scholarly journals Solution to the Time-Dependent Coupled Harmonic Oscillators Hamiltonian with Arbitrary Interactions

2019 ◽  
Vol 1 (1) ◽  
pp. 82-90 ◽  
Author(s):  
Alejandro R. Urzúa ◽  
Irán Ramos-Prieto ◽  
Manuel Fernández-Guasti ◽  
Héctor M. Moya-Cessa
Keyword(s):  
Coupled Oscillators ◽  
Quantum Physics ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Unitary Operators ◽  
Orthogonal Functions ◽  
Molecular Chemistry ◽  
Coupled Harmonic Oscillators ◽  
Generalized Orthogonal

We show that by using the quantum orthogonal functions invariant, we found a solution to coupled time-dependent harmonic oscillators where all the time-dependent frequencies are arbitrary. This system may be found in many applications such as nonlinear and quantum physics, biophysics, molecular chemistry, and cosmology. We solve the time-dependent coupled harmonic oscillators by transforming the Hamiltonian of the interaction using a set of unitary operators. In passing, we show that N time-dependent and coupled oscillators have a generalized orthogonal functions invariant from which we can write a Ermakov–Lewis invariant.

Time-dependent coupled harmonic oscillators: Classical and quantum solutions

2014 ◽  
Vol 23 (09) ◽  
pp. 1450048 ◽  
Author(s):  
D. X. Macedo ◽  
I. Guedes
Keyword(s):  
Canonical Transformation ◽  
Unitary Transformation ◽  
Coupling Parameter ◽  
Wave Functions ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Classical Solutions ◽  
Arbitrary System ◽  
Coupled Harmonic Oscillators ◽  
The Masses

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.

scholarly journals CORRELATIONS IN QUANTUM PHYSICS

2012 ◽  
Vol 27 (01n03) ◽  
pp. 1345017 ◽  
Author(s):  
ROSS DORNER ◽  
VLATKO VEDRAL
Keyword(s):  
Quantum Physics ◽  
Quantum Measurements ◽  
Harmonic Oscillators ◽  
Theoretic Analysis ◽  
Information Theoretic ◽  
Coupled Harmonic Oscillators ◽  
Quantum Mutual Information ◽  
Classical Correlations ◽  
The Difference ◽  
Thermodynamic Transformations

We provide a historical perspective of how the notion of correlations has evolved within quantum physics. We begin by reviewing Shannon's information theory and its first application in quantum physics, due to Everett, in explaining the information conveyed during a quantum measurement. This naturally leads us to Lindblad's information theoretic analysis of quantum measurements and his emphasis of the difference between the classical and quantum mutual information. After briefly summarizing the quantification of entanglement using these ideas, we arrive at the concept of quantum discord, which naturally captures the boundary between entanglement and classical correlations. Finally we discuss possible links between discord, which the generation of correlations in thermodynamic transformations of coupled harmonic oscillators.

scholarly journals Invariant-based inverse engineering of time-dependent, coupled harmonic oscillators

2020 ◽  
Vol 102 (6) ◽  
Author(s):  
A. Tobalina ◽  
E. Torrontegui ◽  
I. Lizuain ◽  
M. Palmero ◽  
J. G. Muga
Keyword(s):  
Harmonic Oscillators ◽  
Time Dependent ◽  
Coupled Harmonic Oscillators

Coupled harmonic oscillators and their application in the dynamics of entanglement and the nonadiabatic Berry phases

2021 ◽  
Author(s):  
A. Abidi ◽  
A. Trabelsi ◽  
S. Krichene
Keyword(s):  
Wave Function ◽  
Coupling Parameter ◽  
Angular Frequency ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Logarithmic Scale ◽  
Initial State ◽  
Coupled Harmonic Oscillators ◽  
Berry Phases ◽  
Time Dependance

In the dynamic description of physical systems, the two coupled harmonic oscillators time-dependent mass, angular frequency and coupling parameter are recognized as a good working example. We present in this work an analytical treatment with a numerical evaluation of the entanglement and the nonadiabatic Berry phases in the vacuum state. On the basis of an exact resolution of the wave function solution of the time-dependent Schr¨odinger’s equation (T DSE) using the Heisenberg picture approach, we derive the wave function of the two coupled harmonic oscillators. At the logarithmic scale, we derive the entanglement entropies and the temperature. We discuss the existence of the cyclical initial state (CIS) based on an instant Hamiltonian and we obtain the corresponding nonadiabatic Berry phases through a period T. Moreover, we extend the result to case of N coupled harmonic oscillators. We use the numerical calculation to follow the dynamic evolution of the entanglement in comparison to the time dependance of the nonadiabatic Berry phases and the time dependance of the temperature. For two coupled harmonic oscillators with time-independent mass and angular frequency, the nonadiabatic Berry phases present a very slight oscillations with the equivalent period as the period of the entanglement. A second model is composed of two coupled harmonic oscillators with angular frequency which change initially as well as lately. Here in, the entanglement and the temperature exhibit the same oscillatory behavior with exponential increase in temperature.

Time-dependent coupled harmonic oscillators: Classical and quantum solutions

2020 ◽  
Vol 29 (05) ◽  
pp. 2075001
Author(s):  
I. Ramos-Prieto ◽  
J. Récamier ◽  
H. M. Moya-Cessa
Keyword(s):  
Harmonic Oscillators ◽  
Time Dependent ◽  
Coupled Harmonic Oscillators

Macedo and Guedes have shown how to solve a system of coupled harmonic oscillators with time-dependent parameters [Int. J. Mod. Phys. 23 (2014) 1450048]. We show that the first transformation they did is not correct. We show how to solve the coupled harmonic oscillators for the cases they treat in their paper, namely, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

scholarly journals Dynamical Invariant and Exact Mechanical Analyses for the Caldirola–Kanai Model of Dissipative Three Coupled Oscillators

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 837
Author(s):  
Salim Medjber ◽  
Salah Menouar ◽  
Jeong Ryeol Choi
Keyword(s):  
Coupled Oscillators ◽  
Three Dimensional ◽  
Original System ◽  
Invariant Operator ◽  
Point Of View ◽  
Wave Functions ◽  
Harmonic Oscillators ◽  
Unitary Operators ◽  
Quantum Invariant ◽  
Full Wave

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.

scholarly journals Simple representation of the quantum entanglement of coupled harmonic oscillators in terms of the reflection coefficient

2021 ◽  
Vol 2086 (1) ◽  
pp. 012171
Author(s):  
Yu V Tsykareva ◽  
D N Makarov
Keyword(s):  
Reflection Coefficient ◽  
Harmonic Oscillator ◽  
Quantum Entanglement ◽  
Simple Form ◽  
Harmonic Oscillators ◽  
Quantum Metrology ◽  
Molecular Chemistry ◽  
Coupled Harmonic Oscillators ◽  
Modern Physics ◽  
Coupled Harmonic Oscillator

Abstract Quantum entanglement of coupled harmonic oscillators is frequently applied in quantum and non-linear physics, molecular chemistry and biophysics, which is why its study is of a great interest for modern physics. In this work a quantum entanglement of a coupled harmonic oscillator in a simple form was found. This simple form is presented as a single parameter – reflection coefficient R. All parameters of the studied system are included in the R coefficient. It is shown that the derivation of the expression can have applications in quantum optics, in particular in quantum metrology.

Sign in / Sign up

Export Citation Format

Share Document