Energy localisation and transfer

This book provides an introduction to localised excitations in spatially discrete systems, from the experimental, numerical and mathematical points of view. Also known as discrete breathers, nonlinear lattice excitations and intrinsic localised modes, these are spatially localised time periodic motions in networks of dynamical units. Examples of such networks are molecular crystals, biomolecules, and arrays of Josephson superconducting junctions. The book also addresses the formation of discrete breathers and their potential role in energy transfer in such systems.

Preface	p. v
Chapter 1 Computational Studies of Discrete Breathers	p. 1
1 Introduction	p. 1
2 A bit on numerics of solving ODEs	p. 6
3 Observing and analyzing breathers in numerical runs	p. 10
3.1 Targeted initial conditions	p. 10
3.2 Breathers in transient processes	p. 17
3.3 Breathers in thermal equilibrium	p. 23
4 Obtaining breathers up to machine precision: Part I	p. 25
4.1 Method No.1 - designing a map	p. 26
4.2 Method No.2 - saddles on the rim with space-time separation	p. 30
4.3 Method No.3 - homoclinic orbits with time-space separation	p. 31
5 Obtaining breathers up to machine precision: Part II	p. 33
5.1 Method No.4 - Newton in phase space	p. 35
5.2 Method No.5 - steepest descent in phase space	p. 37
5.3 Symmetries	p. 38
6 Perturbing breathers	p. 39
6.1 Linear stability analysis	p. 40
6.2 Plane wave scattering	p. 43
7 Breathers in dissipative systems	p. 47
7.1 Obtaining dissipative breathers	p. 48
7.2 Perturbing dissipative breathers	p. 49
8 Computing quantum breathers	p. 51
8.1 The dimer	p. 54
8.2 The trimer	p. 58
8.3 Quantum roto-breathers	p. 65
9 Some applications instead of conclusions	p. 66
Acknowledgments	p. 68
References	p. 68
Chapter 2 Vibrational Spectroscopy and Quantum Localization	p. 73
1 Introduction	p. 74
1.1 Nonlinear dynamics and energy localization	p. 74
1.2 Nonlinear dynamics and vibrational spectroscopy	p. 76
2 Vibrational spectroscopy techniques	p. 78
2.1 Some definitions	p. 78
2.1.1 Spatial resolution	p. 80
2.1.2 Coherence length	p. 80
2.1.3 Energy localization	p. 81
2.1.4 The Franck-Condon principle	p. 81
2.2 Optical techniques	p. 82
2.3 Neutron scattering techniques	p. 84
2.3.1 Nuclear cross-sections	p. 85
2.3.2 Coherent versus incoherent scattering	p. 85
2.3.3 Contrast	p. 86
2.3.4 Penetration depth	p. 86
2.3.5 Wavelength	p. 86
2.3.6 Scattering function	p. 87
2.4 A (not so) simple example	p. 89
3 Molecular vibrations	p. 92
3.1 The harmonic approximation: Normal modes	p. 92
3.2 Anharmonicity	p. 93
3.3 Local modes	p. 93
3.3.1 Diatomic molecules	p. 94
3.3.2 Polyatomic molecules	p. 95
3.4 Local versus normal mode separability	p. 96
3.4.1 Zeroth-order descriptions of the nuclear Hamiltonian	p. 98
3.4.2 Breakdown of the zeroth-order descriptions	p. 99
3.5 The water molecule	p. 100
3.5.1 The normal mode model	p. 100
3.5.2 The local mode model	p. 101
3.5.3 Vibrational wave functions and spectrum	p. 102
3.5.4 Eigenstates and eigenfunctions	p. 103
3.6 The algebraic force-field Hamiltonian	p. 104
3.7 Other molecules	p. 107
3.8 Local modes and energy localization	p. 109
4 Crystals	p. 110
4.1 The harmonic approximation: Phonons	p. 111
4.1.1 The linear single-particle chain	p. 113
4.1.2 The linear di-atom chain	p. 113
