Skip to main content
SpringerLink
Log in

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

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    MATH  Google Scholar 

  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)

  3. See for instance, W P Schleich, Quantum optics in phase space (Wiley-VCH, Weinheim, 2001)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  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 therein

    MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

Download references

Author information

Authors and Affiliations

  1. School of Physics, University of Hyderabad, Hyderabad, 500 046, India

    S. Chaturvedi

  2. Physics Department, University of Bologna, CNISM and INFN, 46 v.Irnerio, I-40126, Bologna, Italy

    E. Ercolessi

  3. Dipartimento di Scienze Fisiche, University of Napoli and INFN, v.Cinzia, I-80126, Napoli, Italy

    G. Marmo

  4. Physics Department, University of Bologna, CNISM and INFN, 6/2 v.le Berti Pichat, I-40127, Bologna, Italy

    G. Morandi

  5. Centre for High Energy Physics, Indian Institute of Science, Bangalore, 560 012, India

    N. Mukunda

  6. The Institute of Mathematical Sciences, C.I.T Campus, Chennai, 600 113, India

    R. Simon

Authors
  1. S. Chaturvedi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Ercolessi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Marmo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Morandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. N. Mukunda
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. R. Simon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Ercolessi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chaturvedi, S., Ercolessi, E., Marmo, G. et al. Ray space ‘Riccati’ evolution and geometric phases for N-level quantum systems. Pramana - J Phys 69, 317–327 (2007). https://doi.org/10.1007/s12043-007-0135-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12043-007-0135-0

Keywords

PACS Nos

Search

Navigation