Kevin Rapedius

Resonance solutions of the nonlinear Schrödinger equation in an open double-well potential

The resonance states and the decay dynamics of the nonlinear Schrodinger (or Gross–Pitaevskii) equation are studied for a simple, but flexible, model system, the double delta-shell potential. This model allows analytical solutions and provides insight into the influence of the nonlinearity on the decay dynamics. The bifurcation scenario of the resonance states is discussed as well as their dynamical stability properties. A discrete approximation using a biorthogonal basis is suggested which allows an accurate description even for only two-basis states in terms of a nonlinear, non-Hermitian matrix problem.

Nonlinear resonant tunneling of Bose-Einstein condensates in tilted optical lattices

Insitut f¨ur theoretische Physik, Universit ¨at Heidelberg, 69120 Heidelberg, Germany(Dated: July 21, 2010)We study the tunneling decay of a Bose-Einstein condensate out of tilted optical lattices withinthe mean-field approximation. We introduce a novel method to calculate also excited resonanceeigenstates of the Gross-Pitaevskii equation, based on a grid relaxation procedure with complex ab-sorbing potentials. This algorithm works efficiently in a wide range of parameters where establishedmethods fail. It allows us to study the effects of the nonlinearity in detail in the regime of resonanttunneling, where the decay rate is enhanced by resonant coupling to excited unstable states.

The nonlinear Schröder equation for the delta-comb potential: quasi-classical chaos and bifurcations of periodic stationary solutions

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.

Multi-barrier resonant tunnelling for the one-dimensional nonlinear Schrödinger Equation

For the stationary one-dimensional nonlinear Schrodinger equation (or Gross–Pitaevskii equation), nonlinear resonant transmission through a finite number of equidistant identical barriers is studied using a (semi-)analytical approach. In addition to the occurrence of bistable transmission peaks known from nonlinear resonant transmission through a single quantum well (respectively a double barrier), complicated (looped) structures are observed in the transmission coefficient which can be identified as the result of symmetry breaking similar to the emergence of self-trapping states in double-well potentials. Furthermore, it is shown that these results are well reproduced by a nonlinear oscillator model based on a small number of resonance eigenfunctions of the corresponding linear system.

Exceptional points in bichromatic Wannier?Stark systems

The resonance spectrum of a tilted periodic quantum system for a bichromatic periodic potential is investigated. For such a bichromatic Wannier–Stark system, exceptional points, degeneracies of the spectrum, can be localized in parameter space by means of an efficient method for computing resonances. Berry phases and Petermann factors are analysed. Finally, the influence of a nonlinearity of the Gross–Pitaevskii type on the resonance crossing scenario is briefly discussed.

Calculating Resonance Positions and Widths Using the Siegert Approximation Method.

Here, we present complex resonance states (or Siegert states) that describe the tunnelling decay of a trapped quantum particle from an intuitive point of view that naturally leads to the easily applicable Siegert approximation method. This can be used for analytical and numerical calculations of complex resonances of both the linear and nonlinear Schrodinger equations. This approach thus complements other treatments of the subject that mostly focus on methods based on continuation in the complex plane or on semiclassical approximations.

Interaction-induced decoherence in non-Hermitian quantum walks of ultracold bosons

We study the decoherence caused by particle interaction for a conceptually simple model, a quantum walk on a bipartite one-dimensional lattice with decay from every second site. The corresponding noninteracting (linear) system has been shown to have a topological transition described by the average displacement before decay. Here, we use this topological quantity to distinguish coherent quantum dynamics from incoherent classical dynamics caused by a breaking of the translational symmetry. Furthermore, we analyze the behavior by means of a rate equation providing a quantitative description of the incoherent nonlinear dynamics.

A heuristic approach to BEC self-trapping in double wells beyond the mean field

We present a technically simple treatment of self-trapping of Bose–Einstein condensates in double well traps based on intuitive semiclassical approximations. Our analysis finally leads to a convenient closed-form approximation for the time-averaged population imbalance valid in both the mean-field case and in the case of finite particle numbers for short times.

Barrier transmission for the nonlinear Schrödinger equation: surprises of nonlinear transport

In this paper we report on a peculiar property of barrier transmission that systems governed by the nonlinear Schr¨ odinger equation share with the linear one: for unit transmission the potential can be divided at an arbitrary point into two sub-potentials, a left and a right one, which have exactly the same transmission. This is a rare case of an exact property of a nonlinear wavefunction which will be of interest, e.g., for studies of coherent transport of Bose‐Einstein condensates through mesoscopic waveguides.