Basic research in meso-nano structures
has boomed in the last two decades in part due to the promise that a
comprehensive understanding of these systems will immediately lead to
the design and construction of devices with broad practical
applications ranging from quantum computing to healthcare. Despite
considerable progress we are still far from a full theoretical
description of the combination of finite size effects, disorder,
interactions and finite temperature.
This is a serious problem for technological applications since
in realistic electronic nanograins it is in general difficult to
disentangle these effects.
The situation is different in ultra cold atoms confined by optical
potentials [9]. In these systems, the
strength of the interactions and the form of the confining potential
can be controlled with great precision. Moreover the recently
introduced speckle potentials [10] act as
an effective random environment suitable to study localization and
other quantum coherence effects in almost free cold atoms.
Another field that has recently
benefited from condensed matter
ideas and techniques is that
of QCD. In the high density limit, the attractive quark-quark pairing [16] leads to the phenomenon of color
superconductivity. Moreover it has been recently reported that
phenomenon of Anderson-Mott localization may also be relevant in in the
description of certain features of the QCD vacuum [17]. In this case the role of
impurities/electrons is played by non perturbative gauge
configurations/quarks. Crossfertilization between different disciplines
promises to be one of the engines for innovation in the years to come.
In the near future I aimed to address the following three broad topics
(for the sake of clarity each of them is divided in three sections: a) what
question I wish to answer, b) how I plan to proceed and c) why
it is relevant.) :
1. Interactions in clean grains.
What?
I plan to carry out a
systematic analysis of the interplay of
interactions and finite size effects in clean grains, namely, how the
relevant observables depend on the size, the shape of the grain,
temperature and magnetic field.
How? The idea is to include finite size effects by using semiclassical techniques originally introduced in the context of quantum chaos and semiclassical physics [18]. Roughly speaking these techniques permit an analytical evaluation of quantities such as the spectral density and eigenstates correlations by using only classical information of the system. In a second stage the outcome of the semiclassical analysis is combined with techniques to tackle the many body problem.
For instance, in the case of a
weak attractive interaction I
combined [6] the BCS formalism [15] with a semiclassical expansion [18] of the spectral density and the
interaction matrix elements to obtain an explicit expression of the
superconducting gap in terms of the energy, number of particles, size
and form of the grain, temperature and magnetic field. I plan to extend
this idea to other type of interactions (BEC-BCS crossover) and spin,
dependence on temperature and magnetic field. Moreover I plan also to
address the combined effect of different type of interactions. This is
necessary to describe experimental results in electronic grains. For
instance in order to describe the experimental results of Tinkham et
al. with superconducting grains [12]
it would be necessary to combine the effect of the repulsive Coulomb
interaction (perturbatively) and the attractive BCS (non perturbative)
interaction.
Why? In highly symmetric grains it is expected the appearance of shell effects and magic numbers. In the context of the BCS theory that means that a slight modification of the number of electrons in the grain modifies substantially the superconducting gap. This is interesting for applications since it would permit to switch superconductivity on and off in a small (~10nm) grains at will and increase dramatically the critical temperature at which superconductivity breaks down.
In electronic systems it is hard
to realize experimentally symmetric
grains. The situation is different in cold atoms. The use of
holographic masks
combined with the recent introduction of spatial light modulators
permits the production of a very
broad range of intensity patterns [21]
which act as effective
spatial potentials for atoms.
2. Localization beyond
condensed matter: QCD and cold atoms.
What? I plan
to investigate whether Anderson localization effects usually discussed
in condensed matter are relevant in other physics problems and how the
current localization theories must be adapted. Specifically I will
address the role of localization in Quantum Chromodynamics and cold
atoms.
How?
QCD: Recent
experimental results from RHIC and soon from LHC and the increasing
reliability
of lattice simulations are providing a much more detailed picture of
both the QCD phase transitions
and the transport properties in the quark-gluon plasma phase.
