Localized modes in the nonlinear Schrodinger equation with...
Transcript of Localized modes in the nonlinear Schrodinger equation with...
Localized modes in the nonlinearSchr odinger equation with periodicnonlinearity and periodic potential
Vladimir Konotop
University of Lisbon, PORTUGAL
Yu. V. Bludov , VVK, Phys. Rev. A 74, 043616 (2006)
Yu. V. Bludov, V.A. Brazhnyi , VVK, Phys Rev A 76 (2007) (in press;
cond-mat:0706.0079)
F. Kh. Abdullaev , Yu. V. Bludov, S. V. Dmitriev , P. G. Kevrekidis , VVK,
(submitted; cond-mat: 0707.2512)Nonlinear Schrodinger equation with periodic coefficients – p. 1/17
Outline
Physical applications
Modulational instability and localized modes
Delocalizing transition
Lattices: "Tight-binding" approximation
Nonlinear Schrodinger equation with periodic coefficients – p. 2/17
EM waves in stratified media
∂2E
∂x2+∂2E
∂z2+ω2
c2n2E = 0,
where n = n0 + n1(x, z) + n2(x, z)|E|2 + n4(x, z)|E|4.In the parabolic approximation E(x, z) = eikzA(x, z), k = ω
cn0,
Az ≪ kA
2ik∂A
∂z+∂2A
∂x2+k2
(
2n1
n0
+n2
1
n20
+2n2
n0
|A|2 +
(
n22
n20
+2n4
n0
)
|A|4)
A = 0
Nonlinear Schrodinger equation with periodic coefficients – p. 3/17
EM waves in stratified media
∂2E
∂x2+∂2E
∂z2+ω2
c2n2E = 0,
where n = n0 + n1(x, z) + n2(x, z)|E|2 + n4(x, z)|E|4.In the parabolic approximation E(x, z) = eikzA(x, z), k = ω
cn0,
Az ≪ kA
2ik∂A
∂z+∂2A
∂x2+k2
(
2n1
n0
+n2
1
n20
+2n2
n0
|A|2 +
(
n22
n20
+2n4
n0
)
|A|4)
A = 0
Nonlinear Schrodinger equation with periodic coefficients – p. 3/17
EM waves in stratified media
∂2E
∂x2+∂2E
∂z2+ω2
c2n2E = 0,
where n = n0 + n1(x, z) + n2(x, z)|E|2 + n4(x, z)|E|4.In the parabolic approximation E(x, z) = eikzA(x, z), k = ω
cn0,
Az ≪ kA
2ik∂A
∂z+∂2A
∂x2+k2
(
2n1
n0
+n2
1
n20
+2n2
n0
|A|2 +
(
n22
n20
+2n4
n0
)
|A|4)
A = 0
Let n1(x, z) ≡ n1(x), n2(x, z) ≡ n2(x), and n4 ≡ 0, then
2ik∂A
∂z+∂2E
∂x2+ 2k2
(
n1(x)
n0
+n2(x)
n0
|A|2)
A = 0
or after renormalization iψt = −ψxx + V (x)ψ +G(x)|ψ|2ψNonlinear Schrodinger equation with periodic coefficients – p. 3/17
BEC in an optical lattice
Heisenberg equation
i~Ψt = − ~2
2m∆Ψ + Vext(r)Ψ + g(r)Ψ†
ΨΨ
Assumption: Ψ = Ψ + ψ with Ψ ≈ 〈N |Ψ|N + 1〉Mean-field approximation (Gross-Pitaevskii equation)
i~Ψt = − ~2
2m∆Ψ + Vext(r)Ψ + g(r)|Ψ|2Ψ
One-dimensional limit
iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψ[Fedichev, Kagan, Shlyapnikov, PRL, 77, 2913 (1996); Abdullaev and Garnier, PRA 72,061605 (2005)]
Nonlinear Schrodinger equation with periodic coefficients – p. 4/17
Boson-fermion mixture in OL
Heisenberg equations
i~Ψt = − ~2
2mb
∆Ψ + VbΨ + g1Ψ†ΨΨ + g2Φ
†ΦΨ ,
i~Φt = − ~2
2mf
∆Φ + Vf Φ + g2Ψ†ΨΦ
Mean-field approximation: Ψ = 〈Ψ〉, 〈Φ†Φ〉 = ρ0(r) + ρ1(r, t)
[Tsurumi, Wadati, J. Phys. Soc. Jap. 69 97 (2000); Bludov, Konotop, PRA 74, 043616 (2006)]
i~∂Ψ
∂t= − ~
2
2mb
∆Ψ + Vb(r)Ψ + gbb|Ψ|2Ψ + gbfρΨ
∂2ρ1
∂t2= ∇
[
ρ0(r)∇(
(6π2)2/3~
2
3m2fρ
1/30 (r)
ρ1 +gbfmf
|Ψ|2)]
.
