Coherent states and localization in a quantized discrete NLS lattice

2016 ◽  
Vol 25 (04) ◽  
pp. 1650047 ◽  
Author(s):  
R. Martinez-Galicia ◽  
Panayotis Panayotaros
Keyword(s):  
Coherent States ◽  
Initial Conditions ◽  
Classical System ◽  
Classical Dynamics ◽  
Nonlinear Schrödinger

We study the evolution of a quantum discrete nonlinear Schrödinger (DNLS) system using as initial conditions coherent states corresponding to points in the vicinity of breather solutions of the classical system. We consider various examples of stable and unstable breathers and examine the distance between exactly evolved states and coherent states with parameters that evolve according to classical dynamics. Initial conditions near stable breathers and their vicinity are seen to lead to recurrences to small distances between the two evolving states. Similar recurrences are not observed for initial conditions near unstable breathers.

scholarly journals ASYMPTOTIC STABILITY OF NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL

2005 ◽  
Vol 17 (10) ◽  
pp. 1143-1207 ◽  
Author(s):  
ZHOU GANG ◽  
I. M. SIGAL
Keyword(s):  
Asymptotic Stability ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Initial Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Generalized Nonlinear Schrödinger Equation

We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.

scholarly journals QUANTUM DYNAMICS OF THE CUBIT SYSTEM IN EXTERNAL FIELDS

2021 ◽  
Vol 26 (4) ◽  
pp. 68-75
Author(s):  
A. V. Gorokhov ◽  
G. I. Eremenko
Keyword(s):  
Symmetry Group ◽  
Time Evolution ◽  
Coherent States ◽  
Quantum Dynamics ◽  
Dynamic Symmetry ◽  
Classical Dynamics ◽  
External Fields

A system of two dipole-dipole interacting two-level elements (qubits) in external fields is considered. It is shown that using the coherent states (CS) of the dynamic symmetry group of the SU(2)SU(2) system, the time evolution can be reduced to the "classical" dynamics of the complex parameters of the CS. The trajectories of the CS are constructed and the time dependences of the probability of finding qubits at the upper levels are calculated.

scholarly journals On the Analytical and Numerical Solutions of the One-Dimensional Nonlinear Schrodinger Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Neveen G. A. Farag ◽  
Ahmed H. Eltanboly ◽  
M. S. EL-Azab ◽  
S. S. A. Obayya
Keyword(s):  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Real Life ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Numerical Approaches

In this paper, four compelling numerical approaches, namely, the split-step Fourier transform (SSFT), Fourier pseudospectral method (FPSM), Crank-Nicolson method (CNM), and Hopscotch method (HSM), are exhaustively presented for solving the 1D nonlinear Schrodinger equation (NLSE). The significance of this equation is referred to its notable contribution in modeling wave propagation in a plethora of crucial real-life applications such as the fiber optics field. Although exact solutions can be obtained to solve this equation, these solutions are extremely insufficient because of their limitations to only a unique structure under some limited initial conditions. Therefore, seeking high-performance numerical techniques to manipulate this well-known equation is our fundamental purpose in this study. In this regard, extensive comparisons of the proposed numerical approaches, against the exact solution, are conducted to investigate the benefits of each of them along with their drawbacks, targeting a broad range of temporal and spatial values. Based on the obtained numerical simulations via MATLAB, we extrapolated that the SSFT invariably exhibits the topmost robust potentiality for solving this equation. However, the other suggested schemes are substantiated to be consistently accurate, but they might generate higher errors or even consume more processing time under certain conditions.

scholarly journals SELECTION OF THE GROUND STATE FOR NONLINEAR SCHRÖDINGER EQUATIONS

2004 ◽  
Vol 16 (08) ◽  
pp. 977-1071 ◽  
Author(s):  
A. SOFFER ◽  
M. I. WEINSTEIN
Keyword(s):  
Excited State ◽  
Ground State ◽  
Bound States ◽  
Initial Conditions ◽  
Bound State ◽  
Large Time Behavior ◽  
Multiple Time ◽  
Time Behavior ◽  
Small Initial Data ◽  
Nonlinear Schrödinger

We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.

SELF-TRAPPING ON A GENERALIZED NONLINEAR TETRAHEDRON

1999 ◽  
Vol 13 (08) ◽  
pp. 225-232 ◽  
Author(s):  
M. I. MOLINA
Keyword(s):  
Exact Solution ◽  
Evolution Equation ◽  
Initial Conditions ◽  
The Self ◽  
Critical Nonlinearity ◽  
Nonlinear Schrödinger ◽  
Equal Distance ◽  
Self Trapping ◽  
Dnls Equation

We analyze the dynamical self-trapping of an excitation propagating on a generalized n-sites tetrahedron, characterized by having every site at equal distance from each other. The evolution equation is given by the Discrete Nonlinear Schrödinger (DNLS) equation. For completely localized initial conditions, we find an exact solution for the critical nonlinearity strength (χ/V) c as a function of the number of sites n of the generalized tetrahedron. This critical nonlinearity, that marks the onset of the self-trapping transition, is always negative for n ≥ 3 and its magnitude increases monotonically with n, always remaining inside the sector delimited by (|χ|/V) = n and (|χ|/V) = 2n.

THE SU(2) SEMI QUANTUM SYSTEMS DYNAMICS AND THERMODYNAMICS

2010 ◽  
Vol 24 (25n26) ◽  
pp. 5037-5049
Author(s):  
C. M. SARRIS ◽  
A. N. PROTO
Keyword(s):  
Specific Heat ◽  
Quantum Chaos ◽  
Initial Conditions ◽  
Coupling Parameter ◽  
Generalized Uncertainty Principle ◽  
Classical System ◽  
Quantum Spin ◽  
Quantum Spin System ◽  
Present Contribution ◽  
Mean Values

The dynamical description of a semi quantum nonlinear systems whose classical limit is not chaotic is still an open question. These systems are characterized by mixing a classical system with a quantum-mechanical one. As some of them lead to an irregular dynamics, the name "semi quantum chaos" arises. In this contribution we study two different Hamiltonians through the Maximum Entropy Principle Approach (MEP). Taking advantage of the MEP formalism, it can be clearly established that the Hamiltonians belonging to the SU(2) Lie algebra have common properties and a common treatment can be developed for them. These Hamiltonians resemble a quantum spin system coupled to a classical cavity. In the present contribution, we show that all of them share the generalized uncertainty principle as an invariant of the motion and other invariants as well. Two different classical potentials V(q) have been studied. Their specific heat are evaluated in terms of the extensive (mean values) and the intensive (Lagrange multipliers) variables. The main result of the present contribution is to show that the specific heat of these systems can be fixed independently of the temperature by setting only the initial conditions on the extensive or intensive variables, as well as the value of the quantum-classical coupling parameter. It could be possible to infer that this result can be extended to generalized forms for the V(q) classical potential.

scholarly journals Semiclassical Aspects of Quantum Mechanics by Classical Fluctuations

1998 ◽  
Vol 12 (08) ◽  
pp. 291-299 ◽  
Author(s):  
Salvatore de Martino ◽  
Silvio de Siena ◽  
Fabrizio Illuminati
Keyword(s):  
Quantum Mechanics ◽  
Classical System ◽  
Quantum Corrections ◽  
Classical Dynamics ◽  
Unit Of Action

Building on a model recently proposed by F. Calogero, we postulate the existence of a universal Keplerian tremor for any stable classical system. Deriving the characteristic unit of action α for each classical interaction, we obtain in all cases α≅h, the Planck action constant, suggesting that quantum corrections to classical dynamics can be simulated through a fluctuative hypothesis of purely classical origin.

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