, Volume 69, Issue 3, pp 317–327 | Cite as

Ray space ‘Riccati’ evolution and geometric phases for N-level quantum systems

  • S. Chaturvedi
  • E. ErcolessiEmail author
  • G. Marmo
  • G. Morandi
  • N. Mukunda
  • R. Simon


We present a simple derivation of the matrix Riccati equations governing the reduced dynamics as one descends from the group \( \mathbb{U} \)(N) describing the Schrödinger evolution of an N-level quantum system to the various coset spaces and Grassmanian manifolds associated with it. The special case pertaining to the geometric phase in N-level systems is described in detail. Further, we show how the matrix Riccati equation thus obtained can be reformulated as an equation describing Hamiltonian evolution in a classical phase space and establish correspondences between the two descriptions.


Quantum dynamics Riccati equations geometric phase 


03.67.Mn 02.40.Yy 03.65.Vf 


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  1. [1]
    M Nielsen and I Chuang, Quantum computation and quantum information (Cambridge University Press, New York, 2000) I Bengtsson and K Życzkowski, Geometry of quantum states: An introduction to quantum entanglement (Cambridge University Press, New York, 2006)zbMATHGoogle Scholar
  2. [2]
    K S Gibbons, M J Hoffman and W K Wootters, Phys. Rev. A70, 062101 (2004) G S Agarwal, Phys. Rev. A24, 2889 (1981) M Ruzzi and D Galetti, J. Phys. A33, 1065 (1999) N Mukunda, S Chaturvedi and R Simon, Phys. Lett. A321, 160 (2004) S Chaturvedi, E Ercolessi, G Marmo, G Morandi, N Mukunda and R Simon, Pramana — J. Phys. 65, 981 (2005) A Vourdas, Rep. Prog. Phys. 67, 267 (2004) D Gross, J. Math. Phys. 47, 122107 (2006)Google Scholar
  3. [3]
    See for instance, W P Schleich, Quantum optics in phase space (Wiley-VCH, Weinheim, 2001)zbMATHGoogle Scholar
  4. [4]
    G Khanna, S Mukhopadhyay, R Simon and N Mukunda, J. Phys. A253, 55 (1997) K Arvind, S Mallesh and N Mukunda, J. Phys. A30, 2417 (1997) L Jakóbiczyk and M Sienicki, Phys. Lett. A286, 383 (2001) G Kimura, Phys. Lett. A314, 3339 (2003) M S Byrd and N Khaneja, Phys. Rev. A68, 062322 (2003)MathSciNetGoogle Scholar
  5. [5]
    See, for instance, J Wei and E Norman, J. Math. Phys. 4, 575 (2001) G Dattoli, J C Gallardo and A Torre, Riv. Nuovo. Cimento 11, 1 (1988) B A Shadwick and W F Buell, Phys. Rev. Lett. 79, 5189 (1997)CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    D B Uskov and A R P Rau, Phys. Rev. A74, 030304(R) (2006) See also, A R P Rau, Phys. Rev. Lett. 81, 4785 (1998) A R P Rau and Weichang Zhao, Phys. Rev. A71, 063822 (2005) A R P Rau, G Selvaraj and D B Uskov, Phys. Rev. A71, 062316 (2005)Google Scholar
  7. [7]
    See, for instance, J F Cariñena, J Grabowski and G Marmo, Lie-Scheffers systems: A geometric approach (Bibliopolis, Naples, 2000) J Grabowski, G Landi, G Marmo and G Vilasi, Fortsch. Phys. 46, 393 (1994)zbMATHGoogle Scholar
  8. [8]
    S Pancharatnam, Proc. Indian Acad. Sci. Section A44, 247 (1956) M V Berry, Proc. Roy. Soc. (London) A392, 45 (1984) Y Aharanov and J Anandan, Phys. Rev. Lett. 58, 1593 (1987) J Samuel and R Bhandari, Phys. Rev. Lett. 60, 2339 (1988) N Mukunda and R Simon, Ann. Phys. (NY) 228, 205 (1993); 228, 269 (1993) Many of the early papers on geometric phase have been reprinted in Geometric Phases in physics by A Shapere and F Wilczek (World Scientific, Singapore, 1989) and in Fundamentals of quantum optics, SPIE Milestone Series, edited by G S Agarwal (SPIE Press, Bellington, 1995) For off-diagonal geometric phase and that associated with mixed states see, for instance, N Mukunda, Arvind, S Chaturvedi and R Simon, Phys. Rev. A65, 012102 (2003) S Chaturvedi, E Ercolessi, G Marmo, G Morandi, N Mukunda and R Simon, Eur. Phys. J. C35, 413 (2004) S Filipp and E Sjöqvist, Phys. Rev. Lett. 90, 050403 (2003) and references cited thereinMathSciNetGoogle Scholar
  9. [9]
    S Berceanu and A Gheorghe, J. Math. Phys. 33, 998 (1992) L Boya, A M Perelomov and M Santander, J. Math. Phys. 42, 5130 (2001) A M Perelomov, Generalized coherent States and their Applications (Springer-Verlag, New York, 1986) G Giavarini and E Onofri, J. Math. Phys. 30, 659 (1989) P Dita, J. Math. Phys. 38, 2657 (2005)zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. [10]
    G Marmo, E J Saletan, A Simoni and B Vitale, Dynamical systems. A differential geometric approach to symmetry and reduction (J Wiley, 1985) A M Perelomov, Integrable systems of classical mechanics and Lie algebras (Birkhäuser, 1990)Google Scholar

Copyright information

© Indian Academy of Sciences 2007

Authors and Affiliations

  • S. Chaturvedi
    • 1
  • E. Ercolessi
    • 2
    Email author
  • G. Marmo
    • 3
  • G. Morandi
    • 4
  • N. Mukunda
    • 5
  • R. Simon
    • 6
  1. 1.School of PhysicsUniversity of HyderabadHyderabadIndia
  2. 2.Physics DepartmentUniversity of Bologna, CNISM and INFNBolognaItaly
  3. 3.Dipartimento di Scienze FisicheUniversity of Napoli and INFNNapoliItaly
  4. 4.Physics DepartmentUniversity of Bologna, CNISM and INFNBolognaItaly
  5. 5.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia
  6. 6.The Institute of Mathematical SciencesC.I.T CampusChennaiIndia

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