ρ0(r) is an unperturbed (Thomas-Fermi) distribution of fermions
Nonlinear Schrodinger equation with periodic coefficients – p. 5/17
BF mixture in OL
1D limit [Bludov, Konotop, PRA 74, 043616 (2006)]
iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψU(x) ≡ U0(x) + U1(x), G(x) = G0 − G1
1/3(x),
Here U0(x) = Vb(x)/(~2κ2/2mb), U1 = 4πκabfmb/m,
G0 = 4abbNb/κa2, and G1 = 2 (6/π)1/3 (abf/a)
2(mfmb/m2)Nb
(x) = ρ0(x)/κ
Λ = κabb
Nonlinear Schrodinger equation with periodic coefficients – p. 6/17
NLS equation with periodic nonlinearity
iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψ,
U(x) = U(x+ π), G(x) = G(x+ π)
The linear eigenvalue problem
−d2ϕ
(σ)n
dx2+ U(x)ϕ(σ)
n = E (σ)n ϕ(σ)
n ,(
E (−)α , E (+)
α
)
is α’s gap
Multiple scale expansion: ψ ≈ ǫA(τ, ξ)ϕ(σ)n (x) where ξ = ǫx
and τ = ǫ2t are the slow variables, ǫ ∼ |µ− E (σ)n | ≪ 1
NLS equation: iAτ = −(2M(σ)n )−1Aξξ + χ
(σ)n |A|2A
M(σ)n = [d2E (σ)
n /dk2]−1 is the effective massχ
(σ)n =
∫ π
0G(x)|ϕ(σ)
n (x)|4dx is the effective nonlinearity
For the modulational instability: M (σ)α χ
(σ)α < 0
Nonlinear Schrodinger equation with periodic coefficients – p. 7/17
NLS equation with periodic nonlinearity
If γ(−)2 < Λ < γ
(+)1 : gap
solitons do not exist
Since Gm = minG(X) < 0,in the limit E → −∞, thereexists a soliton:
φS(X) ≈ e−iET
p
2|E|/|Gm|
cosh(Xp
|E|)
If χ(−)1 > 0 in the semi-infinite
band there exists a minimalnumber of bosons necessaryfor creation of a localizedmode [Sakaguchi, Malomed,Phys. Rev. A 72, 046610(2005) ]
Nonlinear Schrodinger equation with periodic coefficients – p. 8/17
NLS equation with periodic nonlinearity
All bright localized modes are real [Alfimov, VVK, Salerno, Europhys. Lett. 58, 7
(2002), review Brazhnyi, VVK, Mod. Phys. Lett. B 14 627 (2004)]
Nonlinear Schrodinger equation with periodic coefficients – p. 9/17
NLS equation with periodic nonlinearity
The first lowest gap
Nonlinear Schrodinger equation with periodic coefficients – p. 10/17
Delocalizing transition
Decrease of the linear lattice potential in the NLS equation results in delocalizingtransition in 2D and 3D, but no transition occurs in 1D. [Kalosakas, Rasmussen,Bishop, PRL 89, 030402 (2002); Baizakov, Salerno, PRA 69, 013602 (2004).]
iψt = −ψxx + U(x)ψ + |ψ|2ψ
V (x) = −A cos(2x)
k
E
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
A= 3
α=0
α=1
Nonlinear Schrodinger equation with periodic coefficients – p. 11/17
Delocalizing transition
Consider iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψ
Recall M (σ)n = [d2E (σ)
n /dk2]−1, and χ(σ)n =
∫ π
0G(x)|ϕ(σ)
n (x)|4dxIf M (+)
n χ(+)n > 0 and M (−)
n χ(−)n > 0, small amplitude solitons
cannot exist at the both gap edges.
Let
U(x) = −V cos(2x)
G(x) = G− cos(2x)
A stationary solution:
ψ(x, t) → ψ(x)e−iµtG
(σ)n = G
(σ)n (V ) is the value of G at which
χ(σ)n becomes zero.
Nonlinear Schrodinger equation with periodic coefficients – p. 12/17
Delocalizing transition
Let V is changing and G is fixed
The chemical potential is µ = (E + Enl)/N where
E =
∫(
|ψx|2 + U(x)|ψ|2 +1
2
∫
G(x)|ψ|4)
dx
Enl =1
2
∫
G(x)|ψ|4dx
Nonlinear Schrodinger equation with periodic coefficients – p. 13/17
Delocalizing transition
Let G is changing and V is fixed
Nonlinear Schrodinger equation with periodic coefficients – p. 14/17
Tight-binding approximation
Wannier functions
wnα(x) =1√2
∫ 1
−1
ϕαq(x)e−iπnq dq
The expansion: ψ(x, t) =∑
n,α cnα(t)wnα(x) leads to
icnα − cnαω0α − (cn−1,α + cn+1,α)ω1α−∑
n1,n2,n3
cn1αcn2αcn3αWnn1n2n3αααα = 0
where Eαq =∑
n ωnαeiπnq, ωnα = 1
2
∫ 1
−1Eαqe−iπnqdq
W nn1n2n3αα1α2α3
=
∫ ∞
−∞
G(x)wnα(x)wn1α1(x)wn2α2(x)wn3α3(x)dx
Nonlinear Schrodinger equation with periodic coefficients – p. 15/17
Tight-binding approximation
icn = ω0cn + ω1(cn−1 + cn+1) +W0|cn|2cn+W1
(
|cn−1|2cn−1 + σcn−1c2n + 2σ|cn|2cn−1
+2|cn|2cn+1 + cn+1c2n + σ|cn+1|2cn+1
)
+W2
(
2|cn−1|2cn + cnc2n−1 + cnc
2n+1 + 2|cn+1|2cn
)
,
with
Wj =
∫ ∞
−∞
G(x)wj1α(x)w4−j0α (x)dx j = 1, 2
Nonlinear Schrodinger equation with periodic coefficients – p. 16/17
Tight-binding approximation
t
x
0 20 40 60 80 100
−10
−5
0
5
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t
n
0 20 40 60 80 100
−10
−5
0
5
10 0
0.2
0.4
0.6
0.8
cn(t) = A (−i)n exp(−iω0t)Jn(2ω1t)
Nonlinear Schrodinger equation with periodic coefficients – p. 17/17