Quasi-one-dimensional harmonically trapped quantum droplets

We theoretically consider effectively one-dimensional quantum droplets in a symmetric Bose-Bose mixture confined in a parabolic trap. We systematically investigate ground and excited families of localized trapped modes which bifurcate from eigenstates of quantum harmonic oscillator. Families of nonlinear modes have nonmonotonous behavior of chemical potential on number of particles and feature bistability regions. Excited states are unstable close to the linear limit, but become stable as the number of particles increases. In the limit of large density, we derive a modified Thomas-Fermi distribution. Decrease of the trapping strength dynamically transforms the ground state solution to the solitonlike quantum droplet.Comment: 10 pages, 6 figures, to be submitte

ATOMTRONICS: QUANTUM TECHNOLOGY WITH COLD ATOMS IN RING SHAPED OPTICAL LATTICES

Ph.DDOCTOR OF PHILOSOPH

Exact spatiotemporal traveling and solitary wave solutions for the generalized nonlinear Schrödinger equation

Напредак у нелинеарној оптици умногоме зависи од наше способности да нађемо нова решења разних диференцијалних једначина које се природно јављају у системима где светлост интерагује са нелинеарном средином. Иако су рекреирање ових система кроз експеримент и компјутерска симулација система два најчешћа и плодотворна приступа, крајњи циљ остаје да се нађу егзактна решења ових система. Циљ ове тезе је да комбинује раније технике налажења егзактних решења диференцијалних једначина и примени их на нелинеарну Шредингерову диференцијалну једначину (НШДЈ). Конкретно, настао је недавно пробој у применама одређених техника експанзије у налажењу одређених егзактних решења НШДЈ. Упркос ограничењу у комбиновању решења због нелинеарности система и чињенице да не могу општа решења да се нађу, сама чињеница да можемо идентификовати нека егзактна решења је од великог значаја за област, посебно код евалуирања какве су појаве могуће у таквим системима. Ова теза ће се фокусирати на примену технике Ф-експанзије користећи се Јакобијевим елиптичним функцијама (ЈЕФ) да би се решиле разне форме НШДЈ са нелинеарношћу трећег степена. НШДЈ са нелинеарношћу трећег степена је од фундаменталне важности за област нелинеарне оптике јер описује путовање светлости кроз материјал са Керовом нелинеарношћу. Одређеним модификацијама технике Ф-експанзије можемо наћи егзактна решења за широку класу система. Системи које ја презентујем у тези имају одређен скуп заједничких особина. Све једначине имају једну лонгитудиналну промењиву, или просторну или временску, због параксијалне апроксимације, и до три трансферзалне димензије, такође или просротне или временске понаособ. Ако су све трансферзалне вариабле просторне vii онда суму њихових других извода множим са коефицијентом дифракције β, а ако је нека од варијабли темпорална, онда говорим о коефицијенту дифракције/дисперзије. Та два коефицијента (дифракција и дисперзија) могу да се нормализују у један до на знак. У случају аномалне дисперзије коефицијенти имају исти знак, а у случају нормалне дисперзије супротан знак. Осим ова два коефицијента редукована у један, имамо такође и коефицијент χ који одређује јачину нелинеарности трећег степена, и коефицијент γ који одређује добитак (за позитивно γ) или губитак сигнала у нашем систему...The progress of the field of non-linear optics greatly depends on our ability to find solutions of various differential equations that naturally occur in the systems where light interacts with nonlinear media. Though re-creating the systems through experiment and performing computer simulations are the two most common and fruitful approaches, the ultimate goal remains to find exact solutions of these systems. The goal of this Thesis is to combine the work done in the field of finding exact solutions to certain classes of non-linear differential Schrödinger equations (NLSE). Most notably, there has been a breakthrough as of late in applying various expansion techniques in finding certain exact solutions to various NLSE. Despite the limitations of combining said solutions due to the non-linear nature of the solutions and the fact that not all solutions can be found using these techniques, the very fact that we can identify certain exact solutions is of tremendous importance to the field, especially when it comes to evaluating the kinds of functions and behavior that are possible within such systems. This Thesis will focus primarily on applying the F-expansion technique using the Jacobi elliptic functions (JEFs) to solve various forms of the NLSE with the cubic nonlinearity. The NLSE with a cubic nonlinearity is one of fundamental importance in the field of nonlinear optics because it describes the travelling of a light wave through a medium with a Kerr-like nonlinearity. Through certain modification of the technique we can find exact solutions in a very large class of systems. The systems I present in this Thesis will share a certain set of common properties. All of the equations I will tackle have a single longitudinal variable, either temporal or spatial, due to the application of the paraxial wave approximation, and up to three transverse dimensions, again both temporal and spatial. If all the transverse variables are spatial I x assign to the sum of their second derivatives a diffraction coefficient β whereas if one of them is temporal, I speak of the diffraction/dispersion coefficient. The two coefficients can be normalized into one, up to their sign. In the case of anomalous dispersion, the two coefficients have the same sign. In the case of normal dispersion, the two coefficients have the opposite signs. Apart from these terms which are present in the ordinary wave equation of linear optics, we also have the third order nonlinearity whose strength is determined by a parameter χ and we also have the term γ which describes the gain of loss of the signal inside our system..

