Full analytic spectrum of generalized Jaynes–Cummings Hamiltonians

A. J. Adanmitonde   ; G. Y. H. Avossevou

Teoretičeskaâ i matematičeskaâ fizika, Tome 201 (2019) no. 1, pp. 105-117

Résumé

We develop an analytic formalism using basic quantum mechanics techniques to successfully solve the multiphoton Jaynes–Cummings and the generalized Dicke Hamiltonians. For this, we split the Hamiltonians of these models into two operators that have the properties of constants of motion for these systems. We then use some well-known operator properties to obtain complete analytic spectra for the considered models.

Détail

Keywords: quantum mechanics, Jaynes–Cummings Hamiltonian, commuting operator, confluent hypergeometric function.
Mots-clés : constant of motion

Bibliographie

[1] I. I. Rabi, “On the process of space quantization”, Phys. Rev., 49:4 (1936), 324–328 ; “Space quantization in a gyrating magnetic field”, 51:8 (1937), 652–654 | DOI | Zbl | DOI | Zbl

[2] E. T. Jaynes, F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser”, Proc. IEEE, 51:1 (1963), 89–109 | DOI

[3] P. Carbonaro, G. Compagno, F. Persico, “Canonical dressing of atoms by intense radiation fields”, Phys. Lett. A, 73:2 (1979), 97–99 | DOI

[4] J. H. Eberly, N. B. Narozhny, J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model”, Phys. Rev. Lett., 44:20 (1980), 1323–1326 | DOI | MR

[5] R. Krivec, V. B. Mandelzweig, “Nonvariational calculation of the sticking probability and fusion rate for the $\mu\,dt$ molecular ion”, Phys. Rev. A, 52:1 (1995), 221–226 | DOI

[6] K. Wódkiewicz, P. L. Knight, S. J. Buckle, S. M. Barnett, “Squeezing and superposition states”, Phys. Rev. A, 35:6 (1987), 2567–2577 | DOI | MR

[7] A. Imamolg̃lu, S. E. Harris, “Lasers without inversion: interference of dressed lifetime-broadened states”, Opt. Lett., 14:24 (1989), 1344–1346 | DOI

[8] E. A. Kochetov, “Exactly solvable non-linear generalisations of the Jaynes–Cummings model”, J. Phys. A: Math. Gen., 20:9 (1987), 2433–2442 | DOI | MR

[9] V. Buzek, “On the non-linear Jaynes–Cummings model: The path-integral approach”, Czech. J. Phys. B, 39:7 (1989), 757–765 | DOI

[10] S. C. Gou, “Characteristic oscillations of phase properties for pair coherent states in the two-mode Jaynes–Cummings-model dynamics”, Phys. Rev. A, 48:4 (1993), 3233–3241 | DOI

[11] E. Choreño, D. Ojeda-Guillén, M. Salazar-Ramírez, V. D. Granados, “Two-mode generalization of the Jaynes–Cummings and Anti-Jaynes–Cummings models”, Ann. Phys., 387 (2017), 121–134 | DOI | MR

[12] M. Chaichian, D. Ellinas, P. Kulish, “Quantum algebra as the dynamical symmetry of the deformed Jaynes–Cummings model”, Phys. Rev. Lett., 65:8 (1990), 980–983 | DOI | MR

[13] A. F. Dossa, G. Y. H. Avossevou, “Full spectrum of the two-photon and the two-mode quantum Rabi models”, J. Math. Phys., 55:10 (2014), 102104, 18 pp. | DOI | MR | Zbl

[14] L. Lamata, J. Casanova, R. Gerritsma, C. F. Roos, J. J. García-Ripoll, E. Solano, “Relativistic quantum mechanics with trapped ions”, New J. Phys., 13:9 (2011), 095003, 19 pp. | DOI

[15] A. Retzker, E. Solano, B. Reznik, “Tavis–Cummings model and collective multiqubit entanglement in trapped ions”, Phys. Rev. A, 75:2 (2007), 022312 | DOI

[16] W. Kopylov, M. Radonjić, T. Brandes, A. Balaz̆, A. Pelster, “Dissipative two-mode Tavis–Cummings model with time-delayed feedback control”, Phys. Rev. A, 92:6 (2015), 063832 | DOI

[17] B. Buck, C. V. Sukumar, “Exactly soluble model of atom-phonon coupling showing periodic decay and revival”, Phys. Lett. A, 81:2–3 (1981), 132–135 | DOI

[18] C. C. Gerry, “Two-photon Jaynes–Cummings model interacting with the squeezed vacuum”, Phys. Rev. A, 37:7 (1988), 2683–2686 | DOI

[19] A. Joshi, R. R. Puri, “Dynamical evolution of the two-photon Jaynes–Cummings model in a Kerr-like medium”, Phys. Rev. A, 45:7 (1992), 5056–5060 | DOI

[20] V. Bartzis, N. Nayak, “Two-photon Jaynes–Cummings model”, J. Opt. Soc. Amer. B, 8:8 (1991), 1779–1786 | DOI

