60 research outputs found
A nonlinear dynamics approach to Bogoliubov excitations of Bose-Einstein condensates
We assume the macroscopic wave function of a Bose-Einstein condensate as a
superposition of Gaussian wave packets, with time-dependent complex width
parameters, insert it into the mean-field energy functional corresponding to
the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational
principle. In this way the GPE is mapped onto a system of coupled equations of
motion for the complex width parameters, which can be analyzed using the
methods of nonlinear dynamics. We perform a stability analysis of the fixed
points of the nonlinear system, and demonstrate that the eigenvalues of the
Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation
spectrum of a condensate in an axisymmetric trap.Comment: 7 pages, 3 figures, Proceedings of the "8th International Summer
School/Conference Let's Face Chaos Through Nonlinear Dynamics", CAMTP,
University of Maribor, Slovenia, 26 June - 10 July 201
Exceptional Points for Nonlinear Schroedinger Equations Describing Bose-Einstein Condensates of Ultracold Atomic Gases
The coalescence of two eigenfunctions with the same energy eigenvalue is not possible in Hermitian Hamiltonians. It is, however, a phenomenon well known from non-hermitian quantum mechanics. It can appear, e.g., for resonances in open systems, with complex energy eigenvalues. If two eigenvalues of a quantum mechanical system which depends on two or more parameters pass through such a branch point singularity at a critical set of parameters, the point in the parameter space is called an exceptional point. We will demonstrate that exceptional points occur not only for non-hermitean Hamiltonians but also in the nonlinear Schroedinger equations which describe Bose-Einstein condensates, i.e., the Gross-Pitaevskii equation for condensates with a short-range contact interaction, and with additional long-range interactions. Typically, in these condensates the exceptional points are also found to be bifurcation points in parameter space. For condensates with a gravity-like interaction between the atoms, these findings can be confirmed in an analytical way
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.Comment: 5 pages, 4 figures, 1 table, one reference was adde
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
Exactly solvable Wadati potentials in the PT-symmetric Gross-Pitaevskii equation
This note examines Gross-Pitaevskii equations with PT-symmetric potentials of
the Wadati type: . We formulate a recipe for the construction of
Wadati potentials supporting exact localised solutions. The general procedure
is exemplified by equations with attractive and repulsive cubic nonlinearity
bearing a variety of bright and dark solitons.Comment: To appear in Proceedings of the 15 Conference on Pseudo-Hermitian
Hamiltonians in Quantum Physics, May 18-23 2015, Palermo, Italy (Springer
Proceedings in Physics, 2016
Resonance scattering and singularities of the scattering function
Recent studies of transport phenomena with complex potentials are explained
by generic square root singularities of spectrum and eigenfunctions of
non-Hermitian Hamiltonians. Using a two channel problem we demonstrate that
such singularities produce a significant effect upon the pole behaviour of the
scattering matrix, and more significantly upon the associated residues. This
mechanism explains why by proper choice of the system parameters the resonance
cross section is increased drastically in one channel and suppressed in the
other channel.Comment: 4 pages, 3 figure
Analysis technique for exceptional points in open quantum systems and QPT analogy for the appearance of irreversibility
We propose an analysis technique for the exceptional points (EPs) occurring
in the discrete spectrum of open quantum systems (OQS), using a semi-infinite
chain coupled to an endpoint impurity as a prototype. We outline our method to
locate the EPs in OQS, further obtaining an eigenvalue expansion in the
vicinity of the EPs that gives rise to characteristic exponents. We also report
the precise number of EPs occurring in an OQS with a continuum described by a
quadratic dispersion curve. In particular, the number of EPs occurring in a
bare discrete Hamiltonian of dimension is given by ; if this discrete Hamiltonian is then coupled to continuum
(or continua) to form an OQS, the interaction with the continuum generally
produces an enlarged discrete solution space that includes a greater number of
EPs, specifically , in which
is the number of (non-degenerate) continua to which the discrete sector is
attached. Finally, we offer a heuristic quantum phase transition analogy for
the emergence of the resonance (giving rise to irreversibility via exponential
decay) in which the decay width plays the role of the order parameter; the
associated critical exponent is then determined by the above eigenvalue
expansion.Comment: 16 pages, 7 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
PT-Symmetric Dimer in a Generalized Model of Coupled Nonlinear Oscillators
Abstract In the present work, we explore the case of a general PT -symmetric dimer in the context of two both linearly and nonlinearly coupled cubic oscillators. To obtain an analytical handle on the system, we first explore the rotating wave approximation converting it into a discrete nonlinear Schrödinger type dimer. In the latter context, the stationary solutions and their stability are identified numerically but also wherever possible analytically. Solutions stemming from both symmetric and anti-symmetric special limits are identified. A number of special cases are explored regarding the ratio of coefficients of nonlinearity between oscillators over the intrinsic one of each oscillator. Finally, the considerations are extended to the original oscillator model, where periodic orbits and their stability are obtained. When the solutions are found to be unstable their dynamics is monitored by means of direct numerical simulations
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