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arXiv:2212.07643

Formulation of general dynamical invariants and their unitary relations for time-dependent three coupled quantum oscillators

Jeong Ryeol Choi

Published Dec 15, 2022• 0 citations
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Abstract

A general dynamical invariant operator for three coupled time-dependent oscillators is derived. Although the obtained invariant operator satisfies the Liouville-von Neumann equation, its mathematical formula is somewhat complicated due to arbitrariness of time variations of parameters. The parametric conditions required for formulating this invariant are definitely specified. By using the unitary transformation method, the invariant operator is transformed to the one that corresponds to three independent simple harmonic oscillators. Inverse transformation of the well-known quantum solutions associated with such a simplified invariant enables us to identify quantum solutions of the coupled original systems. These solutions are exact since we do not use approximations not only in formulating the invariant operator but in the unitary transformation as well. The invariant operator and its eigenfunctions provided here can be used to characterize quantum properties of the systems with various choices of the types of time-dependent parameters.

DOI
10.48550/arxiv.2212.07643

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A general dynamical invariant operator for three coupled time-dependent oscillators is derived. Although the obtained invariant operator satisfies the Liouville-von Neumann equation, its mathematical formula is somewhat complicated due to arbitrariness of time variations of parameters. The parametric conditions required for formulating this invariant are definitely specified. By using the unitary transformation method, the invariant operator is transformed to the one that corresponds to three independent simple harmonic oscillators. Inverse transformation of the well-known quantum solutions associated with such a simplified invariant enables us to identify quantum solutions of the coupled original systems. These solutions are exact since we do not use approximations not only in formulating the invariant operator but in the unitary transformation as well. The invariant operator and its eigenfunctions provided here can be used to characterize quantum properties of the systems with various choices of the types of time-dependent parameters.
An exact invariant operator of time-dependent coupled oscillators is derived using the Liouville-von Neumann equation. The unitary relation between this invariant and the invariant of two uncoupled simple harmonic oscillators is represented. If we consider the fact that quantum solutions of the simple harmonic oscillator is well-known, this unitary relation is very useful in clarifying quantum characteristics of the original systems, such as entanglement, probability densities, fluctuations of the canonical variables, and decoherence. We can identify such quantum characteristics by inversely transforming the mathematical representations of quantum quantities belonging to the simple harmonic oscillators. As a case in point, the eigenfunctions of the invariant operator in the original systems are found through inverse transformation of the well-known eigenfunctions associated with the simple harmonic 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 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.
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.
By using the Lewis–Riesenfeld invariants theory, we investigate the quantum dynamics of a two-dimensional (2D) time-dependent coupled oscillator. We introduce a unitary transformation and show the conditions under which the invariant operator is uncoupled to describe two simple harmonic oscillators with time-independent frequencies and unit masses. The decouplement allows us to easily obtain the corresponding eigenstates. The generalization to three-dimensional (3D) time-dependent coupled oscillator is briefly discussed where a diagonalized invariant, which is exactly the sum of three simple harmonic oscillators, is obtained under specific conditions on the parameters.
We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.