NETADIS Kick off meeting. Torino,3-6 february Payal Tyagi

Size: px

Start display at page:

Download "NETADIS Kick off meeting. Torino,3-6 february Payal Tyagi"

Ethelbert Hodges
5 years ago
Views:

1 NETADIS Kick off meeting Torino,3-6 february 2013 Payal Tyagi

2 Outline My background Project: 1) Theoretical part a) Haus master equation and Hamiltonian b) First steps: - Modelling systems with disordered couplings - with multi body interactions ( non- linear) - Cavity method 2) Numerical part GPU based Monte Carlo numerical simulation Courses

3 My Background --> Previous research experience: Dr. Pranesh Sengupta, Materials Science Department, Bhabha Atomic Research Centre (BARC), Mumbai : "Crystal-chemical comparative study between naturally occurring sphene crystal". EPMA, TEM, Raman microscopy, UV-VIS spectrography --> MS Physics, Banasthali University (july 2010-may 2012). --> BS Physics, St. Stephen's College, University of Delhi (july may 2010).

4 Project Supervisor : Dr. Luca Leuzzi Aim : Inference of coupling of waves in non linear disordered medium. - understanding random laser through disordered systems - strong interaction in closed cavity - random distribution of scatterers - localization of em modes - scattering -> degree of localization - disorder induced random couplings - non linearity couples the modes - paramagnetic - ferromagnetic - complex glassy regimes - statistical properties-> phase relations C. Conti and L. Leuzzi Complexity of waves in non linear disordered media, Phys Rev B 83, (2011) D.S. Wiersma, Nat Phys 4, 359 (2008)

5 Haus master equation for mode locking lasers in ordered cavities Mode locking condition : Langevin dynamical equation : Haus Master equation T R a t =( g l)a+( 1 ω 2 +id) 2 a τ 2 +(γ i δ) a 2 a ω j +ω k ω l ω m = 0 ȧ (t)= H a n +η n (t) ȧ (t)=( g n l n +id n ) a n +(γ i δ)σ ω j +ω k =ω l +ω m a j a k a l +η n (t) * Hamiltonian : H = R[ j<k G (2) jk a j a + (4) k ω G j +ω k =ω l +ω jklm a* j a* k a l a m ] m Where, g n : gain coefficient l n : loss term D n : group velocity of the wave packet γ : coefficient of saturable absorber δ: coefficient of the Kerr lens η n (t): white noise H. A. Haus, IEEE J. Quantum Electron. 6, 1173 (2000)

6 Hamiltonian Amplitude : i a i 2 = i A i 2 =E, a= A e i ϕ Hamiltonian for closed cavity: 1, N H =Σ' ω j +ω k =ω l +ω m G jklm A j A k A l A m cos(ϕ j +ϕ k ϕ l ϕ m ) 1, N H J [ϕ]= i1 <i 2, i 3 <i 4 J i cos(ϕ i1 +ϕ i 2 ϕ i3 ϕ i 4 ) Effective interaction among mode amplitudes is governed by: G jklm =1/2 ω j ω k ω l ω m V d 3 (3) r χ αβ γ δ (ω j, ω k,ω l, ω m ; r) E α j (r) E β k (r) E γ l (r) E δ m (r) where, G jklm χ (3) : Coefficient of spatial overlap of electromagnetic fields of the modes : non linear susceptibility C. Conti and L. Leuzzi Complexity of waves in non linear disordered media, Phys Rev B 83, (2011)

7 Bethe Lattice we study two different mean field models of Bethe lattice and look into their mean field properties. XY model H J [ϕ]= i1 J <i 2, i 3 <i i cos(ϕ i 4 1 +ϕ i 2 ϕ i3 ϕ i 4 ) - angle variable is the phase of a mode. - mode phases are represented by continuous spins. - 4 body interaction, couplings from short to long range Spherical spin model - fully connected - Ising spins are replaced by real continuous variables σ i, for p=4 - spherical constraint to keep energy finite: Σ i σ 2 i = N, N-dim sphere of N radius. Continuos variable-xorsat Theoretical par N H = i1 >..>i p =1 1, N J i1... i p σ i1...i p where, i={i 1,i 2,i 3, i 4 } These are the optimization problems where we study a diluted p=4 spin model ( both spherical and XY). M. Mezard, G. Parisi, M. A. Virasoro, Spin glass Theory and Beyond, World Scientific Lecture Notes in Physics Vol. 9

