Spreading mechanism of wave packets in one dimensional disordered Klein-Gordon chains

Transcription

1 Spreading mechanism of wave packets in one dimensional disordered Klein-Gordon chains Haris Skokos Max Planck Institute for the Physics of Complex Systems Dresden, Germany URL: Work in collaboration with Sergej Flach, Dima Krimer and Stavros Komineas

2 Outline The quartic Klein-Gordon (KG) disordered lattice Computational methods Three different dynamical behaviors Numerical results Similarities with the disordered nonlinear Schrödinger equation (DNLS) Conclusions H. Skokos Chemnitz, 7 May 009

3 Interplay of disorder and nonlinearity Waves in disordered media Anderson localization (Anderson Phys. Rev. 1958). Experiments on BEC (Billy et al. Nature 008) Waves in nonlinear disordered media localization or delocalization? Theoretical and/or numerical studies (Shepelyansky PRL 1993, Molina Phys. Rev. B 1998, Pikovsky & Shepelyansky PRL 008, Kopidakis et al. PRL 008) Experiments: propagation of light in disordered 1d waveguide lattices (Lahini et al. PRL 008) H. Skokos Chemnitz, 7 May 009 3

4 The Klein Gordon (KG) model N p l εl K l + ul ul+1 - ul l=1 4 W H = + u + ( ) with fixed boundary conditions u 0 =p 0 =u N+1 =p N+1 =0. Usually N= Parameters: W and the total energy E. ε l chosen uniformly from,. Linear case (neglecting the term u l4 /4) Ansatz: u l =A l exp(iωt) Eigenvalue problem: λa l = ε l A l -(A l+1 + A l-1 ) with λ =Wω -W -, ε =W(ε -1) Unitary eigenvectors (normal modes - NMs) A ν,l are ordered according X = to their center-of-norm coordinate: ν ν l N la l=1,l All eigenstates are localized (Anderson localization) having a localization length which is bounded from above. l H. Skokos Chemnitz, 7 May 009 4

5 Scales ω, width of the squared frequency spectrum: ν, + W K 1 Localization volume of eigenstate: Average spacing of squared eigenfrequencies of NMs within the range of a localization volume: Δ Δω = p K ν p = 4 Δ =1+ W For small values of W we have Nonlinearity induced squared frequency shift of a single site oscillator δ = l Δω The relation of the two scales Δω Δ K with the nonlinear frequency shift δ l determines the packet evolution. H. Skokos Chemnitz, 7 May ν 3E ε N l=1 W l l E A 4 ν,l

6 Distribution characterization We consider normalized energy distributions in normal mode (NM) space z ν Eν m m of the νth NM. 1 E with E ( ) ν = A ν +ων Aν N ν ν=1 Second moment: ( ) Participation number:, where A ν is the amplitude m = ν - ν z N with ν = ν z ν P= N z ν=1 ν measures the number of stronger excited modes in z ν. Single mode P=1, Equipartition of energy P=N. H. Skokos Chemnitz, 7 May ν =1

7 Integration scheme We use a symplectic integration scheme developed for Hamiltonians of the form H=A+εB where A, B are both integrable and ε a parameter. Formally the solution of the Hamilton equations of motion for such a system can be written as: n dx { } s slh = H,X = LHX X = LHX =e X ds n! f n i i where X is the full coordinate vector and L H the Poisson operator: n³0 3 H f H f L H f = - p q q p j=1 j j j j H. Skokos Chemnitz, 7 May 009 7

8 Symplectic Integrator SABA C The operator e sl Hcan be approximated by the symplectic integrator (Laskar & Robutel, Cel. Mech. Dyn. Astr., 001): SABA =e e e e e csl 1 A dsl 1 εb csl A dsl 1 εb csl 1 A with c 1 = -, c =, d 1 =. 6 3 The integrator has only small positive steps and its error is of order O(s 4 ε+s ε ). In the case where A is quadratic in the momenta and B depends only on the positions the method can be improved by introducing a corrector C, having a small negative step: 3 c -s ε L {{ A,B },B} C=e - 3 with c=. 4 Thus the full integrator scheme becomes: SABAC = C (SABA ) C and its error is of order O(s 4 ε+s 4 ε ). H. Skokos Chemnitz, 7 May 009 8

9 Symplectic Integrator SABA C We apply the SABAC integrator scheme to the KG Hamiltonian by using the splitting: N p l A= l=1 N ε l B = u + u + 4 W u -u ε =1 l=1 ( ) l l l+1 l with a corrector term which corresponds to the Hamiltonian function: { } N 1 l l l l-1 l+1 l l=1 W { } A,B,B = u ( ε + u ) ( u +u -u ). H. Skokos Chemnitz, 7 May 009 9

10 slope 1/3 E = 0.05, 0.4, W = 4. Single site excitations Regime I: Small values of nonlinearity. δ l < Δω frequency shift is less than the average spacing of interacting modes. Localization as a transient (like in the linear case), with subsequent subdiffusion. slope 1/6 Regime II: Intermediate values of nonlinearity. Δω < δ resonance overlap may happen l < ΔK immediately. Immediate subdiffusion (Molina Phys. Rev. B 1998, Pikovsky & Shepelyansky PRL 008). Regime III: Big nonlinearities. δ l > Δ K frequency shift exceeds the spectrum width. Some frequencies of NMs are tuned out of resonances with the NM spectrum, leading to selftrapping, while a small part of the wave packet subdiffuses (Kopidakis et al. PRL 008). a a/ Subdiffusion: m t, P t Assuming that the spreading is due to heating of the cold exterior, induced by the chaoticity of the wave packet, we theoretically predict α=1/3. H. Skokos Chemnitz, 7 May

