Experimental Demonstration of the Stability of Berry’s Phase for a Spin-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math>Particle

Filipp, Stefan; Klepp, Jürgen; Hasegawa, Yuji; Plonka-Spehr, C.; Schmidt, Ulrich; Geltenbort, P.; Rauch, H.

doi:10.1103/physrevlett.102.030404

Cited by 139 publications

(169 citation statements)

References 34 publications

(36 reference statements)

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“…The geometric phase is nowadays seen as an important tool for implementing robust quantum gates that can be employed in information processing (E. Sjöqvist, 2008). It appears to be noise resilient, as recent experiments seem to confirm (S. Fillip, 2009). Ref.…”

Section: Interferometric Arrangementsupporting

confidence: 58%

The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

Zela¹

2012

Theoretical Concepts of Quantum Mechanics

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14www.intechopen.com Will-be-set-by-IN-TECHAnd in all cases Pancharatnam's definition applies. Pancharatnam's phase bore therefore an anticipation and -at the same time -a generalization of Berry's phase. Indeed, Berry's assumptions about a cyclic, adiabatic and unitary evolution, turned out to be unnecessary for a geometric phase to appear. This was made clear through the contributions of several authors that addressed the issue right after Berry published his seminal results (Y. Aharonov, 1987;J. Samuel, 1988). Pancharatnam's approach, general as it was when viewed as pregnant of so many concepts related to geometric phases, underlay nonetheless two important restrictions. It addressed nonorthogonal and at the same time pure, viz totally, polarized states. Here again the assumed restrictions turned out to be unnecessary. Indeed, it was recently proposed how to decide whether two orthogonal states are in phase or not (H. M. Wong, 2005). Mixed states have also been addressed (A. Uhlmann, 1986;E. Sjöqvist, 2000) in relation to geometric phases which -under appropriate conditions -can be exhibited as well-defined objects underlying the evolution of such states. The present Chapter should provide an overview of the Pancharatnam-Berry phase by introducing it first within Berry's original approach, and then through the kinematic approach that was advanced by Simon and Mukunda some years after Berry's discovery (N. Mukunda, 1993). The kinematic approach brings to the fore the most essential aspects of geometric phases, something that was not fully accomplished when Berry first addressed the issue. It also leads to a natural introduction of geodesics in Hilbert space, and helps to connect Pancharatnam's approach with the so-called Bargmann invariants. We discuss these issues in the present Chapter. Other topics that this Chapter addresses are interferometry and polarimetry, two ways of measuring geometric phases, and some recent generalizations of Berry's phase to mixed states and to non-unitary evolutions. Finally, we show in which sense the relativistic effect known as Thomas rotation can be understood as a manifestation of a Berry-like phase, amenable to be tested with partially polarized states. All this illustrates how -as it has often been the case in physics -a fundamental discovery that is made by addressing a particular issue, can show afterwards to bear a rather unexpected generality and applicability. Berry's discovery ranks among this kind of fundamental advances. The adiabatic and cyclic case: Berry's approachLet us consider a non-conservative system, whose evolution is ruled by a time-dependent Hamiltonian H(t). This occurs when the system is under the influence of an environment. The configuration of the environment can generally be specified by a set of parameters (R 1 , R 2 ,...). For a changing environment the R i are time-dependent, and so also the observables of the system, e.g., the Hamiltonian:The evolution of the quantum system is ruled by the Schrödinger equation, or more generally, by the Liou...

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Section: Interferometric Arrangementsupporting

confidence: 58%

The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

Zela¹

2012

Theoretical Concepts of Quantum Mechanics

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“…This gave results as shown in Fig. 13 [39]. This clearly indicates that the geometric phase becomes better defined when the neutron spends longer time within the noisy field, an effect opposite to the behavior of the dynamical phase.…”

Section: φ(T ) = Arg ψ(T )|ψ(0)supporting

confidence: 50%

“…12 Experimental set-up to measure geometric phases with ultra-cold neutrons by means of a spin-echo method to balance the dynamical phase [39] less sensitive to any fluctuation of external parameters [38]. A related experiment with bottled ultra-cold neutrons has been performed recently [39] (Fig. 12).…”

