• 38 citations
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Abstract

We present the first scheme for producing and measuring an Abelian geometric phase shift in a three-level system where states are invariant under a non-Abelian group. In contrast to existing experiments and proposals for experiments, based on U(1)-invariant states, our scheme geodesically evolves U(2)-invariant states in a four-dimensional SU(3)/U(2) space and is physically realized via a three-channel optical interferometer.

DOI
10.1103/physrevlett.86.369

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We present the first scheme for producing and measuring an Abelian geometric phase shift in a three-level system where states are invariant under a non-Abelian group. In contrast to existing experiments and proposals for experiments, based on U(1)-invariant states, our scheme geodesically evolves U(2)-invariant states in a four-dimensional SU(3)/U(2) space and is physically realized via a three-channel optical interferometer.
Inspired by the optical experimental scheme [Phys. Rev. Lett. 86, 369 (2001)], we propose and implement a NMR experiment to observe an Abelian geometric phase shift arising from the cyclic evolution of U(2)-invariant states. Such a phase shift geometrically refers to the Bargmann invariant in a four-dimensional SU(3)/U(2) parameter space, while the usual geometric phases arise in the evolution of U(1)-invariant states and is related to solid angles in a two-dimensional $\text{SU}(2)/\text{U}(1)\ensuremath{\approx}{S}^{2}$ parameter space (or Poincar\'e sphere). We present the experiment in a four-level system and investigate its topological invariance for different circuits. This may provide a realistic candidate to design complex geometric quantum circuits for future quantum computers.
We describe a recently developed generalization of the Poincaré sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group SU(3) and its generators in the defining representation, and uses geometrical objects and operations in an eight-dimensional real Euclidean space. This construction is then used to develop a generalization of the well known Pancharatnam geometric phase formula, for evolution of a three-level system along a geodesic triangle in state space.
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.
A common strategy to measure the Abelian geometric phase for a qubit is to let it evolve along an orange-slice-shaped path connecting two antipodal points on the Bloch sphere by two different semi-great-circles. Since the dynamical phases vanish for such paths, this allows for direct measurement of the geometric phase. Here, we generalize the ``orange-slice'' setting to the non-Abelian case. The proposed method to measure the non-Abelian geometric phase can be implemented in a cyclic chain of four qubits with controllable interactions.