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

Laboratory of Optoelectronics and Compounds (LOC), Department of Physics, Faculty of Science, University of Ferhat Abbas Setif 1, Setif 19000, Algeria Laboratory of Applied Mathematics (LaMA), Department of Mathematics, Faculty of Science, University of Ferhat Abbas Setif 1, Setif 19000, Algeria Department of Nanoengineering, Kyonggi University, Yeongtong-gu, Suwon, Gyeonggi-do 16227, Republic of Korea


Introduction
Arrays of coupled oscillators are ubiquitous in nature, and their oscillatory motions are usually applied in technological sciences [1][2][3][4][5]. Thanks to this, there has been an increasing interest in the dynamical and statistical behavior of a set of coupled oscillators so far. 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]. Hence, the entanglement in three coupled harmonic oscillators should be analyzed from the fundamental quantum mechanical point of view.
If we consider the above-mentioned importance of coupled oscillatory systems, the investigation of their dynamical properties are highly required. Because the motion of coupled oscillators is complex, oscillator dynamics requires rigorous description especially when the parameters vary over time and/or the number of coupling is more than two [3]. One of the potential tools for studying oscillator dynamics in the time-dependent regime is the dynamical invariant operator [15,16]. The invariant operator theory for timedependent harmonic oscillators is based on the construction of adiabatic invariants which were firstly introduced by Lewis and Riesenfeld [15,16] in describing their quantum features. The analysis of many oscillatory systems was fulfilled by introducing an invariant [13][14][15][16][17][18][19][20][21][22][23][24][25]. The dynamical invariant is also useful when we examine the entanglement for timedependent coupled oscillators based on the Schrödinger equation [14].
The aim of this work is to study dynamical invariant of general time-dependent three coupled nano-optomechanical oscillators [6,7,[26][27][28]. This research is motivated by the usefulness of the dynamical invariant on elucidating classical and quantum mechanical properties of complicated mechanical systems. Considering the powerfulness of the invariant in analyzing mechanical systems, we will formulate a mathematical invariant for general time-dependent three coupled nanooptomechanical oscillators in this work. The obtained invariant will be applied to the analysis of the dynamics of the optomechanical systems. In particular, we will focus on the use of the invariant in unfolding exact quantum theory of the system.
If a time-dependent system is relatively simple, the invariant operator can be directly applied to analyzing the quantum dynamics of the system. However, due to the complexity of the three coupled nano-optomechanical oscillatory systems that we consider, we will diagonalize the invariant via the unitary transformation technique. The diagonalization of the invariant may greatly simplify the analysis of the dynamics of the system. In order to show this, the complete quantum wave functions of the system will be derived using the eigenstates of the invariant operator as an example.
Our work is organized as follows. Our main study starts from Section 2. In Subsection 2.1, we will derive the classical invariant quantity as a preliminary step in our development, starting from the classical equations of motion for the coupled nano-optomechanical oscillators. We will find the quantum invariant operator together with the ladder operators and will simplify it by means of the unitary transformation approach in Subsection 2.2. Matrix representation of this invariant operator will be diognalized in Subsection 2.3 by using the technique of diagonalization in 3D. The eigenfunctions of the invariant operator and wave functions of the system will be evaluated in Section 3. Because the wave functions of the transformed (or diagonalized) system is well known, we can easily find the wave functions of the original system through an inverse transformation. Finally, concluding remarks are given in the last section.

Results and Discussion
2.1. Construction of the Classical Invariant. 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].
where X i and P i are the canonical coordinates and momenta, and k 12 ðtÞ, k 13 ðtÞ, and k 23 ðtÞ are coupling parameters. We will derive the classical invariant for this system in this section. To this end, let us first start from the Hamilton's equations of the form From minor evaluations with these, we easily see that the classical equations of motion are given by There exist in principle many dynamical invariants for any given Hamiltonian H. We consider a quadratic invariant indexed by one or more parameters. Let us assume that its formula is given in the form where the time-dependent parameters A i ðtÞ, B i ðtÞ, C i ðtÞ, D 12 ðtÞ, D 13 ðtÞ, and D 23 ðtÞ are real and differentiable functions, which will be explicitly evaluated later on. The dynamical invariant OðtÞ in classical mechanics satisfy the equation To derive the formulae of time-dependent parameters in Equation (6), we carry out some basic evaluations after inserting Equations (1) and (6) into Equation (7). Then, this gives _ _ Possible solutions for these differential equations are obtained in the following forms 2 Journal of Nanomaterials In terms of the above solutions of the coupled equations, we can describe the classical invariant OðtÞ in the form It is obvious that this invariant is not unique; there may exist many invariants for the given Hamiltonian, because Equations (14)- (19) are not uniquely determined in general.

