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Non-Abelian quantum holonomy of hydrogen-like atoms
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. (Physics)
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics. (Physics)ORCID iD: 0000-0003-4489-7561
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry, Quantum Chemistry) (Quantum information theory.
2011 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, ISSN 1050-2947, Vol. 84, no 3, p. Article ID: 032111-Article in journal (Refereed) Published
Abstract [en]

We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum states for hydrogen-like atoms, where the intrinsic spin and orbital angular momentum are coupled by the spin-orbit interaction and subject to a slowly varying magnetic field. We show that the holonomy for the orbital angular momentum and spin subsystems is non-Abelian, while the holonomy of the whole system is Abelian. Quantum entanglement in the states of the whole system is crucially related to the non-Abelian gauge structure of the subsystems. We analyze the phase of the Wilson loop variable associated with the Uhlmann holonomy, and find a relation between the phase of the whole system with corresponding marginal phases. Based on the result for the model system we provide evidence that the phase of the Wilson loop variable and the mixed-state geometric phase [Phys. Rev. Lett. 85, 2845 (2000)] are in general inequivalent.

Place, publisher, year, edition, pages
2011. Vol. 84, no 3, p. Article ID: 032111-
Keywords [en]
Quantum holonomy, spin-orbit coupling, hydrogen-like atoms, quantum entanglement
National Category
Natural Sciences
Research subject
Natural Science, Physics
Identifiers
URN: urn:nbn:se:lnu:diva-11495DOI: 10.1103/PhysRevA.84.032111Scopus ID: 2-s2.0-80053118405OAI: oai:DiVA.org:lnu-11495DiVA, id: diva2:413438
Available from: 2011-04-28 Created: 2011-04-28 Last updated: 2017-12-11Bibliographically approved
In thesis
1. Quantum Holonomy for Many-Body Systems and Quantum Computation
Open this publication in new window or tab >>Quantum Holonomy for Many-Body Systems and Quantum Computation
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The research of this Ph. D. thesis is in the field of Quantum Computation and Quantum

Information. A key problem in this field is the fragile nature of quantum states. This

becomes increasingly acute when the number of quantum bits (qubits) grows in order to

perform large quantum computations. It has been proposed that geometric (Berry) phases

may be a useful tool to overcome this problem, because of the inherent robustness of such

phases to random noise. In the thesis we investigate geometric phases and quantum

holonomies (matrix-valued geometric phases) in many-body quantum systems, and elucidate

the relationship between these phases and the quantum correlations present in the systems.

An overall goal of the project is to assess the feasibility of using geometric phases and

quantum holonomies to build robust quantum gates, and investigate their behavior when the

size of a quantum system grows, thereby gaining insights into large-scale quantum

computation.

In a first project we study the Uhlmann holonomy of quantum states for hydrogen-like

atoms. We try to get into a physical interpretation of this geometric concept by analyzing its

relation with quantum correlations in the system, as well as by comparing it with different

types of geometric phases such as the standard pure state geometric phase, Wilczek-Zee

holonomy, Lévay geometric phase and mixed-state geometric phases. In a second project we

establish a unifying connection between the geometric phase and the geometric measure of

entanglement in a generic many-body system, which provides a universal approach to the

study of quantum critical phenomena. This approach can be tested experimentally in an

interferometry setup, where the geometric measure of entanglement yields the visibility of

the interference fringes, whereas the geometric phase describes the phase shifts. In a third

project we propose a scheme to implement universal non-adiabatic holonomic quantum

gates, which can be realized in novel nano-engineered systems such as quantum dots,

molecular magnets, optical lattices and topological insulators. In a fourth project we propose

an experimentally feasible approach based on “orange slice” shaped paths to realize non-

Abelian geometric phases, which can be used particularly for geometric manipulation of

qubits. Finally, we provide a physical setting for realizing non-Abelian off-diagonal

geometric phases. The proposed setting 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 geometric phase.

Place, publisher, year, edition, pages
Växjö: Linnaeus University Press, 2013. p. 140
Series
Linnaeus University Dissertations ; 141/2013
Keywords
Quantum holonomy, geometric phase, quantum correlations, quantum phase transitions, quantum computation
National Category
Condensed Matter Physics Other Physics Topics
Research subject
Natural Science; Natural Science, Physics
Identifiers
urn:nbn:se:lnu:diva-28311 (URN)978-91-87427-38-1 (ISBN)
Public defence
2013-08-26, Ny227, Kalmar Nyckel, Kalmar, 13:00 (English)
Opponent
Supervisors
Available from: 2013-09-10 Created: 2013-08-21 Last updated: 2013-09-10Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

Publisher's full textScopushttp://arxiv/1103.2435pdf

Authority records BETA

Azimi Mousolou, VahidCanali, Carlo M.

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