We report on the study of a polariton gas confined in a quasi-periodic one dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of mini-gaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schrödinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.PACS numbers: 71.36.+c,78.55.Cr, 05.45.Df, 61.43.Hv, 71.23.Ft Free quantum particles or waves propagating in a spatially varying potential present modifications of their spectral density, which depend on the symmetry of this potential. The richness of spectral distributions in constrained geometries has long been recognized. The case of a periodic potential described by means of the Bloch theorem is a significant example. The notion of spectral distribution has been deepened in the wake of quasicrystals discovery and it led to a classification of energy spectra into absolutely continuous, pure point and singular continuous spectral distributions [1]. The latter class proved to be surprisingly rich and it encompasses a broad range of potentials, such as quasi-periodic potentials which have been thoroughly studied [2,3].An interesting quasi-periodic potential can be designed using a Fibonacci sequence. The corresponding singular continuous energy spectrum has a fractal structure of the Cantor set type [4][5][6][7], and it displays self-similarity i.e., a symmetry under a discrete scaling transformation. Denoting ρ(ε) the relevant density of states (DOS) in ε (either energy or frequency), a discrete scaling symmetry about a particular value ε u is expressed by the propertywhere µ (ε) = ε −∞ ρ (ε ) dε is the integrated density of states (IDOS), or density measure, and α and β are scaling parameters which usually, depend on ε u . Defining a shifted IDOS by N εu (ε) ≡ µ(ε) − µ (ε u ), the general solution of (1) can be written as [8]where γ = ln α ln β is the local (ε u -dependent) scaling exponent and F(z) is a periodic function of period unity, whose (non-universal) form depends on the problem at hand. Generally, the exponent γ takes values between zero and unity, so that the density ρ (ε) is a singular function. Such scaling properties of a fractal spectrum are expected to modify the behavior of physical quantities [8]. Recently studied examples include thermodynamic properties of photons [9], random walks [10], quantum diffusion of wave packets [11] and spontaneous emission triggered by a fractal vacuum [12]. The diffusion of a wave packet in a quasi-periodic medium is predicted to be neither diffusive, nor ballistic but to present a behavior characterized by non-universal exponents and a logperiodic modulation of its time dynamics. Experimental demonstration of these specific properties of quasiperiodic structures is still missing ...

Anderson localization of matter waves was recently observed with cold atoms in a weak one-dimensional disorder realized with laser speckle potential ͓Billy et al., Nature ͑London͒ 453, 891 ͑2008͔͒. The latter is special in that it does not have spatial frequency components above certain cutoff q c . As a result, the Lyapunov exponent ͑LE͒ or inverse localization length vanishes in Born approximation for particle wave vector k Ͼ 1 2 q c , and higher orders become essential. These terms, up to the fourth order, are calculated analytically and compared with numerical simulations. For very weak disorder, LE exhibits a sharp drop at k = 1 2 q c . For moderate disorder ͑a͒ the drop is less dramatic than expected from the fourth-order approximation and ͑b͒ LE becomes very sensitive to the sign of the disorder skewness ͑which can be controlled in cold atom experi-ments͒. Both observations are related to the strongly non-Gaussian character of the speckle intensity.

We study orbital magnetism of a degenerate electron gas in a number of two-dimensional integrable systems, within linear response theory. There are three relevant energy scales: typical level spacing ∆, the energy Γ, related to the inverse time of flight across the system, and the Fermi energy ε F .

We consider diffusion of a cold Fermi gas in the presence of a random optical speckle potential.The evolution of the initial atomic cloud in space and time is discussed. Analytical and numerical results are presented in various regimes. Diffusion of a Bose-Einstein condensate is also briefly discussed and similarity with the Fermi gas case is pointed out.

Spontaneous emission of a quantum emitter coupled to a QED vacuum with a deterministic fractal structure of its spectrum is considered. We show that the decay probability does not follow a Wigner-Weisskopf exponential decrease but rather an overall power law behavior with a rich oscillatory structure, both depending on the local fractal properties of the vacuum spectrum. These results are obtained by giving first a general perturbative derivation for short times. Then we propose a simplified model which retains the main features of a fractal spectrum to establish analytic expressions valid for all time scales. Finally, we discuss the case of a Fibonacci cavity and its experimental relevance to observe these results.PACS numbers: 05.45.Df, 42.50.PqSpontaneous emission results from the coupling of a quantum system with a discrete energy spectrum (an "atom") to a quantum vacuum. This is an important and widely studied phenomenon both from fundamental and applied points of view [1]. Spontaneous emission allows to probe properties of the quantum vacuum, its dynamics and correlations. The wide zoology of behaviors of spontaneous emission depends on spectral properties of the vacuum and on its coupling to the atom [2]. A standard textbook description [1,3] considers the coupling to a vacuum having a smooth and non-singular density of modes of photons, in which case, the probability for spontaneous emission follows the well-known Wigner-Weisskopf decay law, |U e (t)| 2 = e −Γe(ωe)t .Relevant definitions of the quantum amplitude U e (t) and of the inverse lifetime Γ e (ω e ) will be given below. This description has been further developed towards quantum emitters coupled to more complicated environments such as semiconductors, QED-cavities, photonic crystals and micro-cavities [2]. The existence of singularities in the spectrum of the vacuum leads to a qualitatively different behavior which has been studied in various cases [4]. In this letter, we address the problem of spontaneous emission from an atom coupled to a vacuum whose spectrum is characterized by a discrete scaling symmetry expressed by the propertywhere µ (ω) is the integrated density of modes, or spectral measure, and the dimensionless scaling parameter a and the map T (ω) provide a full characterization of the specific discrete scaling symmetry. Introducing N ωu (ω) ≡ µ(ω)−µ (ω u ), the scaling relation (2) can be written more concisely as N ωu (ω) = 1 a N T (ωu) (T (ω)). A spectrum described by (2) is often called fractal (see [5] and refs.