Nonlinearity of Pancharatnam’s topological phase

Schmitzer, Heidrun; Klein, Susanne; Dultz, W.

doi:10.1103/physrevlett.71.1530

Cited by 40 publications

(24 citation statements)

References 13 publications

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“…The three-vertex geometric phase varies extremely nonlinearly at points α = 3π/4 and 5π/4. This is closely related to the nonlinear behavior of the three-vertex geometric phases of two-state systems [15][16][17][18][19]. The nonlinearity in the two-state systems is due to the geometry of the Bloch sphere and it originates from the drastic change in the geodesic arcs surrounding the spherical triangle.…”

Section: Parameter Dependence Of the Three-vertex Geometric Phasementioning

confidence: 98%

Bloch-sphere representation of three-vertex geometric phases

2011

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The properties of the geometric phases between three quantum states are investigated in a highdimensional Hilbert space using the Majorana representation of symmetric quantum states. We found that the geometric phases between the three quantum states in an N -state quantum system can be represented by N − 1 spherical triangles on the Bloch sphere. The parameter dependence of the geometric phase was analyzed based on this picture. We found that the geometric phase exhibits rich nonlinear behavior in a high-dimensional Hilbert space.

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Section: Parameter Dependence Of the Three-vertex Geometric Phasementioning

confidence: 98%

Bloch-sphere representation of three-vertex geometric phases

2011

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“…This work contained a formal expression for the relative phase between beams in different elliptic polarizations of light, as well as a construction (employing the so-named "Poincare sphere") that related the phase difference to a geometrical, area concept. (For experimental realizations with polarized light beams we quote [115,116]; the issue of any arbitrariness in experimentally pinning down the topological part of the phase was raised in [118].) Regarding the interesting question of any common ground between classical and quantal phases, the relation between the adiabatic (Hannay's) angle in mechanics and the phase in wave functions was the subject of [117].…”

Section: Aspects Of Phase In Moleculesmentioning

confidence: 99%

Complex States of Simple Molecular Systems

Englman¹,

Yahalom²

2002

Advances in Chemical Physics

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A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.When the stationary Born-Oppenheimer description for nearly degenerate state is regarded as "embedded" in a broader dynamic formulation, namely through solution of the time dependent Schrödinger equation, the wave function becomes necessarily complex. The analytic behavior of this function in the complex time-plane can be exploited to gain information about the connection between phases and moduli of the componentamplitudes. One form of this connection are a pair of reciprocal (Kramers-Kronig type) relations in the complex time-domain. These show that certain phase changes in a component amplitude (such as, e.g., lead to a non-zero Berry-phase) require changes in the amplitude-moduli, too, and imply degeneracies of the electronic state.The subject of conical degeneracy, or intersection, of adjacent potential surface is extended to nonlinear nuclear-electronic coupling, which can then result in multiple conical degeneracies on a nuclear coordinate plane. We treat cases of double and fourfold intersections, the latter under trigonal symmetry, and employ an analytic-graphical phase tracing method to obtain the resulting Berry phases as the system circles around some or all of the degeneracy points. These phases can take up values that are all integral multiples of π, with integers that vary with the physical situation (or with the model postulated for it).A further type of invariance, with respect to gauge transformation, is also tied to the complex form of the wave function. For a many-component state this leads to a consideration of tensorial (Yang-Mills type) fields F for molecular systems. It is shown that it is the truncation of the Born-Oppenheimer electron-nuclear superposition that generates the molecular Yang-Mills fields.The equations of motion for the nuclear degrees of freedom are derived from a Lagr...

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“…The behavior of ␥ ABC on varying B can be linear [16], nonlinear [17], or singular [18][19][20], as we have also shown for ␥ AB . However ␥ AB has singularities at B =−A and B =−X.…”

Section: ͑19͒mentioning

confidence: 99%

Geometric interpretation of the Pancharatnam connection and non-cyclic polarization changes

2010

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If the state of polarization of a monochromatic light beam is changed in a cyclical manner, the beam acquires-in addition to the usual dynamic phase-a geometric phase. This geometric or Pancharatnam-Berry phase equals half the solid angle of the contour traced out on the Poincaré sphere. We show that such a geometric interpretation also exists for the Pancharatnam connection, the criterion according to which two beams with different polarization states are said to be in phase. This interpretation offers what is to our knowledge a new and intuitive method to calculate the geometric phase that accompanies non-cyclic polarization changes. © 2010 Optical Society of America OCIS codes: 350.1370, 260.5430, 260.6042. In 1984 Berry pointed out that a quantum system whose parameters are cyclically altered does not return to its original state but acquires, in addition to the usual dynamic phase, a so-called geometric phase [1]. It was soon realized that such a phase is not just restricted to quantum systems, but also occurs in contexts such as Foucault's pendulum [2]. Also the polarization phenomena described by Pancharatnam [3] represent one of its manifestations. The polarization properties of a monochromatic light beam can be represented by a point on the Poincaré sphere [4]. When, with the help of optical elements such as polarizers and retarders, the state of polarization is made to trace out a closed contour on the sphere, the beam acquires a geometric phase. This Pancharatnam-Berry phase, as it is nowadays called, is equal to half the solid angle of the contour subtended at the origin of the sphere [5][6][7][8][9][10].In this work we show that such a geometric relation also exists for the so-called Pancharatnam connection, the criterion according to which two beams with different polarization states are in phase, i.e., their superposition produces a maximal intensity. This relation can be extended to arbitrary (e.g., non-closed) paths on the Poincaré sphere and allows us to study how the phase builds up for such non-cyclic polarization changes. Our work offers a geometry-based alternative to the algebraic work presented in [11,12].The state of polarization of a monochromatic beam can be represented as a two-dimensional Jones vector [13] with respect to an orthonormal basis ͕ê 1 , ê 2 ͖ as E = cos ␣ê 1 + sin ␣ exp͑i͒ê 2 , ͑1͒with 0 Յ ␣ Յ /2, − Յ Յ , and ê i · ê j = ␦ ij ͑i , j =1,2͒. The angle ␣ is a measure of the relative amplitudes of the two components of the electric vector E, and the angle denotes their phase difference. Two different states of polarization, A and B, can hence be written asSince only relative phase differences are of concern, the overall phase of E A in Eq. (2) is taken to be zero. According to Pancharatnam's connection [5] these two states are in phase when their superposition yields a maximal intensity, i.e., whenreaches its greatest value, implying thatThese two conditions uniquely determine the phase ␥ AB , except when the states A and B are orthogonal.Let us now consider a sequence o...

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Nonlinearity of Pancharatnam’s topological phase

Cited by 40 publications

References 13 publications

Bloch-sphere representation of three-vertex geometric phases

Bloch-sphere representation of three-vertex geometric phases

Complex States of Simple Molecular Systems

Geometric interpretation of the Pancharatnam connection and non-cyclic polarization changes

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