Dynamical Invariant Applied on General Time-Dependent Three Coupled Nano-Optomechanical Oscillators

A quadratic invariant operator for general time-dependent three coupled nano-optomechanical oscillators is investigated. We show that the invariant operator that we have established satisfies the Liouville-von Neumann equation and coincides with its classical counterpart. To diagonalize the invarian...

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Bibliographic Details
Published in:Journal of nanomaterials 2021-07, Vol.2021, p.1-9
Main Authors: Hassoul, Sara, Menouar, Salah, Benseridi, Hamid, Choi, Jeong Ryeol
Format: Article
Language:English
Subjects:
Eigenvalues
Eigenvectors
Invariants
Matrix representation
Operators (mathematics)
Oscillators
Time dependence
Transformations (mathematics)
Wave functions
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Summary:A quadratic invariant operator for general time-dependent three coupled nano-optomechanical oscillators is investigated. We show that the invariant operator that we have established satisfies the Liouville-von Neumann equation and coincides with its classical counterpart. To diagonalize the invariant, we carry out a unitary transformation of it at first. From such a transformation, the quantal invariant operator reduces to an equal, but a simple one which corresponds to three coupled oscillators with time-dependent frequencies and unit masses. Finally, we diagonalize the matrix representation of the transformed invariant by using a unitary matrix. The diagonalized invariant is just the same as the Hamiltonian of three simple oscillators. Thanks to such a diagonalization, we can analyze various dynamical properties of the nano-optomechanical system. Quantum characteristics of the system are investigated as an example, by utilizing the diagonalized invariant. We derive not only the eigenfunctions of the invariant operator, but also the wave functions in the Fock state.
ISSN:1687-4110
1687-4129
DOI:10.1155/2021/6903563