Abstract
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.
Highlights
Arrays of coupled oscillators are ubiquitous in nature, and their oscillatory motions are usually applied in technological sciences [1,2,3,4,5]
The dynamical invariant is useful when we examine the entanglement for timedependent coupled oscillators based on the Schrödinger equation [14]
We investigated the diagonalization of the invariant operator in order to show the usefulness of the quantum invariant in analyzing the dynamical properties of the nano-optomechanical system
Summary
Arrays of coupled oscillators are ubiquitous in nature, and their oscillatory motions are usually applied in technological sciences [1,2,3,4,5]. A wide class of physical phenomena of interacting systems is explainable from the model of coupled harmonic oscillators. It includes nano-optomechanical resonances [6, 7], electromagnetically induced transparency [8], stimulated Raman effects [9], Josephson phenomena [10], one-half spin dynamics [11], and trapping of different particles [1]. Among various issues related to coupled harmonic oscillators, the entanglement and mixedness are the most remarkable characters which give a theoretical basis for quantum technologies, such as quantum computing and quantum cryptography [12,13,14]. The entanglement in three coupled harmonic oscillators should be analyzed from the fundamental quantum mechanical point of view
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
