Asymmetric wave propagation in the open, discrete nonlinear Schr - PowerPoint PPT Presentation

Asymmetric wave propagation in the open, discrete nonlinear Schr¨ odinger equation Stefano Lepri Istituto dei Sistemi Complessi ISC-CNR Firenze, Italy Stefano Lepri (ISC-CNR) Asymmetric wave propagation 1 / 26

Introduction Principles of control of wave propagation Control of direction: “wave diode” Simple model: Discrete Nonlinear Schr¨ odinger (DNLS) Applications: BEC, photonic/phononic lattices etc. [S.L., G. Casati, Phys. Rev. Lett. 106, 164101 (2011) ] Stefano Lepri (ISC-CNR) Asymmetric wave propagation 2 / 26

The reciprocity theorem ω A, A, ω Lord Rayleigh ”The theory of sound” : Let A and B be two points ... between which are situated obstacles of any kind. Than a sound originating at A is perceived at B with the same intensity as that with which an equal sound originating at B would be perceived at A. In acoustics ... in consequence of the not insignificant value of the wavelength in comparison with the dimension of ordinary obstacles the reciprocal relation is of considerable interest Stefano Lepri (ISC-CNR) Asymmetric wave propagation 3 / 26

The (ideal) “wave diode” ω A, A, ω To violate the reciprocity theorem (without breaking time-reversal) both asymmetry and nonlinearity are necessary ! Stefano Lepri (ISC-CNR) Asymmetric wave propagation 4 / 26

Acoustic rectifier [Liang et al. Nature Materials (2010)] Stefano Lepri (ISC-CNR) Asymmetric wave propagation 5 / 26

Layered photonic or phononic crystal Nonlinear Linear Linear asymmetric For linear propagation perpendicular to the layers: cos k ( d 1 + d 2 ) = cos( ωd 1 ) cos( ωd 2 ) − 1 2( c 1 + c 2 ) sin( ωd 1 ) sin( ωd 2 ) c 1 c 2 c 2 c 1 c 1 c 2 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 6 / 26

DNLS approximation Thin layers d 1 ≪ d 2 : ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω ( k ) = ω 0 ± 2 C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 7 / 26

DNLS approximation Thin layers d 1 ≪ d 2 : ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω ( k ) = ω 0 ± 2 C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0 Altogether: i ˙ φ n = V n φ n − φ n +1 − φ n − 1 + α n | φ n | 2 φ n Conservation of energy and norm, no harmonics. [A. Kosevich, JETP (2001)] Stefano Lepri (ISC-CNR) Asymmetric wave propagation 7 / 26

DNLS approximation Thin layers d 1 ≪ d 2 : ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω ( k ) = ω 0 ± 2 C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0 Altogether: i ˙ φ n = V n φ n − φ n +1 − φ n − 1 + α n | φ n | 2 φ n Conservation of energy and norm, no harmonics. [A. Kosevich, JETP (2001)] Stefano Lepri (ISC-CNR) Asymmetric wave propagation 7 / 26

DNLS approximation Thin layers d 1 ≪ d 2 : ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω ( k ) = ω 0 ± 2 C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0 Altogether: i ˙ φ n = V n φ n − φ n +1 − φ n − 1 + α n | φ n | 2 φ n Conservation of energy and norm, no harmonics. [A. Kosevich, JETP (2001)] Stefano Lepri (ISC-CNR) Asymmetric wave propagation 7 / 26

Transmission problem Stationary DNLS, φ n = ψ n e − iωt , V n � = 0 and α n � = for 1 ≤ n ≤ N ωψ n = V n ψ n − ψ n +1 − ψ n − 1 + α n | ψ n | 2 ψ n R 0 eikn Teikn Re − ikn ψ 1 . . . ψN ω = − 2 cos k, 0 ≤ k ≤ π Stefano Lepri (ISC-CNR) Asymmetric wave propagation 8 / 26

