Seminars 2016

Models of various physical media based on configurations with two parallel nonlinear waveguides (cores) amount to systems of linearly coupled nonlinear Schrödinger equations (NLSEs), or by a single NLSE with an effective double-well trapping potential [1]. Well-known examples are models of dual-core optical fibers or planar waveguides, with the Kerr self-focusing nonlinearity in each core. In such systems, the competition of the linear inter-core coupling and intra-core nonlinearity gives rise to symmetry-breaking bifurcations (SBBs), which destabilize obvious states that are symmetric with respect to the two cores, and replace them by asymmetric ones, when the total power of the optical wave exceeds a critical value. The same happens with symmetric temporal or spatial solitons (in the dual-core fibers and planar waveguides, respectively), which are destabilized by the respective SBB, when the soliton's energy exceeds a respective critical value. Similar SBBs are known in dual-core traps for matter waves in atomic Bose-Einstein condensates (BECs). The SBB may be of subcritical and supercritical types, the former type giving rise to a narrow region of bistability between the symmetric and asymmetric states, while in the latter case there is no bistability. The SBBs were studied in detail theoretically, and observed experimentally in some optical and BEC settings. The talk aims to present an overview of basic models, results, and physical realizations of the symmetry-breaking phenomena in nonlinear photonic and matter-wave media.

[1] ‍ B. A. Malomed, Symmetry breaking in laser cavities, Nature Photonics 9, 287 (2015).

Bistable systems occur throughout the natural sciences and when such systems are subjected to random noise, one observes probabilistic transitions between co-existing metastable states. Such behavior is found in chemical reaction kinetics, driven nonlinear mechanical systems, nonlinear electronic transport systems, climate variability models, and pulse propagation dynamics in neurons, to name but a few. In this talk, I will discuss recent work carried out in my group on noise-induced transitions in bistable systems that are far from thermal equilibrium. Experimental studies focus on switching transitions between distinct states of electrical current flow in quantum tunneling structures such as semiconductor superlattices and tunnel diodes.

Much of our understanding of the tropospheric dynamics relies on the concept of discrete internal modes. However, discrete modes are the signature of a finite system, while the atmosphere should be modeled as infinite and "is characterized by a single isolated eigenmode and a continuous spectrum" (Lindzen, JAS 2003). Is it then unphysical to use discrete modes? To resolve this issue we obtain an approximate radiation condition at the tropopause---this yields an EBC. We then use this EBC to compute a new set of vertical modes: the leaky rigid lid modes. These modes decay, with decay time-scales for the first few modes ranging from an hour to a week. This suggests that the rate of energy loss through upwards propagating waves may be an important factor in setting the time scale for some atmospheric phenomena. The modes are not orthogonal, but they are complete, with a simple way to project initial conditions onto them.

The EBC formulation requires an extension of the dispersive wave theory. There it is shown that sinusoidal waves carry energy with the group speed $c_g = d\omega/dk$, where both the frequency $\omega$ and wavenumber $k$ are real. However, when there are losses, complex $k$'s and $\omega$'s arise, and a more general theory is required. I will briefly comment on this theory, and on how the Laplace Transform can be used to implement generic EBC.

I will address three topics related to new geometrical/topological aspects in 2D materials. Firstly, I will discuss the magnetic response of general 2D materials and point out that the standard approach pioneered by Peierls and Landau was missing important geometrical contributions related to the Berry curvature. Secondly, I will discuss plasmonic excitations in a thin slab of topological insulators and point out that plasmons consist either of collective charge or spin excitations (spin-charge separation). Last, I will focus on a new, theoretically proposed two-dimensional carbon allotrope that entirely consists of carbon atoms forming pentagrams: penta-graphene.

[1] ‍ G. Gómez-Santos and T. Stauber, Phys. Rev. Lett. 106, 045504 (2011); A. Gutiérrez-Rubio, T. Stauber, G. Gómez-Santos, R. Asgari, and F. Guinea, Phys. Rev. B 93, 085133 (2016).
[2] ‍ T. Stauber, G. Gómez-Santos, and L. Brey, Phys. Rev. B 88, 205427 (2013); to be published.
[3] ‍ T. Stauber, J. I. Beltrán, and J. Schliemann, Sci. Rep. 6, 22672 (2016).

