The nonlinear Schrödinger equation (NLSE)
on metric graphs
Sven Gnutzmann (Nottingham) & Uzy Smilansky (Weizmann)
Workshop on Random-Matrix Theory, Brunel, 19 December 2009
The nonlinear Schrödinger equation on metric graphs
Physical applications of the NLSE
Shallow water waves
The nonlinear Schrödinger equation on metric graphs
Physical applications of the NLSE
Shallow water waves
Bose-Einstein Condensates and
Superfluids
The nonlinear Schrödinger equation on metric graphs
Physical applications of the NLSE
Shallow water waves
Bose-Einstein Condensates and
Superfluids
Nonlinear optics
Fibre optics
The nonlinear Schrödinger equation on metric graphs
Metric Graph
quasi one-dimensional trap or waveguide for ultra-cold atoms
network of optical fibres
The nonlinear Schrödinger equation on metric graphs
Waves on metric graphs
The nonlinear Schrödinger equation on metric graphs
Waves on metric graphs
linear waves: quantum graphs
Schrödinger/Helmholtz equation, Dirac equation,
Bogoliubov-de Gennes equation.
The nonlinear Schrödinger equation on metric graphs
Waves on metric graphs
linear waves: quantum graphs
Schrödinger/Helmholtz equation, Dirac equation,
Bogoliubov-de Gennes equation.
nonlinear waves: nonlinear quantum graph (?)
Nonlinear Schrödinger/Gross-Pitaevski, Burger’s,
Korteweg-de Vries, Sine-Gordon
The nonlinear Schrödinger equation on metric graphs
The stationary NLSE in one dimension
−ψ ′′ (x) + g |ψ(x)|2 ψ(x) = E ψ(x)
The nonlinear Schrödinger equation on metric graphs
The stationary NLSE in one dimension
−ψ ′′ (x) + g |ψ(x)|2 ψ(x) = E ψ(x)
coupling constant g
may be positive (repulsive) or negative (attractive)
’energy’ E
The nonlinear Schrödinger equation on metric graphs
The stationary NLSE in one dimension
−ψ ′′ (x) + g |ψ(x)|2 ψ(x) = E ψ(x)
coupling constant g
may be positive (repulsive) or negative (attractive)
’energy’ E
integrable, stationary solutions are known (Carr et al , PRA
2001)
The nonlinear Schrödinger equation on metric graphs
Classical dynamics picture for NLSE for E > 0
τ=
√
E x,
ψ(x) =
s
2E
r (τ )e iθ(τ ) ,
|g |
σ=
g
|g |
The nonlinear Schrödinger equation on metric graphs
Classical dynamics picture for NLSE for E > 0
τ=
√
E x,
ψ(x) =
s
2E
r (τ )e iθ(τ ) ,
|g |
NLSE is generated by Lagrangian
L=
ṙ 2 + r 2 θ̇2 r 2 + σr 4
−
2
2
σ=
g
|g |
The nonlinear Schrödinger equation on metric graphs
Classical dynamics picture for NLSE for E > 0
τ=
√
ψ(x) =
E x,
s
2E
r (τ )e iθ(τ ) ,
|g |
NLSE is generated by Lagrangian
L=
ṙ 2 + r 2 θ̇2 r 2 + σr 4
−
2
2
conjugate momenta
p=
ℓ=
∂L
∂ ṙ
∂L
∂ θ̇
= ṙ : radial momentum
= r 2 θ̇: angular momentum (conserved)
σ=
g
