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

Asymmetric wave propagation in the open, discrete nonlinear Schr¨

• dinger equation

Stefano Lepri

Istituto dei Sistemi Complessi ISC-CNR Firenze, Italy

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Introduction

Principles of control of wave propagation Control of direction: “wave diode” Simple model: Discrete Nonlinear Schr¨

• dinger (DNLS)

Applications: BEC, photonic/phononic lattices etc. [S.L., G. Casati, Phys. Rev. Lett. 106, 164101 (2011) ]

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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

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The (ideal) “wave diode”

ω A, A,ω

To violate the reciprocity theorem (without breaking time-reversal) both asymmetry and nonlinearity are necessary !

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Acoustic rectifier

[Liang et al. Nature Materials (2010)]

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Layered photonic or phononic crystal

asymmetric Nonlinear Linear Linear

For linear propagation perpendicular to the layers: cos k(d1 + d2) = cos(ωd1 c1 ) cos(ωd2 c2 ) − 1 2(c1 c2 + c2 c1 ) sin(ωd1 c1 ) sin(ωd2 c2 )

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DNLS approximation

Thin layers d1 ≪ d2: ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω(k) = ω0 ± 2C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0

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DNLS approximation

Thin layers d1 ≪ d2: ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω(k) = ω0 ± 2C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0 Altogether: i ˙ φn = Vnφn − φn+1 − φn−1 + αn|φn|2φn Conservation of energy and norm, no harmonics. [A. Kosevich, JETP (2001)]

Stefano Lepri (ISC-CNR) Asymmetric wave propagation 7 / 26

DNLS approximation

Thin layers d1 ≪ d2: ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω(k) = ω0 ± 2C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0 Altogether: i ˙ φn = Vnφn − φn+1 − φn−1 + αn|φn|2φn Conservation of energy and norm, no harmonics. [A. Kosevich, JETP (2001)]

Stefano Lepri (ISC-CNR) Asymmetric wave propagation 7 / 26

DNLS approximation

Thin layers d1 ≪ d2: ”Kronig-Penney model” Approximate dispersion for high-frequency bands: ω(k) = ω0 ± 2C cos kd (single band approx.) Defective layers Kerr nonlinearity Rescale units, band center at ω = 0 Altogether: i ˙ φn = Vnφn − φn+1 − φn−1 + αn|φn|2φn Conservation of energy and norm, no harmonics. [A. Kosevich, JETP (2001)]

Stefano Lepri (ISC-CNR) Asymmetric wave propagation 7 / 26

Transmission problem

Stationary DNLS, φn = ψne−iωt, Vn = 0 and αn = for 1 ≤ n ≤ N ωψn = Vnψn − ψn+1 − ψn−1 + αn|ψn|2ψn

ψ1 . . . ψN Teikn Re−ikn R0eikn

ω = −2 cos k, 0 ≤ k ≤ π

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Transmission problem

Look for complex solutions such that: ψn =

• R0eikn + Re−ikn

n ≤ 1 Teikn n ≥ N ψn complex, current J = 2|T|2 sin k Convention: k < 0 is for (Vn, αn) − → (VN−n+1, αN−n+1) (“flip the sample”) For αn = 0: reciprocity for any Vn To break the mirror symmetry: Vn = VN−n+1 and/or αn = αN−n+1

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Reduction to nonlinear map

Let un = ψn and vn = ψn+1. Back iterating from uN = T exp(ikN), vN = T exp(ik(N + 1)) un−1 = −vn + (Vn − ω + αn|un|2)un, vn−1 = un Map is area preserving. For given T and k R0 = exp(−ik)u0 − v0 exp(−ik) − exp(ik), R = exp(ik)u0 − v0 exp(ik) − exp(−ik)

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Reduction to nonlinear map

Let un = ψn and vn = ψn+1. Back iterating from uN = T exp(ikN), vN = T exp(ik(N + 1)) un−1 = −vn + (Vn − ω + αn|un|2)un, vn−1 = un Map is area preserving. For given T and k R0 = exp(−ik)u0 − v0 exp(−ik) − exp(ik), R = exp(ik)u0 − v0 exp(ik) − exp(−ik) Transmission coefficient t(k, |T|2) = |T|2 |R0|2

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The simplest case: the dimer N = 2

For k > 0: t =

• eik − e−ik

1 + (ν − eik)(eik − δ)

• 2

δ = V2 − ω + α2T 2, ν = V1 − ω + α1T 2[1 − 2δ cos k + δ2]. For k < 0: exchange the subscripts 1 and 2 Symmetric case (V1,2 = V0, α1,2 = α): two nonlinear resonances V0 + αT 2 = 0 (V0 < 0) V0 + αT 2 = ω (V0 < ω)

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The dimer N = 2: reciprocity breaking

V1,2 = −2.5(1 ± ε) α1,2 = 1

1 2 3 4 5

• 3
• 2
• 1

1 2 3 |T|2 k ε= 0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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The dimer N = 2: reciprocity breaking

V1,2 = −2.5(1 ± ε) α1,2 = 1

1 2 3 4 5

• 3
• 2
• 1

1 2 3 |T|2 k ε= 0.050 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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The dimer N = 2: transmission curves

1 2 3

|R0|

2

0.0 1.0 2.0 3.0

|T|

2

ε = 0 (a)

1 2 3

|T|

2

0.2 0.4 0.6 0.8 1

t(k,|T|

2)

ε=0 (b)

V0 = −2.5, α = 1, |k| = 0.1, ε = 0.05

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The dimer N = 2: transmission curves

