# Cubic algebra and averaged Hamiltonian for the resonance 3: (−1) Penning-ioffe trap

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## Abstract

For the 3: (−1) resonance Penning trap, we describe the algebra of symmetries which turns out to be a non-Lie algebra with cubic commutation relations. The irreducible representations and coherent states of this algebra are constructed explicitly. The perturbing inhomogeneous magnetic field of Ioffe type, after double quantum averaging, generates an effective Hamiltonian of the trap. In the irreducible representation, this Hamiltonian becomes a second-order ordinary differential operator of the Heun type.

## Keywords

Irreducible Representation Coherent State Transverse Mode Symmetry Algebra Paul Trap
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## References

- 1.L. S. Brown and G. Gabrielse, “Precision Spectroscopy of a Charged Particle in an Imperfect Penning Trap,” Phys. Rev. A
**25**(4), 2423–2425 (1982).ADSCrossRefGoogle Scholar - 2.G. Gabrielse, “Relaxation Calculation of the Electrostatic Properties of Compensated Penning Traps with Hyperbolic Electrodes,” Phys. Rev. A
**27**(5), 2277–2290 (1983).ADSCrossRefGoogle Scholar - 3.G. Gabrielse, “Detection, Damping, and Translating the Center of the Axial Oscillation of a Charged Particle in a Penning Trap with Hyperbolic Electrodes,” Phys. Rev. A
**29**(2), 462–469 (1984).ADSCrossRefGoogle Scholar - 4.M. Kretzschmar, “Single Particle Motion in a Penning Trap: Description in the Classical Canonical Formalism,” Phys. Scripta
**46**, 544–554 (1992).ADSCrossRefGoogle Scholar - 5.D. Segal and M. Shapiro, “Nanoscale Paul Trapping of a Single Electron,” Nano Letters
**6**(8), 1622–1626 (2006).ADSCrossRefGoogle Scholar - 6.G. Gabrielse and F. C. Mackintosh, “Cylindrical Penning Traps with Orthogonalized Anharmonicity Compensation,” Int. J. Mass Spectrometry and Ion Processes
**57**, 1–17 (1984).CrossRefGoogle Scholar - 7.G. Gabrielse, L. Haarsma, and S. L. Rolston, “Open Endcap Penning Traps for High Precision Experiments,” Int. J. Mass Spectrometry and Ion Processes
**88**, 319–332 (1989).CrossRefGoogle Scholar - 8.G. Gabrielse and H. Dehmelt, “Geonium without a Magnetic Bottle-A New Generation,” in
*Precision Measurement and Fundamental Constants. II*, Ed. by B. N. Taylor and W. D. Phillips Natl. Bur. Stand. (U.S.), Spec. Publ. (617), 219–221 (1984).Google Scholar - 9.V. P. Maslov,
*Theory of Perturbations and Asymptotic Methods*(Moscow State Univ., Moscow, 1965).Google Scholar - 10.V. M. Babič [Babich] and V. S. Buldyrev,
*Short-Wavelength Diffraction Theory*(Nauka, Moscow, 1972; Springer-Verlag, Berlin, 1991),.Google Scholar - 11.M. V. Karasev and E. M. Novikova, “Algebra and Quantum Geometry of Multifrequency Resonance,” Izv. Ross. Akad. Nauk Ser. Mat.
**74**(6), 55–106 (2010) [Izvestiya: Math.**74**(6), 1155–1204 (2010)].MathSciNetGoogle Scholar - 12.M. V. Karasev, “Birkhoff Resonances and Quantum Ray Method,” in
*Proc. Intern. Seminar “Days of Diffraction” 2004*(St. Petersburg University and Steklov Math. Institute, St. Petersburg, 2004), pp. 114–126.Google Scholar - 13.M. V. Karasev, “Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances, I,” in
*Quantum Algebras and Poisson Geometry in Mathematical Physics*, Ed. by M. Karasev, Amer. Math. Soc. Transl. Ser. 2, Vol. 216 (Providence, 2005), pp. 1–18; arXiv: math.QA/0412542. M. V. Karasev, “Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances, II,” Adv. Stud. Contemp. Math.**11**, 33–56 (2005). M. Karasev, “Noncommutative Algebras, Nano-Structures, and Quantum Dynamics Generated by Resonances, III,” ISSN 1061-9208, Russ. J. Math. Phys.**13**(2), 131–150 (2006).Google Scholar - 14.D. J. Fernández and M. Velázquez,“Coherent States Approach to Penning Trap,” J. Phys. A: Math. Theor.
**42**, 085304 (2009).ADSCrossRefGoogle Scholar - 15.M. Genkin and E. Lindroth, “On the Penning Trap Coherent States,” J. Phys. A: Math. Theor.
**42**, 275305 (2009).ADSCrossRefMathSciNetGoogle Scholar - 16.M. Karasev and E. Novikova, “Non-Lie Permutation Relations, Coherent States, and Quantum Embedding”, in:
*Coherent Transform, Quantization, and Poisson Geometry*(M. Karasev, Ed.), Amer. Math. Soc. Transl.**187**(2) (AMS, Providence, RI, 1998), pp. 1–202.Google Scholar - 17.T. M. Squires, P. Yesley, and G. Gabrielse, “Stability of a Charged Particle in a Combined Penning-Ioffe Trap,” Phys. Rev. Lett.
**86**(23), 5266–5269 (2001).ADSCrossRefGoogle Scholar - 18.B. Hezel, I. Lesanovsky, and P. Schmelcher, “Ultracold Rydberg Atoms in a Ioffe-Pritchard Trap”, arXiv: 0705.1299v2.Google Scholar
- 19.M. Karasev and V. P. Maslov, “Asymptotic and Geometric Quantization,” Uspekhi Mat. Nauk
**39**(6), 115–173 (1984) [Russian Math. Surveys**39**(6), 133–205 (1984)].MathSciNetGoogle Scholar - 20.H. Bateman and A. Erdélyi,
*Higher Transcendental Functions*(McGraw-Hill, New York, 1953).Google Scholar

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