Transmission problem Look for complex solutions such that: R 0 e ikn + Re − ikn � n ≤ 1 ψ n = Te ikn n ≥ N ψ n complex, current J = 2 | T | 2 sin k Convention: k < 0 is for ( V n , α n ) − → ( V N − n +1 , α N − n +1 ) (“flip the sample”) For α n = 0: reciprocity for any V n To break the mirror symmetry: V n � = V N − n +1 and/or α n � = α N − n +1 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 9 / 26

Reduction to nonlinear map Let u n = ψ n and v n = ψ n +1 . Back iterating from u N = T exp( ikN ), v N = T exp( ik ( N + 1)) u n − 1 = − v n + ( V n − ω + α n | u n | 2 ) u n , v n − 1 = u n Map is area preserving. For given T and k exp( − ik ) u 0 − v 0 exp( ik ) u 0 − v 0 R 0 = exp( − ik ) − exp( ik ) , R = exp( ik ) − exp( − ik ) Stefano Lepri (ISC-CNR) Asymmetric wave propagation 10 / 26

Reduction to nonlinear map Let u n = ψ n and v n = ψ n +1 . Back iterating from u N = T exp( ikN ), v N = T exp( ik ( N + 1)) u n − 1 = − v n + ( V n − ω + α n | u n | 2 ) u n , v n − 1 = u n Map is area preserving. For given T and k exp( − ik ) u 0 − v 0 exp( ik ) u 0 − v 0 R 0 = exp( − ik ) − exp( ik ) , R = exp( ik ) − exp( − ik ) Transmission coefficient t ( k, | T | 2 ) = | T | 2 | R 0 | 2 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 10 / 26

The simplest case: the dimer N = 2 For k > 0: e ik − e − ik 2 � � � � t = � 1 + ( ν − e ik )( e ik − δ ) � � � δ = V 2 − ω + α 2 T 2 , ν = V 1 − ω + α 1 T 2 [1 − 2 δ cos k + δ 2 ] . For k < 0: exchange the subscripts 1 and 2 Symmetric case ( V 1 , 2 = V 0 , α 1 , 2 = α ): two nonlinear resonances V 0 + αT 2 = 0 ( V 0 < 0) V 0 + αT 2 = ω ( V 0 < ω ) Stefano Lepri (ISC-CNR) Asymmetric wave propagation 11 / 26

The dimer N = 2: reciprocity breaking V 1 , 2 = − 2 . 5(1 ± ε ) α 1 , 2 = 1 ε = 0.000 5 1 0.9 4 0.8 0.7 3 0.6 |T| 2 0.5 2 0.4 0.3 1 0.2 0.1 0 0 -3 -2 -1 0 1 2 3 k Stefano Lepri (ISC-CNR) Asymmetric wave propagation 12 / 26

The dimer N = 2: reciprocity breaking V 1 , 2 = − 2 . 5(1 ± ε ) α 1 , 2 = 1 ε = 0.050 5 1 0.9 4 0.8 0.7 3 0.6 |T| 2 0.5 2 0.4 0.3 1 0.2 0.1 0 0 -3 -2 -1 0 1 2 3 k Stefano Lepri (ISC-CNR) Asymmetric wave propagation 12 / 26

The dimer N = 2: transmission curves 3.0 2.0 ε = 0 2 |T| 1.0 (a) 0.0 0 1 2 3 2 |R 0 | 1 ε=0 0.8 0.6 2 ) t(k,|T| 0.4 0.2 (b) 0 0 1 2 3 2 |T| V 0 = − 2 . 5 , α = 1 , | k | = 0 . 1 , ε = 0 . 05 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 13 / 26

The dimer N = 2: transmission curves 3.0 W 2 2.0 ε = 0 2 k = + 0.1 |T| k = - 0.1 1.0 W 1 (a) 0.0 0 1 2 3 2 |R 0 | 1 ε=0 k = +0.1 0.8 k = - 0.1 0.6 2 ) t(k,|T| 0.4 0.2 (b) 0 0 1 2 3 2 |T| V 0 = − 2 . 5 , α = 1 , | k | = 0 . 1 , ε = 0 . 05 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 13 / 26