Quantum Invariant
Operator. The relation between the pair of classical canonical variables, ðX i , P j Þ, and the corresponding quantum variables, ðX i ,P j Þ, is given by In case we know a classical invariant of the system, we can easily identify the counterpart quantum invariant on account of the above simple relation. By replacing the canonical variables in the classical invariant with the corresponding quantum operators, we have the quantum invariant operator such that We can show that this invariant obeys the Liouville-von Neumann equation which is of the form Because the quantum invariant operator given in Equation (22) is a somewhat complicated form, it may be necessary to simplify it forits use in the dynamical analysis of the system. For this purpose, we will use the unitary transformation method by introducing appropriate unitary operators.
At first, we consider the following unitary operator whereÛ 1 ðtÞ andÛ 2 ðtÞ are given bŷ Using this operator, we transform the invariantÔðtÞ aŝ Then, after a straightforward evaluation, we get We see that the coefficients m −1 i ofP 2 i terms in the original invariant operator have been eliminated through the above transformation. However,ÔðtÞ involves interacting 3 Journal of Nanomaterials termsX iX j yet. For further simplification of the invariant operator with the removal of the interacting terms, we consider the diagonalization of its matrix representation in the subsequent subsection.

Diagonalization of the Invariant Operator.
Removing of the interacting terms in Equation (28) may be very helpful for evaluating, for example, the eigenfunctions of the invariant operator. Such a removal can be fulfilled by diagonalizing the invariant operator after converting it into an equivalent matrix. We easily confirm that the matrix form of Equation (28) is given bŷ where δ ij = 1 for i = j while δ ij = 0 otherwise, k ij are ith row and jth column elements of the matrix k, and the two matrices are represented as whereas the involved parameters are given by It is known that not only a diagonal but a diagonalizable matrixes are square matrixes as well [30,31]. In general, this property is equivalent to the existence of a basis of eigenvectors and makes it possible to define in a diagonalizable endomorphism in a vector space. Therefore, to diagonalize the first matrix in Equation (30), it is necessary to seek square forms of its eigenvalues, Ω 2 i , and the corresponding eigenvectors V ! i . Hence the diagonalized representation of k can be represented in the form We will evaluate Ω 2 i together with the eigenvectors V ! i from now on. For this, we consider the eigenvalue equation where I is the identity matrix. Considering that the eigenvalues of the matrix k are values of α, we represent the secular equation as Then, the eigenvalues of the matrix k are obtained to be where From a straightforward evaluation, each eigenvector related to Ω 2 i is given by Journal of Nanomaterials Similar eigenvectors are introduced in Ref. [14] but for a Hamiltonian instead of the invariant. On the other hand, the matrix form of ℝ is written as a block of the column vectors Now, in terms of the new coordinates the diagonalized invariant operatorÔ takes the form Here, the full expressions ofx i are given bŷ p i are also represented in the same manners. Equation (48) is a sum of three simple harmonic oscillators with constant frequencies and unit masses. We can easily manage the diagonalized invariant due to its simplicity. For instance, it is possible to solve the eigenvalue equation of it without difficulty. This means that the diagonalized invariant can be usefully applied to analyzing various dynamical properties of the system. In the subsequent section, we will apply the diagonalized invariant in the investigation of quantum characteristics of the system.

Application in Quantum Mechanics
Until now, we have constructed an invariant operator for the three coupled nano-optomechanical oscillators and diago-nalized it. We can easily find the solutions of the eigenvalue equation of the diagonalized invariant operator. We will apply this, in this section, in the quantum mechanical problem of the system as an example. Exact wave functions of the system will be derived by taking advantage of the eigenfunctions of the invariant operator.

