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

Chaotic Quantum Decay in Driven Biased Optical Lattices

Quantum decay in an ac driven biased periodic potential modeling cold atoms in optical lattices is studied for a symmetry broken driving. For the case of fully chaotic classical dynamics the classical exponential decay is quantum mechanically suppressed for a driving frequency \omega in resonance with the Bloch frequency \omega_B, q\omega=r\omega_B with integers q and r. Asymptotically an algebraic decay ~t^{-\gamma} is observed. For r=1 the exponent \gamma agrees with $q$ as predicted by non-Hermitian random matrix theory for q decay channels. The time dependence of the survival probability can be well described by random matrix theory. The frequency dependence of the survival probability shows pronounced resonance peaks with sub-Fourier character.Comment: 7 pages, 5 figure

Beyond mean-field dynamics of small Bose-Hubbard systems based on the number-conserving phase space approach

The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for the illustrative case of two and three modes, as well as the generalization of the two-mode case to an open quantum system. The phase-space description based on generalized SU(M) coherent states yields a Liouvillian flow in the macroscopic limit, which can be efficiently simulated using Monte Carlo methods even for large systems. We show that this description clearly goes beyond the common mean-field limit. In particular it resolves well-known problems where the common mean-field approach fails, like the description of dynamical instabilities and chaotic dynamics. Moreover, it provides a valuable tool for a semi-classical approximation of many interesting quantities, which depend on higher moments of the quantum state and are therefore not accessible within the common approach. As a prominent example, we analyse the depletion and heating of the condensate. A comparison to methods ignoring the fixed particle number shows that in this case artificial number fluctuations lead to ambiguities and large deviations even for quite simple examples.Comment: Significantly enhanced and revised version (20 pages, 20 figures

Quantum transport and localization in biased periodic structures under bi- and polychromatic driving

We consider the dynamics of a quantum particle in a one-dimensional periodic potential (lattice) under the action of a static and time-periodic field. The analysis is based on a nearest-neighbor tight-binding model which allows a convenient closed form description of the transport properties in terms of generalized Bessel functions. The case of bichromatic driving is analyzed in detail and the intricate transport and localization phenomena depending on the communicability of the two excitation frequencies and the Bloch frequency are discussed. The case of polychromatic driving is also discussed, in particular for flipped static fields, i.e. rectangular pulses, which can support an almost dispersionless transport with a velocity independent of the field amplitude.Comment: 18 pages, 11 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

Quasiclassical analysis of Bloch oscillations in non-Hermitian tight-binding lattices

Many features of Bloch oscillations in one-dimensional quantum lattices with a static force can be described by quasiclassical considerations for example by means of the acceleration theorem, at least for Hermitian systems. Here the quasiclassical approach is extended to non-Hermitian lattices, which are of increasing interest. The analysis is based on a generalised non-Hermitian phase space dynamics developed recently. Applications to a single-band tight-binding system demonstrate that many features of the quantum dynamics can be understood from this classical description qualitatively and even quantitatively. Two non-Hermitian and $PT$-symmetric examples are studied, a Hatano-Nelson lattice with real coupling constants and a system with purely imaginary couplings, both for initially localised states in space or in momentum. It is shown that the time-evolution of the norm of the wave packet and the expectation values of position and momentum can be described in a classical picture.Comment: 20 pages, 8 figures, typos corrected, slightly extended, accepted for publication in New Journal of Physics in Focus Issue on Parity-Time Symmetry in Optics and Photonic

Wannier-Stark resonances in optical and semiconductor superlattices

In this work, we discuss the resonance states of a quantum particle in a periodic potential plus a static force. Originally this problem was formulated for a crystal electron subject to a static electric field and it is nowadays known as the Wannier-Stark problem. We describe a novel approach to the Wannier-Stark problem developed in recent years. This approach allows to compute the complex energy spectrum of a Wannier-Stark system as the poles of a rigorously constructed scattering matrix and solves the Wannier-Stark problem without any approximation. The suggested method is very efficient from the numerical point of view and has proven to be a powerful analytic tool for Wannier-Stark resonances appearing in different physical systems such as optical lattices or semiconductor superlattices.Comment: 94 pages, 41 figures, typos corrected, references adde