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.
Author(s): V. M. (Nitant) Kenkre
Series: Lecture Notes in Physics, 997
Publisher: Springer
Year: 2022
Language: English
Pages: 327
City: Cham
Preface
Outline of the Book
Acknowledgments
Contents
About the Author
1 The Discrete Nonlinear Schrödinger Equation and the Two-State System (Dimer)
1.1 Introduction
1.2 A Numerical Study Indicating the Appearance of a Self-trapping Transition
1.3 Density Matrix Equations for a System of Very Small Size
1.4 Considerations via Potentials
1.5 Arming Oneself with Elliptic Functions
1.5.1 Definition of the Elliptic Sine and Related Functions
1.5.2 Some Useful Properties of the Jacobian Elliptic Functions
1.5.3 Trigonometric and Hyperbolic Approximations
1.5.4 Shifting the Elliptic Modulus
1.5.5 Jacobi's Imaginary Transformation
1.5.6 Weierstrass and Other Elliptic Functions
1.6 Application of Elliptic Functions to a Familiar System
1.6.1 The Displacement of the Physical Pendulum
1.6.2 The Angular Velocity of the Pendulum
1.7 Application to Bacterial Populations in a Petri Dish
1.8 Chapter 1 in Summary
2 Dimer Solutions, Mobility Reduction, and Neutron Scattering
2.1 Self-trapping as the cn-dn Transition
2.1.1 Kinship to Two Linear Systems
2.1.2 Reduction of Mobility from the DNLSE
2.2 Neutron Scattering Lineshapes
2.2.1 Introduction to the Experiment and Basic Formalism
2.2.2 Calculation of the Lineshape
2.2.3 Transition Behavior and Motional Narrowing
2.3 Comparison of the Motional Narrowing to the Linear Damped Result
2.4 Solution for Arbitrary Initial Conditions
2.5 Stationary States and Stability Analysis
2.6 Chapter 2 in Summary
3 Initial Delocalization, Phase-Nonlinearity Interplay, and Fluorescence Depolarization
3.1 Real Initial Conditions
3.1.1 In-Phase Case: r0 = +1-p02, Expected Behavior
3.1.2 Out-of-Phase Case: r0 = -1-p02, a New Transition
3.1.3 Meaning of the Amplitude Transition
3.2 Complex Initial Conditions
3.3 Fluorescence Depolarization in Stick Dimers
3.3.1 The Observable and Relation to the Dimer Density Matrix
3.3.2 Observable in the Absence of the DNSLE
3.3.3 Self-Trapping Effects on Fluorescence Depolarization
3.4 Chapter 3 in Summary
4 What Polarons Owe to Their Harmonic Origins
4.1 A Graphical Understanding of How the DNLSE Could Arise
4.2 Rotational Coordinates and Nonlinear Dependence
4.3 Surprises in the Dynamics of the Rotational Polaron
4.4 Non-monotonicity and Stationary States
4.4.1 Potential Considerations
4.4.2 Stationary States of the Rotational Polaron
4.4.3 Some General Comments
4.5 Further Work on the Nature of the Transitions and Additional Examples
4.5.1 General Considerations
4.5.2 Illustrative Applications for and Beyond Rotational Polarons
4.5.2.1 Rotational Polarons
4.5.2.2 Logarithmically Hard and Soft Oscillators
4.6 Chapter 4 in Summary
5 Static Energy Mismatch in the Nonlinear Dimer: Nondegeneracy
5.1 Evolution of the Nondegenerate Nonlinear Dimer
5.1.1 Potential Shapes and Physical Arguments to Gain Insight into the Time Dependence
5.2 Specifics of the Weierstrass Calculation
5.3 Analytic Results in Terms of Jacobian Functions
5.3.1 Jacobian Results
5.3.2 Reduction to the Case of the Degenerate Dimer
5.4 Systematic Study of the Effect of Nondegeneracy
5.4.1 Difference in Positive and Negative Static Mismatch in Site Energy
5.4.2 Graphical Perspective of Potential Plots
5.5 The Blend of Static and Dynamic Mismatch
5.6 Behavior at the Critical Point
5.7 Stationary States
5.8 Alternative Method for the Analysis of the Nonlinear Dimer
5.9 Chapter 5 in Summary
6 Extended Systems with Global Interactions, and Nonlinear Trapping
6.1 Trimers and N-mers with any Pair of Sites in Equal Communication