1 2 3

|R0|

2

0.0 1.0 2.0 3.0

|T|

2

ε = 0 k = + 0.1 k = - 0.1 W1 W2 (a)

1 2 3

|T|

2

0.2 0.4 0.6 0.8 1

t(k,|T|

2)

ε=0 k = +0.1 k = - 0.1 (b)

V0 = −2.5, α = 1, |k| = 0.1, ε = 0.05

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The dimer N = 2: transmission curves

1 2 3

|R0|

2

0.0 1.0 2.0 3.0

|T|

2

ε = 0 k = + 0.1 k = - 0.1

0.2 0.4 0.6 0.2 0.4 0.6

W1 W2 (a)

1 2 3

|T|

2

0.2 0.4 0.6 0.8 1

t(k,|T|

2)

ε=0 k = +0.1 k = - 0.1 (b)

V0 = −2.5, α = 1, |k| = 0.1, ε = 0.05

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The dimer N = 2: multistability

In the window W1

1 2 3 4 5

• 100
• 50

50 100

lattice site n

1 2 3 4 5

k=+0.1 k=-0.1 |ψn|

2

V0 = −2.5, α = 1, |k| = 0.1, ε = 0.05

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The dimer N = 2: multistability

In the window W1

1 2 3 4 5

• 100
• 50

50 100

lattice site n

1 2 3 4 5

k=+0.1 k=-0.1 |ψn|

2

V0 = −2.5, α = 1, |k| = 0.1, ε = 0.05

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Rectification factor

f = t(k, |T|2) − t(−k, |T|2) t(k, |T|2) + t(−k, |T|2)

1 2 3 4 5 |T|2 ε= 0.05 ε= 0.10

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1 2 3 4 5 0.5 1 1.5 2 2.5 3 |T|2 k ε= 0.20 0.5 1 1.5 2 2.5 3 k ε= 0.40

V = −2.5, α = 1

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Several layers

1 2 3 |R0|

2

0.5 1 1.5 2 2.5 3

|T|

2

ε = 0 k = - 0.1 k = +0.1 1 2 3 4 5 |T|2 ε= 0.05 ε= 0.10

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1 2 3 4 5 0.5 1 1.5 2 2.5 3 |T|2 k ε= 0.20 0.5 1 1.5 2 2.5 3 k ε= 0.40

N = 4, Vn = −2.5[1 + ε(1 − 2(n − 1)/(N − 1)], α = 1 Consequence of the mixed phase-space of the map!

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Nonlinear Anderson model (in progress)

N = 10; Vn random in [−ε, ε] α1,2 = 1

0.5 1 1.5 2

• 3
• 2
• 1

1 2 3 |T|2 k ε= 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Wavepacket transmission

Numerical simulation on a finite lattice |n| < M i ˙ φn = Vnφn − φn+1 − φn−1 + αn|φn|2φn Initial condition: Gaussian φn(0) = I exp

• −(n − n0)2

w2 + ik0n

• Transmission coefficient (for n0 < 0)

tp =

• n>N |φn(tfin)|2
• n<0 |φn(0)|2

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Wavepacket transmission

50 100 150 200

• 400 -200

200 400 t n

• 400 -200

200 400 n 0.5 1 1.5 2 2.5 3 3.5 4

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Wavepacket transmission

1 2 3 4 5

|R0|

2

0.2 0.4 0.6 0.8 1 wave transmission coefficient t

k=+π/2 k=-π/2 N=2 ε=0.05 ω=0 Stefano Lepri (ISC-CNR) Asymmetric wave propagation 20 / 26

Wavepacket transmission

1 2 3 4 5

|I|

2

0.2 0.4 0.6 0.8 1

tp

Right-incoming Left-incoming

V0 = −2.5, |k0| = 1.57, ε = 0.05, M = 500, n0 = ±250, w = 102

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Nonreciprocal chaotic scattering?

Sensitive dependence on scattering parameters (e.g. the width w)

1 2 3 4 5

|I|

2

0.1 0.2 0.3 0.4 0.5 0.6

tp w=5

1 2 3 4 5

|I|

2

0.2 0.4 0.6 0.8

tp w=100 V0 = −2.5, |k0| = 1.57, ε = 0.05, M = 500, n0 = ±250

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Wavepacket transmission

1 2 3 0.1 0.2 0.3 0.4 Spectrum at t=0 Spectrum at t=250 1 2 3

V0=-2.5 I

2=3 k=1.57 ε=0.05

k

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Qualitative explanation

Integrate out the ψn, n ≤ 0 and n > N the problem is equivalent to the boundary conditions ψ0(t) = F0(t) − i t G(t − s)ψ1(s)ds ψN+1(t) = −i t G(t − s)ψN(s)ds Memory term: G(t) ≡ J1(2t)

t

. F0(t) is determined by the inital condition i.e. by the incoming wavepacket. From open Hamiltonian problem to driven, dissipative. Qualitatively: small w changes the initial condition only; large w closer to steady driving (transient chaos?).

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PT symmetric DNLS (Vn = V ∗

N−n+1)

V1 = iγ, V2 = −iγ

0.5 1 1.5 2 2.5 3 0.5 1 1.5 |R0|2 t k0 T −2 2 0.5 1 1.5 2 0.5 1 1.5 k0 T 1 2 3 0.5 1 1.5 2 −0.5 0.5

[J. D’Ambroise, P.G. Kevrekidis, S.L. (2012)]

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Conclusions

Simple modeling of “wave diode” Application to layered photonic or phononic crystals Nonlinear resonances and multistability Nonreciprocal wavepacket trasmission (chaotic?) Dynamical stability and bifurcations?

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