arXiv subjects

Jörg Main

Publications and source records attributed to Jörg Main.

At least 109 records · Page 6Linked to original sources

arXiv2011

Variational calculations for anisotropic solitons in dipolar Bose-Einstein condensates

We present variational calculations using a Gaussian trial function to calculate the ground state of the Gross-Pitaevskii equation and to describe the dynamics of the quasi-two-dimensional solitons in dipolar Bose-Einstein condensates. Furthermore we extend the ansatz to a linear superposition of Gaussians improving the results for the ground state to exact agreement with numerical grid calculations using imaginary time and split-operator method. We are able to give boundaries for the scattering length at which stable solitons may be observed in an experiment. By dynamical calculations with coupled Gaussians we are able to describe the rather complex behavior of the thermally excited solitons. The discovery of dynamically stabilized solitons indicates the existence of such BECs at experimentally accessible temperatures.

cond-mat.quant-gas↗

arXiv2010

Semiclassical quantization of the diamagnetic hydrogen atom with near action-degenerate periodic-orbit bunches

The existence of periodic orbit bunches is proven for the diamagnetic Kepler problem. Members of each bunch are reconnected differently at self-encounters in phase space but have nearly equal classical action and stability parameters. Orbits can be grouped already on the level of the symbolic dynamics by application of appropriate reconnection rules to the symbolic code in the ternary alphabet. The periodic orbit bunches can significantly improve the efficiency of semiclassical quantization methods for classically chaotic systems, which suffer from the exponential proliferation of orbits. For the diamagnetic hydrogen atom the use of one or few representatives of a periodic orbit bunch in Gutzwiller's trace formula allows for the computation of semiclassical spectra with a classical data set reduced by up to a factor of 20.

nlin.CD↗

arXiv2010

Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. I. General concept

The variational method of coupled Gaussian functions is applied to Bose-Einstein condensates with long-range interactions. The time-dependence of the condensate is described by dynamical equations for the variational parameters. We present the method and analytically derive the dynamical equations from the time-dependent Gross-Pitaevskii equation. The stability of the solutions is investigated using methods of nonlinear dynamics. The concept presented in this paper will be applied to Bose-Einstein condensates with monopolar 1/r and dipolar 1/r^3 interaction in the subsequent paper [S. Rau et al., Phys. Rev. A, submitted], where we will present a wealth of new phenomena obtained by using the ansatz with coupled Gaussian functions.

cond-mat.quant-gas↗

arXiv2010

Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. II. Applications

Bose-Einstein condensates with an attractive 1/r interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A, submitted]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both, stable and unstable states of the Gross-Pitaevskii equation, and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov-de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.

cond-mat.quant-gas↗

arXiv2010

Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems

The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations.

nlin.CD↗

arXiv2010

Variational calculations on multilayer stacks of dipolar Bose-Einstein condensates

We investigate a multilayer stack of dipolar Bose-Einstein condensates in terms of a simple Gaussian variational ansatz and demonstrate that this arrangement is characterized by the existence several stationary states. Using a Hamiltonian picture we show that in an excited stack there is a coupled motion of the individual condensates by which they exchange energy. We find that for high excitations the interaction between the single condensates can induce the collapse of one of them. We furthermore demonstrate that one collapse in the stack can force other collapses, too. We discuss the possibility of experimentally observing the coupled motion and the relevance of the variational results found to full numerical investigations.

cond-mat.quant-gas↗

arXiv2010

Resonance wave functions located at the Stark saddle point

We calculate quantum mechanically exact wave functions of resonances in spectra of the hydrogen atom in crossed external fields and prove the existence of long-lived decaying quantum states localized at the Stark saddle point. A spectrum of ground and excited states reproducing the nodal patterns expected from simple quadratic and cubic expansions of the potential in the vicinity of the saddle point can be identified. The results demonstrate the presence of resonances in the vicinity of the saddle predicted by simple approximations.

physics.atom-ph↗

arXiv2010

Pitchfork bifurcations in blood-cell shaped dipolar Bose-Einstein condensates

We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation of condensates with electromagnetically induced attractive 1/r interaction, or with dipole-dipole interaction. Moreover, Gaussian wave packets are superior in that they are capable of producing both stable and unstable stationary solutions, and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. We apply the method to clarify the theoretical nature of the collapse mechanism of blood-cell shaped dipolar condensates: On the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.

cond-mat.quant-gas↗

arXiv2009

Fictitious-time wave-packet dynamics in atomic systems

Gaussian wave packets (GWPs) are well suited as basis functions to describe the time evolution of arbitrary wave functions in systems with nonsingular smooth potentials. They are less so in atomic systems on account of the singular behavior of the Coulomb potential. We present a time-dependent variational method that makes the use of GWPs possible in the description of propagation of quantum states also in these systems. We do so by a regularization of the Coulomb potential, and by introduction of a fictitious time coordinate in which the evolution of an initial state can be calculated exactly and analytically for a pure Coulomb potential. Therefore in perturbed atomic systems variational approximations only arise from those parts of the potentials which deviate from the Coulomb potential. The method is applied to the hydrogen atom in external magnetic and electric fields. It can be adapted to systems with definite symmetries, and thus allows for a wide range of applications.

