Spontaneous emission from a fractal vacuum

Abstract

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.

Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential

“…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.…”

Regularized Laplacian determinants of self-similar fractals

“…Our long-term motivation comes from quantum physics, in particular such recent papers as [ 1 – 5 , 14 , 15 , 25 , 28 , 36 ] and more classical works [ 16 , 17 , 20 , 23 ]. Our immediate mathematical motivation is twofold.…”

Topological boundary states in 1D: An effective Fabry-Perot model

“…It is extensively used in a variety of problems such as the Casimir effects (static and dynamic) [33][34][35], cavity optomechanics [36], surface physics, molecular or atomic (e.g. van der Waals) interactions or the amplification of spontaneous emission (Purcell effect) [12].…”

“…Here we discuss the case of a dielectric modulation, such that the letters {A, B} represent different values of refractive index {n A , n B }, and different layer thickness {d A , d B }, as depicted in Figure 3. The DOS and the transmission spectrum of a Fibonacci dielectric structure, calculable using the scattering approach, has a fractal structure [12][13][14] made out of a rich variety of bands and gaps (see Fig. 4).…”

“…Periodic dielectric structures are, to some extent, a photonic counterpart of a solid state crystal whose spectrum contains both transmission bands and gaps [1]. Random dielectric structures have been used to study Anderson localization of light [2][3][4][5][6][7][8][9], and quasiperiodic structures have been shown to be a photonic analogue of a solid state quasicrystal with a fractal spectrum containing infinitely many gaps [10][11][12][13][14]. Gap states, which may occur in any gapped spectrum are very useful due to their relatively high spectral isolation and spatial localization.…”