Quantum Approach to Damped Three Coupled Nano-Optomechanical Oscillators

We investigate quantum features of three coupled dissipative nano-optomechanical oscillators. The Hamiltonian of the system is somewhat complicated due not only to the coupling of the optomechanical oscillators but to the dissipation in the system as well. In order to simplify the problem, a spatial unitary transformation approach and a matrix-diagonalization method are used. From such procedures, the Hamiltonian is eventually diagonalized. In other words, the complicated original Hamiltonian is transformed to a simple one which is associated to three independent simple harmonic oscillators. By utilizing such a simplification of the Hamiltonian, complete solutions (wave functions) of the Schrödinger equation for the optomechanical system are obtained. We confirm that the probability density converges to the origin of the coordinate in a symmetric manner as the optomechanical energy dissipates. The wave functions that we have derived can be used as a basic tool for evaluating diverse quantum consequences of the system, such as quadrature fluctuations, entanglement entropy, energy evolution, transition probability, and the Wigner function.

A quantum moat barrier, realized with a finite square well

The notion of a double well potential typically involves two regions of space separated by a repulsive potential barrier. The ground state is a wave function that is suppressed in the barrier region and localized in the two surrounding regions. We illustrate that an attractive potential well (a quantum moat) with a finite non-zero width also acts as a barrier, using a simple square well model. We also show how the pseudopotential method both explains the role of the well as a barrier, and greatly improves the efficiency of constructing wave functions for this system using matrix diagonalization. With this simplified model we provide an introduction to the ideas typically used to simplify calculations in solids, where in place of the double well potential, multiple potentials occur in a periodic array.

Solution of the Bloch Equations including Relaxation

The magnetization differential equations of Bloch are integrated using a matrix diagonalization method. The solution describes several limiting cases and leads to compact expressions of wide validity for a spin ensemble initially at equilibrium.

Solvability of integral boundary value problems at resonance in $R^{n}$

Under a resonance condition involving integral boundary value problems for a second-order nonlinear differential equation in $\mathbb{R}^{n}$ R n , we show its solvability by using the coincidence degree theory of Mawhin and the theory of matrix diagonalization in linear algebra.