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Table of contents
I Quantum walks: from synthetic gauge fields
1 HOMOGENEOUS QUANTUM WALKS
1.1 Quantum walks
1.1.1 Introduction
1.1.2 FromClassical to Quantumrandom walks
1.1.3 General Setup of a Discrete Time QuantumWalks
1.1.4 Qualitative description
1.1.5 Quantitative description
1.2 Connections between QuantumWalks and RelativisticWave Equations
1.2.1 Quantum walks and Feynman’s Checkerboard
1.2.2 Homogeneous DTQWs andWeyl equation
1.2.3 Publication: « MasslessDirac Equation fromFibonacciDiscrete-TimeQuantumWalk »
2 INHOMOGENEOUS QUANTUM WALKS AND CONTINUOUS LIMITS
2.1 Inhomogeneous QuantumWalks as synthetic gauge fields simulators
2.1.1 Quantum Simulation
2.1.2 What is a synthetic gauge field?
2.1.3 FromInhomogeneous QuantumWalks to synthetic gauge fields
2.2 A synthetic gravitational gauge field
2.2.1 Simulating the effects of a gravitational gauge field
2.2.2 Curved spacetime and chiral field theory: an introduction
2.2.3 A formal general setup
2.2.4 Dirac equation in curved spacetime in (1+1) dimensions
2.2.5 Publication: « Quantum walks asmasslessDirac fermions in curved spacetime »
2.3 A synthetic electric gauge field
2.3.1 Publication: « QuantumWalks in artificial electric and gravitational gauge fields. »
3 QUANTUM WALKS, DECOHERENCE AND RANDOM SYNTHETIC GAUGE FIELD
3.1 Quantum Decoherence: An introduction
3.2 QuantumWalks and decoherence
3.2.1 An overview
3.2.2 A qualitative picture
3.2.3 Projections cause spin and spatial decoherence
3.3 Publication : « Discrete-time QuantumWalks in random artificial Gauge Fields. »
II … to spontaneous equilibration.
4 THERMALIZATION AND QUANTUM WALKS
4.1 Absolute equilibrium in conservative systems
4.1.1 A general introduction
4.1.2 Thermalization and absolute equilibria in Galerkin truncated PDEs
4.1.3 Frommicrocanonical to grand canonical ensemble
4.2 Nonlinear QW-like models and thermalization
4.2.1 Thermalization in closed quantumsystems
4.2.2 QWs on N-cycle and limiting distribution
4.2.3 A Nonlinear QuantumWalk-like model on N-cycle
4.3 Publication: « Nonlinear Optical Galton Board: thermalization and continuous limit »
III Conclusions and Perspectives
5 CONCLUSIONS AND PERSPECTIVES
5.1 Conclusions
5.2 Perspectives
6 ACADEMIC PUBLISHING AND SCIENTIFIC COMMUNICATIONS
6.1 Academic publishing
6.1.1 Submitted
6.1.2 Published
6.1.3 Scientific Projects
6.2 Awards and Fellowship
6.3 Workshop
Appendices
Appendix A NUMERICAL METHODS
A.1 SpectralMethods
A.1.1 Fundamentals
A.2 Convergence in spectral methods
A.3 Approximate a PDE by spectralmethod
A.3.1 Galerkin method
A.3.2 Pseudo-spectral method
A.3.3 De-aliasing
A.3.4 Time-stepping
A.4 Discrete Fourier Transform
Appendix B TRUNCATED EULER-VOIGT-Æ EQUATION AND THERMALIZATION
B.1 Absolute equilibria in truncated Euler equation
B.2 Eddy-damped quasi-normalMarkovian theory (EDQNM)
B.3 Self-truncation
B.4 Publication A1: « Self-truncation and scaling in Euler-Voigt-Æ and related fluid models »
