An integrable (classical and quantum) four-wave mixing Hamiltonian system. (English) Zbl 1454.81085
Summary: A four-wave mixing Hamiltonian system on the classical as well as on the quantum level is investigated. In the classical case, if one assumes the frequency resonance condition of the form \(\omega_0 - \omega_1 + \omega_2 - \omega_3 = 0\), this Hamiltonian system is integrated in quadratures, and the explicit formulas of solutions are presented. Under the same condition, the spectral decomposition of quantum Hamiltonian is found, and thus, the Heisenberg equation for this system is solved. Some applications of the obtained results in non-linear optics are discussed.
©2020 American Institute of Physics
©2020 American Institute of Physics
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 70H05 | Hamilton’s equations |
| 81S08 | Canonical quantization |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81V80 | Quantum optics |
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