|g |
The nonlinear Schrödinger equation on metric graphs
Classical dynamics picture for NLSE for E > 0
τ=
√
ψ(x) =
E x,
s
2E
r (τ )e iθ(τ ) ,
|g |
NLSE is generated by Lagrangian
L=
ṙ 2 + r 2 θ̇2 r 2 + σr 4
−
2
2
conjugate momenta
p=
ℓ=
∂L
∂ ṙ
∂L
∂ θ̇
= ṙ : radial momentum
= r 2 θ̇: angular momentum (conserved)
angular momentum → flux
3/2
ψ ∗ ψ ′ = 2E|g | (rp + iℓ)
σ=
g
|g |
The nonlinear Schrödinger equation on metric graphs
Classical dynamics picture for NLSE for E > 0
τ=
√
ψ(x) =
E x,
s
2E
r (τ )e iθ(τ ) ,
|g |
NLSE is generated by Lagrangian
L=
ṙ 2 + r 2 θ̇2 r 2 + σr 4
−
2
2
conjugate momenta
p=
ℓ=
∂L
∂ ṙ
∂L
∂ θ̇
= ṙ : radial momentum
= r 2 θ̇: angular momentum (conserved)
angular momentum → flux
3/2
ψ ∗ ψ ′ = 2E|g | (rp + iℓ)
If ℓ = 0 one may choose ψ to be real
σ=
g
|g |
The nonlinear Schrödinger equation on metric graphs
Integration of the stationary NLSE
Two constants of motion
ℓ = r 2 θ̇
EHam =
ṙ 2
+ Veff (r , ℓ)
2
The nonlinear Schrödinger equation on metric graphs
Integration of the stationary NLSE
Two constants of motion
ℓ = r 2 θ̇
EHam =
with effective potential
Veff (r ; ℓ) =
ℓ2
2r 2
+
r2
2
4
− σ r2
ṙ 2
+ Veff (r , ℓ)
2
The nonlinear Schrödinger equation on metric graphs
Integration of the stationary NLSE
Two constants of motion
ℓ = r 2 θ̇
EHam =
ṙ 2
+ Veff (r , ℓ)
2
with effective potential
Veff (r ; ℓ) =
ℓ2
2r 2
+
r2
2
4
− σ r2
τ − τ0 =
Z
1
p
dr
2(H − Veff (r ; ℓ))
The nonlinear Schrödinger equation on metric graphs
Integration of the stationary NLSE
Two constants of motion
ℓ = r 2 θ̇
EHam =
ṙ 2
+ Veff (r , ℓ)
2
with effective potential
Veff (r ; ℓ) =
ℓ2
2r 2
+
r2
2
4
− σ r2
τ − τ0 =
Z
θ − θ0 = ℓ
Z
1
p
dr
2(H − Veff (r ; ℓ))
r2
1
p
dr
2(H − Veff (r ; ℓ))
The nonlinear Schrödinger equation on metric graphs
Amplitude dependence – attractive case
ℓ = 0: → real solution
|g |
max (|ψ(x)|2 ) = max (r 2 )
One parameter: I0 = 2E
³p
´
s s
I0
sn
(2I
+
1)E
x,
0
2I0 +1
2E I0 (I0 + 1)
´
³p
ψ(x) =
|g |
2I0 + 1 dn
(2I + 1)E x, I0
0
2I0 +1
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
1
2
3
4
5
6
The nonlinear Schrödinger equation on metric graphs
Amplitude dependence – repulsive case
ψ(x) =
s
1.2
´
 ³ p
I0

sn
2
(1
−
I
)E
x,
0
 “
1−I0
”
2EI0
×

|g |

√
2I0 +1
sn 2 I0 E x, 4I
0
”
“ √
2I0 +1
1+cn 2 I0 E x, 4I
0
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2
4
6
8
10
12
14
16
18
20
for I0 < 1/2
for I0 > 1/2
The nonlinear Schrödinger equation on metric graphs
NLSE on a metric graph
vertices i = 1, 2, . . . , V
edges may either be finite bonds b ≡ (i, j) that connect two
vertices i and j
or infinite leads l = (i, ∞) that start a vertex i and go to
infinity.
