Analogue gravity in nonlocal fluids of light

Analogue gravity designates the study of curved spacetime in a laboratory environment and allows to test concepts of General Relativity. This analogy is established via a conformal identity between the flow of curved spacetime and inhomogeneous flows in hydrodynamics, which predicts that small waves on a fluid behave exactly as scalar fields in a curved spacetime metric. Atomic quantum fluids such as Bose-Einstein Condensates (BEC) are a widespread workbench for studying artificial black holes and many-body physics but face considerably large experimental challenges. In recent years, quantum fluids of light became a promising alternative at less technical expense, where the many-body dynamics in a laser beam are established via photon-photon interactions mediated through an optical nonlinearity. Whereas recent works considered strongly confined laser fields in microcavities, this work presents a photon fluid in a propagating geometry, i.e. a paraxially propagating laser beam in a bulk nonlinear medium. In this scenario, the propagating direction maps onto a time coordinate and the photon fluid is established in the transverse beam profile. The thermal nonlinearity is excited through heating of the absorbed laser power that introduces a nonlocal response of the medium and adds another level of complexity. It is experimentally shown that the dynamics of small amplitude excitations are governed by the Bogoliubov dispersion relation and allows to observe superfluidity at sufficiently large wavelengths. This is confirmed by the onset of persistent currents and the nucleation of quantized vortices in sub- and supercritical flows around an extended obstacle, which is a direct observation of superfluidity in a room-temperature system. The superfluid regime is a requirement for building analogue spacetime metrics and is thus of paramount importance. The spacetime of a rotating black and white whole was then created by shaping the topology of the spatial phase using diffractive phase masks. The experimental measurements of the inhomogeneous flows revealed, for the first time conclusive evidence of a (2+1) dimensional acoustic horizon and ergosphere. Such a system promises to study Penrose superradiance, where first experimental and numerical results for its observation are presented. Finally, nonlinear wave dynamics such as self-steepening and shock formation are studied where the dynamics can be interpreted in terms of a self-induced spacetime. Furthermore, the dynamics of a sea of incoherent waves is studied with respect to the long-range interactions provided by the nonlocality, where a novel transition from individual dispersive shock waves towards a collective giant shock wave is observed

Solitons in nonlinear lattices

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation (BEC). The solitons are considered in one, two, and three dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of 1D solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike. In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.Comment: 169 pages, 35 figures, a comprehensive survey of results on solitons in purely nonlinear and mixed lattices, to appear in Reviews of Modern Physic

Hydrodynamics

The phenomena related to the flow of fluids are generally complex, and difficult to quantify. New approaches - considering points of view still not explored - may introduce useful tools in the study of Hydrodynamics and the related transport phenomena. The details of the flows and the properties of the fluids must be considered on a very small scale perspective. Consequently, new concepts and tools are generated to better describe the fluids and their properties. This volume presents conclusions about advanced topics of calculated and observed flows. It contains eighteen chapters, organized in five sections: 1) Mathematical Models in Fluid Mechanics, 2) Biological Applications and Biohydrodynamics, 3) Detailed Experimental Analyses of Fluids and Flows, 4) Radiation-, Electro-, Magnetohydrodynamics, and Magnetorheology, 5) Special Topics on Simulations and Experimental Data. These chapters present new points of view about methods and tools used in Hydrodynamics

Dynamics of bright solitons in Bose–Einstein condensates: investigations of soliton behaviour in the vector Gross–Pitaevskii equation and applications to enhanced matter-wave interferometry

Bright solitons in a quasi-1D Bose–Einstein condensate can be used to enhance precision in matter-wave interferometry, due to their inherent robustness and support against dispersion. Such a soliton interferometer typically relies on a potential barrier used to split a single soliton into two smaller coherent solitons which can then be recombined on the same barrier. In this thesis we examine two extensions to this scheme. Firstly, we investigate a binary BEC system consisting of two bright solitons which are coupled through a mutual nonlinear interaction term. We derive a set of conditions under which the two components can be separated on a potential barrier and use numerical simulations to probe the regimes beyond which this mathematical treatment is applicable. We then use the numerical simulations to look at the effect of the nonlinear coupling on the dynamics of the binary solitons interacting with the barrier. We also look at the interference behaviour found by doubling the simulation time in either a ring trap or a harmonic trapping potential (to ensure recombination on the barrier); as well as the case where the solitons start spatially separated on either side of the barrier in order to find conditions under which the solitons will combine on the barrier. We find a good agreement between the analytical predictions and the results of simulations. Beyond the regions of parameter space where the predictions are expected to hold, we find complex transmission and interference behaviour as a result of nonlinear effects. The second part of this thesis consists of an examination of the prospect of using a subwavelength barrier scheme in a soliton interferometry experiment. This involves using two resonant coupling beams in a Λ-system with a spatially varying intensity. Under certain conditions, this can be used to form an effective potential barrier with a width which is not diffraction-limited. We look at suitable parameter regimes for such a barrier to split and recombine solitons in an interferometer and probe the effects of possible complications such as misalignment in the beams and different scattering lengths in the different states. We simulate the soliton interferometer using the full three component GPE as well as the single component analogue with the effective potential in order to characterise the soliton behaviour and the dependence of the interferometer sensitivity on the system parameters. We find a trend towards an idealised soliton interferometer with a decreasing value of the parameter, w, controlling the barrier characteristics. Also, we demonstrate an agreement between the relevant three component GPE and the analogous single component GPE, in the limit of strong coupling fields

