66 research outputs found
Barrier transmission for the one-dimensional nonlinear Schr\"odinger equation: resonances and transmission profiles
The stationary nonlinear Schr\"odinger equation (or Gross-Pitaevskii
equation) for one-dimensional potential scattering is studied. The nonlinear
transmission function shows a distorted profile, which differs from the
Lorentzian one found in the linear case. This nonlinear profile function is
analyzed and related to Siegert type complex resonances. It is shown, that the
characteristic nonlinear profile function can be conveniently described in
terms of skeleton functions depending on a few instructive parameters. These
skeleton functions also determine the decay behavior of the underlying
resonance state. Furthermore we extend the Siegert method for calculating
resonances, which provides a convenient recipe for calculating nonlinear
resonances. Applications to a double Gaussian barrier and a square well
potential illustrate our analysis.Comment: 9 pages, 6 figures, 1 tabl
Calculating resonance positions and widths using the Siegert approximation method
Here we present complex resonance states (or Siegert states), that describe
the tunneling decay of a trapped quantum particle, from an intuitive point of
view which naturally leads to the easily applicable Siegert approximation
method that can be used for analytical and numerical calculations of complex
resonances of both the linear and nonlinear Schr\"odinger equation. Our
approach thus complements other treatments of the subject that mostly focus on
methods based on continuation in the complex plane or on semiclassical
approximations.Comment: 15 pages, 1 figure, contains MATLAB source code; new version with
additional illustration
Barrier transmission for the Nonlinear Schr\"odinger Equation: Surprises of nonlinear transport
In this communication we report on a peculiar property of barrier
transmission that systems governed by the nonlinear Schroedinger 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 wave function which will be of interest, e.g., for studies of
coherent transport of Bose-Einstein condensates through mesoscopic waveguideComment: 7 pages, 2 figure
Nonlinear resonant tunneling of Bose-Einstein condensates in tilted optical lattices
We study the tunneling decay of a Bose-Einstein condensate out of tilted
optical lattices within the mean-field approximation. We introduce a novel
method to calculate also excited resonance eigenstates of the Gross-Pitaevskii
equation, based on a grid relaxation procedure with complex absorbing
potentials. This algorithm works efficiently in a wide range of parameters
where established methods fail. It allows us to study the effects of the
nonlinearity in detail in the regime of resonant tunneling, where the decay
rate is enhanced by resonant coupling to excited unstable states.Comment: Revised and enlarged version, including 1 additional figur
Resonance solutions of the nonlinear Schr\"odinger equation in an open double-well potential
The resonance states and the decay dynamics of the nonlinear Schr\"odinger
(or Gross-Pitaevskii) equation are studied for a simple, however 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, nonhermitian matrix problem.Comment: 21 pages, 14 figure
Interaction-induced decoherence in non-Hermitian quantum walks of ultracold Bosons
We study the influence of particle interaction on a quantum walk on a
bipartite one-dimensional lattice with decay from every second site. The
corresponding non-interacting (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. We furthermore analyze the behavior by means of a rate equation
providing a quantitative description of the incoherent nonlinear dynamics.Comment: Revised and extended version, 5 pages, 5 figure
Nonlinear Schr\"odinger equation for a PT symmetric delta-functions double well
The time-independent nonlinear Schr\"odinger equation is solved for two
attractive delta-function shaped potential wells where an imaginary loss term
is added in one well, and a gain term of the same size but with opposite sign
in the other. We show that for vanishing nonlinearity the model captures all
the features known from studies of PT symmetric optical wave guides, e.g., the
coalescence of modes in an exceptional point at a critical value of the
loss/gain parameter, and the breaking of PT symmetry beyond. With the
nonlinearity present, the equation is a model for a Bose-Einstein condensate
with loss and gain in a double well potential. We find that the nonlinear
Hamiltonian picks as stationary eigenstates exactly such solutions which render
the nonlinear Hamiltonian itself PT symmetric, but observe coalescence and
bifurcation scenarios different from those known from linear PT symmetric
Hamiltonians.Comment: 16 pages, 9 figures, to be published in Journal of Physics
State-Dependent Network Connectivity Determines Gating in a K+ Channel
YesX-ray crystallography has provided tremendous insight into the different structural states of membrane proteins and, in particular, of ion channels. However, the molecular forces that determine the thermodynamic stability of a particular state are poorly understood. Here we analyze the different X-ray structures of an inwardly rectifying potassium channel (Kir1.1) in relation to functional data we obtained for over 190 mutants in Kir1.1. This mutagenic perturbation analysis uncovered an extensive, state-dependent network of physically interacting residues that stabilizes the pre-open and open states of the channel, but fragments upon channel closure. We demonstrate that this gating network is an important structural determinant of the thermodynamic stability of these different gating states and determines the impact of individual mutations on channel function. These results have important implications for our understanding of not only K+ channel gating but also the more general nature of conformational transitions that occur in other allosteric proteins.Wellcome Trus
Multi-barrier resonant tunneling for the one-dimensional nonlinear Schr\"odinger Equation
For the stationary one-dimensional nonlinear Schr\"odinger 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.Comment: 22 pages, 11 figure
- …