The Geometric Phase in Quantum Systems

Böhm, Arno; Mostafazadeh, Alí; Koizumi, Hiroyasu; Niu, Qian; Zwanziger, Joseph

doi:10.1007/978-3-662-10333-3

Cited by 472 publications

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“…This "Hall Effect for light" is a manifestation of the Magnus-type interaction between the refractive medium and the photon's polarization [4]. It can be derived in a semiclassical framework, which also includes a Berry-type term [5,6,7].In this Rapid Communication, we argue that the deviation of polarized light from the trajectory predicted by ordinary geometrical optics is indeed analogous to the deviation of a spinning particle from geodesic motion in General Relativity. The resulting equations are reminiscent of those of Papapetrou and Souriau [8].…”

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confidence: 77%

Fermat principle for spinning light

2006

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Mimicking the description of spinning particles in General Relativity, the Fermat Principle is extended to spinning photons. Linearization of the resulting Papapetrou-Souriau type equations yields the semiclassical model used recently to derive the "Optical Hall Effect" for polarized light (alias the "Optical Magnus Effect").PACS numbers: 42.25. Bs, 03.65.Sq, 03.65.Vf, Light is an electromagnetic wave, whose propagation is described by Maxwell's theory. It can also be viewed, however, as a particle (a "photon"). Here we adhere to the second approach: we describe light by a bona fide mechanical model in that we use a Lagrangian.In traditional geometrical optics the spin degree of freedom is neglected, and the light rays obey the Fermat Principle [1]. In the intermediate model advocated by Landau and Lifchitz [2], the photon is polarized, but the polarization is simply carried along by the light rays, and has no influence on the trajectory of light. Recent approaches [3, 4] go one step further : the feedback from the polarization deviates the trajectory from that given by the Fermat Principle. A dramatic consequence is that, for polarized light, the Snel(-Descartes) law of refraction requires correction : the plane of the refracted (or reflected) ray is shifted perpendicularly to that of the incident ray [3]. This "Hall Effect for light" is a manifestation of the Magnus-type interaction between the refractive medium and the photon's polarization [4]. It can be derived in a semiclassical framework, which also includes a Berry-type term [5,6,7].In this Rapid Communication, we argue that the deviation of polarized light from the trajectory predicted by ordinary geometrical optics is indeed analogous to the deviation of a spinning particle from geodesic motion in General Relativity. The resulting equations are reminiscent of those of Papapetrou and Souriau [8].In detail, the Fermat Principle of geometrical optics * UMR 6207 du CNRS associée aux Universités d'Aix-Marseille I et II et Université du Sud Toulon-Var; Laboratoire affiliéà la FRUMAM-FR2291.; Electronic address: duval@cpt.univ-mrs.fr † Electronic address: e-mail:zalanh@ludens.elte.hu ‡ Electronic address: horvathy@lmpt.univ-tours.fr says that light in an isotropic medium of refractive index n = n(r) propagates along curves that minimize the optical length. Light rays are hence geodesics of the "optical" metric g ij = n 2 (r)δ ij of 3-space. To extend this theory to spin we consider the bundle of positively oriented orthonormal frames over a 3-manifold endowed with a Riemannian metric g ij . At each point, such a "Dreibein" is given by three orthogonal vectors U i , V i , W i of unit length that span unit volume. We stress that the [6-dimensional] orthonormal frame bundle we are using here is a mere artifact that allows us to define a variational formalism. Eliminating unphysical degrees of freedom will leave us with 4 independent physical variables.Introducing the covariant exterior derivative associated with the Levi-Civita connection, DU k = dU k + Γ k ij d...

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confidence: 77%

Fermat principle for spinning light

2006

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“…Interest in such modes goes back to the first studies of the Quantum Hall Effect (QHE) where it was found that the edge current was quantized [1][2][3]. There has also been interesting research on the connection of the existence of edge states to the geometry of eigenspaces of Schrödinger operators [4][5][6][7][8][9]. Recently, theoretical results gave support to the possible existence of unidirectional modes in optical honeycomb lattices [10,11].…”

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confidence: 99%

Linear and nonlinear traveling edge waves in optical honeycomb lattices

2014

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Traveling unidirectional localized edge states in optical honeycomb lattices are analytically constructed. They are found in honeycomb arrays of helical waveguides designed to induce a periodic pseudo-magnetic field varying in the direction of propagation. Conditions on whether a given pseudofield supports a traveling edge mode are discussed; a special case of the pseudo-fields studied agrees with recent experiments. Interesting classes of dispersion relations are obtained. Envelopes of nonlinear edge modes are described by the classical one-dimensional nonlinear Schrödinger equation along the edge. Novel nonlinear states termed edge solitons are predicted analytically and found numerically.

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“…Our implementation does not require secondary techniques such as double-loop or spin-echo technique and is expected to be physically straightforward on NMR. We remark that the Lie-algebraic method employed above can be used to construct non-adiabatic quantum gates based on non-abelian holonomies [37,38] exhibiting a richer geometrical structure [39], which we leave for future work. …”

Section: Discussionmentioning

confidence: 99%

Non-Adiabatic Universal Holonomic Quantum Gates Based on Abelian Holonomies

2014

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We implement a non-adiabatic universal set of holonomic quantum gates based on abelian holonomies using dynamical invariants, by Lie-algebraic methods. Unlike previous implementations, presented scheme does not rely on secondary methods such as double-loop or spin-echo and avoids associated experimental difficulties. It turns out that such gates exist purely in the non-adiabatic regime for these systems.

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The Geometric Phase in Quantum Systems

Cited by 472 publications

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Fermat principle for spinning light

Fermat principle for spinning light

Linear and nonlinear traveling edge waves in optical honeycomb lattices

Non-Adiabatic Universal Holonomic Quantum Gates Based on Abelian Holonomies

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