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

SIGMA 7 (2011), 084, 20 pages      arXiv:1108.5005

Para-Grassmannian Coherent and Squeezed States for Pseudo-Hermitian q-Oscillator and their Entanglement

Yusef Maleki
Department of Physics, University of Mohaghegh Ardabili, Ardabil, 179, Iran

Received May 27, 2011, in final form August 19, 2011; Published online August 25, 2011

In this paper, q-deformed oscillator for pseudo-Hermitian systems is investigated and pseudo-Hermitian appropriate coherent and squeezed states are studied. Also, some basic properties of these states is surveyed. The over-completeness property of the para-Grassmannian pseudo-Hermitian coherent states (PGPHCSs) examined, and also the stability of coherent and squeezed states discussed. The pseudo-Hermitian supercoherent states as the product of a pseudo-Hermitian bosonic coherent state and a para-Grassmannian pseudo-Hermitian coherent state introduced, and the method also developed to define pseudo-Hermitian supersqueezed states. It is also argued that, for q-oscillator algebra of k+1 degree of nilpotency based on the changed ladder operators, defined in here, we can obtain deformed SUq2(2) and SUq2k(2) and also SUq2k(1,1). Moreover, the entanglement of multi-level para-Grassmannian pseudo-Hermitian coherent state will be considered. This is done by choosing an appropriate weight function, and integrating over tensor product of PGPHCSs.

Key words: para-Grassmann variables; coherent state; squeezed state; pseudo-Hermiticity; entanglement.

pdf (412 Kb)   tex (20 Kb)