8 - SK model - infinite range spin glass model - fully connected - Couplings J are random and non zero - distribution of local fields is Gaussian Replica Symmetric treatment: First Steps Hamiltonian: H J [ s]= Σ 1 i k N J i,k s i s k Σ i hs i Free energy : f n = ( 1 β nn )ln Z n Replicated partition function: (Z J ) n n =Σ s 1 Σ s 2... Σ s nexp Σ a=1 β H J [ s a ] f = β/4(1 q) 2 dz /(2π) 1 /2exp( z 2 /2)ln(2cosh (β q 1/2 z+βh)) Magnetisation: m= dz/(2π) 1/ 2 exp( z 2 /2) tanh(βq 1/2 z+β h) Order parameter : q= dz/(2π) 1/2 exp( z 2 /2) tanh 2 (β q 1 /2 z+β h) Entropy : S(0)= 0.17

9 - Replica Symmetry Breaking solution - Order parameter : probability distribution function P(q): P(q)=m δ(q q 0 )+(1 m)δ(q q 1 ) q k =lim n 0 1/(n(n 1))/2 Σ {a,b} [Q a,b ] k Matrix : Q ab =q 1 if I (a/m)=i (b/m) Q ab =q 0 if I (a/m) I (b/m) Free energy : f (q 0, q 1, m)= β/4 [1+mq 0 2 +(1 m) q 1 2 2q 1 ] ln2 dp q0 ( z)m 1 ln { dp (q1 q 0 ) ( y)cosh m [β( z+ y+h)]} q 1 = dp q0 ( z h) dp q1 q 0 ( y)cosh m [β( z+ y)] tanh 2 (β( z+ y))/ D (z) q 0 = dp q0 ( z h){ dp q1 q 0 ( y)cosh m [β(z+ y)] tanh(β(z+ y))/ D(z)} 2

10 Finite dimensional model System of N Ising spins, Bethe Lattice σ i =±1, i ϵ{1, N } E= Σ i, j J i, j σ i σ j J i, j : independent random variable with probability distribution P(J) Cayley tree: - inhomogeneous - 1 st shell :at i=0, k+1 neighbours --> 2 nd shell each spin connected to k new neighbours... --> L'th shell (boundary). Random graph with fluctuating connectivity: Erdos Renyi graphs - link present : c/n and absent : 1- c/n - where c : number of links connected to a point with Poisson distribution Bethe lattice : -Random graph with fixed connectivity k+1 - its a subset of Cayley tree containing only first few shells - statistically homogeneous - locally tree like, with fixed branching ratio - No small loops, size of loop ~ O ( log N) M. Mezard, G. Parisi, The Bethe lattice spin glass revisited, Eur. Phys. J. B 20, 217 (2001)

11 Cavity Method h 2 h 3 σ 3 τ 3 g 3 σ 2 J 3 K 3 h 1 J 2 σ 1 σ 2 J 1 σ 0 h 2 J 2 σ 0 τ 0 K 2 τ 2 g 2 J 4 σ 4 J 3 σ 3 h 3 h 1 σ 1 J 1 K 1 τ 1 site link g 1 h 4 - mean field approach - equivalent to Replica approach - iterative method on tree like structures - comparison of a N --> N+1 spin system. - adding a new spin at site 0. - site and link contribution. - reshuffling of spins. - reweighing of configurational weights for contribution to new equilibrium state

