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

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

  • S. Chaturvedi
  • E. Ercolessi
  • 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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Copyright information

© Indian Academy of Sciences 2007

Authors and Affiliations

  • S. Chaturvedi
    • 1
  • E. Ercolessi
    • 2
  • 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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