Abstract
Pseudo-Hermitian Hamiltonians and pseudo-Hermitian coupling between two electromagnetic modes are analyzed by using similarity transformations of Hermitian Hamiltonians or of Hermitian operators, including a special metric and biorthogonal relations replacing the usual orthogonal relations used in quantum mechanics. The coupling between two electromagnetic (em) modes including certain decay and amplification processes is related to a coupling matrix G which has parity-time (PT) symmetry and which obeys the pseudo-Hermiticity condition ηGη−1 = G † where η is a metric. The linear equations representing the pseudo-Hermitian coupling between the two em modes are diagonalized, in the interaction picture, by introducing ‘dressed’ α¯ and β~ operators which have real or pure imaginary eigenfrequencies. The commutation-relations (CR) for the α~ and β~ operators and for the two-mode operators ā and b~, in the interaction picture and under the condition of real eigenfrequencies are obtained by the use of the pseudo-Hermiticity property of the G matrix. These CR for real eigenfrequencies, are preserved in time without any Langevin noise terms.
Similar content being viewed by others
References
1. M.C. Pease, Methods of Matrix Algebra (Academic, New York, 1965).
2. Y. Ben-Aryeh, A. Mann, and I. Yaakov, J. Phys. A: Math. Gen. 37, 12059 (2004).
3. P.A. M. Dirac, Proc. Roy. Soc. A 180, 1 (1942).
4. W. Pauli, Rev. Mod. Phys. 15, 175 (1943).
5. T. D. Lee and G. C. Wick, Nuc. Phys. B 9, 209 (1969).
6. F. G. Scholtz, H.B. Geyer, and F. J. W. Hahne, Ann. Phys. 213, 74 (1992).
7. Y. Ben-Aryeh and R. Barak, Phys. Lett. A 351, 388 (2006).
8. A. Mostafazadeh, J. Math. Phys. 43, 205, 3944 (2002).
9. Y. Ben-Aryeh, J. Opt B: Quantum Semiclass. Opt 7, S452 (2005).
10. W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964).
11. C. M. Bender, D. C. Brody, and H. F.Jones, Phys. Rev. Lett. 89, 270401 (2002).
12. C. M. Bender, D.C. Brody, and H. F. Jones, Am. J. Phys. 71, 1095 (2003).
13. V. V. Dodonov and V. I. Man'ko, Invariants and the Evolution of Nonstationary Quantum Systems, M. A. Markov, ed. (Nova Science, Commack, NY, 1989).
14. O. Man'ko, V. I. Man'ko, and G. Marmo, J. Phys. A: Math. Gen. 35, 699 (2002).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ben-Aryeh, Y. Similarity Transformations of Hermitian Hamiltonians and Pseudo-Hermitian Coupling Between Two Electromagnetic Modes. Found Phys Lett 19, 747–756 (2006). https://doi.org/10.1007/s10702-006-1062-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10702-006-1062-x