[21] P. Zhou, J. S. Peng, “Dipole squeezing in the two-photon Jaynes–Cummings model with superposition state preparation”, Phys. Rev. A, 44:5 (1991), 3331–3335 | DOI

[22] T. Nasreen, M. S. K. Razmi, “Atomic emission and cavity field spectra for a two-photon Jaynes–Cummings model in the presence of the Stark shift”, J. Opt. Soc. Amer. B, 10:7 (1993), 1292–1300 | DOI

[23] K. M. Ng, C. F. Lo, K. L. Liu, “Exact eigenstates of the two-photon Jaynes–Cummings model with the counter-rotating term”, Eur. Phys. J. D, 6:1 (1999), 119–126 | DOI | MR

[24] C. V. Sukumar, B. Buck, “Multi-phonon generalisation of the Jaynes–Cummings model”, Phys. Lett. A., 83:5 (1981), 211–213 | DOI

[25] H. R. Baghshahi, M. K. Tavassoly, A. Behjat, “Entropy squeezing and atomic inversion in the $k$-photon Jaynes–Cummings model in the presence of the Stark shift and a Kerr medium: A full nonlinear approach”, Chinese Phys. B, 23:7 (2014), 074203 | DOI

[26] R. Koc, H. Tütünculer, M. Koca, E. Olgar, “Algebraic treatments of the problems of the spin-1/2 particles in the one- and two-dimensional geometry: A systematic study”, Ann. Phys., 319:2 (2005), 333–347 | DOI | MR

[27] H. Panahi, S. A. Rad, “Two and $k$-photon Jaynes–Cummings models and Dirac oscillator problem in Bargmann–Segal representation”, Int. J. Theor. Phys., 52:11 (2013), 4068–4073 | DOI | MR

[28] F. Cooper, A. Khare, U. Sukhateme, “Supersymmetry and quantum mechanics”, Phys. Rep., 251:5–6 (1995), 267–385 | DOI | MR

[29] R. Dutt, A. Khare, U. Sukhatme, “Supersymmetry, shape invariance, and exactly solvable potentials”, Amer. J. Phys., 56:2 (1988), 163–168 | DOI

[30] Yu. A. Golfand, E. P. Likhtman, “Rasshirenie algebry generatorov gruppy Puankare i narushenie $P$-invariantnosti”, Pisma v ZhETF, 13:8 (1971), 452–455

[31] D. V. Volkov, V. P. Akulov, “O vozmozhnom universalnom vzaimodeistvii neitrino”, Pisma v ZhETF, 16:11 (1972), 621–624

[32] P. Ramond, “Dual theory for free fermions”, Phys. Rev. D, 3:10 (1971), 2415–2418 | DOI | MR

[33] A. Neveu, J. Schwarz, “Factorizable dual model of pions”, Nucl. Phys. B, 31:1 (1971), 86–12 | DOI

[34] J. Wess, B. Zumino, “Supergauge transformations in four dimensions”, Nucl. Phys. B, 70:1 (1974), 39–50 | DOI | MR

[35] H.-Y. Fan, L.-S. Li, “Supersymmetric unitary operator for some generalized Jaynes–Cummings models”, Commun. Theor. Phys., 25:1 (1996), 105–110 | DOI | MR

[36] Zh. V. Ungvu, F. A. Dosa, G. I. Avosvu, “Biortogonalnaya kvantovaya mekhanika dlya neermitovykh mnogomodovoi i mnogofotonnoi modelei Dzheinsa–Kammingsa”, TMF, 193:1 (2017), 66–83 | DOI

[37] H.-X. Lu, X.-Q. Wang, “Multiphoton Jaynes–Cummings model solved via supersymmetric unitary transformation”, Chinese Phys., 9:8 (2000), 1009–1963

[38] E. Choreño, D. Ojeda-Guillén, V. D. Granados, “Matrix diagonalization and exact solution of $k$-photon Jaynes–Cummings model”, Eur. Phys. J. D, 72:8 (2018), 142, 7 pp., arXiv: 1803.03206 | DOI

[39] G. Petiau, “Sur la détermination des fonctions d'ondes du corpuscule de spin $\hbar$ en interaction avec un champ magnitique ou electrique constant”, J. Phys. Radium, 17:11 (1956), 956–964 | DOI | MR

[40] M. Abramovich, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979 | MR | MR | Zbl

[41] A. Maggitti, M. Radonjić, B. M. Jelenković, “Dark-polariton bound pairs in the modified Jaynes–Cummings–Hubbard model”, Phys. Rev. A, 93:1 (2016), 013835, 10 pp. | DOI

[42] Ts. Gantsog, A. Joshi, R. Tanas, “Phase properties of one- and two-photon Jaynes–Cummings models with a Kerr medium”, Quantum Semiclass. Opt., 8:3 (1996), 445–456 | DOI

[43] B. M. Rodríguez-Lara, H. M. Moya-Cessa, “The exact solution of generalized Dicke models via Susskind–Glogower operators”, J. Phys. A: Math. Theor., 46:9 (2013), 095301, 10 pp., arXiv: 1207.6551 | DOI | MR