Eigenfunctions.
For quantum mechanical description of the nano-optomechanical system, it may be necessary to see the eigenfunctions of the invariant operator because they are closely related to the quantum wave functions. More clearly speaking, quantum wave functions for time-varying oscillatory systems are represented in terms of the eigenfunctions according to the invariant operator theory [15,16].
Equation (48) is the same as the Hamiltonian associated to the three independent oscillators. Hence, the eigenfunctions of the diagonalized invariant operator are simply the same as those of the Hamiltonian for the uncoupled 3D oscillator. From the inverse transformation of the well-known form of such eigenfunctions, it may be possible to obtain the eigenfunctions of the original invariant operator. Through a rigorous mathematical procedure, we will demonstrate it in this section. In addition, we will also show that the wave functions of the system can be obtained by using such eigenfunctions.
In order to obtain the eigenvalues of the invariant opera-torÔðtÞ, we first introduce the canonical annihilation and creation operatorsb i andb † i associated to the transformed invariant operator: Notice that these operators obey the usual boson 5 Journal of Nanomaterials canonical commutation rule, which is ½b i ,b † i = 1. Then, Equation (48) can be expressed in terms ofb i andb † i aŝ We now consider the inverse unitary transformation: whereâ i ðtÞ andâ † i ðtÞ are the time-dependent canonical annihilation and creation operators associated with the original invariant. Note that this transformation changesX i andP i to beX Thanks to this, the invariant operatorÔðtÞ, Equation (20), can be rewritten in the form whereas the full expressions ofâ i ðtÞ andâ † i ðtÞ are given bŷ witĥ Let us denote the eigenstates ofâ † iâi as jn i , ti. Then, the eigenvalue equations of the number operators are given bŷ where n i = 0, 1, 2, ⋯. We now easily confirm that the eigenvalues of Equation (58) are represented in terms of n i as λ n 1 ,n 2 ,n 3 The complete eigenstates ofÔ can be written as a product of three individual eigenstates that can be evaluated from the zero-point eigenstates: According to the general convention, we choose the amplitudes of jn 1 , n 2 , n 3 , tiin a way that the states obey the normalization condition, i.e., n 1 , n 2 , n 3 , t n 1 ′ , n 2 ′ , n 3 ′ , t D E = δ n 1 ,n 1 ′ δ n 2 ,n 2 ′ δ n 3 ,n 3 ′ : The eigenstates in Equation (71) are necessary in unfolding quantum theory of the system.

Wave Functions.
According to the Lewis-Riesenfeld theory, the wave functions jψ n 1 ,n 2 ,n 3 ðtÞi of the nanooptomechanical system are represented in terms of the eigenstates derived in the previous section: 6 Journal of Nanomaterials ψ n 1 ,n 2 ,n 3 t ð Þ E = e iζ n 1 ,n 2 ,n 3 t ð Þ n 1 , n 2 , n 3 , where ζ n 1 ,n 2 ,n 3 ðtÞ are phases. We can determine ζ n 1 ,n 2 ,n 3 ðtÞ from the Schrödinger equation, which is of the form A minor evaluation after inserting Equation (73) into the above equation leads to By solving this equation in a straightforward way, we have Finally, the wave functions in the configuration space are written in the form

Conclusion
The invariant operator of time-dependent three coupled nano-optomechanical oscillators was investigated. Starting from the formulation of the classical invariant, we have obtained the quantum invariant operator of the system. It was shown that the invariant operator obeys the Liouvillevon Neumann equation.
We also 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. As a preliminary step before the practical diagonalization of the invariant operator, we have carried out the unitary transformation of it; this procedure allowed us to transform the quantal invariant operator to an equal, but a simple one, which is represented by unit masses.
However, the transformed invariant operator still involve the coupling termsX iX j . Such terms were eliminated by the eventual diagonalization of the matrix representation of the invariant.
The diagonalized invariant is no longer a function of time and it can be represented simply in terms of the creation and annihilation operators ðb † i ,b i Þ (see Equation (54)). Hence, by virtue of its diagonalization, the invariant can be effectively used in analyzing diverse dynamical properties of the nanooptomechanical oscillatory systems. For instance, we can easily identify the eigenstates of the original invariant operator based on fundamental unitary relations. With the help of these eigenstates, the wave functions satisfying the Schrödinger equation have been obtained as shown in Equation (77). Apparently, this shows the powerfulness of the invariant operator when we treat the nano-optomechanical system quantum mechanically. If we think of the fact that quantum characteristics of a mechanical system can be usually examined starting from the interpretation of wave functions, the wave functions obtained here may open a way for clarifying quantum effects of the nano-optomechanical system from fundamental mathematical point of view. This is the main contribution of our research in this work.
In particular, entanglement dynamics between elements of the three or multicoupled nano-optomechnical oscillators can be investigated based on our research. The creation and control of entanglement in coupled quantum optomechanical oscillators are crucial in the realization of quantum information processing in quantum cryptography and quantum computing [32]. Because the technologies related to them 7 Journal of Nanomaterials utilize novel nonlocal quantum effects, clarification of fundamental quantum characteristics of the system is necessary. Notably, our quantum formalism in this context based on the dynamical invariant does not require any approximation and perturbation method even when the parameters vary adiabatically in time.
We know that the size of optomechanical and electromechanical systems becomes gradually smaller from day to day according to the rapid advance of nanotechnology. As the size of optomechanical devices reaches nanoscale, classical mechanics can no longer be applicable in describing mechanical properties of the systems. This is due to the prominence of quantum effects in the realm of nanoscale devices [33][34][35][36]. Hence, we should inevitably use quantum mechanics in order to analyze the dynamics of nano-optomechanical oscillatory systems. The feasibility of quantum description of nanooptomechanical systems based on invariant operator theory that we have treated here may provide an actual solution along this line.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.