arXiv subjects

Fabrice Debbasch

Publications and source records attributed to Fabrice Debbasch.

32 records · Page 2Linked to original sources

arXiv2018

Action Principles for Quantum Automata and Lorentz Invariance of Discrete Time Quantum Walks

A discrete action principle for general quantum automata is proposed. This action principle is particularized to Discrete Time Quantum Walks (DTQWs) and then extended into an energy and momentum preserving, manifestly covariant formulation. Space-time coordinates are introduced as new variables of the action and their equations of motion enforce energy and momentum conservation. This guarantees that the proposed action can be used to build future, DTQW-based self-consistent models of spinors interacting with gauge fields. A discrete stress-energy tensor for the DTQW is also obtained by functional differentiation of the action with respect to the gradients of the coordinates viewed as functions of the discrete grid points. The manifest covariance of the formulation highlights the special role played by the grid reference frame in the DTQW dynamics. The main discussion is complemented by three appendices.

quant-ph↗

arXiv2018

Dirac quantum walks on triangular and honeycomb lattices

In this paper, we present a detailed study on discrete-time Dirac quantum walks (DQWs) on triangular and honeycomb lattices. At the continuous limit, these DQWs coincide with the Dirac equation. Their differences in the discrete regime are analyzed through the dispersion relations, with special emphasis on Zitterbewegung. An extension which couples these walks to arbitrary discrete electromagnetic field is also proposed and the resulting Bloch oscillations are discussed.

quant-ph↗

arXiv2017

Quantum walk hydrodynamics

A simple Discrete-Time Quantum Walk on the line is revisited and given an hydrodynamic interpretation through a novel relativistic generalization of the Madelung transform. Numerical results are presented which show that suitable initial conditions indeed produce hydrodynamical shocks. An analytical computation of the asymptotic quantum shock structure is presented. The non-relativistic limit is explored in the Supplementary Material (SM).

quant-ph↗

arXiv2017

Quantum walks and gravitational waves

A new family of discrete-time quantum walks (DTQWs) propagating on a regular $(1+2)$D spacetime lattice is introduced. The continuous limit of these DTQWs is shown to coincide with the dynamics of a Dirac fermion interacting with an arbitrary relativistic gravitational field. This family is used to model the influence of arbitrary linear gravitational waves (GWs) on DTQWs. Pure shear GWs are studied in detail. On large spatial scales, pure shear GWs do not modify the polarizations of the fermion eigen-modes, but rescale all energies by a common factor. On smaller scales typically comparable to two or three lattice steps, both polarizations and energies are modified in a non-trivial way. This effect is particularly salient on interference patterns between the fermion eigen-modes.

quant-ph↗

arXiv2016

Quantum walks and non-Abelian discrete gauge theory

A new family of discrete-time quantum walks (DTQWs) on the line with an exact discrete $U(N)$ gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual $U(N)$ gauge fields in $2D$ spacetime. A discrete generalization of the usual $U(N)$ curvature is also constructed. An alternate interpretation of these results in terms of superimposed $U(1)$ Maxwell fields and $SU(N)$ gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e. non-quantum) motions in classical $SU(2)$ fields. The results presented in this article constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.

quant-ph↗

arXiv2016

Quantum Walks and discrete Gauge Theories

A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A new family of $2D$ walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local $U(1)$ gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called ${\bf E} \times {\bf B}$ drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.

quant-ph↗

arXiv2015

Nonlinear Optical Galton Board: thermalization and continuous limit

The nonlinear optical Galton board (NLOGB), a quantum walk like (but nonlinear) discrete time quantum automaton, is shown to admit a complex evolution leading to long time thermalized states. The continuous limit of the Galton Board is derived and shown to be a nonlinear Dirac equation (NLDE). The (Galerkin truncated) NLDE evolution is shown to thermalize toward states qualitatively similar to those of the NLOGB. The NLDE conserved quantities are derived and used to construct a stochastic differential equation converging to grand canonical distributions that are shown to reproduce the (micro canonical) NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are thus demonstrated to exhibit spontaneous thermalization.

