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kenkre v interplay quantum mechanics nonlinearity 2022

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Textbook in PDF format This book presents an in-depth study of the discrete nonlinear Schrödinger equation (DNLSE), with particular emphasis on spatially small systems that permit analytic solutions. In many quantum systems of contemporary interest, the DNLSE arises as a result of approximate descriptions despite the fundamental linearity of quantum mechanics. Such scenarios, exemplified by polaron physics and Bose-Einstein condensation, provide application areas for the theoretical tools developed in this text. The book begins with an introduction of the DNLSE illustrated with the dimer, development of fundamental analytic tools such as elliptic functions, and the resulting insights into experiment that they allow. Subsequently, the interplay of the initial quantum phase with nonlinearity is studied, leading to novel phenomena with observable implications in fields such as fluorescence depolarization of stick dimers, followed by analysis of more complex and/or larger systems. Specific examples analyzed in the book include the nondegenerate nonlinear dimer, nonlinear trapping, rotational polarons, and the nonadiabatic nonlinear dimer. Phenomena treated include strong carrier-phonon interactions and Bose-Einstein condensation. This book is aimed at researchers and advanced graduate students, with chapter summaries and problems to test the reader’s understanding, along with an extensive bibliography. The book will be essential reading for researchers in condensed matter and low-temperature atomic physics, as well as any scientist who wants fascinating insights into the role of nonlinearity in quantum physics. Outline of the Book About the Author The Discrete Nonlinear Schrödinger Equation and the Two-State System (Dimer) A Numerical Study Indicating the Appearance of a Self-trapping Transition Density Matrix Equations for a System of Very Small Size Considerations via Potentials Arming Oneself with Elliptic Functions Definition of the Elliptic Sine and Related Functions Some Useful Properties of the Jacobian Elliptic Functions Trigonometric and Hyperbolic Approximations Shifting the Elliptic Modulus Jacobi's Imaginary Transformation Weierstrass and Other Elliptic Functions Application of Elliptic Functions to a Familiar System The Displacement of the Physical Pendulum The Angular Velocity of the Pendulum Application to Bacterial Populations in a Petri Dish Chapter in Summary Dimer Solutions, Mobility Reduction, and Neutron Scattering Self-trapping as the cn-dn Transition Kinship to Two Linear Systems Reduction of Mobility from the DNLSE Neutron Scattering Lineshapes Introduction to the Experiment and Basic Formalism Calculation of the Lineshape Transition Behavior and Motional Narrowing Comparison of the Motional Narrowing to the Linear Damped Result Solution for Arbitrary Initial Conditions Stationary States and Stability Analysis Chapter in Summary Initial Delocalization, Phase-Nonlinearity Interplay, and Fluorescence Depolarization Real Initial Conditions In-Phase Case: r = +-p, Expected Behavior Out-of-Phase Case: r = --p, a New Transition Meaning of the Amplitude Transition Complex Initial Conditions Fluorescence Depolarization in Stick Dimers The Observable and Relation to the Dimer Density Matrix Observable in the Absence of the DNSLE Self-Trapping Effects on Fluorescence Depolarization Chapter in Summary What Polarons Owe to Their Harmonic Origins A Graphical Understanding of How the DNLSE Could Arise Rotational Coordinates and Nonlinear Dependence Surprises in the Dynamics of the Rotational Polaron Non-monotonicity and Stationary States Potential Considerations Stationary States of the Rotational Polaron Some General Comments Further Work on the Nature of the Transitions and Additional Examples General Considerations Illustrative Applications for and Beyond Rotational Polarons Rotational Polarons Logarithmically Hard and Soft Oscillators Chapter in Summary Static Energy Mismatch in the Nonlinear Dimer: Nondegeneracy Evolution of the Nondegenerate