M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392 (1984), 45-57.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(3.8) that not only occurs in connection with an Egorov theorem, but also in a WKB type framework. In this context Littlejohn and Flynn [LF91] introduced a splitting of the analogue to H ;1 (de ned in equation (3. 7) into two contributions, one of which is related to a Berry connection [Ber84]. Subsequently Emmrich and Weinstein [EW96] generalised the approach of [LF91] and gave a geometrical interpretation for the second contribution, which they related to a Poisson curvature. We now want to identify the two contributions in the present situation, i.e. in H ;1 . To this end we ....

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392 (1984), 45-57.

....can be identified with submanifolds of density operators. By the introduction of the Bures metric [18] this identification becomes an isometric embedding onto geodesic submanifolds of the space of density operators. This enables an unambigious way to introduce the parallel transport a la Berry [15] in its extension to density operators [37] Explicit expressions for the parallel transport along geodesic arcs and polygons are derived, and possible implications for describing experiments are mentioned. The remainder of this section is devoted to review and to arrange some facts in order to ....

....one to one to the pure states. Clearly, 1 On leave of absence from University of Leipzig 1 the possibility to handle all physical relevant questions of the theory within CP m is principally known since long. The renewed interest raised from insights in the geometric nature of the Berry phase [15], 35] 44] 37] 38] and from the interesting role of the Study Fubini metric [2] 8] 31] Let Omega = Omega Gamma H) denote the convex (affine) set of all density operators. Let us identify every density operator with the state it defines, i.e. A (A) tr(A ) A point of Omega is ....

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Royal. Soc. Lond. A 392 (1984) 45

....and Information Physics UniversityofTokyo, Bunkyo ku,Tokyo 113, Japan 1 1 Introduction Berry s phase, discovered by M. V,Berry in 1982, convienced by many experiments, is naturally interpreted as a curvature of natural connection introduced on the line bundle over the space of pure states [1][2][19] 20] Berry s phase is a manifestation of non commutative nature. Actulally, various quantum mechanical phenomina, such as quantum hall effect, Aharanov Bohm effect, Yang Tailor effect, and so on, cannot be predicted by naive analogy of classical mechanics, and are explained in terms of ....

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. Roy. Soc. London A392, 45--57 (1984).

....q V(V ) 1 fi 2 0 1 1=2 Tr q V(V ) 1: 24) It should be remarked that fi is geometrical scalar in the sense that it is unchanged by the change of the parameterization of the model, and that it takes value from 0 to 1. It is also remarkable that fi is nothing but Berry s phase [1] [14] per unit area. 8 Theorem 3 If fi( 0 ; M 0 ) fi( M) and J S ( 0 ; M 0 ) J S ( M) then for any weight matrix G, CR(G; 0 ; M 0 ) CR(G; M) 25) Proof The inequality (24) implies that if fi( 0 ; M 0 ) fi( M) V (M) aeV 0 (M 0 ....

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. Roy. Soc. London A392, 45--57 (1984).

....rank density matrices. One is Uhlmann s parallelism and the other is Nagaoka s information geometry. Uhlmann s parallelism is generalization of Berry s phase, which, by far confirmed by several experiments, is a holonomy of a natural connection in the line bundle over the space of pure states [1][3][12] 13] In 1986, Uhlmann generalized the theory to include mixed states in the Hilbert space H [14] 15] 16] It is pointed out that the Uhlmann s parallelism is deeply concerned with quantum estimation theory. Concretely speaking, iff Uhlmann s curvature vanishes, SLD CR bound is attained, ....

....i . SLD is closely related to the horizontal lift by the following equation: M h W X = L S X W: 5) 5 3 Definition of Uhlmann s parallelism Berry s phase, by far confirmed by several experiments, is a holonomyofa natural connection in the line bundle over the space of pure states [1][3]. In 1986, Uhlmann generalized the theory to include mixed states in the Hilbert space H [14] 15] 16] For notational simplicity, the argument is omitted, as long as the omission is not misleading. Define a horizontal lift of a curve C = fae(t)jt 2 Rg in P as a curve C h = fW (t)jt 2 Rg ....

