Abstract
Based on fundamental principles of quantum mechanics, we present a method for a rigorous analytic construction of the spectra of the one- and two-photon Jaynes-Cummings models in a Kerr medium. To obtain an idea of the method, we consider a first generalized Jaynes-Cummings model with a real, linear superpotential using techniques of supersymmetric quantum mechanics. The Hamiltonian of this model is written as a combination of operators generating the underlying superalgebra whose elements are defined as differential matrix operators. Based on the formalism of supersymmetric quantum mechanics and the properties of sets of common observables, we derive solutions of the one- and two-photon models expressed in terms of confluent hypergeometric functions. Finally, using numerical analysis, we study the influence of the Kerr effect on the energy spectra of the physical system.
Similar content being viewed by others
References
E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE, 51, 89–109 (1963)
P. Meystre, E. Geneux, A. Quattropani, and A. Faist, “Long-time behaviour of a two-level system in interaction with an electromagnetic field,” Nuovo Cimento B, 25, 521–537 (1975).
R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev., 93, 99–110 (1954).
J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes-Cummings model,” Phys. Rev. Lett., 105, 263603 (2010); arXiv:1008.1240v3 [quant-ph] (2010).
A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum Rabi model,” Phys. Rev. Lett., 108, 163601 (2012); arXiv:1111.6424v1 [quant-ph] (2011).
D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, and J. Vucković, “Controlling cavity reflectivity with a single quantum dot,” Nature, 450, 857–861 (2007).
P. Forn-Dáz, J. Lisenfeld, D. Marcos, J. J. Garcia-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett., 105, 237001 (2010); arXiv:1005.1559v2 [cond-mat.supr-con] (2010).
T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys., 6, 772–776 (2010).
Ts. Gantsog, A. Joshi, and R. Tanas, “Phase properties of one- and two-photon Jaynes-Cummings models with a Kerr medium,” Quantum Semiclass. Opt., 8, 445–456 (1996).
J. V. Hounguevou, F. A. Dossa, and G. Y. Avossevou, “Biorthogonal quantum mechanics for non-Hermitian multimode and multiphoton Jaynes-Cummings models,” Theor. Math. Phys., 193, 1464–1479 (2017).
I. Travĕnec, “Solvability of the two-photon Rabi Hamiltonian,” Phys. Rev. A, 85, 043805 (2012); arXiv:1201.3717v1 [math-ph] (2012).
B. Gardas and J. Dajka, “Generalized parity in multi-photon Rabi model,” Phys. Lett. A, 377, 3205–3208 (2013); arXiv:1301.3747v1 [quant-ph] (2013).
B. F. Samsonov and J. Negro, “Darboux transformations of the Jaynes-Cummings Hamiltonian,” J. Phys. A: Math. Gen., 37, 10115–10127 (2004); arXiv:quant-ph/0401092v1 (2004).
R. Dutt, A. Khare, and U. Sukhatme, “Supersymmetry, shape invariance, and exactly solvable potentials,” Amer. J. Phys., 56, 163–168 (1988).
F. Cooper, A. Khare, and U. Sukhateme, “Supersymmetry and quantum mechanics,” Phys. Rep., 251, 267–385 (1995); arXiv:hep-th/9405029v2 (1994).
I. Aref’eva, D. J. Fernández, V. Hussin, J. Negro, L. M. Nieto, and B. F. Samsonov, “Progress in supersymmetric quantum mechanics,” J. Phys. A: Math. Gen., 37, 10007–10458 (2004).
A. A. Andrianov and M. V. Ioffe, “Nonlinear supersymmetric quantum mechanics: Concepts and realizations,” J. Phys. A: Math. Gen., 45, 503001 (2012); arXiv:1207.6799v2 [hep-th] (2012).
H.-X. Lu and X.-Q. Wang, “Multiphoton Jaynes-Cummings model solved via supersymmetric unitary transformation,” Chinese Phys., 9, 1009–1963 (2000).
B. M. Rodríguez-Lara, and H. M. Moya-Cessa, “The exact solution of generalized Dicke models via Susskind-Glogower operators,” J. Phys. A: Math. Theor., 46, 095301 (2013); arXiv:1207.6551v2 [quant-ph] (2012).
A. D. Alhaidari, “The supersymmetric Jaynes-Cummings model and its solutions,” J. Phys. A: Math. Gen., 39, 15391–15401 (2006).
C. Buzano, M. G. Rasetti, and M. L. Rastello, “Dynamical superalgebra of the “dressed” Jaynes-Cummings model,” Phys. Rev. Lett., 62, 137–139 (1989)
M. Chaichian, D. Ellinas, and P. Kulish, “Quantum algebra as the dynamical symmetry of the deformed Jaynes-Cummings model,” Phys. Rev. Lett., 65, 980–983 (1990).
A. Maggitti, M. Radonjić, and B. M. Jelenković, “Dark-polariton bound pairs in the modified Jaynes-Cummings-Hubbard model,” Phys. Rev. A, 93, 013835 (2016).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Natl. Bur. Stds. Appl. Math. Ser., Vol. 55), Dover, New York (1972).
E. Choreño, D. Ojeda-Guillén, and V. D. Granados, “Matrix diagonalization and exact solution of k-photon Jaynes-Cummings model,” Eur. Phys. J. D, 72, 142 (2018); arXiv:1803.03206v1 [quant-ph] (2018).
C. B. C. Gomes, F. A. G. Almeida, and A. M. C. Souza, “Influence of the Kerr effect in a Mott insulator on the superfluid transition from the point of view of the Jaynes-Cummings-Hubbard model,” Phys. Lett. A, 380, 1799–1803 (2016).
M. Hohenadler, M. Aichhorn, L. Pollet, and S. Schmidt, “Polariton Mott insulator with trapped ions or circuit QED,” Phys. Rev. A, 85, 013810 (2012); arXiv:1108.5035v2 [cond-mat.str-el] (2011).
J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in acavity,” Rev. Modern Phys., 73, 565–582 (2001).
Acknowledgments
The authors thank a referee for the relevant comments that helped improve the work.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Conflicts of interest
The authors declare no conflicts of interest.
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 3, pp. 451–466, June, 2020.
Rights and permissions
About this article
Cite this article
Adanmitonde, A.J., Avossevou, G.Y.H. & Dossa, F.A. Quantization of Some Generalized Jaynes-Cummings Models in a Kerr-Like Medium. Theor Math Phys 203, 824–836 (2020). https://doi.org/10.1134/S0040577920060082
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577920060082