180 research outputs found
Effect of the lattice alignment on Bloch oscillations of a Bose-Einstein condensate in a square optical lattice
We consider a Bose-Einstein condensate of ultracold atoms loaded into a
square optical lattice and subject to a static force. For vanishing atom-atom
interactions the atoms perform periodic Bloch oscillations for arbitrary
direction of the force. We study the stability of these oscillations for
non-vanishing interactions, which is shown to depend on an alignment of the
force vector with respect to the lattice crystallographic axes. If the force is
aligned along any of the axes, the mean field approach can be used to identify
the stability conditions. On the contrary, for a misaligned force one has to
employ the microscopic approach, which predicts periodic modulation of Bloch
oscillations in the limit of a large forcing.Comment: 4 pages, 3 figure
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
An analytical study of resonant transport of Bose-Einstein condensates
We study the stationary nonlinear Schr\"odinger equation, or Gross-Pitaevskii
equation, for a one--dimensional finite square well potential. By neglecting
the mean--field interaction outside the potential well it is possible to
discuss the transport properties of the system analytically in terms of ingoing
and outgoing waves. Resonances and bound states are obtained analytically. The
transmitted flux shows a bistable behaviour. Novel crossing scenarios of
eigenstates similar to beak--to--beak structures are observed for a repulsive
mean-field interaction. It is proven that resonances transform to bound states
due to an attractive nonlinearity and vice versa for a repulsive nonlinearity,
and the critical nonlinearity for the transformation is calculated
analytically. The bound state wavefunctions of the system satisfy an
oscillation theorem as in the case of linear quantum mechanics. Furthermore,
the implications of the eigenstates on the dymamics of the system are
discussed.Comment: RevTeX4, 16 pages, 19 figure
Mean-field dynamics of a Bose-Einstein condensate in a time-dependent triple-well trap: Nonlinear eigenstates, Landau-Zener models and STIRAP
We investigate the dynamics of a Bose--Einstein condensate (BEC) in a
triple-well trap in a three-level approximation. The inter-atomic interactions
are taken into account in a mean-field approximation (Gross-Pitaevskii
equation), leading to a nonlinear three-level model. New eigenstates emerge due
to the nonlinearity, depending on the system parameters. Adiabaticity breaks
down if such a nonlinear eigenstate disappears when the parameters are varied.
The dynamical implications of this loss of adiabaticity are analyzed for two
important special cases: A three level Landau-Zener model and the STIRAP
scheme. We discuss the emergence of looped levels for an equal-slope
Landau-Zener model. The Zener tunneling probability does not tend to zero in
the adiabatic limit and shows pronounced oscillations as a function of the
velocity of the parameter variation. Furthermore we generalize the STIRAP
scheme for adiabatic coherent population transfer between atomic states to the
nonlinear case. It is shown that STIRAP breaks down if the nonlinearity exceeds
the detuning.Comment: RevTex4, 7 pages, 11 figures, content extended and title/abstract
change
On two-dimensional Bessel functions
The general properties of two-dimensional generalized Bessel functions are
discussed. Various asymptotic approximations are derived and applied to analyze
the basic structure of the two-dimensional Bessel functions as well as their
nodal lines.Comment: 25 pages, 17 figure
The nonlinear Schroedinger equation for the delta-comb potential: quasi-classical chaos and bifurcations of periodic stationary solutions
The nonlinear Schroedinger 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 Schroedinger 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 of 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.Comment: Enhanced and revised version, to appear in J. Nonlin. Math. Phy
Nonlinear Schroedinger equation with two symmetric point interactions in one dimension
We consider a time-dependent one-dimensional nonlinear Schroedinger equation
with a symmetric potential double well represented by two delta interactions.
Among our results we give an explicit formula for the integral kernel of the
unitary semigroup associated with the linear part of the Hamiltonian. Then we
establish the corresponding Strichartz-type estimate and we prove local
existence and uniqueness of the solution to the original nonlinear problem
Bound and resonance states of the nonlinear Schroedinger equation in simple model systems
The stationary nonlinear Schroedinger equation, or Gross-Pitaevskii equation,
is studied for the cases of a single delta potential and a delta-shell
potential. These model systems allow analytical solutions, and thus provide
useful insight into the features of stationary bound, scattering and resonance
states of the nonlinear Schroedinger equation. For the single delta potential,
the influence of the potential strength and the nonlinearity is studied as well
as the transition from bound to scattering states. Furthermore, the properties
of resonance states for a repulsive delta-shell potential are discussed.Comment: 19 pages, 10 figure
Open data base analysis of scaling and spatio-temporal properties of power grid frequencies
The electrical energy system has attracted much attention from an increasingly diverse research community. Many theoretical predictions have been made, from scaling laws of fluctuations to propagation velocities of disturbances. However, to validate any theory, empirical data from large-scale power systems are necessary but are rarely shared openly. Here, we analyse an open database of measurements of electric power grid frequencies across 17 locations in 12 synchronous areas on three continents. The power grid frequency is of particular interest, as it indicates the balance of supply and demand and carries information on deterministic, stochastic, and control influences. We perform a broad analysis of the recorded data, compare different synchronous areas and validate a previously conjectured scaling law. Furthermore, we show how fluctuations change from local independent oscillations to a homogeneous bulk behaviour. Overall, the presented open database and analyses constitute a step towards more shared, collaborative energy research
Tunnelling rates for the nonlinear Wannier-Stark problem
We present a method to numerically compute accurate tunnelling rates for a
Bose-Einstein condensate which is described by the nonlinear Gross-Pitaevskii
equation. Our method is based on a sophisticated real-time integration of the
complex-scaled Gross-Pitaevskii equation, and it is capable of finding the
stationary eigenvalues for the Wannier-Stark problem. We show that even weak
nonlinearities have significant effects in the vicinity of very sensitive
resonant tunnelling peaks, which occur in the rates as a function of the Stark
field amplitude. The mean-field interaction induces a broadening and a shift of
the peaks, and the latter is explained by analytic perturbation theory
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