Feynman, in a little example in his path integral book, formulated a stochastic action, so that the most probable path from point q to point h in configuration space minimizes it. Hence, the physics at most probable path resembles geometric optics or geodesic motion. But with a twist: The path from q to h is not the path from h to q. This is the "geometric" interpretation of breaking detailed balance. It is actually equivalent to the conventional notion, in which detailed balance means that there are equilibrium solutions of the Fokker-Plank equation so the probability current vanishes identically. For linear stochastic dynamics, you can detect the breaking of detailed balance by direct measurements of trajectories: Project the trajectory into the plane of any two independent variables. If detailed balance is broken, the expected area swept out by the trajectory is nonzero. You can do this diagnosis without knowing details of the flow or the noise.

Quantum dots are constrictions in a two-dimensional electron gas by which the electronic states become discrete. Thus, quantum dots represent a solid-state realization of an atom which, in addition, can be contacted with an electron source and drain. When two quantum dots form a "molecule", the resulting current is affected by quantum phenomena such as delocalization and coherence which can be controlled by external fields. For instance, if an energy level of one dot is swept such that it crosses a level of the other dot, one observes Landau-Zener transitions. Repeated sweeps lead to the so-called Landau-Zener-Stückelberg-Majorana interference visible in a characteristic pattern as a function of the detuning and the amplitude of the sweeps. Comparison of experimental results with theoretical predictions provides the corresponding decoherence rate [1], which is a crucial parameter for applications in the context of quantum information processing.
When driving the system with two frequencies, the symmetry of the interference pattern depends on their commensurability. In particular, for commensurable frequencies, the symmetry depends on the relative phase between the two components, while in the incommensurable case, one finds the higher symmetry which otherwise is only found for certain phases. These predictions are confirmed by measurements [2].

[1] ‍ F.Forster et al., PRL 112, 116803 (2014).
[2] ‍ F.Forster et al., PRB 92, 245422 (2015).

A better understanding of the underlying dynamical mechanisms which determine the macroscopic laws of heat conduction may lead to potentially interesting applications based on the ability to control the heat flow. In this talk I will discuss thermal rectification and thermoelectric energy conversion from the perspective of nonequilibrium statistical mechanics and dynamical systems theory. This yields possible microscopic mechanisms for the design of thermal diodes and for the increase of thermoelectric efficiency.

Rippling patterns of myxobacteria appear in starving colonies before they aggregate to form fruiting bodies. These periodic traveling cell density waves arise from the coordination of individual cell reversals, resulting from an internal clock regulating them, and from contact signaling during bacterial collisions. One of the field of interest in this research work focuses on the numerical approximation with high order accuracy in space of the solutions of mathematical model proposed for myxobacteria rippling. We reconsider the papers of Igoshin and coauthors [Proc. Natl. Acad. Sci, USA 98, 14913 (2001) and Phys. Rev. E 70, 041911 (2004)] which describes the rippling phenomena of myxobacteria as a system of hyperbolic conservation law (when the diffusion is zero). Since the properties of the solution of systems of conservation laws develop jump discontinuities and spurious oscillations in time and space, it is important to use accurate numerical simulators in order to explain and predict the natural biological process. Previously, patterns for this model were obtained only by numerical methods of low order of accuracy and it was not possible to find their wavenumber analytically.

The outline of this seminar is the following. First, we revisit a mathematical model of rippling in myxobacteria due to Igoshin et al. Next, we make a description of the high accuracy numerical methods employed in this work and a detailed explanation of the algorithms that we have developed and used for the numerical simulations. We present various numerical tests for the mathematical model of myxobacteria rippling formation. We show that in the absence of white noise sources introduced in the original model, the hyperbolic system of two coupled partial differential equations can reproduce the reversal process on a regular basis. We have used high-precision numerical tools in order to maintain the amplitude of the travelling waves. We derive an evolution equation for the reversal point density that selects the pattern wavenumber in the weak signaling limit. We show the validity of the selection rule by solving numerically the model equations and describe other stable patterns in the strong signaling limit. The nonlocal mean-field coupling tends to decohere and confine patterns. Under appropriate circumstances, it can anihilate the patterns leaving a constant density state via a nonequilibrium phase transition reminiscent of destruction of synchronization in the Kuramoto model.

[1] ‍ O. A. Igoshin, A. Mogilner, R.D. Welch, D. Kaiser and G. Oster, Pattern formation and traveling waves in myxobacteria: Theory and modeling. Proc. Natl. Acad. Sci. USA 98, 14913-14918 (2001).