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

SIGMA 6 (2010), 096, 11 pages      arXiv:1009.5293      https://doi.org/10.3842/SIGMA.2010.096
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Supersymmetric Extension of Non-Hermitian su(2) Hamiltonian and Supercoherent States

Omar Cherbal a, Mahrez Drir a, Mustapha Maamache b and Dimitar A. Trifonov c
a) Faculty of Physics, Theoretical Physics Laboratory, USTHB, B.P. 32, El Alia, Algiers 16111, Algeria
b) Laboratoire de Physique Quantique et Systemes Dynamiques, Department of Physics, Setif University, Setif 19000, Algeria
c) Institute of Nuclear Research, 72 Tzarigradsko chaussée, 1784 Sofia, Bulgaria

Received September 29, 2010, in final form December 04, 2010; Published online December 15, 2010

Abstract
A new class of non-Hermitian Hamiltonians with real spectrum, which are written as a real linear combination of su(2) generators in the form H=ωJ3+αJ−+βJ+, α≠β, is analyzed. The metrics which allows the transition to the equivalent Hermitian Hamiltonian is established. A pseudo-Hermitian supersymmetic extension of such Hamiltonians is performed. They correspond to the pseudo-Hermitian supersymmetric systems of the boson-phermion oscillators. We extend the supercoherent states formalism to such supersymmetic systems via the pseudo-unitary supersymmetric displacement operator method. The constructed family of these supercoherent states consists of two dual subfamilies that form a bi-overcomplete and bi-normal system in the boson-phermion Fock space. The states of each subfamily are eigenvectors of the boson annihilation operator and of one of the two phermion lowering operators.

Key words: pseudo-Hermitian quantum mechanics; supersymmetry; supercoherent states.

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References

  1. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, math-ph/9712001.
  2. Mostafazadeh A., Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002), 205-214, math-ph/0107001.
  3. Swanson M.S., Transition elements for a non-Hermitian quadratic Hamiltonian, J. Math. Phys. 45 (2004), 585-601.
  4. Geyer H.B., Scholtz F.G., Snyman I., Quasi-Hermiticity and the role of a metric in some boson Hamiltonians, Czechoslovak J. Phys. 54 (2004), 1069-1073.
  5. Jones H.F., On pseudo-Hermitian Hamiltonians and their Hermitian counterparts, J. Phys. A: Math. Gen. 38 (2005), 1741-1746, quant-ph/0411171.
  6. Bagchi B., Quesne C., Roychoudhury R., Pseudo-Hermiticity and some consequences of a generalized quantum condition, J. Phys. A: Math. Gen. 38 (2005), L647-L652, quant-ph/0508073.
  7. Musumbu D.P., Geyer H.B., Heiss W.D., Choice of a metric for the non-Hermitian oscillator, J. Phys. A: Math. Theor. 40 (2007), F75-F80, quant-ph/0611150.
  8. Quesne C., A non-Hermitian oscillator Hamiltonian and su(1,1): a way towards generalizations, J. Phys. A: Math. Theor. 40 (2007), F745-F751, arXiv:0705.2868.
  9. Quesne C., Quasi-Hermitian supersymmetric extensions of a non-Hermitian oscillator Hamiltonian and of its generalizations, J. Phys. A: Math. Theor. 41 (2008), 244022, 10 pages, arXiv:0710.2453.
  10. Bagchi B., Quesne C., Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework, Phys. Lett. A 300 (2002), 18-26, math-ph/0205002.
  11. Assis P.E.G., Fring A., Non-Hermitian Hamiltonians of Lie algebraic type, J. Phys. A: Math. Theor. 42 (2009), 015203, 23 pages, arXiv:0804.4677.
  12. Allen L., Eberly J.H., Optical resonance and two-level atoms, Dover Publications, Inc., New York, 1987.
  13. Wódkiewicz K., Eberly J.H., Coherent states, squeezed fluctuations, and the SU(2) and SU(1,1) groups in quantum-optics applications, J. Opt. Soc. Am. B 2 (1985), 458-466.
  14. Arecchi F.T., Courtens E., Gilmore R., Thomas H., Atomic coherent states in quantum optics, Phys. Rev. A 6 (1972), 2211-2237.
  15. Zhang W.M., Feng D.H., Gilmore R., Coherent states: theory and some applications, Rev. Modern Phys. 62 (1990), 867-927.
  16. Mostafazadeh A., Statistical origin of pseudo-Hermitian supersymmetry and pseudo-Hermitian fermions, J. Phys. A: Math. Gen. 37 (2004), 10193-10207, quant-ph/0404025.
  17. Cherbal O., Drir M., Maamache M., Trifonov D.A., Invariants and coherent states for a nonstationary fermionic forced oscillator, Phys. Lett. A 374 (2010), 535-538, arXiv:0912.4820.
  18. Mostafazadeh A., Pseudo-supersymmetric quantum mechanics and isospectral pseudo-Hermitian Hamiltonians, Nuclear Phys. B 640 (2002), 419-434, math-ph/0203041.
  19. Mostafazadeh A., Pseudo-Hermitian supersymmetry: a brief review, Czechoslovak J. Phys. 54 (2004), 1371-1374.
  20. Gendenshtein L.E., Krive I.V., Supersymmetry in quantum mechanics, Soviet Phys. Uspekhi 28 (1985), 645-666.
  21. Cooper F., Khare A., Sukhatme U., Supersymmetry in quantum mechanics, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
  22. Aragone C., Zypman F., Supercoherent states, J. Phys. A: Math. Gen. 19 (1986), 2267-2279.
  23. Fatyga B.W., Kostelecký V.A., Nieto M.M., Truax D.R., Supercoherent states, Phys. Rev. D 43 (1991), 1403-1412.
  24. Berube-Lauziere Y., Hussin V., Comments of the definitions of coherent states for the SUSY harmonic oscillator, J. Phys. A: Math. Gen. 26 (1993), 6271-6275.
  25. Cherbal O., Drir M., Maamache M., Trifonov D.A., Fermionic coherent states for pseudo-Hermitian two-level systems, J. Phys. A: Math. Gen. 40 (2007), 1835-1844, quant-ph/0608177.
  26. Trifonov D.A., Pseudo-boson coherent and Fock states, in Proceedings of the 9th International Workshop on "Complex Structures, Integrability and Vector Fields" (August 25-29, 2008, Sofia), Editors K. Sekigawa, V. Gerdjikov and S. Dimiev, World Scientific Publishing Co., 2009, 241-250, arXiv:0902.3744.
  27. Bagarello F., Pseudobosons, Riesz bases, and coherent states, J. Math. Phys. 51 (2010), 023531, 10 pages, arXiv:1001.1136.
  28. Berezin F.A., The method of second quantization, Academic Press, New York, 1966.
  29. Berezin F.A., Marinov M.S., Particle spin dynamics as the Grassmann variant of classical mechanics, Ann. Physics 104 (1977), 336-362.
  30. Cahill K.E., Glauber R.J., Density operators for fermions, Phys. Rev. A 59 (1999), 1538-1555, physics/9808029.
  31. Junker G., Klauder J.R., Coherent-state quantization of constrained fermion systems, Eur. Phys. J. C 4 (1998), 173-183, quant-ph/9708027.
  32. Gilmore R., Baker-Campbell-Hausdorff formulas, J. Math. Phys. 15 (1974), 2090-2092.
  33. Glauber R.J., Coherent and incoherent states of the radiation field, Phys. Rev. 131 (1963), 2766-2788.

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