Non-perturbative effects in field theory and gravity

Nonperturbative effects are crucial to fully understand the dynamics of quantum field theories including important topics such as confinement or black hole evaporation. In this thesis we investigate two systems where nonperturbative effects are of paramount importance. In the first part we study the dynamics of non-abelian gauge theories, while in the second part we try to shed light on mysterious properties of black holes using a model proposed earlier by Dvali and Gomez.\\ Non-abelian gauge theories are the central element in the standard model of particle physics and many dynamical aspects remain elusive. $\mathcal{N}=1$ supersymmetric Yang-Mills theories with $SU(N_C)$ allows for domain walls with several curious properties. They are expected to have gauge fields with a Chern-Simons (CS) term living on their worldvolume, while in the 't Hooft limit of a large number of colors many of their properties seem reminiscent of string theoretic D-Branes. Similar domain walls were also conjectured to be present in non supersymmetric Yang Mills theories. In our work, we investigate this problem from several points of view. We construct a toy model of how to localize a gauge field with a CS term on a domain wall extending earlier work by Dvali and Shifman. We then derive the peculiar properties of CS terms in terms of effects of the underlying microscopic dynamics. Then we look at the actual theory of interest. Here the main novelty is the focus on the topological part of the Yang-Mills theory allowing us to make robust statements despite working in a strongly coupled theory. We construct the low energy effective action of both the non-supersymmetric as well as the supersymmetric Yang Mills theory, which due to the presence of a mass gap is a topological field theory. This topological field theory encodes the Aharanov-Bohm phases in the theory as well as phases due to intersection of flux tubes. In this topological field theory we see that the worldvolume theory of domain walls contains a level $N_C$ CS term. The presence of this term was already conjectured in ealier works based on string theoretic constructions. Here we give its first purely field theoretical construction. Within this construction we also illuminate differences between domain walls in the supersymmetric and non-supersymmetric case.\\ Lastly we try to relate the effects observed to similar effects in critical string theories and we also speculate on whether the behaviour of these domain walls is due to an analog of the fractional quantum hall effect.\\ In the second part of this thesis we investigate non-perturbative aspects of black hole physics. Here we consider a model for a low energy description of black holes due to Dvali and Gomez, where black holes are described in terms of a Bose-Einstein condensate (BEC) of weakly interacting gravitons near a quantum critical point. We focus on nonperturbative properties of a system of attractively self-interacting non-relativistic bosons, which was proposed as a toy model for graviton BECs by Dvali and Gomez. In this thesis we investigate this system mostly relying on a fully non-perturbative approach called exact diagonalization. We first investigate entanglement properties of the ground state of the system, showing that the ground state becomes strongly entangled as one approaches the quantum critical point. In order to make this notion precise we introduce the notion of fluctuation entanglement. We then compute it in a Bogoliubov analysis and extract it from the exact diagonlization procedure as well. We also consider the real time evolution of the system. Here we are interested in finding an analog of the conjectured fast scrambling property of black holes originally introduced by Hayden and Preskill. We only consider the weaker notion of quantum breaking and show that the toy model has a quantum break time consistent with the fast scrambling time scale conjectured in the black hole context. We then conclude by pointing out several possible extensions of these results

