485 research outputs found
Born Rule and Noncontextual Probability
The probabilistic rule that links the formalism of Quantum Mechanics (QM) to
the real world was stated by Born in 1926. Since then, there were many attempts
to derive the Born postulate as a theorem, Gleason's being the most prominent.
The Gleason derivation, however, is generally considered rather intricate and
its physical meaning, in particular in relation with the noncontextuality of
probability (NP), is not quite evident. More recently, we are witnessing a
revival of interest in possible demonstrations of the Born rule, like Zurek's
and Deutsch's based on the decoherence and on the theory of decisions,
respectively. Despite an ongoing debate about the presence of hidden
assumptions and circular reasonings, these have the merit of prompting more
physically oriented approaches to the problem. Here we suggest a new proof of
the Born rule based on the noncontextuality of probability. Within the theorem
we also demonstrate the continuity of probability with respect to the
amplitudes, which has been suggested to be a gap in Zurek's and Deutsch's
approaches, and we show that NP is implicitly postulated also in their
demonstrations. Finally, physical motivations of NP are given based on an
invariance principle with respect to a resolution change of measurements and
with respect to the principle of no-faster-than-light signalling.Comment: 10 page
Nonlinear Tight-Binding Approximation for Bose-Einstein Condensates in a Lattice
The dynamics of Bose-Einstein condensates trapped in a deep optical lattice
is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity
and the intersite hopping rates are retrieved from a nonlinear tight-binding
approximation taking into account the effective dimensionality of each
condensate. We derive analytically the Bloch and the Bogoliubov excitation
spectra, and the velocity of sound waves emitted by a traveling condensate.
Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of
localized wavepackets. We calculate the ground-state atomic distribution in the
presence of an harmonic confining potential, and the frequencies of small
amplitude dipole and quadrupole oscillations. We finally quantize the DNL,
recovering an extended Bose-Hubbard model
Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates
We study the dynamical phase diagram of a dilute Bose-Einstein condensate
(BEC) trapped in a periodic potential. The dynamics is governed by a discrete
non-linear Schr\"odinger equation: intrinsically localized excitations,
including discrete solitons and breathers, can be created even if the BEC's
interatomic potential is repulsive. Furthermore, we analyze the
Anderson-Kasevich experiment [Science 282, 1686 (1998)], pointing out that mean
field effects lead to a coherent destruction of the interwell Bloch
oscillations
- …