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

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.

Comment on: "Quantum dynamics of a general time-dependent coupled oscillator"

“…3 Three-dimensional time-dependent harmonic oscillators A generalization of the 2D coupled oscillator to a 3D one has been incorrectly made by Hassoul et al in [2] where the authors study general time-dependent three coupled nano-optomechanical oscillators. The Hamiltonian operator of the 3D system reads…”

“…The time-dependent coupled oscillator that has been a point of interest in the research field for a few years now [3]- [8], modele various physical systems [9]- [19] and helped in explaining numerous physical interacting systems including trapped atoms [20], nano-optomechanical resonances [21,22], electromagnetically induced transparency [23], stimulated Raman effects [24], time-dependent Josephson phenomena [25], and systems of three isotropically coupled spins 1/2 [26]. Coupled oscillator are fundamental for quantum technologies such as quantum computing and quantum cryptography [27,28,29] In what follows, we highlight many basic and mathematical flaws made by Hassoul et al in their recent papers [1,2] while investigating the quantum dynamical properties of a general time-dependent coupled oscillator using two and three-dimensional dynamical invariants. From there, we proceeded to reevaluate the entire study and approach the whole subject in a more scientifically coherent manner.…”

“…The harmonic oscillator is one of the most important models in quantum mechanics, and one of the few ones that has an exact analytical solution what made it applicable in the study of the dynamical properties of different physical systems. Recently, Hassoul et al [1,2] have investigated quantum dynamical properties of a two (2D) and three (3D) dimensional time-dependent coupled oscillator. Their studies are based on the theory of two-dimensional (2D) and three-dimensional (3D) dynamical invariants.…”

“…In section 2, we recapitulate the basic errors made in [1] where a 2D quantum invariant operator is introduced for the study of a 2D time-dependent coupled oscillator system, then we show that the linear canonical transformation defined in [1] can not be adapted to the considered problem without a constraint on the mass and to solve the problem in question, we introduce more adapting canonical transformations. In Section 3, we consider the 3D case investigated in [2] and highlight with some simple mathematical calculation all the flawed results found in [2].…”

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By using dynamical invariants theory, Hassoul et al [1,2] investigate the quantum dynamics of two (2D) and three (3D) dimensional time-dependent coupled oscillators. They claim that, in the 2D case, introducing two pairs of annihilation and creation operators uncouples the original invariant operator so that it becomes the one that describes two independent subsystems.

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