Spin coherent states as anticipators of the geometric phase

American Journal of Physics

Volumn 67, Issue 10, 1999, Pages 899-904

  Aravind, P K ^a,b  

^a WORCESTER POLYTECHNIC INSTITUTE   (United States)

^b CENTRE DE PHYSIQUE THÉORIQUE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033440843     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19145     Document Type: Article

References (30)

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4. J. R. Klauder and B. S. Skagerstam, Eds., Coherent States (World Scientific, Singapore, 1985). This is an extensive collection of reprints on coherent states and their applications to many fields.
5. M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. London, Ser. A 392, 45-57 (1984).
6. Y. Aharonov and J. Anandan, "Phase change during a cyclic quantum evolution," Phys. Rev. Lett. 58, 1593-1596 (1987).
7. S. Pancharatnam, "Generalized theory of interference and its applications," Proc.-Indian Acad. Sci., Sect. A 44, 247-262 (1956).
8. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, NJ, 1957).
9. This result for the area of a spherical triangle was first obtained (in non-vector form) by Euler and Lagrange. For a proof of this formula, and some of its history, see F. Eriksson, "On the measure of solid angles," Math. Mag. 63, 184 (1990).
10. A small caution needs to be observed in Fig. 1 if the ensuing discussion is to be free of ambiguity: the spherical triangle ABC must always be chosen to be an Euler triangle. An Euler triangle is one in which no angle or side exceeds π (and consequently the area does not exceed 2π). This restriction is made to prevent any quadrant ambiguities from arising when trigonometric quantities have to be inverted. As will become evident later, this restriction does not in any way limit the generality of the results to be derived concerning geometric phases.
11. S. Ramaseshan and R. Nityananda, Curr. Sci., India 55, 1225-1226 (1986).
12. M. V. Berry, "The adiabatic phase and Pancharatnam's phase for polarized light," J. Mod. Opt. 34, 1401-1407 (1987).
13. The advent of matter-wave interferometry now makes it possible to justify this criterion for particles (such as neutrons) as well.
14. In particular, any non-Euler triangle can always be divided into a finite number of Euler triangles; thus the restriction to Euler triangles made earlier is seen to involve no loss of generality.
15. This of course ignores the dynamical phase that accumulates during the evolution and also the important practical problem of disentangling the dynamical and geometrical phases. However our purpose here is simply to demonstrate how the geometrical phase can arise during dynamical evolution and to stress that it arises even if the evolution is not adiabatic.
16. This result is easily established for a geodesic triangle; it then follows for an arbitrary closed curve if the latter is decomposed into a sufficiently large number of oriented geodesic triangles. Let us prove the result for a geodesic triangle. For this purpose we consider the geodesic triangle of Fig. 1, referring to its angles simply by the letters of the corresponding vertices. Consider a tiny vector at A that is directed along the tangent to the side AB and suppose that this vector is parallel transported around the triangle in the sense ABCA. By "parallel transported" we mean that the angle between the vector and the local tangent to the geodesic along which it is moved is always a constant. After the vector is parallel transported from A to B, it makes the counterclockwise angle (π-B) with side BC. After it is transported from B to C, it makes the clockwise angle (B +C) with side CA. And when it is transported from C back to its starting position at A, the counterclockwise angle it makes with its original direction is (B+C)-(π-A)=A+B+C-π. The entire treatment up to this point is equally valid for Euclidean, spherical, and hyperbolic triangles. For a Euclidean triangle, A+B+C=π and there is no rotation of the vector. For a spherical triangle, A+B+C>π and the vector rotates counterclockwise (i.e., in the same sense as its motion) by an angle equal to the spherical excess, or area, of the triangle. For a hyperbolic triangle, A+B +C<π and the vector rotates clockwise (i.e., in the opposite sense to its motion) by an angle equal to the "defect," or area, of the triangle. For a more sophisticated treatment of parallel transport that is capable of generalization to quantum dynamics on curved manifolds, see M. Berry, "The quantum phase, five years after" in Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989), pp. 7-28.
17. See the paper by Berry quoted at the end of Ref. 16.
18. E. Layton, Y. Huang, and S. I. Chu, "Cyclic quantum evolution and Aharonov-Anandan geometric phases in SU(2) spin-coherent states," Phys. Rev. A 41, 42-48 (1990).
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20. J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: Geometric Phases in Physics," Am. J. Phys. 65, 180-185 (1997).
21. Many other reviews of geometric phases have appeared over the years. Some of them are: A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989). This is a collection of reprints on all aspects of geometric phases, together with an introduction and commentary by the editors; C. A. Mead, "The geometric phase in molecular systems," Rev. Mod. Phys. 64, 51-85 (1992); J. W. Zwanziger, M. Koenig, and A. Pines, "Berry's phase," Annu. Rev. Phys. Chem. 41, 601-646 (1990).
22. Many other reviews of geometric phases have appeared over the years. Some of them are: A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989). This is a collection of reprints on all aspects of geometric phases, together with an introduction and commentary by the editors; C. A. Mead, "The geometric phase in molecular systems," Rev. Mod. Phys. 64, 51-85 (1992); J. W. Zwanziger, M. Koenig, and A. Pines, "Berry's phase," Annu. Rev. Phys. Chem. 41, 601-646 (1990).
23. Many other reviews of geometric phases have appeared over the years. Some of them are: A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989). This is a collection of reprints on all aspects of geometric phases, together with an introduction and commentary by the editors; C. A. Mead, "The geometric phase in molecular systems," Rev. Mod. Phys. 64, 51-85 (1992); J. W. Zwanziger, M. Koenig, and A. Pines, "Berry's phase," Annu. Rev. Phys. Chem. 41, 601-646 (1990).
24. J. Samuel and R. Bhandari, "General setting for Berry's phase," Phys. Rev. Lett. 60, 2339-2342 (1988).
25. A thorough treatment of SU(2) and SU(1,1) coherent states, and their applications to many problems in physics, can be found in the book by A. Inomata, H. Kuratsuji, and C. C. Gerry, Path Integrals and Coherent States of SU(2) and SU(1,1) (World Scientific, Singapore, 1992).
26. R. Y. Chiao and T. F. Jordan, "Lorentz group Berry phases in squeezed light," Phys. Lett. A 132, 77-81 (1988); see also T. F. Jordan, "Berry phases and unitary transformations," J. Math. Phys. 29, 2042-2052 (1988).
27. R. Y. Chiao and T. F. Jordan, "Lorentz group Berry phases in squeezed light," Phys. Lett. A 132, 77-81 (1988); see also T. F. Jordan, "Berry phases and unitary transformations," J. Math. Phys. 29, 2042-2052 (1988).
28. R. Simon, "Optical phases and the sympectic group," Curr. Sci. 59, 1168-1174 (1988), and references therein.
29. M. Kitano and T. Yabuzaki, "Observation of Lorentz-group Berry phases in polarization optics," Phys. Lett. A 142, 321-325 (1989).
30. P. K. Aravind, "The Wigner angle as an anholonomy in rapidity space," Am. J. Phys. 65, 634-636 (1997).