Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 050, 10 pages      quant-ph/0605091      https://doi.org/10.3842/SIGMA.2006.050

Perturbative Treatment of the Evolution Operator Associated with Raman Couplings

Benedetto Militello a, Paolo Aniello b and Antonino Messina a
a) INFM, MIUR and Dipartimento di Scienze Fisiche ed Astronomiche dell'Università di Palermo, Via Archirafi, 36 - 90123 Palermo, Italy
b) Dipartimento di Scienze Fisiche dell'Università di Napoli `Federico II' and INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cintia - 80126 Napoli, Italy

Received November 02, 2005, in final form April 13, 2006; Published online May 10, 2006

Abstract
A novel perturbative treatment of the time evolution operator of a quantum system is applied to the model describing a Raman-driven trapped ion in order to obtain a suitable 'effective model'. It is shown that the associated effective Hamiltonian describes the system dynamics up to a certain transformation which may be interpreted as a 'dynamical dressing' of the effective model.

Key words: perturbation theory; time-dependent problems; Raman couplings.

pdf (234 kb)   ps (173 kb)   tex (16 kb)

References

  1. Wineland D.J., Monroe C., Itano W.M., Leibfried D., King B.E., Meekhof D.M., Experimental issues in coherent quantum-state manipulation of trapped atomic ions, J. Res. Natl. Inst. Stand. Tech., 1998, V.103, 259-328, quant-ph/9710025.
  2. Leibfried D., Blatt R., Monroe C., Wineland D., Quantum dynamics of single trapped ions, Rev. Modern Phys., 2003, V.75, 281-324.
  3. Ghosh P.K., Ion traps, Oxford, Clarendon Press, 1995.
  4. Steinbach J., Twamley J., Knight P.L., Generation of nonclassical motional states of a trapped atom, Phys. Rev. A, 1997, V.56, 4815-4825.
  5. Aniello P., A new perturbative expansion of the time evolution operator associated with a quantum system, J. Opt. B Quantum Semiclass. Opt., 2005, V.7, S507-S522, quant-ph/0508017.
  6. Aniello P., Man'ko V.I., Marmo G., Porzio A., Solimeno S., Zaccaria F., Trapped ions interacting with laser fields: a perturbative analysis without the rotating wave approximation, J. Russ. Las. Res., 2004, V.25, 30-53, quant-ph/0301138.
  7. Aniello P., Man'ko V., Marmo G., Porzio A., Solimeno S., Zaccaria F., Ion traps in interaction with laser fields: a RWA-free perturbative approach, in Proceedings of the 8-th ICSSUR, Editors H. Moya-Cessa, R. Járengui R., S. Hacyan and O. Castaños, Princeton, Rinton Press, 2003, 17-22.
  8. Aniello P., Perturbative solutions of differential equations in Lie groups, Int. J. Geom. Methods Mod. Phys., 2005, V.2, 111-126.
  9. Militello B., Aniello P., Messina A., Coarse grained and fine dynamics in trapped ion Raman schemes, J. Phys. A: Math. Gen., 2004, V.37, 8177-8187, quant-ph/0310162.
  10. Magnus W., On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 1954, V.7, 649-673.
  11. Amerio L., Prouse G., Almost periodic functions and functional equations, Van Nostrand Reinhold, 1971.
  12. Wilcox R.M., Exponential operators and parameter differentiation in quantum physics, J. Math. Phys., 1967, V.8, 962-982.
  13. Meekhof D.M., Monroe C., King B.E., Itano W.M., Wineland D.J., Generation of nonclassical motional states of a trapped atom, Phys. Rev. Lett., 1996, V.76, 1796-1799.
  14. Leibfried D., Meekhof D.M., King B.E., Monroe C., Itano W.M., Wineland D.J., Experimental preparation and measurement of the state of motion of a trapped atom, J. Modern Opt., 1997, V.44, 2485-2505.
  15. Di Fidio C., Vogel W., Damped Rabi oscillations of a cold trapped ion, Phys. Rev. A, 2000, V.62, 031802, 4 pages.
  16. Militello B., Aniello P., Messina A., Trapped ion Raman schemes: a novel time-dependent perturbative approach, in preparation.

Previous article   Next article   Contents of Volume 2 (2006)