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Quantum dynamics; Riccati equations; geometric phase.

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 Schrõdinger 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.

S Chaturvedi¹ E Ercolessi² G Marmo³ G Morandi⁴ N Mukunda⁵ R Simon⁶

1. School of Physics. University of Hyderabad, Hyderabad 500 046, India
2. Physics Department, University of Bologna, CNISM and INFN, 46 v.Irnerio, I-40126, Bologna, Italy
3. Dipartimento di Scienze Fisiche, University of Napoli and INFN, v.Cinzia, I-80126, Napoli, Italy
4. Physics Department, University of Bologna, CNISM and INFN, 6/2 v.le Berti Pichat, I-40127, Bologna, Italy
5. Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India
6. The Institute of Mathematical Sciences, C.I.T Campus, Chennai 600 113, India

Published

September 2007

Manuscript received

7 June 2007

Manuscript revised

3 July 2007

Accepted

6 July 2007