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arXiv:2409.08179 · Journal of the Optical Society of America B

SU (1,1) × SU (2) approach and the Mandel parameter to the Hamiltonian of two oscillators with weak coupling

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

Published Jan 16, 2025• 2 citations
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Abstract

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.

DOI
10.1364/josab.542294

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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.
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 .
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.
We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Speciflcally, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former.
We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. The unitary operator U, which is found to be able to transform the Fock space of a three–dimensional isotropic harmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum–mechanical rotation in three–mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.
We present a perturbative analysis of a Rabi model where the coupling between the quantized single-mode electromagnetic field and the two-level atom depends on the field intensity. Upon modeling the matter–radiation coupling through the Holstein–Primakoff realization of algebra su(1,1), we evaluate first- and second-order eigenenergies and eigenstates both in the weak-coupling regime (atom transition frequency smaller than the coupling strength) and in the strong-coupling regime. In the first case, among various effects, we observe a quadratic dependence on the photon number of energy eigenvalues and the possible formation of level doublets. In the strong-coupling case, the perturbative analysis becomes considerably complex due to the su(1,1)-valued form of the unperturbed Hamiltonian. The critical condition for the transition to an almost continuous spectrum is found in terms of the model parameters.