4.2 Phonon-phonon interaction	p. 113
4.3 Phonon-electron interaction	p. 116
4.4 Local modes	p. 119
4.5 Nonlinear dynamics	p. 123
4.5.1 Quantum rotational dynamics for infinite chains of coupled rotors	p. 123
4.5.2 Strong vibrational coupling: Hydrogen bonding	p. 131
4.5.3 Davydov's model	p. 139
5 Conclusion	p. 141
5.1 Vibrational spectroscopy and nonlinear dynamics	p. 141
5.2 Optical vibrational spectroscopy and energy localization	p. 141
5.2.1 Molecules	p. 142
5.2.2 Crystals	p. 142
5.3 Inelastic neutron scattering spectroscopy of solitons	p. 142
5.4 Vibrational spectroscopy and dynamical models	p. 143
References	p. 143
Chapter 3 Slow Manifolds	p. 149
1 Introduction	p. 149
2 Normally Hyperbolic versus General Case	p. 152
3 Hamiltonian versus General Case	p. 154
4 Improving a slow manifold	p. 160
5 Symplectic slow manifolds	p. 162
6 The Methods of Collective Coordinates	p. 169
7 Velocity Splitting	p. 171
8 Poisson slow manifolds	p. 173
9 Slow manifolds with Internal Oscillation	p. 173
10 Internal oscillation: U(1)-symmetric Hamiltonians	p. 176
11 Internal oscillation: General Hamiltonians	p. 181
12 Bounds on time evolution	p. 185
13 Weak Damping	p. 187
Acknowledgements	p. 187
References	p. 188
Chapter 4 Localized Excitations in Josephson Arrays. Part I: Theory and Modeling	p. 193
1 Introduction	p. 193
2 The single Josephson junction	p. 194
2.1 Josephson effect	p. 194
2.2 Superconducting tunnel junctions	p. 194
2.3 Long Josephson junctions	p. 199
2.4 Quantum effects in Josephson junctions	p. 200
3 Modeling Josephson arrays	p. 201
3.1 Series arrays	p. 203
3.2 rf-SQUID	p. 204
3.3 dc-SQUID	p. 204
3.4 JJ parallel array	p. 205
3.5 JJ ladder array	p. 206
3.6 2D arrays	p. 207
4 Localized excitations in Josephson arrays: Vortices and kinks	p. 209
4.1 Vortices in 2D arrays	p. 209
4.1.1 Single vortex properties at zero temperature	p. 210
4.1.2 Array properties at non-zero temperatures	p. 211
4.2 2D arrays with small junctions	p. 211
4.3 Kinks in parallel arrays	p. 212
4.3.1 Fluxon ratchet potentials	p. 215
4.4 Charge solitons in 1D arrays	p. 217
5 Discrete breathers in Josephson arrays	p. 217
5.1 Oscillobreather in an ac biased parallel array	p. 219
5.2 Rotobreathers in Josephson arrays	p. 220
5.3 The ladder array	p. 220
5.4 Rotobreathers in a dc biased ladder	p. 221
5.4.1 Analysis of the breather solutions using a dc model	p. 225
5.4.2 Simulations	p. 226
5.4.3 Breather existence diagrams	p. 229
5.4.4 Different [lambda] regimes	p. 233
5.4.5 Breather-vortex collision in the Josephson ladder	p. 236
5.5 Single-plaquette arrays	p. 238
5.6 DBs in two-dimensional Josephson junction arrays	p. 238
Acknowledgments	p. 240
References	p. 241
Chapter 5 Localized Excitations in Josephson Arrays. Part II: Experiments	p. 247
1 Introduction	p. 247
2 Fabrication of Josephson arrays	p. 248
2.1 Materials	p. 249
2.1.1 Low-temperature superconducting technology	p. 249
2.1.2 High-temperature superconducting technology	p. 251
2.2 Layout	p. 252
2.3 Junction parameters	p. 254
3 Measurement techniques	p. 255
3.1 Generation of localized excitations	p. 256
3.2 Hot probe imaging techniques	p. 257
4 Experiments in the classical regime	p. 259
4.1 Fluxons in Josephson arrays	p. 259
4.1.1 Parallel 1-D arrays	p. 259
4.1.2 Ladders	p. 261
4.1.3 2-D arrays	p. 261
4.2 Rotobreathers in Josephson ladders	p. 262
