58 research outputs found
Comparing algebraic and numerical solutions of classical diffusion process equations in computational financial mathematics
We revise the interrelations between the classical Black Scholes equation, the diffusion equation and Burgers equation. Some of the algebraic properties the diffusion equation shows are elaborated and qualitatively presented. The related numerical elementary recipes are briefly elucidated in context of the diffusion equation. The quality of the approximations to the exact solutions is compared throughout the visualizations. The article mainly is based on the pedagogical style of the presentations to the Novacella Easter School 2000 on Financial Mathematics
Inhomogeneous Light Shift Effects on Atomic Quantum State Evolution in Non-Destructive Measurements
Various parameters of a trapped collection of cold and ultracold atoms can be
determined non--destructively by measuring the phase shift of an off--resonant
probe beam, caused by the state dependent index of refraction of the atoms. The
dispersive light--atom interaction, however, gives rise to a differential light
shift (AC Stark shift) between the atomic states which, for a nonuniform probe
intensity distribution, causes an inhomogeneous dephasing between the atoms. In
this paper, we investigate the effects of this inhomogeneous light shift in
non--destructive measurement schemes. We interpret our experimental data on
dispersively probed Rabi oscillations and Ramsey fringes in terms of a simple
light shift model which is shown to describe the observed behavior well.
Furthermore, we show that by using spin echo techniques, the inhomogeneous
phase shift distribution between the two clock levels can be reversed.Comment: 9 pages, 7 figures, updated introduction and reference lis
Echo Spectroscopy of Atomic Dynamics in a Gaussian Trap via Phase Imprints
We report on the collapse and revival of Ramsey fringe visibility when a
spatially dependent phase is imprinted in the coherences of a trapped ensemble
of two-level atoms. The phase is imprinted via the light shift from a Gaussian
laser beam which couples the dynamics of internal and external degrees of
freedom for the atoms in an echo spectroscopy sequence. The observed revivals
are directly linked to the oscillatory motion of atoms in the trap. An
understanding of the effect is important for quantum state engineering of
trapped atoms
- …