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arXiv:2312.15114 · Journal of the Optical Society of America B

Berry phase and the Mandel parameter of the non-degenerate parametric amplifier

J. C. Vega, E. Choreño, D. Ojeda-Guillén, R. D. Mota

Published Mar 15, 2024• 4 citations
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Abstract

We study the non-degenerate parametric amplifier problem from an algebraic approach of the SU(1,1) group. We write the Hamiltonian of this problem in terms of the boson generators of the SU(1,1) group and the difference operator. We apply the tilting transformation to our results to exactly solve this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by assuming that our Hamiltonian is an explicit function of time, we calculate its Berry phase. Finally we obtain the Mandel Q -parameter of the photon numbers n a and n b .

DOI
10.1364/josab.517533

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We study the non-degenerate parametric amplifier problem from an algebraic approach of the SU(1,1) group. We write the Hamiltonian of this problem in terms of the boson generators of the SU(1,1) group and the difference operator. We apply the tilting transformation to our results to exactly solve this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by assuming that our Hamiltonian is an explicit function of time, we calculate its Berry phase. Finally we obtain the Mandel Q -parameter of the photon numbers n a and n b .
We study the Hamiltonian of two isotropic oscillators with weak coupling from an algebraic approach. We write the Hamiltonian of this problem in terms of the boson generators of the S U (1,1) and S U (2) groups. This allows us to apply two tilting transformations based on both group similarity transformations to obtain their energy spectrum and eigenfunctions. Then, we obtain the Mandel Q -parameter and the second-order correlation function g 2 (0) of the photon numbers n a and n b . It is important to note that, in our procedure, we consider the case of weak coupling.
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We construct the Perelomov number coherent states for an arbitrary su(1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the non-degenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the two-dimensional harmonic oscillator.
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