Nonlinear Dimer Potential Shapes and Physical Arguments to Gain Insight into the Time Dependence Specifics of the Weierstrass Calculation Analytic Results in Terms of Jacobian Functions Jacobian Results Reduction to the Case of the Degenerate Dimer Systematic Study of the Effect of Nondegeneracy Difference in Positive and Negative Static Mismatch in Site Energy Graphical Perspective of Potential Plots The Blend of Static and Dynamic Mismatch Behavior at the Critical Point Stationary States Alternative Method for the Analysis of the Nonlinear Dimer Chapter in Summary Extended Systems with Global Interactions, and Nonlinear Trapping Trimers and N-mers with any Pair of Sites in Equal Communication Rescaling to Connect the N-mer to the Nonlinear Nondegenerate Dimer Additional Results from Considerations of N-mers Nonlinear Trapping from a Linear Lattice Antenna Arbitrary Initial Conditions, Analytic Solution Stationary States of the Reaction Center Dynamics for Initial Zero Occupation of the Reaction Center Further Directions of Research and Remarks Chapter in Summary Slow Relaxation: The Nonadiabatic Nonlinear Dimer Preliminary Considerations Relaxation at Finite Rates Numerical Explorations Localized Initial Conditions Delocalized Initial Conditions Some Exact Calculations in the Non-adiabatic Regime But in the Absence of Damping The Essence of the Technique and Solutions Tiptoeing Around Regions with Chaos Averaging Approximation Chapter in Summary Thermal Effects: Phase-Space and Langevin Formulations Introduction: Background on the Davydov Soliton Stability Against Thermal Fluctuations Phase-Space Considerations and Partition Function Analysis Choice of Observable, Basic Expression, and Primary Behavior Low Temperature Behavior Extension Beyond the Dimer Langevin/Fokker-Planck Analysis of Brownian Motion Kramer's Escape Time as Representative of Thermal Stability The Ecumenical Equation and Its Unification Capabilities Onset of Bifurcations and Limit Cycles Linear Stability Analysis Bursts and Limit Cycles of Time-Dependent Fluorescence Depolarization Remarks About the Non-equilibrium Considerations of Thermal Effects Chapter in Summary Microscopic Origin Issues About the DNLSE for Polarons Preliminary Concepts Dressing Transformations and the Memory Function Success of the Memory Approach Nature of the Memory Function and Hierarchy of Time Scales Criticism of the Semiclassical Treatment/DNLSE, Numerical Confirmation, and Timescale Hierarchy Additional Investigations into the Validity Question A Linear Four-State Model Extreme Limits of the Transformation: Bare and Fully Dressed An Infinite Number of Semiclassical Approximations Relations to Other Approximation Programs Leggett et al's Noninteracting Blip Approximation and Its Equivalence to our Memory Method Grigolini's Analysis, Its Importance and Risks of Its Misinterpretation A Brief Return to Davydov Solitons Chapter in Summary Bose-Einstein Condensate Tunneling: The Gross-Pitaevskii Equation A Lookalike of the DNLSE in Bose Einstein Condensate (BEC) Dynamics Transitions and Tunneling in Condensates Analyzed via DNLSE Techniques Validity of the DNLSE in the Light of Quantum Dynamics Formulation of the Problem Initial Condition Specification and the Numeric Exercise Recent Results in Quantum Oscillations in BEC Condensates Non-resonance Effects in the Time Evolution of Condensates How to Make the Self-trapping Transition Coincide with the Amplitude Transition Movement in Parameter Space of Critical Points and the Critical Line Chapter in Summary Miscellaneous Topics and Summary of the Book Assorted Subjects and Directions Application of Projection Techniques to the Nonlinear Dimer: Generalized Master Equations External Fields Considered via Time-dependent System Parameters The Semiclassical Approximation for Anharmonic Vibrations Boson-Fermion Mixtures and Soliton Propagation Nonlinear Impurity in an Extended Chain Excimer Formation as a Nonlinear Problem: A Classical Treatment Review of Topics Covered in the Book Parting Words Chapter in Summary

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kenkre v interplay quantum mechanics nonlinearity 2022