11 Different spreading regimes The fraction of the wave packet that spreads decreases with increasing nonlinearity. Second moment m Participation number P The detrapping time increases with increasing W. H. Skokos Chemnitz, 7 May

12 W=4, E=10 W=4, E=1.5 W=4, E=0.4 W=4, E=0.05 H. Skokos Chemnitz, 7 May 009 1

13 W=4, E=0.05 W=6, E=0.05 W=1, E=0.05 H. Skokos Chemnitz, 7 May

14 W=4, E=0.4 W=16, E=0.4 W=18, E=0.4 H. Skokos Chemnitz, 7 May

15 W=4, E=1.5 W=6, E=1.5 W=15, E=1.5 H. Skokos Chemnitz, 7 May

16 Accuracy H. Skokos Chemnitz, 7 May

17 Accuracy H. Skokos Chemnitz, 7 May

18 Relation of regimes I and II We start a single site excitation in the intermediate regime II, measure the distribution at some time t d, and relaunch the distribution as an initial condition at time t=0. m(t)=m(t+t ) d Detrapping time τ d t d 1/3 slope 1/3 H. Skokos Chemnitz, 7 May

19 Detrapping Nonlocal excitations of the KG chain corresponding to initial homogeneous distributions of energy E=0.4 (regime II of single site excitation) over L neighboring sites. Assuming that m ~t 1/3 and that spreading is due to some diffusion process we conclude for the diffusion rate D that D=τ -1 d ~n 4 where n is the average energy density of the excited NMs, i.e. L~n -1. Thus we expect: τ d ~L 4 slope 4 α = τ d =105 H. Skokos Chemnitz, 7 May

20 The discrete nonlinear Schrödinger We also consider the system: (DNLS) equation β H = ε ψ + ψ - ψ ψ + ψ ψ N 4 ( * * ) D l l l l+1 l l+1 l l=1 W W where ε l chosen uniformly from, and β is the nonlinear parameter. The parameters of the KG and the DNLS models was chosen so that the linear parts of both Hamiltonians would correspond to the same eigenvalue problem. Conserved quantities: The energy and the norm of the wave packet. H. Skokos Chemnitz, 7 May 009 0

21 Similar behavior of DNLS DNLS KG slope 1/3 slope 1/3 slope 1/6 slope 1/6 Single site excitations Regimes I, II, III In regime II we averaged the measured exponent α over 0 realizations: α=0.33±0.05 (KG) α=0.33±0.0 (DLNS) H. Skokos Chemnitz, 7 May 009 1

22 Similar behavior of DNLS DNLS slope 1/3 KG slope 1/3 Single mode excitations Regimes I, II, III slope 1/6 slope 1/6 H. Skokos Chemnitz, 7 May 009

23 Evolution of the second moment We use the DNLS model for theoretical considerations. Equations of motion in normal mode space: * i ϕ = λ ϕ + β I ϕ ϕ ϕ μ μ μ ν,ν,ν,ν ν ν ν ν,ν,ν with the overlap integral: I ν,ν,ν,ν = Av,l Av,l Av,l Av,l Assume that at some time t the wave packet contains and each mode on average has a norm ϕν Then for the second moment we have: l 1/n >> P ν n<<1. m = Dt 1/n. The heating of an exterior mode φ μ should evolve as: 3/ i ϕ λ ϕ + Fβn f(t) where f(t)f(t ) = δ (t - t ) μ μ μ modes with F being the fraction of resonant modes inside the packet, and φ μ being a mode at the cold exterior. Then 3 ϕ μ F β nt. H. Skokos Chemnitz, 7 May 009 3

24 Dephasing Let us assume that all modes in the packet are chaotic, i.e. F=1. The momentary diffusion rate of packet equals the inverse time the exterior mode needs to heat up to the packet level: 1 D β n m 1/ t T We test this prediction by additionally dephasing the normal modes: DNLS W=4, β=3 W=7, β=4 W=10, β=6 slope1/ slope1/ KG W=10, E=0.5 W=7, E=0.3 W=4, E=0.4 slope 1/3 slope 1/3 H. Skokos Chemnitz, 7 May 009 4

25 Evolution of the second moment Thus not all modes in the wave packet evolve chaotically. We numerically estimated the probability F for a mode, which is excited to a norm n (the average norm density in the packet), to be resonant with at least one other mode and found: So we get: F βn 1 D β n m t T 4 4 1/3 H. Skokos Chemnitz, 7 May 009 5

26 Conclusions Chart of different dynamical behaviors: Weak nonlinearity: Anderson localization on finite times. After some detrapping time the wave packet delocalizes (Regime I) Intermediate nonlinearity: wave packet delocalizes without transients (Regime II) Strong nonlinearity: partial localization due to selftrapping, but a (small) part of the wave packet delocalizes (Regime III) Subdiffusive spreading induced by the chaoticity of the wave packet Second moment of wave packet ~ t α with α=1/3 Spreading is universal due to nonintegrability and the exponent α does not depend on strength of nonlinearity and disorder Conjecture: Anderson localization is eventually destroyed by the slightest amount of nonlinearity? S. Flach, D.O. Krimer, Ch. Skokos, 009, PRL, 10, Ch. Skokos, D.O. Krimer, S. Komineas, S. Flach, 009, PRE, 79, H. Skokos Chemnitz, 7 May 009 6