4.3 Meandered states in 2-D Josephson arrays	p. 264
5 Experiments in the quantum regime	p. 265
5.1 Single Josephson junction	p. 265
5.2 Coupled Josephson junctions	p. 268
6 Conclusions and outlook	p. 269
Acknowledgments	p. 269
References	p. 270
Chapter 6 Protein Functional Dynamics: Computational Approaches	p. 273
1 Introduction	p. 273
2 Protein structure	p. 273
3 Energetics of protein stabilisation	p. 275
4 Protein folding	p. 276
4.1 On-lattice models	p. 277
4.2 Off-lattice models	p. 283
4.3 More detailed models	p. 285
5 Protein conformational changes	p. 285
5.1 Functional motions	p. 285
5.2 Collective motions	p. 286
5.3 Low-frequency normal modes	p. 289
5.3.1 Normal mode analysis	p. 289
5.3.2 The RTB approximation	p. 291
5.3.3 Comparison with crystallographic B-factors	p. 292
5.3.4 Comparison with conformational changes	p. 293
5.3.5 Simplified potentials	p. 296
6 Dissipation of energy in proteins	p. 297
7 Conclusion	p. 298
Acknowledgments	p. 299
References	p. 299
Chapter 7 Nonlinear Vibrational Spectroscopy: A Method to Study Vibrational Self-Trapping	p. 301
1 Introduction: The Story of Davidov's Soliton	p. 301
2 Nonlinear Spectroscopy of Vibrational Modes	p. 303
2.1 Harmonic and Anharmonic Potential Energy Surfaces	p. 303
2.2 Linear and Nonlinear Spectroscopy	p. 305
3 Proteins and Vibrational Excitons	p. 307
3.1 Theoretical Background	p. 307
3.2 Experimental Observation	p. 309
4 Hydrogen Bonds and Anharmonicity	p. 310
4.1 Theoretical Background	p. 310
4.2 Experimental Observation	p. 312
5 Vibrational Self-Trapping	p. 314
5.1 Theoretical Background	p. 314
5.2 Experimental Observation	p. 315
6 Conclusion and Outlook	p. 319
Acknowledgments	p. 320
Appendix Feynman Diagram Description of Linear and Nonlinear Spectroscopy	p. 321
References	p. 323
Chapter 8 Breathers in Biomolecules?	p. 325
1 Introduction	p. 325
2 Classical vibrations	p. 326
2.1 Local modes in small molecules	p. 326
2.2 Local modes in large molecules	p. 327
2.3 Local modes in crystals	p. 329
2.4 Localisation of vibrations and chemical reaction rates	p. 330
2.5 Fluctuational opening in DNA	p. 331
3 Quantum self-trapping	p. 333
4 Discussion	p. 337
Acknowledgments	p. 339
References	p. 339
Chapter 9 Statistical Physics of Localized Vibrations	p. 341
1 Introduction/Outlook	p. 341
2 Thermal DNA denaturation: A domain-wall driven transition?	p. 343
3 ILMs in DNA dynamics?	p. 345
4 Helix formation and melting in polypeptides	p. 348
4.1 Definitions, Notation	p. 348
4.2 Thermodynamics	p. 350
Acknowledgments	p. 351
References	p. 352
Chapter 10 Localization and Targeted Transfer of Atomic-Scale Nonlinear Excitations: Perspectives for Applications	p. 355
1 Introduction	p. 355
2 Discrete Breathers	p. 359
2.1 DBs in periodic lattices	p. 360
2.2 DBs in random systems	p. 371
3 Targeted energy transfer	p. 376
3.1 Nonlinear resonance	p. 378
3.2 Targeted energy transfer in a nonlinear dimer	p. 380
3.3 Targeted energy transfer through discrete breathers	p. 383
4 Ultrafast Electron Transfer	p. 387
4.1 Nonlinear dynamical model for ET	p. 390
4.2 ET in the Dimer	p. 394
4.3 Catalytic ET in a trimer	p. 395
4.4 The example of bacterial photosynthetic reaction center	p. 396
5 Conclusions and perspectives	p. 400
Acknowledgments	p. 402
References	p. 402
Index	p. 405

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