Abstract
We develop a biorthogonal formalism for non-Hermitian multimode and multiphoton Jaynes–Cummings models. For these models, we define supersymmetric generators, which are especially convenient for diagonalizing the Hamiltonians. The Hamiltonian and its adjoint are expressed in terms of supersymmetric generators having the Lie superalgebra properties. The method consists in using a similarity dressing operator that maps onto spaces suitable for diagonalizing Hamiltonians even in an infinite-dimensional Hilbert space. We then successfully solve the eigenproblems related to the Hamiltonian and its adjoint. For each model, the eigenvalues are real, while the eigenstates do not form a set of orthogonal vectors. We then introduce the biorthogonality formalism to construct a consistent theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Moiseyev, “Quantum theory of resonances: Calculating energies, widths, and cross-sections by complex scaling,” Phys. Rep., 302, 211–293 (1998).
J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep., 374, 271–383 (2003).
N. Moiseyev, Non-Hermitian Quantum Mechanics, Cambridge Univ. Press, Cambridge (2011).
F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne, “Quasi-Hermitian operators in quantum mechanics and the variational principle,” Ann. Phys., 213, 74–101 (1992).
H. B. Geyer, W. D. Heiss, and F. G. Scholtz, “The physical interpretation of non-Hermitian Hamiltonians and other observables,” Canadian J. Phys., 86, 1195–1201 (2008).
C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Progr. Phys., 70, 947–1018 (2007).
A. Mostafazadeh, “Pseudo-Hermitian representation of quantum mechanics,” Internat. J. Geom. Methods Mod. Phys., 7, 1191–1306 (2010).
M. Znojil, “Three-Hilbert-space formulation of quantum mechanics,” SIGMA, 5, 001 (2009).
T. Curtright and L. Mezincescu, “Biorthogonal quantum systems,” J. Math. Phys., 48, 092106 (2007).
D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A: Math. Theor., 47, 035305 (2014).
F. Bagarello, “Transition probabilities for non self-adjoint Hamiltonians in infinite dimensional Hilbert spaces,” Ann. Phys., 362, 424–435 (2015).
F. Bagarello and A. Fring, “Generalized Bogoliubov transformations versus D-pseudo-bosons,” J. Math. Phys., 56, 103508 (2015).
F. Bagarello, “Some results on the dynamics and transition probabilities for non self-adjoint Hamiltonians,” Ann. Phys., 356, 171–184 (2015).
F. Bagarello, M. Luttuca, R. Passante, L. Rizzuto, and S. Spagnolo, “Non-Hermitian Hamiltonian for a modulated Jaynes–Cummings model with PT symmetry,” Phys. Rev. A, 91, 042134 (2015).
F. Bagarello, J.-P. Gazeau, F. H. Szafraniec, and M. Znojil, eds., Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Wiley, New York (2015).
F. Bagarello, “Pseudobosons, Riesz bases, and coherent states,” J. Math. Phys., 51, 023531 (2010).
F. Bagarello, “Pseudo-bosons from Landau levels,” SIGMA, 6, 093 (2010).
F. Bagarello, “Pseudo-bosons, so far,” Rep. Math. Phys., 68, 175–210 (2011).
P. Carbonaro, G. Compagno, and F. Persico, “Canonical dressing of atoms by intense radiation fields,” Phys. Lett. A, 73, 97–99 (1979).
J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett., 44, 1323–1326 (1980).
R. Krivec and V. B. Mandelzweig, “Nonvariational calculation of the sticking probability and fusion rate for the µdt molecular ion,” Phys. Rev. A, 52, 221–226 (1995).
K. Wódkiewicz, P. L. Knight, S. J. Buckle, and S. M. Barnett, “Squeezing and superposition states,” Phys. Rev. A, 35, 2567–2577 (1987).
A. Imamol˜glu and S. E. Harris, “Lasers without inversion: Interference of dressed lifetime-broadened states,” Opt. Lett., 14, 1344–1346 (1989).
I. I. Rabi, “On the process of space quantization,” Phys. Rev., 49, 324–328 (1936)
“Space quantization in a gyrating magnetic field,” Phys. Rev., 51, 652–654 (1937).
A. F. Dossa and G. Y. H. Avossevou, “Full spectrum of the two-photon and the two-mode quantum Rabi models,” J. Math. Phys., 55, 102104 (2014).
F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep., 251, 267–385 (1995).
R. Dutt, A. Khare, and U. Sukhatme, “Supersymmetry, shape invariance, and exactly solvable potentials,” Am. J. Phys., 56, 163–168 (1988).
H.-Y. Fan and L.-S. Li, “Supersymmetric unitary operator for some generalized Jaynes–Cummings models,” Commun. Theor. Phys., 25, 105–110 (1996).
H.-Y. Fan, “Representation and transformation theory in quantum mechanics,” in: Progress of Dirac’s Symbolic Method, Scientific and Technical Univ. Press, Shanghai (1997), p.181.
E. Brian Davies, Linear Operators and their Spectra (Cambridge Stud. Adv. Math., Vol. 106), Cambridge Univ. Press, Cambridge (2007).
P. E. G. Assis, “Non-Hermitian Hamiltonians in field theory,” Doctoral dissertation, City Univ. London, London (2009).
H.-X. Lu, X.-Q. Wang, and Y.-D. Zhang, “Exact solution for super–Jaynes–Cummings model,” Chinese Phys., 9, 325–328 (2000).
J. Yang, W.-L. Yu, and A.-P. Xiang, “Exact solution for Jaynes–Cummings model with bosonic field nonlinearity and strong boson–fermion coupling,” Commun. Theor. Phys., 45, 143–146 (2006).
T.-Q. Song and Y.-J. Zhu, “Solving generalized non-degenerate two-mode two-photon Jaynes–Cummings model by supersymmetric unitary transformation,” Commun. Theor. Phys., 38, 85–88 (2002).
H.-X. Lu and X.-Q. Wang, “Multiphoton Jaynes–Cummings model solved via supersymmetric unitary transformation,” Chinese Phys., 9, 568–571 (2000).
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of J. V. Hounguevou is supported by the Ministère de l’Enseignement Supérieur et de la Recherche Scientifique of the Republic of Benin.
Rights and permissions
About this article
Cite this article
Hounguevou, J.V., Dossa, F.A. & Avossevou, G.Y.H. Biorthogonal quantum mechanics for non-Hermitian multimode and multiphoton Jaynes–Cummings models. Theor Math Phys 193, 1464–1479 (2017). https://doi.org/10.1134/S0040577917100051
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577917100051