Direct measurement of Chern numbers in the diffraction pattern of a - PowerPoint PPT Presentation

Direct measurement of Chern numbers in the diffraction pattern of a Fibonacci chain. JMC15 – Bordeaux 25 th August 2016 Dareau et al. , arXiv 1607.00901 Alexandre Dareau , E. Levy (*) , M. Bosch, R.Bouganne, E. Akkermans (*) , F . Gerbier & J. Beugnon Laboratoire Kastler Brossel , Collège de France, CNRS, ENS, UPMC. (*) Technion Israel Institute of Technology , Department of Physics (Israel)

1 Fibonacci Chain Constructing the Fibonacci chain Cut and Project (C&P) A slope = B A A B irrational A B A A aperiodic structure B A (quasicrystal) ABAABABAABA... (in this case : Fibonacci) origin NB : aperiodic order comes from projection of a periodic structure of higher dimension

2 Fibonacci Chain Diffraction from a Fibonacci chain Peaks positions given by two integers a

3 Fibonacci Chain Topological properties of the 1D Fibonacci chain From density of states → multi-gap system → gap labeling theorem (Bellissard, 1982) NB : gaps open at the position of the gaps position in reciprocal space : diffraction peaks. with p and q : integers q is a Chern number Connected to structural properties Levy et al. , arXiv 1509.04028 “phason” degree of freedom

4 Fibonacci Chain Phason degree of freedom additional degree cut and project (C&P) of freedom = “phason” slope : A B ... new origin origin scanning phason spatial shift Φ for a finite chain ΔX

5 Fibonacci Chain Effect of the phason ? scanning Φ → spatial shift : spatial shift → phase shift phason affects the phase (real space) (reciprocal space) of the diffracted field for a diffraction peak at the phase shift is Dareau et al. , arXiv 1607.00901 F = 1 4 4 Example : for n θ : phase of the Fibonacci diffracted field (units of 2π) q = -1 q = 2 q = 2 q = -1

6 Optical diffraction by a Fibonacci chain Our experimental setup Digital Micromirror Device (DMD) – mirror (“pixel”) size ~ 14 µm – 1024 × 768 pixels 532nm laser lens (focal : f) f located at the Fourier plane of the DMD image CCD camera Fraunhofer (far-field) diffraction pattern

6 Optical diffraction by a Fibonacci chain Our experimental setup Digital Micromirror Device (DMD) – mirror (“pixel”) size ~ 14 µm – 1024 × 768 pixels 532nm laser Fibonacci encoding : A = (pixel OFF) B = (pixel ON) DMD front view outside outside lens (OFF) (OFF) (focal : f) Fibonacci chain A B A A B A B A A B A A B . . . f located at the Fourier plane of the DMD image CCD camera Fraunhofer (far-field) diffraction pattern

7 Optical diffraction by a Fibonacci chain Diffraction by a single Fibonacci chain DMD pattern peaks located at (in units of 2π/a) a Diffraction pattern q = + 2 q = - 4 q = + 1 Ex : main peaks (q=±1) 0 0.38 0.5 0.62 (units of 2π/a)

8 Optical diffraction by a Fibonacci chain Scanning the phason : results Dareau et al. , arXiv 1607.00901 ← Φ=0 DMD Pattern Fibonacci Fibonacci 89 letters No effect of the phason scan !

8 Optical diffraction by a Fibonacci chain Scanning the phason : results Dareau et al. , arXiv 1607.00901 ← Φ=0 DMD Pattern Fibonacci 89 letters q = -3 q = 2 q = 4 q = 1 Peaks are crossed by holes Slope / number of crossings gives the Chern number q

8 Optical diffraction by a Fibonacci chain Scanning the phason : results Dareau et al. , arXiv 1607.00901 ← Φ=0 DMD Pattern Fibonacci 89 letters q = -3 q = 2 q = 4 q = 1 q = 4 q = 1 k x cuts at initial peak position : oscillation with period π/q

9 Optical diffraction by a Fibonacci chain Scanning the phason : discussion spatial Fibonacci shift x → -x Fibonacci Fibonacci no Φ dependence Fibonacci → sinusoidal variation with Φ, period T = π/q

10 Optical diffraction by a Fibonacci chain Diffraction from 2D (x,Φ) pattern 89 letters L = F n × a y peaks located at peak for (89 chains) = same as before Φ x a Dareau et al. , arXiv 1607.00901 q = 4 q = 1 Peak position along y is proportional to the Chern number q = -1 q = -7

10 Optical diffraction by a Fibonacci chain Diffraction from 2D (x,Φ) pattern 89 letters L = F n × a y peaks located at peak for (89 chains) = same as before Φ x a

11 Optical diffraction by a Fibonacci chain Testing robustness : effect of noise randomly flip average N noise lines diffraction peaks (Φ=0) Φ scans a number of c “noisy” lines b a b total number of lines c (units of 2π/a) (units of 2π/a) Hole crossing visible even for weak peak signal (and number of crossings unchanged)

12 Conclusion and outlook Experimental measurements Diffraction on a optical 1D Reveals underlying topological properties of Fibonacci grating or a 2D Fibonacci quasicrystals set of Fibonacci chains → Stresses the importance of the “phason” degree of freedom Kraus et al. , PRL (2012), Levy et al. , arXiV (2015) How to extend this method ? → Directly applicable to any quasicrystal generated with the “Cut & Project” method → Study effect of “phason” on 2D quasiperiodic tilings ? → Matter-waves diffraction / propagation in 1D quasiperiodic potential DMD can be used to project the grating on an gas of cold atoms

Direct measurement of Chern numbers in the diffraction pattern of a Fibonacci chain. JMC15 – Bordeaux 25 th August 2016 Dareau et al. , arXiv 1607.00901 Alexandre Dareau , E. Levy (*) , M. Bosch, R.Bouganne, E. Akkermans (*) , F . Gerbier & J. Beugnon Laboratoire Kastler Brossel , Collège de France, CNRS, ENS, UPMC. (*) Technion Israel Institute of Technology , Department of Physics (Israel)

+2 Fibonacci Chain Phason degree of freedom cut and project (C&P) characteristic function Kraus et al. , PRL (2012) A B A A B A B A A B A additional degree of freedom = “phason” line slope :

+3 Fibonacci Chain Effect of the phason ? Scanning Ф over 2π generates For a finite chain of length F F n different configurations n NB : The generated configurations are segments of the infinite chain Example : for F = 8 n Infinite chain : ABAABABAABAABABAABABA… AABABAAB ABAABAAB A B ABAABABA Spatial shift : ( F = 5 ) n - 1