physics.atom-ph↗

arXiv2009

Exceptional points in the spectra of atoms in external fields

We investigate exceptional points, which are branch point singularities of two resonance eigenstates, in spectra of the hydrogen atom in crossed external electric and magnetic fields. A procedure to systematically search for exceptional points is presented, and their existence is proven. The properties of the branch point singularities are discussed with effective low-dimensional matrix models, their relation with avoided level crossings is analyzed, and their influence on dipole matrix elements and the photoionization cross section is investigated. Furthermore, the rare case of a connection between three resonances almost forming a triple-degeneracy in the form of a cubic root branch point is discussed.

nlin.CD↗

arXiv2009

Signatures of the classical transition state in atomic quantum spectra

We perform quantum mechanically exact calculations of resonances in the spectrum of the hydrogen atom in crossed external fields and establish a close connection between the classical transition state in phase space and features in the quantum spectrum. By varying the external field strengths, structures are revealed which are surprisingly similar to the quantized energy levels of the classical electron motion in the vicinity of the saddle point obtained with an approximation of the potential. The results give clear evidence for signatures of the transition state in quantum spectra.

physics.atom-ph↗

arXiv2008

Lifetime statistics in chaotic dielectric microresonators

We discuss the statistical properties of lifetimes of electromagnetic eigenmodes in dielectric microresonators with fully chaotic ray dynamics. Using the example of a resonator of stadium geometry, we find that a recently proposed random-matrix model very well describes the lifetime statistics of long-lived resonances, provided that two effective parameters are appropriately renormalized. This renormalization is linked to the formation of anomalously short-lived resonances, a mechanism also known from the fractal Weyl law and the resonance trapping phenomenon.

physics.optics↗

arXiv2008

Bifurcations, order, and chaos in the Bose-Einstein condensation of dipolar gases

We apply a variational technique to solve the time-dependent Gross-Pitaevskii equation for Bose-Einstein condensates in which an additional dipole-dipole interaction between the atoms is present with the goal of modelling the dynamics of such condensates. We show that universal stability thresholds for the collapse of the condensates correspond to bifurcation points where always two stationary solutions of the Gross-Pitaevskii equation disappear in a tangent bifurcation, one dynamically stable and the other unstable. We point out that the thresholds also correspond to "exceptional points," i.e. branching singularities of the Hamiltonian. We analyse the dynamics of excited condensate wave functions via Poincare surfaces of section for the condensate parameters and find both regular and chaotic motion, corresponding to (quasi-) periodically oscillating and irregularly fluctuating condensates, respectively. Stable islands are found to persist up to energies well above the saddle point of the mean-field energy, alongside with collapsing modes. The results are applicable when the shape of the condensate is axisymmetric.

quant-ph↗

arXiv2008

Dynamics and stability of Bose-Einstein condensates with attractive 1/r interaction

The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses.

quant-ph↗

arXiv2007

Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering

We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium. We show that the conjectured fractal Weyl law for open chaotic systems [W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)] is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.

nlin.CD↗

arXiv2007

Discovery of exceptional points in the Bose-Einstein condensation of gases with attractive 1/r-interaction

The extended Gross-Pitaevskii equation for the Bose-Einstein condensation of gases with attractive 1/r-interaction has a second solution which is born together with the ground state in a tangent bifurcation. At the bifurcation point both states coalesce, i.e., the energies and the wave functions are identical. We investigate the bifurcation point in the context of exceptional points, a phenomenon known for linear non-Hermitian Hamiltonians. We point out that the mean field energy, the chemical potential, and the wave functions show the same behavior as an exceptional point in a linear, non-symmetric system. The analysis of the analytically continued Gross-Pitaevskii equation reveals complex waves at negative scattering lengths below the tangent bifurcation. These solutions are interpreted as a decay of the condensate caused by an absorbing potential.

nlin.CD↗

arXiv2007

Exceptional Points in Atomic Spectra

We report the existence of exceptional points for the hydrogen atom in crossed magnetic and electric fields in numerical calculations. The resonances of the system are investigated and it is shown how exceptional points can be found by exploiting characteristic properties of the degeneracies, which are branch point singularities. A possibility for the observation of exceptional points in an experiment with atoms is proposed.

nlin.CD↗

arXiv2006

Hydrogen atom in crossed electric and magnetic fields: Phase space topology and torus quantization via periodic orbits

A hierarchical ordering is demonstrated for the periodic orbits in a strongly coupled multidimensional Hamiltonian system, namely the hydrogen atom in crossed electric and magnetic fields. It mirrors the hierarchy of broken resonant tori and thereby allows one to characterize the periodic orbits by a set of winding numbers. With this knowledge, we construct the action variables as functions of the frequency ratios and carry out a semiclassical torus quantization. The semiclassical energy levels thus obtained agree well with exact quantum calculations.

physics.atom-ph↗