quant-ph↗

arXiv2015

Landau levels for discrete-time quantum walks in artificial magnetic fields

A new family of 2D discrete-time quantum walks (DTQWs) is presented and shown to coincide, in the continuous limit, with the Dirac dynamics of a spin 1/2 fermion coupled to a constant and uniform magnetic field. Landau levels are constructed, not only in the continuous limit, but also for the DTQWs i.e. for finite non-vanishing values of the time- and position-step, by a perturbative approach in the step. Numerical simulations support the above results and suggest that the magnetic interpretation is valid beyond the scope of the continuous limit. The possibility of quantum simulation of condensed-matter systems by DTQWs is also discussed.

quant-ph↗

arXiv2013

Quantum Walks in artificial electric and gravitational Fields

The continuous limit of quantum walks (QWs) on the line is revisited through a recently developed method. In all cases but one, the limit coincides with the dynamics of a Dirac fermion coupled to an artificial electric and/or relativistic gravitational field. All results are carefully discussed and illustrated by numerical simulations.

quant-ph↗

arXiv2013

Quantum walks as massless Dirac Fermions in curved Space-Time

A particular family of time- and space-dependent discrete-time quantum walks (QWs) is considered in one dimensional physical space. The continuous limit of these walks is defined through a new procedure and computed in full detail. In this limit, the walks coincide with the propagation of a massless Dirac fermion in an arbitrary gravitational field. A QW mimicking the radial propagation of a fermion outside and inside the event horizon of a Schwarzschild black hole is explicitly constructed and simulated numerically. Finally, the limiting procedure and the main result itself are carefully discussed.

quant-ph↗

arXiv2013

Propagation in quantum walks and relativistic diffusions

Propagation in quantum walks is revisited by showing that very general 1D discrete-time quantum walks with time- and space-dependent coefficients can be described, at the continuous limit, by Dirac fermions coupled to electromagnetic fields. Short-time propagation is also established for relativistic diffusions by presenting new numerical simulations of the Relativistic Ornstein-Uhlenbeck Process. A geometrical generalization of Fick's law is also obtained for this process. The results suggest that relativistic diffusions may be realistic models of decohering or random quantum walks. Links with general relativity and geometrical flows are also mentioned.

quant-ph↗

arXiv2011

Discrete-time quantum walks: continuous limit and symmetries

The continuous limit of one dimensional discrete-time quantum walks with time- and space-dependent coefficients is investigated. A given quantum walk does not generally admit a continuous limit but some families (1-jets) of quantum walks do. All families (1-jets) admitting a continuous limit are identified. The continuous limit is described by a Dirac-like equation or, alternately, a couple of Klein-Gordon equations. Variational principles leading to these equations are also discussed, together with local invariance properties.

math-ph↗

arXiv2007

Invariance of the relativistic one-particle distribution function

The one-particle distribution function is of importance both in non-relativistic and relativistic statistical physics. In the relativistic framework, Lorentz invariance is possibly its most fundamental property. The present article on the subject is a contrastive one: we review, discuss critically, and, when necessary, complete, the treatments found in the standard literature.

astro-ph↗

arXiv2006

Classical Gravitation as free Membrane Dynamics

The formulation of General Relativity in which the 4-dimensional space-time is embedded in a flat host space of higher dimension is reconsidered. New classes of embeddings (modeled after Nash's classical free embeddings) are introduced. They present the important advantage of being deformable and therefore physically realistic. Explicit examples of embeddings whose deformations DO describe gravitational waves around their respective backgrounds are given for several space-times, including the Schwarzschild black hole. New variational principles which give back Einstein's General Relativity are proposed. In this framework, the 4-D space-time is a membrane moving in a flat host space.

gr-qc↗