scholarly journals Quantum Approach to Damped Three Coupled Nano-Optomechanical Oscillators

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Jeong Ryeol Choi ◽  
Salah Menouar
Keyword(s):  
Transition Probability ◽  
Entanglement Entropy ◽  
Wave Functions ◽  
Harmonic Oscillators ◽  
Energy Evolution ◽  
Basic Tool ◽  
Optomechanical System ◽  
Quantum Approach ◽  
Diagonalization Method ◽  
Matrix Diagonalization

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.

scholarly journals Target space entanglement in quantum mechanics of fermions and matrices

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Sotaro Sugishita
Keyword(s):  
Mutual Information ◽  
Ground State ◽  
Entanglement Entropy ◽  
Algebraic Approach ◽  
Upper Bounds ◽  
Wave Functions ◽  
Target Space ◽  
Single Interval ◽  
Area Law ◽  
Formula Expression

Abstract We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general Rényi entropies are N log 2 for N particles or an N × N matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as $$ \frac{1}{3} $$ 1 3 log N in the large N model. We obtain an analytical $$ \mathcal{O}\left({N}^0\right) $$ O N 0 expression of the mutual information for two intervals in the large N expansion.

CASE STUDY OF THE CONVERGENCY OF NONLINEAR PERTURBATION SERIES: MORSE–FESHBACH NONLINEAR SERIES

1999 ◽  
Vol 14 (13) ◽  
pp. 2103-2115 ◽  
Author(s):  
BISWANATH RATH
Keyword(s):  
New York ◽  
Energy Levels ◽  
Perturbation Series ◽  
Nonlinear Perturbation ◽  
Theoretical Physics ◽  
The Matrix ◽  
Diagonalization Method ◽  
Matrix Diagonalization ◽  
Divergent Behavior

We study the divergent behavior of the Morse–Feshbach nonlinear perturbation series (MFNS) [P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953)] for producing convergent energy levels using the ground state of a quartic anharmonic oscillator (AHO) in the strong coupling limit. Numerical calculations have been done up to tenth order. Further comparison of the MFNS convergent result has been made with the matrix diagonalization method.

scholarly journals Solution of the Bloch Equations including Relaxation

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Eric R. Johnston
Keyword(s):  
Differential Equations ◽  
Bloch Equations ◽  
Diagonalization Method ◽  
Matrix Diagonalization

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.

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.

INVARIANT EIGEN-OPERATOR METHOD FOR SOLVING ENERGY GAP FOR SOME HAMILTONIANS IN MOLECULAR PHYSICS

2007 ◽  
Vol 21 (12) ◽  
pp. 1961-1969 ◽  
Author(s):  
HONG-YI FAN ◽  
TONG-TONG WANG
Keyword(s):  
Energy Gap ◽  
Molecular Model ◽  
Energy Levels ◽  
Linear Chain ◽  
Operator Method ◽  
Harmonic Oscillators ◽  
Molecular Physics ◽  
Coupled Harmonic Oscillators ◽  
Diagonalization Method

We show that the recently proposed invariant eigen-operator method is particularly applicable to solving the energy levels for some Hamiltonians in molecular physics. These are tri-atom molecules, the identical d-dimensional coupled harmonic oscillators and the dissipative linear-chain molecular model etc. The calculation is more direct and simpler than the usual diagonalization method for dynamic Hamiltonians.

Zur Theorie der Auger-Prozesse in III — V-Halbleitern

1969 ◽  
Vol 24 (11) ◽  
pp. 1752-1759
Author(s):  
Dieter Schöne
Keyword(s):  
Transition Probability ◽  
Wave Functions ◽  
Matrix Elements ◽  
Bloch Functions ◽  
Radiative Transitions ◽  
Excess Electrons ◽  
Approximate Wave ◽  
Auger Effect

Abstract This paper presents a calculation of the lifetimes of excess electrons in the III -V compounds InSb, InAs and GaSb, assuming the Auger effect between bands. Following the theory of Beattie and Landsberg matrix elements are calculated by using approximate wave functions instead of Bloch functions. The ninefold integration of the transition probability can be reduced to a four-fold one which then is numerically calculated with the aid of a computer. The results are compared with the lifetimes of radiative transitions. It is shown that the Auger processes are dominant in small gap semiconductors, but not in semiconductors with larger gaps (more than about 0.5 eV).

scholarly journals A quantum moat barrier, realized with a finite square well

2021 ◽  
Author(s):  
A. Ibrahim ◽  
F. Marsiglio
Keyword(s):  
Ground State ◽  
Wave Functions ◽  
Attractive Potential ◽  
Repulsive Potential ◽  
Well Model ◽  
Barrier Region ◽  
Square Well ◽  
Matrix Diagonalization ◽  
Double Well Potential

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.

scholarly journals Entanglement entropy in long-range harmonic oscillators

2012 ◽  
Vol 100 (6) ◽  
pp. 60011 ◽  
Author(s):  
M. Ghasemi Nezhadhaghighi ◽  
M. A. Rajabpour
Keyword(s):  
Long Range ◽  
Entanglement Entropy ◽  
Harmonic Oscillators

scholarly journals An accurate solution of the Poisson equation by the Legendre Tau method

1997 ◽  
Vol 20 (4) ◽  
pp. 713-718
Author(s):  
Muhammad I. Syam
Keyword(s):  
Poisson Equation ◽  
Test Problem ◽  
Accurate Solution ◽  
Tau Method ◽  
Chebyshev Collocation Method ◽  
The Poisson Equation ◽  
The Matrix ◽  
Diagonalization Method ◽  
Matrix Diagonalization ◽  
Comparison Of The Results

A new Tau method is presented for the two dimensional Poisson equation Comparison of the results for the test problemu(x,y)=sin(4πx)sin(4πy)with those computed by Haidvogel and Zang, using the matrix diagonalization method, and Dang-Vu and Delcarte, using the Chebyshev collocation method, indicates that our method would be more accurate.

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