The nonlinear Schrödinger equation on metric graphs
NLSE on a metric graph
vertices i = 1, 2, . . . , V
edges may either be finite bonds b ≡ (i, j) that connect two
vertices i and j
or infinite leads l = (i, ∞) that start a vertex i and go to
infinity.
metric: each bond b has a coordinate 0 ≤ xb ≤ Lb < ∞ and
each lead a coordinate 0 ≤ xl < ∞
The nonlinear Schrödinger equation on metric graphs
NLSE on a metric graph
vertices i = 1, 2, . . . , V
edges may either be finite bonds b ≡ (i, j) that connect two
vertices i and j
or infinite leads l = (i, ∞) that start a vertex i and go to
infinity.
metric: each bond b has a coordinate 0 ≤ xb ≤ Lb < ∞ and
each lead a coordinate 0 ≤ xl < ∞
wave function Ψ(x) = {ψb (xb ), ψl (xl )} solve NLSE on the
bonds and leads
The nonlinear Schrödinger equation on metric graphs
NLSE on a metric graph
vertices i = 1, 2, . . . , V
edges may either be finite bonds b ≡ (i, j) that connect two
vertices i and j
or infinite leads l = (i, ∞) that start a vertex i and go to
infinity.
metric: each bond b has a coordinate 0 ≤ xb ≤ Lb < ∞ and
each lead a coordinate 0 ≤ xl < ∞
wave function Ψ(x) = {ψb (xb ), ψl (xl )} solve NLSE on the
bonds and leads
one may choose the coupling constant g differently on each
edge.
If g = 0 on an edge we call the edge (bond, lead) linear.
The nonlinear Schrödinger equation on metric graphs
Matching conditions
Consider one vertex with v incident edges
and coordinates x1 , x2 , . . . , xv
such that xi = 0 at the vertex and gi = gj = g .
The nonlinear Schrödinger equation on metric graphs
Matching conditions
Consider one vertex with v incident edges
and coordinates x1 , x2 , . . . , xv
such that xi = 0 at the vertex and gi = gj = g .
Continuity:
ψi (0) = ψj (0) = ψ0
The nonlinear Schrödinger equation on metric graphs
Matching conditions
Consider one vertex with v incident edges
and coordinates x1 , x2 , . . . , xv
such that xi = 0 at the vertex and gi = gj = g .
Continuity:
ψi (0) = ψj (0) = ψ0
Generalised Robin-type condition
v
X
ψi′ (0) = λψ0
i=1
where λ is real – vertex potential strength
The nonlinear Schrödinger equation on metric graphs
Flux conservation
Robin-type matching
pconditions in terms of the classical dynamics
picture (with ψ0 = 2E /|g |r0 e iθ0 )
The nonlinear Schrödinger equation on metric graphs
Flux conservation
Robin-type matching
pconditions in terms of the classical dynamics
picture (with ψ0 = 2E /|g |r0 e iθ0 )
v ³
´
√ X
ṙj + irj θ˙j e tiθj = λr0 e iθ0
E
j=1
where the left-hand side is evaluated at the vertex xj = 0 (τj = 0).
The nonlinear Schrödinger equation on metric graphs
Flux conservation
Robin-type matching
pconditions in terms of the classical dynamics
picture (with ψ0 = 2E /|g |r0 e iθ0 )
v ³
´
√ X
ṙj + irj θ˙j e tiθj = λr0 e iθ0
E
j=1
where the left-hand side is evaluated at the vertex xj = 0 (τj = 0).
v
X
λ
pj = √
E
j=1
v
X
ℓj =0
j=1
Sum of fluxes (outgoing) at each vertex vanishes.
The nonlinear Schrödinger equation on metric graphs
How to find the solution?
Consider a finite graph with V vertices and B bonds (no leads)
and fix an energy E > 0.
The nonlinear Schrödinger equation on metric graphs
How to find the solution?
Consider a finite graph with V vertices and B bonds (no leads)
and fix an energy E > 0.
The wave function ψb (xb ) on bond b is characterised by two
parameters (e.g. ℓb , EHam,b )
The nonlinear Schrödinger equation on metric graphs
How to find the solution?
Consider a finite graph with V vertices and B bonds (no leads)
and fix an energy E > 0.
The wave function ψb (xb ) on bond b is characterised by two
parameters (e.g. ℓb , EHam,b )
1
Given a set of V complex numbers φi = find ℓb and EHam, b
such that ψb (xb = 0) = φi and ψb (Lb ) = φj .
Classical picture: find trajectories that connect to two points
in configuration space in a given time.
The nonlinear Schrödinger equation on metric graphs
How to find the solution?