Unfortunately we are still far from a theoretical understanding of the
microscopical origin of these phase transitions,
and the transport properties in the quark-gluon plasma. In this
proposal I investigate to what extent the universal quantum phenomenon
of localization is relevant
for the understanding of some of these problems.
I plan to explore the
role of Anderson localization
on: a) the chiral and deconfinement transition in QCD, b) the transport
properties of hot non perturbative gauge theories above the
deconfinement transition, c) the combined effect of disorder and
interactions on the quark pairing leading to color superconductivity,
d) characterization of the confinement deconfinement transition by the
entanglement entropy [19].
For a
preliminary account of a) see [17].
Part
a) will be investigated by the analysis of data from lattice
simulations. In part b) I plan to combine lattice calculations (In
collaborations with James Osborn) with analytical results obtained with
the help of the AdS-CFT correspondence (in collaboration with Diego
Rodriguez). In part c) an analytical treatment is in principle possible
by adapting recent results [20]
in
condensed matter on the interplay between disorder and pairing. In part
d) an analytical treatment (in collaboration with Igor Klebanov) is
possible by using the AdS-CFT correspondence.
Cold
atoms: The
universal phenomenon of Anderson
localization (the arrest of quantum transport in a random potential
caused by quantum interference) has been intensively investigated for
more than fifty years. Surprisingly enough it has not been until the
last few years that cold atoms [9] (and
also optics [14])
techniques have made
possible to carry out detailed experimental tests of localization [10,11].
Disorder is simulated either by the use of speckle potentials [10] or by switching on and off and regular
interval an optical lattice [11]. I
plan
to study how theories utilized to describe localization can be adapted
to the peculiarities of the speckle potentials [10]
(in collaboration with G. Shlyapnikov and A. Aspect group) or kicked
rotors [11] (in collaboration with W.
Jiao and J. Gong) used to mimic disorder. Moreover in order to compare
with experimental results I plan to investigate how localization
effects depend on a) weak interactions, b)decoherence affects and
c)deviations of the experimental potential from a purely random one.
Why? In the case of cold atoms the motivation is to compare theoretical models of the phenomenon of Anderson localization with experimental results [10,11] in which, unlike in condensed matter, other factors such as interactions or decoherence can be kept under control. In this way, since Anderson localization is universal, these experiments could be used to study both the quantum-classical transition (by increasing decoherence) and the limits of validity of quantum mechanics itself (by comparing precise theoretical predictions with experimental results).
In the case of QCD the
motivation is to provide a qualitative
microscopic picture of the physical mechanisms leading to both the QCD
phase transitions and the suppression of color superconductivity in the
high temperature limit of QCD.
3. Mesoscopic statistical
mechanics
What? I
plan to investigate how the properties of fermionic/bosonic gases
depend on the size and shape of the cavity containing them. I would
like to determine: a) the range of sizes and temperatures in which
these finite size effects are relevant experimentally, b) the effect of
weak interactions on these results, c) the effect of decoherence, d)
the differences between the canonical and the grand-canonical ensemble.
How? The
idea is to use semiclassical analytical techniques to obtain explicit
expressions of the spectral density and matrix elements (if
interactions are taken into account) as a function of the size and
shape of the cavity. I anticipate that the magnitude of the deviations
from the bulk increases with the symmetry of the cavity. It will be
also a priority to identify the range of parameters (temperature, size
and shape of the cavity) for which these effects may be accessible to
experimental verification.
Why? In
the bosonic case (a blackbody) these results may be relevant in the
physics of the cosmic microwave background [23],
sonoluminescence [22] and
in the
determination of temperature standards [13].
We note recent [22]
experiments in
sonoluminiscence provide the first example of a meso-nano blackbody.
Other motivation to study finite size effects in bosonic gases is to
determine to what extent the form of a real grain deviates from some
given "ideal" geometry. This is relevant for industrial applications
aimed to manufacture cavities of a given size and shape.