M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. Roy. Soc. London A392, 45--57 (1984).

....the adiabatic scheme is sensitive to the way one chooses the phases of the adiabatic states: the phase of the n th eigenstate n ( Delta ) should be adjusted so that h n ( j n ( Delta )i = ae is a real positive number. There is a possibility of accumulating geometric Berry phases [51] at each state. If this happens the eigenstate at the end of the pulse has a Berry phase of Omega B = relative to the ground state at time t=0, in other words: h n (0)j n (T f )i = Gamma1. In the calculations for the set of parameters chosen in this study, no such phase effect has been ....

....are: what is the role of laser induced states when longer ranged potentials are present, and when lower frequencies are used. In addition, an exciting new direction is the construction of polarized pulses (interacting with electrons in more than one dimension) where a geometric Berry phase effect [51] can be measured on these light induced resonances. 88 Chapter 5 Introduction to Markov Decision Processes This chapter serves as an introduction to Markov decision problems. It also describes standard methods (i.e. dynamic programming) for solving Markov decision problems. In the next chapter, ....

: Berry M.V., 1984. Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London A 392:45.

....by the early 1950 s although the geometric roots to these problems go back to MacCullagh [1840] and Thomson and Tait [1879, x123 126] See Berry [1990] and Marsden and Ratiu [1994] for additional historical information. More recently this subject became better understood, through the work of Berry [1984, 1985] and Simon [1983] whose papers first brought into clear focus the ubiquity of, and the geometry behind, all these phenomena. It was quickly realized that the phenomenon occurs in essentially the same way in both classical and quantum mechanics (see Hannay [1985] It was also realized by ....

Berry, M. [1984] Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London A 392, 45--57.

.... It was observed by Simon [17] that such projected connections, which are standard in differential geometry, especially the geometry of submanifolds (see for example [3] are just the ones whose holonomy in certain situations of physical interest is popularly called Berry s phase, after [2]. Thus, corresponding expressions in the transport equation are named Berry terms in [13] We turn next to the matrix valued function in the second term on the left hand side of (2) It corresponds to the no name terms in [13] but we will denote it by C and call it the curvature term, for ....

Berry, M.V., Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392 (1984), 45-57.

....the local Riemannian geometry of the space of nonsingular, normalized n Theta n density matrices D 1 : f 2 M n;n (C ) j = 0; T r = 1g equipped with the Riemannian Bures metric g B . This Riemannian metric appears on the background of a generalization of the Berry phase (see [11, 4, 16, 12, 1]) to mixed states in quantum systems proposed by Uhlmann in a series of papers ( 12, 13, 14] Moreover, this metric is just the infinitesimal version of the distance function d( s 2 Gamma 2T r i 1 2 1 2 j 1 2 (1.1) given by Araki many years ago ( 2] A2) So it is natural ....

Berry, M.V., Quantal phase factors accompanying adiabatic changes, Proc. Royal. Soc. Lond. A 392 (1984), 45--57.

....the floor [12] and when a Foucault pendulum oscillates through a 24 hour period as long as it is not situated on the Equator or one of the Poles [5, 6] Phase anholonomies have become the focus of intensive research since the discovery of such an effect in quantum mechanics by M. Berry in 1984 [2, 3, 4]. Classical analogues were then discovered by J. Hannay [10] and since that time, several review papers have been written and many new insights have been gained [8, 12, 18] Our purpose in this paper is to describe how such phase anholonomies arise in vortex dynamics problems, and in particular ....

Berry, M.V., "Quantal Phase Factors accompanying Adiabatic Changes," Proc. Roy. Soc. Lond., A392, pp. 45, 1984.

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M. V. Berry. Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. Lond. A 392, 45 (1984).

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Berry, M.V., Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392 (1984), 45-57.

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M.V. Berry, quantal phase factors accompanying adiabatic changes, Proc.Roy.Soc.Lond.A 392: 45 (1984).

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