Quantum dynamics of a general time-dependent coupled oscillator

Abstract

Quantum dynamical properties of a general time-dependent coupled oscillator are investigated based on the theory of two-dimensional (2D) dynamical invariants. The quantum dynamical invariant of the system satisfies the Liouville–von Neumann equation and it coincides with its classical counterpart. The mathematical formula of this invariant involves a cross term which couples the two oscillators mutually. However, we show that, by introducing two pairs of annihilation and creation operators, it is possible to uncouple the original invariant operator so that it becomes the one that describes two independent subsystems. The eigenvalue problem of this decoupled quantum invariant can be solved by using a unitary transformation approach. Through this procedure, we eventually obtain the eigenfunctions of the invariant operator and the wave functions of the system in the Fock state. The wave functions that we have developed are necessary in studying the basic quantum characteristics of the system. In order to show the validity of our theory, we apply our consequences to the derivation of the fluctuations of canonical variables and the uncertainty products for a particular 2D oscillatory system whose masses are exponentially increasing.

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

“…We consider three coupled nano-optomechanical oscillators with different masses m i ðtÞði = 1, 2, 3Þ and frequencies ω i ðtÞ, respectively; the convention for i (including j and k) given here will be used throughout the paper. Such a system is described by the Hamiltonian [29].…”

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

“…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.…”

“…From there, we proceeded to reevaluate the entire study and approach the whole subject in a more scientifically coherent manner. 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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