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Communications in Theoretical Physics

Non-Abelian Geometric Manipulation of a Crystal of Molecular Nanomagnet Mn 12

Lixian Yu, Gang Chen, Jiu‐Qing Liang

Published Aug 1, 2011• 1 citation
Publisher

Abstract

Based on the strong magnetic anisotropy along the symmetry of the crystal, we construct a U(2) non-Abelian gauge potential for the molecular nanomagnet Mn 12 by varying the external magnetic field adiabatically. Moreover, the non-Abelian geometric phase and the unitary matrix operation, which are the key steps to realize the universal holonomic quantum computing in the degenerate subspace, are also obtained by means of choosing an evolution path properly.

DOI
10.1088/0253-6102/56/2/32

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Based on the strong magnetic anisotropy along the symmetry of the crystal, we construct a U(2) non-Abelian gauge potential for the molecular nanomagnet Mn 12 by varying the external magnetic field adiabatically. Moreover, the non-Abelian geometric phase and the unitary matrix operation, which are the key steps to realize the universal holonomic quantum computing in the degenerate subspace, are also obtained by means of choosing an evolution path properly.
Holonomic quantum computing functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold Rb87 atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in SU(2) , characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems. Published by the American Physical Society 2024
In 1984 Berry pointed out that abelian gauge structures exist naturally in slowly varying quantum system. Since then, a great deal of interest both theoretically and experimentally has been devoted to the interpretation, generalization and detection of geometric phases. In particular, relevant for the present work, it has been shown that a non-Abelian gauge structure emerges if a set of quantum states remains degenerate as the Hamiltonian varies. Non-Abelian phases have been originally investigated in Nuclear Quadrupole Resonance. More recently, they have found a renewed interest within the field of Quantum Computation, since it has been proved that quantum gates can be also implemented by geometrical means using both Abelian and non-Abelian phases. In this novel geometric approach, known as Holonomic Quantum Computation (HQC), information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and the computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops, the non trivial curvature of the geometry of the control manifold gives rise to unitary transformations, i.e. holonomies, that implement desired quantum gates. Until now only one scheme has been proposed to realize HQC in trapped ions by means of pulsed coherent excitations. On the other hand, it would be of great interest to have an implementation for solid-state systems, in view of their scalability and integrability. In particular, the impressing development of nanotechnologies with the consequent control of artificial two-level system in superconducting devices stimulates a more detailed investigation of geometric interference in mesoscopic systems. Here we show that non-Abelian phases can appear in the quantum charge dynamics of Josephson networks. Besides their possible detection, we shall see how they may be employed both in adiabatic charge pumping and as an implementation of HQC with Josephson junctions. 2. Josephson network for non abelian phases
Geometric phase that manifests itself in number of optic and nuclear experiments is shown to be a useful tool for realization of quantum computations in so called holonomic quantum computer model (HQCM). This model is considered as an externally driven quantum system with adiabatic evolution law and finite number of the energy levels. The corresponding evolution operators represent quantum gates of HQCM. The explicit expression for the gates is derived both for one-qubit and for multi-qubit quantum gates as Abelian and non-Abelian geometric phases provided the energy levels to be time-independent or in other words for rotational adiabatic evolution of the system. Application of non-adiabatic geometric-like phases in quantum computations is also discussed for a Caldeira-Legett-type model (one-qubit gates) and for the spin 3/2 quadrupole NMR model (two-qubit gates). Generic quantum gates for these two models are derived. The possibility of construction of the universal quantum gates in both cases is shown.