Geometric phase for mixed states: a differential geometric approach

  • S. Chaturvedi
  • E. Ercolessi
  • G. Marmo
  • G. Morandi
  • N. Mukunda
  • R. Simon
theoretical physics

Abstract.

A new definition and interpretation of the geometric phase for mixed state cyclic unitary evolution in quantum mechanics are presented. The pure state case is formulated in a framework involving three selected principal fiber bundles, and the well-known Kostant-Kirillov-Souriau symplectic structure on (co-) adjoint orbits associated with Lie groups. It is shown that this framework generalizes in a natural and simple manner to the mixed state case. For simplicity, only the case of rank two mixed state density matrices is considered in detail. The extensions of the ideas of null phase curves and Pancharatnam lifts from pure to mixed states are also presented.

Keywords

Mixed State Pure State Density Matrice Fiber Bundle State Case 

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References

  1. 1.
    M.V. Berry, Proc. Roy. Soc. A 392, 45 (1984). Many of the early papers on geometric phase have been reprinted in Geometric Phases in Physics, edited by A. Shapere, F. Wilczek, (World Scientific, Singapore 1989) and in Fundamentals of Quantum Optics, SPIE Milestone Series, edited by G.S. Agarwal (SPIE Press, Bellington 1995)MathSciNetGoogle Scholar
  2. 2.
    Y. Aharanov, J. Anandan, Phys. Rev. Lett. 58, 1593 (1987)CrossRefGoogle Scholar
  3. 3.
    J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988)CrossRefMathSciNetGoogle Scholar
  4. 4.
    N. Mukunda, R. Simon, Ann. Phys. (NY) 228, 205 (1993); 228, 269 (1993)CrossRefMATHGoogle Scholar
  5. 5.
    B. Simon, Phys. Rev. Lett. 51, 2167 (1983)CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Uhlmann, Rep. Math. Phys. 24, 229 (1986); 36, 461 (1995)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    E. Sjöqvist, A.K. Pati, A. Ekert, J.S. Anandan, M. Ericsson, D.K. Loi, V. Vedral, Phys. Rev. Lett. 85, 2845 (2000)CrossRefGoogle Scholar
  8. 8.
    L. Dabrowski, A. Jadczyk, J. Phys. A 22, 3167 (1989)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    A.A. Kirillov, Bull. Am. Math. Soc. 36, 433 (1999)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    E. Ercolessi, G. Marmo, G. Morandi, N. Mukunda, Int. J. Mod. Phys. A 16, 5007 (2001)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    A.P. Balachandran, G. Marmo, B.-S. Skagerstam, A. Stern, Gauge symmetry and fibre bundles - Applications to particle dynamics (Springer, Berlin 1983)Google Scholar
  12. 12.
    A.P. Balachandran, G. Marmo, B.-S. Skagerstam, A. Stern, Classical topology and quantum states (World Scientific, Singapore 1991)Google Scholar
  13. 13.
    E.M. Rabei, Arvind, N. Mukunda, R. Simon, Phys. Rev. A 60, 3397 (1999)CrossRefMathSciNetGoogle Scholar
  14. 14.
    N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, R. Simon, Phys. Rev. A 67, 042114 (2003)CrossRefGoogle Scholar
  15. 15.
    Y. Choquet-Bruhat, C. Dewitt-Morette, M. Dillard-Bleick, Analysis, manifolds and physics - Part 1: Basics, revised edition (North Holland, Amsterdam 1991)Google Scholar
  16. 16.
    S. Kobayashi, K. Nomizu, Foundations of differential geometry (Interscience, New York 1969)Google Scholar
  17. 17.
    C. Nash, S. Sen, Topology and geometry for physicists (Academic Press, 1983)Google Scholar
  18. 18.
    N. Mukunda, Geometrical methods for physics in geometry, fields and cosmology, edited by B.R. Iyer, C.V. Vishveshwara (Kluwer, Dordrecht 1997)Google Scholar
  19. 19.
    V.I. Arnold, Mathematical methods of classical mechanics (Springer, Berlin 1978), Appendices 2 and 5Google Scholar
  20. 20.
    M.A. Nielsen, Phys. Rev. A 62, 052308 (2000)CrossRefGoogle Scholar
  21. 21.
    F. Pistolesi, N. Manini, Phys. Rev. Lett. 85, 1585 (2000); 85, 3067 (2000)CrossRefGoogle Scholar
  22. 22.
    N. Mukunda, Arvind, S. Chaturvedi, R. Simon, Phys. Rev. A 65, 012102 (2001)CrossRefGoogle Scholar
  23. 23.
    F. Wilczek, A. Zee, Phys. Rev. Lett. 52, 2111 (1984)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • S. Chaturvedi
    • 1
  • E. Ercolessi
    • 2
  • G. Marmo
    • 3
  • G. Morandi
    • 4
  • N. Mukunda
    • 5
  • R. Simon
    • 6
  1. 1.Department of PhysicsUniversity of HyderabadHyderabadIndia
  2. 2.Dipartimento di FisicaUniversita di Bologna, INFM and INFNBolognaItaly
  3. 3.Dipartimento di Scienze FisicheUniversita di Napoli Federico II and INFNNapoliItaly
  4. 4.Dipartimento di FisicaUniversita di Bologna, INFM and INFNBolognaItaly
  5. 5.Centre for Theoretical StudiesIndian Institute of ScienceBangaloreIndia
  6. 6.The Institute of Mathematical SciencesTharamaniIndia

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