12 Cavity method equivalent to 1RSB Existence of several pure α : 1 merge k sites to a site i=0, h 0 α field at i=0 :, free energy iid exponential distribution : Initially various states have weights : Field at i=0 calculated as: h 0 = i=1 states are ordered in increasing order of free energy and only lowest M states are considered. When we change the site i, the probability distribution of fields fluctuates and the new probability distribution Q(h) is to be calculated. h i h i α, where i ϵ1,...., N, αϵ 1,..., M N local fields, of which k fields are chosen, k F α W α = exp( β F α ) γ exp( β F γ ) u( J i, h i ) u( j, h)=1/β atanh(tanh (β J ) tanh(β h)) i 1,..,i k of 1,.., N

13 Free Energy σ 1 σ 6 σ 5 σ 1 σ 2 σ 3 σ 4 σ3 σ 4 σ 2 σ 6 σ 5 F = k+1 2 Δ F(link) Δ F ( site) coshβ J ite contribution: β Δ F ( site) ( J 1.. J k +1,h 1.. h k +1 )=ln[2cosh(β Σ k+1 i=1 u ( J i,h i ))]+Σ k+1 i=1 ln[ i cosh(β u( J i, h i )) ] Link contibution : β Δ F (link ) k (J 1.. J k, K 1.. K k,h 1.. h k, g 1.. g k )=Σ i=1 ln k +ln(σ σ 0, τ 0 (β J 0 σ 0 τ 0 + βσ 0 Σ i=1 k u( J i, h i )+ β τ 0 Σ i=1 cosh (β J i )cosh (β K i ) cosh(β u(j i, h i ))cosh (β u( K i, g i )) k u(j i, h i )+ β τ 0 Σ i=1 u( K i, g i )))

14 Site contribution : Field : h 0 =Σ k+1 j=1 u(j j, h j ) Free energy : F (1) = 1 β x ln(1/ M Σ αexp[ β x Δ F α ]) Order parameter : q 1 = 1 M Σ α tanh 2 (β h α ) 1 Order parameter: q 0 = M (M 1) Σ α γ tanh(β h α )tanh(β h γ ) x derivative : d (1) = 1 Σ α exp( β x Δ F α )Δ F α x Σ α exp( β x Δ F α ) Link contribution: Fields: h α k 0 =Σ i=1 u(j i, h i ), g α k 0 =Σ i=1 u(k i, g i ) Free enrgey : F (2) = 1 β x ln(1/ M Σ α exp[ β x Δ F α ]) Order parameter : q 0 (l ) = 1 M Σ α tanh 2 (β v α ) Order parameter : q 1 (l) = x derivative : d (2) = 1 x 1 M (M 1) Σ α γ tanh (β v α ) tanh(β v γ ) Σ α exp( β x Δ F α )Δ F α Σ α exp( β x Δ F α )

15 Numerical par finite dimensional model, finite number of modes in a given sample induced geometry. => simulations. Intensity spectra of random laser--> space and frequency correlations among amplitude and phases. Continuous variables --> Monte Carlo simulations --> GPU's Bethe lattice was mean field model --> modes distance independent. When distance taken into consideration --> computations are complex. Levy graph : P ij = 1 i j ρ i j : distance between the sites ρ 1: short range ρ=0: Bethe lattice

16 Furher developments As a further step--> apart from random lasers --> study interaction between radiative modes in photonics (linear interaction and disorder). Inverse problem --> four point correlation --> recover interaction between modes from experimental data --> correlations between modes. Also, possiblity of secondment in generalisation of inverse problem methods to continuous variables and 4- point correlation Torino

17 Course 1) Coursework in Sapienza University (January- July) : - Structural glasses - Spin glasses - Random graphs - Monte Carlo method 2) Course on GPGPU and CUDA programming, CINECA Bologna (9-10 May). 3) 22nd summer school on Parallel computing, CINECA Bologna (20-31 May). 4) International summer school, fundamental problems in Statistical Mechanics XIII, Leuven, Belgium (16-29 June). 5) Advanced Workshop on Nonlinear Photonics, Disorder and Wave Turbulence ICTP, Trieste (15-19 July).

18 Thank you :)

Information, Physics, and Computation

Information, Physics, and Computation Marc Mezard Laboratoire de Physique Thdorique et Moales Statistiques, CNRS, and Universit y Paris Sud Andrea Montanari Department of Electrical Engineering and Department