Analysis and control of rogue waves in fibre lasers and in hydrodynamics: integrable turbulence framework

Understanding mechanisms underlying the formation of extreme events is the problem of primary importance in various domains of study including hydrodynamics, optics, forecasting natural disasters etc. In these domains, extreme events are known as RogueWaves (RWs). RWs are localised coherent structures of unusually large amplitude spontaneously emerging in nonlinear random wave fields, and as such, can have damaging effect on the environment (e.g. offshore engineering structures) or on the medium they propagate through (e.g. optical fibres). Within this PhD project several problems related to the emergence, control and manipulation of RWs in fibre optics and in hydrodynamics have been investigated. The particular emphasis is on the study of RWs emerging in the propagation of the so-called partially coherent waves described by the focusing nonlinear Schr¨odinger equation (fNLSE), the universal model for the propagation of modulationally unstable quasi-monochromatic wavepackets in a broad range of physical media. fNLSE belongs to the class of the completely integrable equations possessing deep mathematical structure and amenable to analytical methods such as Inverse Scattering Transform and Finite-Gap Integration. We use recent mathematical discoveries related to the semi classical, or small-dispersion, limit of fNLSE to investigate analytically, numerically and experimentally the formation of RWs within the framework of integrable turbulence—the emerging theory of random waves in integrable systems. The study of the RW formation in this project has also prompted the investigation of a closely related problem concerned with dynamics of soliton and breather gases as special types of integrable turbulence. The project’s findings fall in five categories: (i) the analytical description of the emergence of the so-called “heavy tails” in the probability distribution for the field intensity at the early stage of the development of integrable turbulence; (ii) the development and experimental realisation in a water tank of nonlinear spectral engineering, the IST-based method of control and manipulation of RWs; (iii) the development of the spectral theory of bidirectional soliton gases; (iv) numerical synthesis of breather gases and the verification of the recently developed spectral kinetic theory for such gases; (v) the investigation of the RWformation in the compression of broad optical pulses in the highly nonlinear propagation regimes, when the higher order effects such as self steepening, third order dispersion and Raman scattering need to be taken into account

Solitons, Breathers and Rogue Waves in Nonlinear Media

In this thesis, the solutions of the Nonlinear Schrödinger equation (NLSE) and its hierarchy are studied extensively. In nonlinear optics, as the duration of optical pulses get shorter, in highly nonlinear media, their dynamics become more complex, and, as a modelling equation, the basic NLSE fails to explain their behaviour. Using the NLSE and its hierarchy, this thesis explains the ultra-short pulse dynamics in highly nonlinear media. To pursue this purpose, the next higher-order equations beyond the basic NLSE are considered; namely, they are the third order Hirota equation and the fifth order quintic NLSE. Solitons, breathers and rogue wave solutions of these two equations have been derived explicitly. It is revealed that higher order terms offer additional features in the solutions, namely, ‘Soliton Superposition’, ‘Breather Superposition’ and ‘Breather-to-Soliton’ conversion. How robust are the rogue wave solutions against perturbations? To answer this question, two types of perturbative cases have been considered; one is odd-asymmetric and the other type is even-symmetric. For the odd-asymmetric perturbative case, combined Hirota and Sasa-Satsuma equations are considered, and for the latter case, fourth order dispersion and a quintic nonlinear term combined with the NLSE are considered. Indeed, this thesis shows that rogue waves survive these perturbations for specific ranges of parameter values. The integrable Ablowitz-Ladik (AL) equation is the discrete counterpart of the NLSE. If the lattice spacing parameter goes to zero, the discrete AL becomes the continuous NLSE. Similar rules apply to their solutions. A list of corresponding solutions of the discrete Ablowitz-Ladik and the NLSE has been derived. Using associate Legendre polynomial functions, sets of solutions have been derived for the coupled Manakov equations, for both focusing and defocusing cases. They mainly explain partially coherent soliton (PCS) dynamics in Kerr-like media. Additionally, corresponding approximate solutions for two coupled NLSE and AL equations have been derived. For the shallow water case, closed form breathers, rational and degenerate solutions of the modified Kortweg-de Vries equation are also presented