# Inhomogeneous QuantumWalks as synthetic gauge fields simulators

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## Fourier methods for the Hadamard walk

The analytical description of QWs dynamics has represented over the last few years an important field of investigation. Two methods have been extensively used: (i) the Fourier method and (ii) the combinatorial approach. In the former, the QW dynamics is studied in Fourier space in order to get a closed-formof the coin amplitude equations and, therefore, computing statistical properties of the walker. It was first introduced by Nayak and Vishwanath  and later by Kosik . In the combinatorial approach, we shall compute the amplitude for a particular position m component by summing up the amplitudes of all the paths which begin in the given initial condition and up in the same positionm. This approach is also called discrete path integral approach and was developed mainly byMachida and Konno . Here, we briefly recall the first method for the family of QWs obeying to Eqs. (1.12) and (1.13). Note that this could easily be extended to Eqs. (1.15) and (1.16). The closed-formof the coin amplitude is straightforward if we consider Eqs. (1.12) and (1.13) in Fourier space. Then let us look at ˆ™ j ,k = Pm™j ,meikm Fourier transformed state of ™j ,m. The finite difference equations read: √ˆ√ Lj+1,kˆ√Rj+1,k= eiÆ√ ei (ª°k) cosμ ei (≥°k) sinμ °e°i (≥°k) sinμ e°i (ª°k) cosμ !√ ˆ√ Lj ,k ˆ√ Rj ,k !.

### Homogeneous DTQWs andWeyl equation

Differently to what Feynman did, in this section we want to investigate the formal continuous limit of DTQWs with homogeneous and static quantumcoin operator BÆ,ª,≥,μ. In order to investigate the continuous limit we first introduce a time step ¢t and a space step ¢x. We then introduce for any quantity a appearing in Eq. (1.15) and (1.16), a function ˜ a defined on R+ £R such that the number aj ,m is the value taken by ˜ a at the spacetime point (t j = j¢t ,xm = m¢x). We then suppose ™(t j ,xm) to be at least C2 and that its characteristic length § to be much larger than the lattice parameter ¢x. Let us consider here the equations (1.15) and (1.16):√√L(t j +¢t ,xm)√R(t j +¢t ,xm)!= BÆ,ª,≥,μ√√L(t j ,xm °¢x) √R(t j ,xm +¢x) ! (1.35).
If the formal continuous limit exists, it will be obtained formally: (i) expanding each terms of the equations in ¢t and ¢x at fixed t j and xm and finally (ii) moving ¢t and ¢x to zero. Let us now introduce a time-scale ø 2 R and a length-scale Π 2 R and an ≤ 2 R so that ø≤ ø 1 and Π≤ ø 1. Define ¢t = ø≤ and ¢x = Π≤±, where ± is a strictly positive real number, which traces the fact that ¢t and ¢x may tend to zero differently. The Taylor expansion of each spacetime dependent function in (1.35), up to order O(≤2), reads: ™L,R(t j +¢t j ,xm) =™L,R(t j ,xm)+ø≤@t™L,R(t j ,xm)+O(≤2) (1.36).

#### From Inhomogeneous QuantumWalks to synthetic gauge fields

The inhomogeneous QWs are usually characterized by a spatial dependence of the coin. Here we extend this definition to a dependence on each spacetime point of the lattice. Over the past decade many theoretical and experimental researches have been addressed to this particular family of walks.
Time dependent quantum coins have been introduced by Ribeiro et al. . They defined a walk with several biased step operators applied aperiodically, and Banuls et al.  proposed a model with a time dependent coin as a control mechanism over a possible phase arising during the walk (as for example a consequence of an additional interaction). Albertini and D’Alessandro , extended the analysis to a d-dimensional lattice and to the cycle.
RecentlyMontero  unveiled some unexplored invariance in quantumwalks with timedependent coin. Space dependent quantum coins have been first introduced by Linden and Sharam  who investigated a spatial inhomogeneous quantumwalk, where the coin operator depended periodically on the position; Shikano and Katsura  numerically and analytically showed the energy spectrum of such walks and Konno et al. ,  have investigated the localization problems and the limit measures. Cedzich et al.  studied propagation and spectral properties of a QWin an electric field introduced as a space dependent phase in the quantum coin.
One of the main contribution of this thesis is to show that inhomogeneous QWs can be used to provide the gauge invariance with respect to certain gauge transformations, and therefore code for the corresponding gauge field, much in the same way as explained in the previous subsection for the continuum. In the continuumlimit, the QWgauge field tend toward the Dirac equation gauge fields; as we will show for both the electric and the gravitational field in subsection 2.2.5 and as well as their attached papers.