Section: φ(T ) = Arg ψ(T )|ψ(0)mentioning

confidence: 99%

Neutron Matter Wave Quantum Optics

Rauch

2011

Found Phys

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Neutron matter-wave optics provides the basis for new quantum experiments and a step towards applications of quantum phenomena. Most experiments have been performed with a perfect crystal neutron interferometer where widely separated coherent beams can be manipulated individually. Various geometric phases have been measured and their robustness against fluctuation effects has been proven, which may become a useful property for advanced quantum communication. Quantum contextuality for single particle systems shows that quantum correlations are to some extent more demanding than classical ones. In this case entanglement between external and internal degrees of freedom offers new insights into basic laws of quantum physics. Non-contextuality hidden variable theories can be rejected by arguments based on the Kochen-Specker theorem.

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“…The acquired abelian phase can be divided into two parts: The dynamical phase which is proportional to the evolution time and the energy of the system, and the geometric phase which depends only on the path of the system in Hilbert space. This characteristic feature leads to a resilience of the geometric phase to certain fluctuations during the evolution [17][18][19] , a property which has attracted particular attention in the field of quantum information processing 20 . However, universal quantum computation cannot be based on simple phase gates, which modify only the relative phase of a superposition state, unless they act on specific basis states 21 been proposed for holonomic quantum computation fully based on geometric concepts 8 .…”

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confidence: 99%

Experimental realization of non-Abelian non-adiabatic geometric gates

Abdumalikov

Fink

Júlíusson

et al. 2013

Nature

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The geometric aspects of quantum mechanics are underlined most prominently by the concept of geometric phases, which are acquired whenever a quantum system evolves along a closed path in Hilbert space. The geometric phase is determined only by the shape of this path [1][2][3][4] and is -in its simplest form -a real number. However, if the system contains degenerate energy levels, matrix-valued geometric phases, termed non-abelian holonomies, can emerge 5 . They play an important role for the creation of synthetic gauge fields in cold atomic gases 6 and the description of non-abelian anyon statistics 7 . Moreover, it has been proposed to exploit non-abelian holonomic gates for robust quantum computation [8][9][10] . In contrast to abelian geometric phases 11 , nonabelian ones have been observed only in nuclear quadrupole resonance experiments with a large number of spins and without fully characterizing the geometric process and its non-commutative nature 12,13 . Here, we realize non-abelian holonomic quantum operations 14,15 on a single superconducting artificial three-level atom 16 by applying a well controlled two-tone microwave drive. Using quantum process tomography, we determine fidelities of the resulting non-commuting gates exceeding 95%. We show that a sequence of two paths in Hilbert space traversed in different order yields inequivalent transformations, which is an evidence for the non-abelian character of the implemented holonomic quantum gates. In combination with two-qubit operations, they form a universal set of gates for holonomic quantum computation.A cyclic evolution of a non-degenerate quantum system is in general accompanied by a phase change of its wave function. The acquired abelian phase can be divided into two parts: The dynamical phase which is proportional to the evolution time and the energy of the system, and the geometric phase which depends only on the path of the system in Hilbert space. This characteristic feature leads to a resilience of the geometric phase to certain fluctuations during the evolution 17-19 , a property which has attracted particular attention in the field of quantum information processing 20 . However, universal quantum computation cannot be based on simple phase gates, which modify only the relative phase of a superposition state, unless they act on specific basis states 21 . Furthermore, geometric operations acting on degenerate subspaces have * abdumalikov@phys.ethz.ch † Now at Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA been proposed for holonomic quantum computation fully based on geometric concepts 8 . In this scheme, quantum bits are encoded in a doubly degenerate eigenspace of the system hamiltonian h( λ). The parameters λ are varied to induce a cyclic evolution of the system. When the system returns back to its initial state, it can acquire not only a simple geometric phase factor, but also undergoes a path-dependent unitary transformation, a non-abelian holonomy, which causes a transition bet...

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Experimental Demonstration of the Stability of Berry’s Phase for a Spin-1/2Particle

Cited by 139 publications

References 34 publications

The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

Neutron Matter Wave Quantum Optics

Experimental realization of non-Abelian non-adiabatic geometric gates

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