Eur. Phys. J. C (2004) 35: 413-423
https://doi.org/10.1140/epjc/s2004-01814-5

theoretical physics

Geometric phase for mixed states: a differential geometric approach

¹ Department of Physics, University of Hyderabad, 500046, Hyderabad, India
² Dipartimento di Fisica, Universita di Bologna, INFM and INFN, Via Irnerio 46, 40126, Bologna, Italy
³ Dipartimento di Scienze Fisiche, Universita di Napoli Federico II and INFN, Via Cinzia, 80126, Napoli, Italy
⁴ Dipartimento di Fisica, Universita di Bologna, INFM and INFN, Viale Berti-Pichat 6/2, 40127, Bologna, Italy
⁵ Centre for Theoretical Studies, Indian Institute of Science, 560012, Bangalore, India
⁶ The Institute of Mathematical Sciences, C.I.T. Campus, 600113, Tharamani, India

^* e-mail: scsp@uohyd.ernet.in

Received: 16 December 2003
Revised: 8 March 2004
Published online: 18 May 2004

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.