Simulating the effects of a gravitational gauge field

How to create synthetic gravity is a question often related to the reproduction of the gravitational force. This physical and engineering question can pertain both to zero-gravity environments and to the Earth. This task can be accomplished by several methods: for example by mechanical procedures , or by producing electric  and magnetic  forces. However, in this section, synthetic gravitational field has a quite differentmeaning and refers to mimicking the effect of gravity on the spacetime curvature of a specific particle motion. As we know, curved spacetime is not accessible in the labs, what we can look for is an analogue curved spacetime.
An interesting challenge is represented, for instance, by the simulation and study of QFTs in curved spacetime. Cold atoms have become a very useful tool for simulating gauge fields. Zohar et al.  and Boada et al.  were the first to prove that using ultra-cold atoms in optical lattices makes possible to mimic various types of statics and dynamical curved spacetime through simple manipulation of the optical setup.
Here we propose a new idea of simulators that reproduces the propagation of massless particle with two internal states on a curved (1+1) spacetime. In particular, the correction to the spinor propagation due to the gravitational deformation of the spacetime, is not real, but emerge as a geometric property of the QW. However, before we introduce the model, let us define several important elements of general relativity and curved field theory.

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Curved spacetime and chiral field theory: an introduction

In the pre-relativistic context the framework of scientific researchers was the flat space of Euclidean geometry, grounded on the five Euclid’s postulates. In 1827, in the Disquitiones generales circa superficies curvas, Gauss distinguished for the first time the inner or intrinsic properties of a surface from the outer or extrinsic ones. The former is measured by an observer living on the surface. The latter is derived from embedding the surface in a higher-dimensional space.
According to Gauss, the most fundamental inner property of a surface was the shortest path between two points. He realized, for instance, that a distance measured over a conic surface was the same as that one measured over a cylinder. These two surfaces, in fact, share the same intrinsic properties and in particular, their metric is flat. We cannot say the same for a sphere, as a sphere cannot bemapped onto a plane without distortions. However, since a cylinder or a cone are round in radial direction, we might be led to think they are curved surfaces.
This is due to the fact that we consider them as 2-dimensional surfaces in a 3-dimensional space, and we intuitively compare the curvature of the lines, which are on the surfaces, with straight lines in the flat 3-dimensional space.
The interplay between the inner and the outer properties of surfaces is one of the key point of the Einstein’s theory of general relativity. To bemore clear we need first to introduce the formalism and the main assumptions of the theory. Then we will be prepared to define properly the Dirac equation in curved spacetime.

Affine connection and Spin connection

To introduce the notion of connection, let us describe the Fig. 2.1. Consider two spacetime points xÆ ¥ A and xÆ + ±xÆ ¥ B and denote by XÆ(x) a vector at point A. If we perform a translation of ±XÆ(x) in spacetime, the transformed vector, at point B, (XÆ(x))0, is given by XÆ(x) + ±XÆ(x). In general, the pseudo-Riemann manifold possesses a non-vanishing curvature and, thus, we should consider a distortion effect in respect to the vector XÆ +±XÆ as in a flat spacetime. From the Fig. 2.1 we can geometrically observe that the curvature of the manifold is measured by the quantity [XÆ+±XÆ]°[XÆ+±XÆ] = ±XÆ°±XÆ. The affine connection represents the discrepancy between the variation ±XÆ and the vector ±XÆ. It represents a multiplicative factor between XØ itself and the finite displacement ±x∞. Let us denote this factor by °Æ Ø∞XØ.

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
6.1.1 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 »

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