"Gap labelling in quasicrystals: from microwaves to ultracold atoms"
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Quasicrystals can be modelled with a collection of polygons (tiles) that 
cover the whole plane, so that
each pattern (a sub-collection of tiles) appears up to translation with 
a positive frequency, but the tiling is not periodic. The frequency of 
presence of each pattern determines the spectrum of the system, and this 
is the subject of the gap labelling theory [1].
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Using a microwave realization of a tight-binding Penrose-tiled 
quasicrystal [2], we measured the gap labelling and the spatial energy 
distribution of each eigenstate [3]. The energy-scaling behaviour of the 
hopping terms in this particular system, allowed us to identify the main 
patterns that determine the first hierarchical structure of the spectrum 
. Our energy-scaling analysis enabled us not only a straightforward 
interpretation of the gap labelling but also a full understanding of the 
wavefunction behaviour observed in each band.
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The next goal will be the realization of a light Penrose-tiled 
quasicrystal for ultracold atoms, the underlying idea being to have 
access to samples with a lager number of tiles and thus to have the 
opportunity to observe the hierarchical effect of larger and larger 
patterns in the gap labelling. We have shown that this may be possible 
by mapping the gap labelling on a Brillouin zone (BZ) labelling and 
measure the areas of the extended BZs or the Bragg peak intensity 
distribution via different time-of-flight techniques [4].
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[1] J. Bellissard in "From Number Theory to Physics", 538 (Springer, 1993).
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[2] M. Bellec, et al., Phys. Rev. Lett. 110, 033902 (2013).
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[3] P. Vignolo et al. , in preparation.
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[4] J.M. Gambaudo, P. Vignolo, New J. Phys. 16, 043013 (2014).