Consider a finite graph with V vertices and B bonds (no leads)
and fix an energy E > 0.
The wave function ψb (xb ) on bond b is characterised by two
parameters (e.g. ℓb , EHam,b )
1
Given a set of V complex numbers φi = find ℓb and EHam, b
such that ψb (xb = 0) = φi and ψb (Lb ) = φj .
Classical picture: find trajectories that connect to two points
in configuration space in a given time.
2
The Robin-type matching conditions are then a set of v
equations for v unknowns φi .
Each solution {φi } yields a wave function Ψ on the nonlinear
graph.
The nonlinear Schrödinger equation on metric graphs
Normalisation and quantisation
NLSE solutions Ψ at a given energy E > 0 usually exist for
arbitrary energy.
The nonlinear Schrödinger equation on metric graphs
Normalisation and quantisation
NLSE solutions Ψ at a given energy E > 0 usually exist for
arbitrary energy.
Ψ cannot be normalised arbitrarily – the value
XZ
b
Lb
0
|ψb |2 dxb = N
Fixing N implies energy quantisation.
The nonlinear Schrödinger equation on metric graphs
Normalisation and quantisation
NLSE solutions Ψ at a given energy E > 0 usually exist for
arbitrary energy.
Ψ cannot be normalised arbitrarily – the value
XZ
b
Lb
0
|ψb |2 dxb = N
Fixing N implies energy quantisation.
For N → 0 the equations reduce to simple matrix equations
and yield an explicit secular equation.
This may be exploited numerically and analytically.
The nonlinear Schrödinger equation on metric graphs
Star graphs
Only one vertex and no fluxes
→ ℓb = 0 and ψb = ψb∗
For fixed normalisation the Robin-type matching condition
yields a secular equation for the ’spectrum’
The nonlinear Schrödinger equation on metric graphs
Scattering from a nonlinear graph
The physical relevance of the ’spectrum’ is questionable apart
from the ground state energy
in a fibre optics application one may consider to inject a laser
beam into a fibre network and measure response or transport
The nonlinear Schrödinger equation on metric graphs
Scattering from a nonlinear graph
The physical relevance of the ’spectrum’ is questionable apart
from the ground state energy
in a fibre optics application one may consider to inject a laser
beam into a fibre network and measure response or transport
model this by adding linear (g = 0) leads which are attached
to some vertex of a (otherwise finite) nonlinear quantum graph
The nonlinear Schrödinger equation on metric graphs
Scattering from a nonlinear graph
The physical relevance of the ’spectrum’ is questionable apart
from the ground state energy
in a fibre optics application one may consider to inject a laser
beam into a fibre network and measure response or transport
model this by adding linear (g = 0) leads which are attached
to some vertex of a (otherwise finite) nonlinear quantum graph
solution follows basically the same steps as for a finite graph
scattering coefficients depends on intensities of incoming
waves
The nonlinear Schrödinger equation on metric graphs
Scattering from a nonlinear graph
The physical relevance of the ’spectrum’ is questionable apart
from the ground state energy
in a fibre optics application one may consider to inject a laser
beam into a fibre network and measure response or transport
model this by adding linear (g = 0) leads which are attached
to some vertex of a (otherwise finite) nonlinear quantum graph
solution follows basically the same steps as for a finite graph
scattering coefficients depends on intensities of incoming
waves
at low intensities the first order is given by the scattering
matrix of the corresponding linear quantum graph
The nonlinear Schrödinger equation on metric graphs
Scattering phase and delay
One lead attached to a star graph with incoming wave at energy
E = k2
r
I
cos(kx + δ/2)
(1)
ψlead (x) = 2
k
Scattering phase: δ = δ(I , k)
∂δ
delay time: ∂k
The nonlinear Schrödinger equation on metric graphs
Multistabilities & hysteresis
The nonlinear Schrödinger equation on metric graphs
Outlook
transport in one-dimensional (periodic) graphs such as the
chain, or a comb
Ground state energy for BEC in a non-trivial trap, e.g. in
figure eight shape
(chaotic) scattering from nonlinear quantum graphs
theory for time-dependent NLSE on graphs
solitons in a non-linear graph