Nonlinear dynamics of exciton-polariton Bose-Einstein condensate

Exciton-polariton Bose-Einstein condensates (BECs) are newly emerged quantum systems that are capable of showing macroscopic quantum phenomena with intrinsic open-dissipative nature. The spatial distribution of the polariton density, without any external potential, can be controlled by the geometric shape of the pumping laser, enabling the investigation of polariton dynamics with topologically non-trivial configurations. Meanwhile, exciton-polaritons have spin degrees of freedom inherited from excitons and photons, making it a candidate for the realization of quantum logic gates. In this thesis, we will investigate theoretically the nonlinear dynamics of exciton-polariton BECs involving both polaritons' spatial degrees of freedom and spin degrees of freedom, and interactions between them. This thesis is organised as follows: In Chapter 1, we will present an overall review of exiton-polariton systems and important properties of polariton BECs and then introduce the dynamical equations with various interactions that will serve as the main theoretical tool for subsequent chapters. Several polariton pumping and trapping techniques appearing in later chapters will also be introduced. In Chapter 2, we will investigate the superfluidity properties of a single-component polariton condensate under an incoherent annular pumping configuration. By studying the stability properties of polariton persistent currents, we find that the persistent currents can exhibit dynamical instability and energetic-like instability according to different parameter region. A stability phase diagram will be given and its relation with the Landau's criterion will be discussed. In Chapter 3, we will investigate the spin dynamics of a two-component polariton condensate under a homogeneous pumping configuration. Owing to the Josephson coupling, there exist multiple steady state solutions that allow of controlled spin state switching. A desynchronized region where there exists no stable steady solution is found. In the desynchronized region, a desynchronized state beating periodically over time can exist, which will serve as a building block of spin waves presented in the next chapter. In Chapter 4, by combining results from the previous two chapters we will investigate generally the nonlinear dynamics of polariton condensates under an annular pumping configuration. The spin-orbit interaction provided by the Josephson coupling supports azimuthon states that have simultaneous modulations in both amplitude and phase. The azimuthon states, when viewed in a different polarization basis, form rotating spin waves that can be referred to as the optical ferris wheel. In Chapter 5, results from previous chapters will be extended to micocavities that support the anisotropic TE-TM splitting interaction. Rotating singularities (small-scale vortices) are found as a result. Their properties and experimental observation techniques will be discussed. Chapter 2-5 provide a theoretical framework for the nonlinear dynamics of polariton condensates. They rely mostly on optical trapping techniques and are ready to be tested in experiments. In Chapter 6, polaritons trapped by an engineered periodic mesa potential will be discussed. We will investigate the band structure of polaritons under the influence of the periodic potential together with discussions on the phase-modulated interference pattern which corresponds to the polariton Talbot patterns observed in experiments

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time

The multi-scale nature of the solar wind

The solar wind is a magnetized plasma and as such exhibits collective plasma behavior associated with its characteristic spatial and temporal scales. The characteristic length scales include the size of the heliosphere, the collisional mean free paths of all species, their inertial lengths, their gyration radii, and their Debye lengths. The characteristic timescales include the expansion time, the collision times, and the periods associated with gyration, waves, and oscillations. We review the past and present research into the multi-scale nature of the solar wind based on in-situ spacecraft measurements and plasma theory. We emphasize that couplings of processes across scales are important for the global dynamics and thermodynamics of the solar wind. We describe methods to measure in-situ properties of particles and fields. We then discuss the role of expansion effects, non-equilibrium distribution functions, collisions, waves, turbulence, and kinetic microinstabilities for the multi-scale plasma evolution.Comment: 155 pages, 24 figure

Long Time Dynamics of Resonant Systems

This thesis studies the long time dynamics of resonant systems in the weakly nonlinear regime. It is divided into two main parts. In the first one, we consider the resonant equation, which captures the energy transfer between normal modes of the system. Different tools to extract analytic information from the resonant equation are developed. After that, we apply them to a large number of resonant models. Some of them consist of a scalar field in different geometries as well as the Gross-Pitaevskii equation. In the second part of this thesis, asymptotically anti-de Sitter geometries subject to time-periodic boundary conditions are studied. The phenomenology allowed by these conditions is explored through the environment of time-periodic geometries. In particular, we construct their phase-space and delimit the regions of linear stability. We also present a protocol to dynamically construct time-periodic geometries