The dimer N = 2: transmission curves 3.0 W 2 0.6 0.4 2.0 0.2 ε = 0 2 k = + 0.1 |T| 0 0 0.2 0.4 0.6 k = - 0.1 1.0 W 1 (a) 0.0 0 1 2 3 2 |R 0 | 1 ε=0 k = +0.1 0.8 k = - 0.1 0.6 2 ) t(k,|T| 0.4 0.2 (b) 0 0 1 2 3 2 |T| V 0 = − 2 . 5 , α = 1 , | k | = 0 . 1 , ε = 0 . 05 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 13 / 26

The dimer N = 2: multistability In the window W 1 k=+0.1 5 2 | ψ n | 4 3 2 1 0 k=-0.1 5 4 3 2 1 0 -100 -50 0 50 100 lattice site n V 0 = − 2 . 5 , α = 1 , | k | = 0 . 1 , ε = 0 . 05 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 14 / 26

The dimer N = 2: multistability In the window W 1 k=+0.1 5 2 | ψ n | 4 3 2 1 0 k=-0.1 5 4 3 2 1 0 -100 -50 0 50 100 lattice site n V 0 = − 2 . 5 , α = 1 , | k | = 0 . 1 , ε = 0 . 05 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 14 / 26

Rectification factor f = t ( k, | T | 2 ) − t ( − k, | T | 2 ) t ( k, | T | 2 ) + t ( − k, | T | 2 ) ε = 0.05 ε = 0.10 5 4 1 3 0.8 |T| 2 2 0.6 1 0.4 0.2 0 ε = 0.20 ε = 0.40 0 5 -0.2 4 -0.4 3 |T| 2 -0.6 2 -0.8 1 -1 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 k k Stefano Lepri (ISC-CNR) Asymmetric wave propagation 15 / 26 V = − 2 . 5, α = 1

Several layers ε = 0.05 ε = 0.10 5 4 1 3 0.8 |T| 2 2 0.6 1 0.4 0.2 0 ε = 0.20 ε = 0.40 0 5 3 -0.2 ε = 0 4 2.5 k = - 0.1 -0.4 k = +0.1 3 2 |T| 2 -0.6 2 1.5 2 |T| -0.8 1 1 -1 0.5 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0 1 2 3 k k 2 |R 0 | N = 4 , V n = − 2 . 5[1 + ε (1 − 2( n − 1) / ( N − 1)] , α = 1 Consequence of the mixed phase-space of the map! Stefano Lepri (ISC-CNR) Asymmetric wave propagation 16 / 26

Nonlinear Anderson model (in progress) N = 10; V n random in [ − ε, ε ] α 1 , 2 = 1 ε = 1 2 1 0.9 0.8 1.5 0.7 0.6 |T| 2 1 0.5 0.4 0.3 0.5 0.2 0.1 0 0 -3 -2 -1 0 1 2 3 k Stefano Lepri (ISC-CNR) Asymmetric wave propagation 17 / 26

Wavepacket transmission Numerical simulation on a finite lattice | n | < M i ˙ φ n = V n φ n − φ n +1 − φ n − 1 + α n | φ n | 2 φ n Initial condition: Gaussian − ( n − n 0 ) 2 � � φ n (0) = I exp + ik 0 n w 2 Transmission coefficient (for n 0 < 0) n>N | φ n ( t fin ) | 2 � t p = n< 0 | φ n (0) | 2 � Stefano Lepri (ISC-CNR) Asymmetric wave propagation 18 / 26

Wavepacket transmission 4 200 3.5 3 150 2.5 2 t 100 1.5 1 50 0.5 0 -400 -200 0 200 400 -400 -200 0 200 400 n n Stefano Lepri (ISC-CNR) Asymmetric wave propagation 19 / 26