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

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.

Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians

The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned.

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

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.

A Liouville’s Formula for Systems with Reflection

In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillators.