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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 8, Iss. 2 — Jan. 15, 2001
  • pp: 123–130

Symmetries and solutions of the three-dimensional Paul trap

Michael Martin Nieto and D. Rodney Truax  »View Author Affiliations


Optics Express, Vol. 8, Issue 2, pp. 123-130 (2001)
http://dx.doi.org/10.1364/OE.8.000123


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Abstract

Using the symmetries of the three-dimensional Paul trap, we derive the solutions of the time-dependent Schrödinger equation for this system, in both Cartesian and cylindrical coordinates. Our symmetry calculations provide insights that are not always obvious from the conventional viewpoint.

© Optical Society of America

OCIS Codes
(020.7010) Atomic and molecular physics : Laser trapping
(270.5570) Quantum optics : Quantum detectors

ToC Category:
Focus Issue: Quantum control of photons and matter

History
Original Manuscript: November 15, 2000
Published: January 15, 2001

Citation
Michael Nieto and D. Truax, "Symmetries and solutions of the three-dimensional Paul trap," Opt. Express 8, 123-130 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-123


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References

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  2. P. H. Dawson, Quadrupole Mass Spectrometry and its Applications (Elsevier, Amsterdam, 1976), Chaps. I-IV. Reprinted by (AIP, Woodbury, NY, 1995).
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  7. M. M. Nieto and D. R. Truax, "Coherent states sometimes look like squeezed states, and visa versa: The Paul trap," New J. Phys. 2, 18.1-18.9 (2000). Eprint quant-ph/0002050. [CrossRef]
  8. G. Schrade, V. I. Man'ko, W. P. Schleich, and R. J. Glauber, "Wigner functions in the Paul trap," Quantum Semiclass. Opt. 7, 307-325 (1995). [CrossRef]
  9. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1977).
  10. M. M. Nieto and D. R. Truax (in preparation).
  11. V. A. Kostelecky, V. I. Man'ko, M. M. Nieto, and D. R. Truax, "Supersymmetry and a time-dependent Landau system," Phys. Rev. A 48, 951-963 (1993). [CrossRef] [PubMed]
  12. J. R. Klauder, private communication.
  13. The phase factor can also be obtained [14, 15], by solving the eigenvalue equation 3 nz = (nz � ) nz, where 3 = {3 t � (Z z � ) - i/4 3z2} Then, solving the equation Jz-0=0 will yield the extremal state function up to a factor of (pi)^-1/4.
  14. D. R. Truax, "Symmetry of time-dependent Schr� odinger equations. II. Exact solutions for the equation {xx 2 t - 2 2(t)x2 - 2 1(t)x-2 0(t) = 0," J. Math. Phys. 23, 43-54 (1982). [CrossRef]
  15. M. M. Nieto and D. R. Truax, "Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras," J. Math. Phys. 38, 84-97 (1997). [CrossRef]
  16. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).
  17. V. A. Kostelecky, M. M. Nieto, and D. R. Truax, "Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions," Phys. Rev. D 32, 2627-2633 (1985). See. Eqs. (2.7) and (2.8). [CrossRef]
  18. M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

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