Some integrable systems in nonlinear quantum optics

The relationship between classical and quantum three one-mode systems interacting in a non-linear way is described. We investigate the integrability of these systems by using the reduction procedure. The reduced coherent states for the quantum system are constructed. We find the explicit formulas for the reproducing measure for these states. Examples of some applications of the obtained results in non-linear quantum optics are presented.

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“…3) see [15,12]. The Hamiltonian (3.1) expressed in terms of A 0 , A 1 , A 2 , A and A * takes the form…”

Section: Correspondence Between Quantum and Classical Systemsmentioning

confidence: 99%

Integrability and correspondence of classical and quantum non-linear three-mode systems

Odzijewicz

Wawreniuk

2018

Journal of Mathematical Physics

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“…The algebras A R find application for the integration of quantum optical models, see [6]. For the rational R they also can be considered as the symmetry algebras in the theory of the basic hypergeometric series, see [17] In the paper [11] one investigates the case when R is invertible map.…”

Section: Examplementioning

confidence: 99%

Noncommutative Kähler-like structures in quantization

Odzijewicz

2007

Journal of Geometry and Physics

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“…In order to construct the coherent states, we first investigate the Hamiltonian which describes this effect. We use the techniques developed in [5,6], which is strictly related to the theory of the orthogonal polynomials, and we make the reduction of the Hamiltonian. The Hamiltonian of the down conversion can be written down…”

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confidence: 99%

Three boson interaction process: spectra and coherent states

Chadzitaskos

Daskaloyannis

Smotlacha

2013

Journal of Modern Optics

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We use the methods of constructions of su(2) and deformed su(2) coherent states in order to construct the coherent states for down conversion processes. The down conversion process can be understood as a quasi-exactly solvable model of quantum mechanics. After the reduction of the Hamiltonian, we use the Turbiner polynomials approach, and the eigenvalues of the Hamiltonian for low number of photons are calculated and the approximation formula is found out. After the discussion on the time evolution and the entanglement, the coherent states are constructed as the eigenstates of the reduced annihilation operator. IntroductionThe down conversion process and the second harmonic generation are some kinds of the nonlinear optical processes [1] which can be generated by a linear combination of powers of a 'beam splitter' Hamiltonian. These processes have a wide scale of applications: for instance, in the case of the parametric down conversion and using a nonlinear crystal, the entangled photons can be produced. This can be used in the area of quantum teleportation [2]. The next possible use is in optical communication, where the intensity of light in an optical fibre ensures the routing switch [3]. It can be also exploited in the field of quantum information, state preparation, quantum control and so on.During recent years, the down conversion process and the second harmonic generation have been intensively investigated, because of the development of the experimental methods. It is the simplest nonlinear model in which the entangled photons are produced [4]. In this process a photon of frequency ω 2 is converted into two photons of frequency ω 1 , or vice versa in the second harmonic generation process. The conservation of energy imposes that ω 1 = ω and ω 2 = 2ω. In order to construct the coherent states, we first investigate the Hamiltonian which describes this effect. We use the techniques developed in [5,6], which is strictly related to the theory of the orthogonal polynomials, and we make the reduction of the Hamiltonian. The Hamiltonian of the down conversion can be written down

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Some integrable systems in nonlinear quantum optics

Cited by 24 publications

References 22 publications

Integrability and correspondence of classical and quantum non-linear three-mode systems

Integrability and correspondence of classical and quantum non-linear three-mode systems

Noncommutative Kähler-like structures in quantization

Three boson interaction process: spectra and coherent states

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