129 research outputs found
Uniform -bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation
In this paper, we consider an energy-conserving continuous Galerkin
discretization of the Gross-Pitaevskii equation with a magnetic trapping
potential and a stirring potential for angular momentum rotation. The
discretization is based on finite elements in space and time and allows for
arbitrary polynomial orders. It was first analyzed in [O. Karakashian, C.
Makridakis; SIAM J. Numer. Anal. 36(6):1779-1807, 1999] in the absence of
potential terms and corresponding a priori error estimates were derived in 2D.
In this work we revisit the approach in the generalized setting of the
Gross-Pitaevskii equation with rotation and we prove uniform -bounds
for the corresponding numerical approximations in 2D and 3D without coupling
conditions between the spatial mesh size and the time step size. With this
result at hand, we are in particular able to extend the previous error
estimates to the 3D setting while avoiding artificial CFL conditions
A two level approach for simulating Bose-Einstein condensates by localized orthogonal decomposition
In this work, we consider the numerical computation of ground states and
dynamics of single-component Bose-Einstein condensates (BECs). The
corresponding models are spatially discretized with a multiscale finite element
approach known as Localized Orthogonal Decomposition (LOD). Despite the
outstanding approximation properties of such a discretization in the context of
BECs, taking full advantage of it without creating severe computational
bottlenecks can be tricky. In this paper, we therefore present two
fully-discrete numerical approaches that are formulated in such a way that they
take special account of the structure of the LOD spaces. One approach is
devoted to the computation of ground states and another one for the computation
of dynamics. A central focus of this paper is also the discussion of
implementation aspects that are very important for the practical realization of
the methods. In particular, we discuss the use of suitable data structures that
keep the memory costs economical. The paper concludes with various numerical
experiments in 1d, 2d and 3d that investigate convergence rates and
approximation properties of the methods and which demonstrate their performance
and computational efficiency, also in comparison to spectral and standard
finite element approaches
Physics with Coherent Matter Waves
This review discusses progress in the new field of coherent matter waves, in
particular with respect to Bose-Einstein condensates. We give a short
introduction to Bose-Einstein condensation and the theoretical description of
the condensate wavefunction. We concentrate on the coherence properties of this
new type of matter wave as a basis for fundamental physics and applications.
The main part of this review treats various measurements and concepts in the
physics with coherent matter waves. In particular we present phase manipulation
methods, atom lasers, nonlinear atom optics, optical elements, interferometry
and physics in optical lattices. We give an overview of the state of the art in
the respective fields and discuss achievements and challenges for the future
Quantum information dynamics
Presented is a study of quantum entanglement from the perspective of the theory of quantum information dynamics. We consider pairwise entanglement and present an analytical development using joint ladder operators, the sum of two single-particle fermionic ladder operators. This approach allows us to write down analytical representations of quantum algorithms and to explore quantum entanglement as it is manifested in a system of qubits.;We present a topological representation of quantum logic that views entangled qubit spacetime histories (or qubit world lines) as a generalized braid, referred to as a super-braid. The crossing of world lines may be either classical or quantum mechanical in nature, and in the latter case most conveniently expressed with our analytical expressions for entangling quantum gates. at a quantum mechanical crossing, independent world lines can become entangled. We present quantum skein relations that allow complicated superbraids to be recursively reduced to alternate classical histories. If the superbraid is closed, then one can decompose the resulting superlink into an entangled superposition of classical links. Also, one can compute a superlink invariant, for example the Jones polynomial for the square root of a knot.;We present measurement-based quantum computing based on our joint number operators. We take expectation values of the joint number operators to determine kinetic-level variables describing the quantum information dynamics in the qubit system at the mesoscopic scale. We explore the issue of reversibility in quantum maps at this scale using a quantum Boltzmann equation. We then present an example of quantum information processing using a qubit system comprised of nuclear spins. We also discuss quantum propositions cast in terms of joint number operators.;We review the well known dynamical equations governing superfluidity, with a focus on self-consistent dynamics supporting quantum vortices in a Bose-Einstein condensate (BEC). Furthermore, we review the mutual vortex-vortex interaction and the consequent Kelvin wave instability. We derive an effective equation of motion for a Fermi condensate that is the basis of our qubit representation of superfluidity.;We then present our quantum lattice gas representation of a superfluid. We explore aspects of our model with two qubits per point, referred to as a Q2 model, particularly its usefulness for carrying out practical quantum fluid simulations. We find that it is perhaps the simplest yet most comprehensive model of superfluid dynamics. as a prime application of Q2, we explore the power-law regions in the energy spectrum of a condensate in the low-temperature limit. We achieved the largest quantum simulations to date of a BEC and, for the first time, Kolmogorov scaling in superfluids, a flow regime heretofore only obtainably by classical turbulence models.;Finally, we address the subject of turbulence regarding information conservation on the small scales (both mesoscopic and microscopic) underlying the flow dynamics on the large hydrodynamic (macroscopic) scale. We present a hydrodynamic-level momentum equation, in the form of a Navier-Stokes equation, as the basis for the energy spectrum of quantum turbulence at large scales. Quantum turbulence, in particular the representation of fluid eddies in terms of a coherent structure of polarized quantum vortices, offers a unique window into the heretofore intractable subject of energy cascades
The Mathematics of the Bose Gas and its Condensation
This book surveys results about the quantum mechanical many-body problem of
the Bose gas that have been obtained by the authors over the last seven years.
These topics are relevant to current experiments on ultra-cold gases; they are
also mathematically rigorous, using many analytic techniques developed over the
years to handle such problems. Some of the topics treated are the ground state
energy, the Gross-Pitaevskii equation, Bose-Einstein condensation,
superfluidity, one-dimensional gases, and rotating gases. The book also
provides a pedagogical entry into the field for graduate students and
researchers.Comment: 213 pages. Slightly revised and extended version of Oberwolfach
Seminar Series, Vol. 34, Birkhaeuser (2005
Making, probing and understanding Bose-Einstein condensates
Contribution to the proceedings of the 1998 Enrico Fermi summer school on
Bose-Einstein condensation in Varenna, Italy.Comment: Long review paper with ~90 pages, ~20 figures. 2 GIF figures in
separate files (4/5/99 fixed figure
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