6.2 Rescaling to Connect the N-mer to the Nonlinear Nondegenerate Dimer
6.3 Additional Results from Considerations of N-mers
6.4 Nonlinear Trapping from a Linear Lattice Antenna
6.4.1 Arbitrary Initial Conditions, Analytic Solution
6.4.2 Stationary States of the Reaction Center
6.4.3 Dynamics for Initial Zero Occupation of the Reaction Center
6.5 Further Directions of Research and Remarks
6.6 Chapter 6 in Summary
7 Slow Relaxation: The Nonadiabatic Nonlinear Dimer
7.1 Preliminary Considerations
7.2 Relaxation at Finite Rates
7.3 Numerical Explorations
7.3.1 Localized Initial Conditions
7.3.2 Delocalized Initial Conditions
7.4 Some Exact Calculations in the Non-adiabatic Regime But in the Absence of Damping
7.4.1 The Essence of the Technique and Solutions
7.4.2 Tiptoeing Around Regions with Chaos
7.5 Averaging Approximation
7.6 Chapter 7 in Summary
8 Thermal Effects: Phase-Space and Langevin Formulations
8.1 Introduction: Background on the Davydov Soliton Stability Against Thermal Fluctuations
8.2 Phase-Space Considerations and Partition Function Analysis
8.2.1 Choice of Observable, Basic Expression, and Primary Behavior
8.2.2 Low Temperature Behavior
8.2.3 Extension Beyond the Dimer
8.3 Langevin/Fokker-Planck Analysis of Brownian Motion
8.3.1 Kramer's Escape Time as Representative of Thermal Stability
8.3.2 The Ecumenical Equation and Its Unification Capabilities
8.3.3 Onset of Bifurcations and Limit Cycles
8.3.4 Linear Stability Analysis
8.4 Bursts and Limit Cycles of Time-Dependent Fluorescence Depolarization
8.5 Remarks About the Non-equilibrium Considerations of Thermal Effects
8.6 Chapter 8 in Summary
9 Microscopic Origin Issues About the DNLSE for Polarons
9.1 Preliminary Concepts
9.1.1 Dressing Transformations and the Memory Function
9.1.2 Success of the Memory Approach
9.1.3 Nature of the Memory Function and Hierarchy of Time Scales
9.2 Criticism of the Semiclassical Treatment/DNLSE, Numerical Confirmation, and Timescale Hierarchy
9.3 Additional Investigations into the Validity Question
9.3.1 A Linear Four-State Model
9.3.2 Extreme Limits of the Transformation: Bare and Fully Dressed
9.3.3 An Infinite Number of Semiclassical Approximations
9.4 Relations to Other Approximation Programs
9.4.1 Leggett et al.'s Noninteracting Blip Approximation and Its Equivalence to our Memory Method
9.4.2 Grigolini's Analysis, Its Importance and Risks of Its Misinterpretation
9.5 A Brief Return to Davydov Solitons
9.6 Chapter 9 in Summary
10 Bose-Einstein Condensate Tunneling: The Gross-Pitaevskii Equation
10.1 A Lookalike of the DNLSE in Bose Einstein Condensate (BEC) Dynamics
10.2 Transitions and Tunneling in Condensates Analyzed via DNLSE Techniques
10.3 Validity of the DNLSE in the Light of Quantum Dynamics
10.3.1 Formulation of the Problem
10.3.2 Initial Condition Specification and the Numeric Exercise
10.4 Recent Results in Quantum Oscillations in BEC Condensates
10.4.1 Non-resonance Effects in the Time Evolution of Condensates
10.4.2 How to Make the Self-trapping Transition Coincide with the Amplitude Transition
10.4.3 Movement in Parameter Space of Critical Points and the Critical Line
10.5 Chapter 10 in Summary
11 Miscellaneous Topics and Summary of the Book
11.1 Assorted Subjects and Directions
11.1.1 Application of Projection Techniques to the Nonlinear Dimer: Generalized Master Equations
11.1.2 External Fields Considered via Time-dependent System Parameters
11.1.3 The Semiclassical Approximation for Anharmonic Vibrations
11.1.4 Boson-Fermion Mixtures and Soliton Propagation
11.1.5 Nonlinear Impurity in an Extended Chain
11.1.6 Excimer Formation as a Nonlinear Problem: A Classical Treatment
11.2 Review of Topics Covered in the Book
11.3 Parting Words
11.4 Chapter 11 in Summary
References
Index
