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arXiv:1307.5315 · Physical Review A

Non-Abelian geometric phases in a system of coupled quantum bits

Vahid Azimi Mousolou, Erik Sjöqvist

Published Feb 20, 2014• 27 citations
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Abstract

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.

DOI
10.1103/physreva.89.022117

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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.
The concept of off-diagonal geometric phase (GP) has been introduced in order to recover interference information about the geometry of quantal evolution where the standard GPs are not well-defined. In this Letter, we propose a physical setting for realizing non-Abelian off-diagonal GPs. The proposed non-Abelian off-diagonal GPs can be implemented in a cyclic chain of four qubits with controllable nearest-neighbor interactions. Our proposal seems to be within reach in various nano-engineered systems and therefore opens up for first experimental test of the non-Abelian off-diagonal GP.
Geometric phases encode the holonomy acquired under cyclic control of a quantum Hamiltonian: for nondegenerate eigenstates the effect is Abelian (a scalar phase), while in degenerate manifolds the Wilczek-Zee generalization yields a matrix-valued holonomy that can mix states and depends on path ordering. We present an overview of measuring Abelian and non-Abelian geometric phases in nitrogen-vacancy (NV) centers in diamond and propose a non-rotating measurement protocol enabled by the intrinsic14 N{ }^{14} ~\mathrm{N}14 Nhyperfine manifold. As a first step, we propose to measure the Abelian Berry phase on a single hyperfine-resolved two-level transition using microwaveengineered closed loops with dynamical-phase cancellation by loop reversal. We then propose a non-Abelian protocol that engineers a two-dimensional degenerate dark subspace in an NV-14 N{ }^{14} ~\mathrm{N}14 Nmanifold via RF dressing and three phase-coherent microwave couplings (a tripod Hamiltonian). Two closed control loops implemented by winding independent microwave phases generate non-commuting SU(2) holonomies, which can be verified using final-only population readout and minimal tomography. We also present timing trade-off curves, full unitary Schrödinger evolution of the four-level tripod, and projected dark-subspace diagnostics that quantify both leakage and non-commuting holonomy.
We present and implement a method for the experimental measurement of geometric phase of non-geodesic (small) circles on any SU(2) parameter space. This phase is measured by subtracting the dynamic phase contribution from the total phase accumulated. Our design does not require theoretical anticipation of this dynamic phase value and the methods are generally applicable to any system accessible to interferometric and projection measurements. Experimental implementations are presented for two settings: (1) the sphere of modes of orbital angular momentum, and (2) the Poincaré sphere of polarizations of Gaussian beams.
The spin vector of a spin-1 system, unlike that of a spin-1/2 system, can lie anywhere on or inside the Bloch sphere representing the phase space. As a consequence, the geometrical and topological properties of the spin-1 phase space of quantum states are richer and require a generalization of Berry's phase. For special trajectories passing through the center of the Bloch sphere (singular loops), the geometric phase has a non-Abelian nature. Here, we experimentally explore this geometric phase for singular loops in a spin-1 quantum system using ultracold ^{87}Rb atoms confined in an optical trap using microwave and rf control fields.
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.
We present a split-beam neutron interferometric experiment to test the non-cyclic geometric phase tied to the spatial evolution of the system: the subjacent two-dimensional Hilbert space is spanned by the two possible paths in the interferometer and the evolution of the state is controlled by phase shifters and absorbers. A related experiment was reported previously by some of the authors to verify the cyclic spatial geometric phase. The interpretation of this experiment, namely to ascribe a geometric phase to this particular state evolution, has met severe criticism. The extension to non-cyclic evolution manifests the correctness of the interpretation of the previous experiment by means of an explicit calculation of the non-cyclic geometric phase in terms of paths on the Bloch-sphere. The theoretical treatment comprises the cyclic geometric phase as a special case, which is confirmed by experiment.