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.

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.

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.

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

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 COHERENT STATES AND GEOMETRIC PHASES IN THE CALOGERO–SUTHERLAND MODEL

2002 ◽  
Vol 17 (02) ◽  
pp. 259-267 ◽  
Author(s):  
DAE-YUP SONG ◽  
JEONGHYEONG PARK
Keyword(s):  
Coherent State ◽  
Coherent States ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Periodic Systems ◽  
Classical Solutions ◽  
Geometric Phases ◽  
Sutherland Model ◽  
Inverse Square Potentials ◽  
Time Periodic

Exact coherent states in the Calogero–Sutherland models (of time-dependent parameters) which describe identical harmonic oscillators interacting through inverse-square potentials are constructed, in terms of the classical solutions of a harmonic oscillator. For quasi-periodic coherent states of the time-periodic systems, geometric phases are evaluated. For the AN-1 Calogero–Sutherland model, the phase is calculated for a general coherent state. The phases for other models are also considered.

scholarly journals The evolution and two-mode squeezed states of the time-dependent two coupled harmonic oscillators

2005 ◽  
Vol 54 (2) ◽  
pp. 522
Author(s):  
Li Jiang-Fan ◽  
Huang Chun-Jia ◽  
Jiang Zong-Fu ◽  
Huang Zu-Hong
Keyword(s):  
Harmonic Oscillators ◽  
Time Dependent ◽  
Squeezed States ◽  
Coupled Harmonic Oscillators
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