  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. Bender C.M., Boettcher S., Meisenger P.N., PT-symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201-2229, quant-ph/9809072.
  3. Bender C.M., Dunne G.V., Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian, J. Math. Phys. 40 (1999), 4616-4621, quant-ph/9812039.
  4. Mostafazadeh A., Pseudo-Hermiticity and generalized PT- and CPT-symmetries, J. Math. Phys. 44 (2003), 974-989, math-ph/0209018.
  5. 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.
  6. Mostafazadeh A., Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. 43 (2002), 2814-2816, math-ph/0110016.
  7. Perlomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  8. Fujii K., Introduction to coherent states and quantum information theory, quant-ph/0112090.
  9. Najarbashi G., Maleki Y., Maximal entanglement of two-qubit states constructed by linearly independent coherent states, Internat. J. Theoret. Phys. 50 (2011), 2601-2608, arXiv:1007.1387.
  10. Najarbashi G., Maleki Y., Entanglement of Grassmannian coherent states for multi-partite n-level systems, SIGMA 7 (2011), 011, 11 pages, arXiv:1008.4836.
  11. Najarbashi G., Maleki Y., Entanglement in multi-qubit pure fermionic coherent states, arXiv:1004.3703.
  12. Fu H., Wang X., Solomon A.I., Maximal entanglement of nonorthogonal states: classification, Phys. Lett. A 291 (2001), 73-76, quant-ph/0105099.
  13. Wang X., Sanders B.C., Multipartite entangled coherent states, Phys. Rev. A 65 (2001), 012303, 7 pages, quant-ph/0104011.
  14. Wang X., Bipartite entangled non-orthogonal states, J. Phys. A: Math. Gen. 35 (2002), 165-173, quant-ph/0102011.
  15. Wang X., Sanders B.C., Pan S.-H., Entangled coherent states for systems with SU(2) and SU(1,1) symmetries, J. Phys. A: Math. Gen. 33 (2000), 7451-7467, quant-ph/0001073.
  16. El Baz M., Hassouni Y., On the construction of generalized Grassmann representatives of state vectors, J. Phys. A: Math. Gen. 37 (2004), 4361-4368, math-ph/0409038.
  17. El Baz M., On the construction of generalized Grassmann coherent states, math-ph/0511028.
  18. El Baz M., Hassouni Y., Madouri F., Z3-graded Grassmann variables, parafermions and their coherent states, Phys. Lett. B 536 (2002), 321-326, math-ph/0206017.
  19. Cahill K.E., Glauber R.J., Density operators for fermions, Phys. Rev. A 59 (1999), 1538-1555, physics/9808029.
  20. Mansour T., Schork M., On linear differential equations with variable coefficients involving a para-Grassmann variable, J. Math. Phys. 51 (2010), 043512, 21 pages.
  21. Mansour T., Schork M., On linear differential equations involving a paragrassmann variable, SIGMA 5 (2009), 073, 26 pages, arXiv:0907.2584.
  22. Cabra D.C., Moreno E.F., Tanasa A., Para-Grassmann variables and coherent states, SIGMA 2 (2006), 087, 8 pages, hep-th/0609217.
  23. Trifonov D.A., Pseudo-boson coherent and Fock states, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics, Editors K. Sekigawa et al., World Scientific, 2009, 241-250, arXiv:0902.3744.
  24. Cherbal O., Drir M., Maamache M., Trifonov D.A., Fermionic coherent states for pseudo-Hermitian two-level systems, J. Phys. A: Math. Theor. 40 (2007), 1835-1844, quant-ph/0608177.
  25. Najarbashi G., Fasihi M.A., Fakhri H., Generalized Grassmannian coherent states for pseudo-Hermitian n-level systems, J. Phys. A: Math. Theor. 43 (2010), 325301, 10 pages, arXiv:1007.1392.
  26. Daoud M., Hassouni Y., Kibler M., On generalized super-coherent states, Phys. Atomic Nuclei 61 (1998), 1821-1824, quant-ph/9804046.
  27. Daoud M., Kibler M., A fractional supersymmetric oscillator and its coherent states, math-ph/9912024.
  28. Nieto M.M., Coherent states and squeezed states, supercoherent states and supersqueezed states, in On Klauder's Path: a Field Trip, World Sci. Publ., River Edge, NJ, 1994, 147-155, hep-th/9212116.
  29. Nielsen M.A., Chuang I.L., Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000.
  30. Petz D., Quantum information theory and quantum statistics, Springer-Verlag, Berlin, 2008.
  31. Majid S., Rodríguez-Plaza M.J., Random walk and the heat equation on superspace and anyspace, J. Math. Phys. 35 (1994), 3753-3760.
  32. Kerner R., Z3-graded algebras and the cubic root of the supersymmetry translations, J. Math. Phys. 33 (1992), 403-411.
  33. Filippov A.T., Isaev A.P., Kurdikov A.B., Para-Grassmann differential calculus, Theoret. and Math. Phys. 94 (1993), 150-165, hep-th/9210075.
  34. Isaev A.P., Para-Grassmann integral, discrete systems and quantum groups, Internat. J. Modern Phys. A 12 (1997), 201-206, q-alg/9609030.
  35. Cugliandolo L.F., Lozano G.S., Moreno E.F., Schaposnik F.A., A note on Gaussian integrals over para-Grassmann variables, Internat. J. Modern Phys. A 19 (2004), 1705-1714, hep-th/0209172.
  36. Ilinski K.N., Kalinin G.V., Stepanenko A.S., q-functional Wick's theorems for particles with exotic statistics, J. Phys. A: Math. Gen. 30 (1997), 5299-5310, hep-th/9704181.
  37. El Baz M., Fresneda R., Gazeau J.P., Hassouni Y., Coherent state quantization of paragrassmann algebras, J. Phys. A: Math. Theor. 43 (2010), 385202, 15 pages, arXiv:1004.4706.
  38. Chaichian M., Demichev A.P., Polynomial algebras and higher spins, Phys. Lett. A 222 (1996), 14-20, hep-th/9602008.
  39. Daoud M., Kibler M., Fractional supersymmetric quantum mechanics as a set of replicas of ordinary supersymmetric quantum mechanics, Phys. Lett. A 321 (2004), 147-151, math-ph/0312019.
  40. Cherbal O., Drir M., Maamache M., Trifonov D.A., Supersymmetric extension of non-Hermitian su(2) Hamiltonian and supercoherent states, SIGMA 6 (2010), 096, 11 pages, arXiv:1009.5293.

Previous article   Next article   Contents of Volume 7 (2011)