3,044 research outputs found
The Kapitza - Dirac effect
The Kapitza - Dirac effect is the diffraction of a well - collimated particle
beam by a standing wave of light. Why is this interesting? Comparing this
situation to the introductory physics textbook example of diffraction of a
laser beam by a grating, the particle beam plays the role of the incoming wave
and the standing light wave the role of the material grating, highlighting
particle - wave duality. Apart from representing such a beautiful example of
particle - wave duality, the diffracted particle beams are coherent. This
allows the construction of matter interferometers and explains why the Kapitza
- Dirac effect is one of the workhorses in the field of atom optics. Atom
optics concerns the manipulation of atomic waves in ways analogous to the
manipulation of light waves with optical elements. The excitement and activity
in this new field of physics stems for a part from the realisation that the
shorter de Broglie wavelengths of matter waves allow ultimate sensitivities for
diffractive and interferometric experiments that in principle would far exceed
their optical analogues. Not only is the Kapitza - Dirac effect an important
enabling tool for this field of physics, but diffraction peaks have never been
observed for electrons, for which is was originally proposed in 1933. Why has
this not been observed? What is the relation between the interaction of laser
light with electrons and the interaction of laser light with atoms, or in other
words what is the relation between the ponderomotive potential and the
lightshift potential? Would it be possible to build interferometers using the
Kapitza - Dirac effect for other particles? These questions will be addressed
in this paper.Comment: 17 pages, 13 figure
Non-standard Hubbard models in optical lattices: a review
Originally, the Hubbard model has been derived for describing the behaviour
of strongly-correlated electrons in solids. However, since over a decade now,
variations of it are also routinely being implemented with ultracold atoms in
optical lattices. We review some of the rich literature on this subject, with a
focus on more recent non-standard forms of the Hubbard model. After an
introduction to standard (fermionic and bosonic) Hubbard models, we discuss
briefly common models for mixtures, as well as the so called extended
Bose-Hubbard models, that include interactions between neighboring sites,
next-neighboring sites, and so on. The main part of the review discusses the
importance of additional terms appearing when refining the tight-binding
approximation on the original physical Hamiltonian. Even when restricting the
models to the lowest Bloch band is justified, the standard approach neglects
the density-induced tunneling (which has the same origin as the usual on-site
interaction). The importance of these contributions is discussed for both
contact and dipolar interactions. For sufficiently strong interactions, also
the effects related to higher Bloch bands become important even for deep
optical lattices. Different approaches that aim at incorporating these effects,
mainly via dressing the basis Wannier functions with interactions, leading to
effective, density-dependent Hubbard-type models, are reviewed. We discuss also
examples of Hubbard-like models that explicitly involve higher -orbitals, as
well as models that couple dynamically spin and orbital degrees of freedom.
Finally, we review mean-field nonlinear-Schr\"odinger models of the Salerno
type that share with the non-standard Hubbard models the nonlinear coupling
between the adjacent sites. In that part, discrete solitons are the main
subject of the consideration. We conclude by listing some future open problems.Comment: expanded version 47pp, accepted in Rep. Prog. Phy
Static and dynamic properties of a few spin interacting fermions trapped in an harmonic potential
We provide a detailed study of the properties of a few interacting spin
fermions trapped in a one-dimensional harmonic oscillator potential. The
interaction is assumed to be well represented by a contact delta potential.
Numerical results obtained by means of exact diagonalization techniques are
combined with analytical expressions for both the non-interacting and strongly
interacting regime. The case is used to benchmark our numerical
techniques with the known exact solution of the problem. After a detailed
description of the numerical methods, in a tutorial-like manner, we present the
static properties of the system for and 5 particles, e.g.
low-energy spectrum, one-body density matrix, ground-state densities. Then, we
consider dynamical properties of the system exploring first the excitation of
the breathing mode, using the dynamical structure function and corresponding
sum-rules, and then a sudden quench of the interaction strength
Calculating principal eigen-functions of non-negative integral kernels: particle approximations and applications
Often in applications such as rare events estimation or optimal control it is
required that one calculates the principal eigen-function and eigen-value of a
non-negative integral kernel. Except in the finite-dimensional case, usually
neither the principal eigen-function nor the eigen-value can be computed
exactly. In this paper, we develop numerical approximations for these
quantities. We show how a generic interacting particle algorithm can be used to
deliver numerical approximations of the eigen-quantities and the associated
so-called "twisted" Markov kernel as well as how these approximations are
relevant to the aforementioned applications. In addition, we study a collection
of random integral operators underlying the algorithm, address some of their
mean and path-wise properties, and obtain error estimates. Finally,
numerical examples are provided in the context of importance sampling for
computing tail probabilities of Markov chains and computing value functions for
a class of stochastic optimal control problems.Comment: 38 pages, 4 figures, 1 table; to appear in Mathematics of Operations
Researc
- …