Research
Bose-Einstein Condensates in Optical Lattices
While the physics of the last century was mainly characterized by great advances in
understanding the properties of one-particle systems, recent experimental developments
have put the effects of interacting bosons at the top of the research agenda. In fact,
the growing theoretical interest was further enhanced by recent experimental achievements.
The most fascinating of these was the realization of Bose-Einstein condensation (BEC) of
ultra-cold atoms in periodic optical lattices, which allows for novel concrete applications
of quantum mechanics such as atom interferometers and atom lasers.
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Quantum Graphs as Models of Mesoscopic Physics and Quantum Chaos
In quantum chaos one tries to understand the implications of a chaotic classical limit
for a quantum system. Despite the large variety of such systems, they have a number of
universal physical properties in common. For example, a statistical analysis of the energy
levels invariably shows striking similarities with the "energies" obtained from the
diagonalization of random matrices. In the semiclassical approach to this problem the
quantum density of states is expressed as a sum over classical periodic orbits by
Gutzwiller's trace formula. One can then describe fluctuations in quantum spectra on
the basis of information about the classical dynamics (sum rules and action correlations
of periodic orbits). In most cases, only the semi-classical approximation for the
trace formulae are known and their application is not only hampered by the intrinsic
complexity of the set of periodic orbits, but also by doubts about the ability of
the semi-classical trace formulae to provide an accurate enough basis for further
developments.
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Wavepacket Dynamics, Quantum Irreversibility and the Applicability of RMT
The past mesoscopic physics/quantum chaos literature was strongly focused on understanding
the interplay between universal (RMT-like) and non-universal (semiclassical) features as far
as spectral properties are concerned. However, the study of spectral statistics is just the
lower level in the hierarchy of challenges to understand quantum systems. The two other levels
are: studies of the shape of the eigenstates, and studies of the generated dynamics. The latter
two aspects had been barely treated prior to our studies.
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Parametric Evolution of Wavefunctions (LDoS)
The analysis of the structural changes that the eigenstates of a mesoscopic/chaotic system
exhibit as one varies a parameter $x$ of the Hamiltonian ${\cal H}(x)$ has sparked a great
deal of research activity for many years. Physically the change of $x$ may represent the
effect of some externally controlled field (like electric field, magnetic flux, gate voltage)
or a change of an effective-interaction (as in molecular dynamics). Thus, these studies are
relevant for diverse areas of physics ranging from nuclear and atomic physics to quantum
chaos and mesoscopics.
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Driven Mesoscopic Systems
Driven systems are described by Hamiltonian H(Q,P,x(t)), where x(t) is a
time dependent parameter and Q,P is a set of canonical co-ordinates. Such
systems are of interest in mesoscopic physics (quantum dots), as well as
in nuclear, atomic and molecular physics. The time dependent parameter
x(t) may have the significance of an external electric field or magnetic
flux or gate voltage. Due to the time dependence of x(t), the energy of
the system is not a constant of motion. Rather the system makes
"transitions" between energy levels, and therefore the energy distribution
evolves with time. The aim of these studies is to develop a general theory
for this evolution. The possibility to present a general quantum
mechanical theory follows from the simple fact that chaos leads to
universality. This universality is captured to some extend by RMT.
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Transport Properties of Random Media
Descriptions and pictures
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Control of Chaos and Pattern Formation by Impurities
Coupled arrays of oscillators are studied extensively in many fields of
science because of their prevalence in nature. They are used as models for
coupled arrays of neurons, chemical reactions, coupled lasers or Josephson
junctions, charge-density wave conductors, crystal dislocations in metals,
and proton conductivity in hydrogen-bonded chains. Various models and
coupling schemes have been proposed and analyzed previously. A particular
class are arrays of coupled oscillators, which exhibit chaotic motion when
uncoupled. This class includes the forced Frenkel-Kontorova model, which
finds a straightforward physical realization in an array of diffusively
coupled Josephson junctions, in which the applied current of each junction
is modulated by a common frequency. Motivated by ideas that emerged in the
theory of disordered systems, we have studied the appearance of
synchronized motion in one and two-dimensional arrays of coupled chaotic
pendula, due to the presence of a single impurity.
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