Fabrice Debbasch

Quantum walks in artificial electric and gravitational fields

The continuous limit of quantum walks (QWs) on the line is revisited through a new, 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. Possible experimental realizations are also addressed.

Relativistic hydrodynamics of semiclassical quantum fluids

Abstract The description in terms of hydrodynamical variables of a relativistic superfluid, modeled by a semiclassical wave equation, is given using a generalized Madelung transformation. The Galilean limit is shown (for both wavefunction and fluid variables) to be the well known Landau-Pitaevski model of superflows at T = 0 K. The special relativistic elementary classical acoustic and vortex excitations are explicited. A model for a relativistic self-gravitating superfluid is obtained by minimally coupling the wave equation to Einsteins gravity. The equations corresponding to a static star (using fluid variables) and an isotropic cosmology are derived.

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 (microcanonical) NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are thus demonstrated to exhibit spontaneous thermalization.

Generic inflationary and noninflationary behavior in toy-cosmology

Cosmological solutions of a toy homogeneous isotropic universe filled with a superfluid Bose condensate described by a complex scalar field (with relativistic barotropic fluid interpretation) are studied. The eigenvalues of the tangent map for the resulting Hamiltonian system are used to classify the phase space regions and to understand the typical toy-universe evolution. After a transient, inflation is obtained in the hyperbolic (real eigenvalues) region. This new independent eigenvalue-based inflationary criterion is shown to be compatible and complementary to the standard slow roll-over conditions. For the later evolution of the toy-universe, a family of adiabatic trajectories oscillating about a conventional cosmology filled with a relativistic fluid is obtained once the system falls into the elliptic (imaginary eigenvalue) regions. The corresponding thermodynamic functions are computed.

Global potentials for general relativistic flows of ideal charged fluids

It is shown that general relativistic nondissipative flows of a possibly charged one component fluid, submitted to the action of its own electromagnetic field, admit a representation in terms of global potentials. A corresponding variational principle is presented, from which the equations of motion for the fluid and the fields can be obtained.

Galilean and Relativistic Nonlinear Wave equations: an Hydro dynamical Tool?

The connexion of nonlinear wave equations with the dynamics of barotropic fluids by Madelung’s transformation is reviewed in the case of fluids with arbitrary equations of state. Numerical simulations of the Nonlinear Schrodinger Equation (NLSE) reproducing the instabilities of non rotating and rotating cylindrical jets are presented. It is shown that NLSE, a dynamical model of superflows, reproduces many flow features usually obtained in the context of Euler or Navier-Stokes equations.

Non-linear acoustics in Galilean and relativistic barotropic fluids

Abstract Non-linear waves described by the defocusing non-linear Schroedinger (NLS) equation admit a hydrodynamical representation in terms of Galilean potential flows and, using this correspondence, an autonomous equation for potential flows non-linear acoustic has been recently derived by Nore et al. However, this equation does not contain simple solutions of the original one such as (dark) solitons. The purpose of the present article is to characterize the reasons behind this failure and to present an original method to build separate equations describing all different types of acoustic solutions (but one). For reasons of generality, we work in a framework adapted to special relativistic hydrodynamics. All the results we derive have Galilean counterparts which are also discussed. In particular, we argue that there exist an infinity of different acoustic sectors for relativistic barotropic fluids, and we prove this result for fluids with a particularly simple equation of state. Solitons are naturally captured by our approach and a few explicit examples are worked out. Conserved quantities for the acoustic regime are also derived.