unperturbed
perturbed
effective dynamics
ray optics
conclusion
Approximate Dynamics in
Slowly Modulated Photonic Crystals
for Physical States
Max Lein
in collarboration with Giuseppe De Nittis
University of Toronto
2014.03.18@Warwick
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Talk based on
Collaboration with Giuseppe De Nittis
The Perturbed Maxwell Operator as Pseudodifferential Operator
Documenta Mathematica 19, 2014
Effective Light Dynamics in Perturbed Photonic Crystals
to appear in Comm. Math. Phys.
On the Role of Symmetries in the Theory of Photonic Crystals
in preparation
Semiclassical dynamics in Photonic Crystals
in preparation
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
1.
Photonic crystals
2.
Physical states in perturbed photonic crystals
3.
Effective light dynamics
4.
Ray optics equations for physical states
5.
Conclusion
.
.
.
conclusion
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
1.
Photonic crystals
2.
Physical states in perturbed photonic crystals
3.
Effective light dynamics
4.
Ray optics equations for physical states
5.
Conclusion
.
.
.
conclusion
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
.
Assumption
.
Material weights w = (ε, µ)
(
)
1. ε, µ ∈ L∞ R3 , Mat (3)
C
. ε∗ = ε, µ∗ = µ
3. 0 < c ≤ ε, µ ≤ C < ∞
2
. ε, µ periodic wrt Γ ∼
= Z3
5. Im ε, Im µ = 0
4
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
.
Assumption
.
Material weights w = (ε, µ)
(
)
1. ε, µ ∈ L∞ R3 , Mat (3)
C
. ε∗ = ε, µ∗ = µ
3. 0 < c ≤ ε, µ ≤ C < ∞
2
. ε, µ periodic wrt Γ ∼
= Z3
5. Im ε, Im µ = 0
4
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
.
Assumption
.
Material weights w = (ε, µ)
(
)
1. ε, µ ∈ L∞ R3 , Mat (3)
C
. ε∗ = ε, µ∗ = µ
3. 0 < c ≤ ε, µ ≤ C < ∞
2
. ε, µ periodic wrt Γ ∼
= Z3
5. Im ε, Im µ = 0
4
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Maxwell equations as Schrödinger equation
. Field energy
1
(
) 1
E E, H =
2
∫
R3
(
)
dx E(x) · ε(x) E(x) + H(x) · µ(x) H(x)
. Dynamical equations
2
ε ∂t E(t) = −∇x × H(t),
E(0) = E
µ ∂t H(t) = +∇x × E(t),
H(0) = H
. No sources
3
∇x · εE(t) = 0
∇x · µH(t) = 0
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Maxwell equations as Schrödinger equation
. Field energy
1
(
)
(
)
E E, H = E E(t), H(t)
. Dynamical equations
2
ε ∂t E(t) = −∇x × H(t),
E(0) = E
µ ∂t H(t) = +∇x × E(t),
H(0) = H
. No sources
3
∇x · εE(t) = 0
∇x · µH(t) = 0
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Maxwell equations as Schrödinger equation
. Field energy
1
(
)
1
E E, H =
2
∫
R3
(
)
dx E(x) · ε(x) E(x) + H(x) · µ(x) H(x)
. Dynamical equations
2
ε ∂t E(t) = −∇x × H(t),
E(0) = E
µ ∂t H(t) = +∇x × E(t),
H(0) = H
. No sources
3
∇x · εE(t) = 0
∇x · µH(t) = 0
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Maxwell equations as Schrödinger equation
. Field energy
1
(
)
1
E E, H =
2
∫
R3
(
)
dx E(x) · ε(x) E(x) + H(x) · µ(x) H(x)
. Dynamical equations
2
ε ∂t E(t) = −∇x × H(t),
E(0) = E
µ ∂t H(t) = +∇x × E(t),
H(0) = H
. No sources
3
∇x · εE(t) = 0
∇x · µH(t) = 0
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Maxwell equations as Schrödinger equation
. Field energy (E, H) ∈ L2 (R3 , C6 ) with norm
w
∫
(
)
(E, H)22 :=
dx E(x) · ε(x) E(x) + H(x) · µ(x) H(x)
L
1
w
R3
. Dynamical equations ⇝ »Schrödinger equation«
)
(
) (
)(
d E(t)
0
+ε−1 (−i∇x )×
E(t)
i
=
−µ−1 (−i∇x )×
0
H(t)
dt H(t)
|
{z
}
2
=M
. No sources
{
}
J := (E, H) ∈ L2w (R3 , C6 ) ∇x · εE = 0 ∧ ∇x · µH = 0
3
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Maxwell equations as Schrödinger equation
. Field energy (E, H) ∈ L2 (R3 , C6 ) with norm
w
1
(
)
(E, H)22 = 2 E E, H
L
w
. Dynamical equations ⇝ »Schrödinger equation«
(
) (
)(
)
d E(t)
0
+ε−1 (−i∇x )×
E(t)
i
=
−µ−1 (−i∇x )×
0
H(t)
dt H(t)
|
{z
}
2
=M
. No sources
{
}
J := (E, H) ∈ L2w (R3 , C6 ) ∇x · εE = 0 ∧ ∇x · µH = 0
3
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Maxwell equations as Schrödinger equation
. Field energy (E, H) ∈ L2 (R3 , C6 ) with norm
w
∫
(
)
(E, H)22 :=
dx E(x) · ε(x) E(x) + H(x) · µ(x) H(x)
L
1
w
R3
. Dynamical equations ⇝ »Schrödinger equation«
(
)(
)
) (
d E(t)
0
+ε−1 (−i∇x )×
E(t)
i
=
−µ−1 (−i∇x )×
0
H(t)
dt H(t)
|
{z
}
2
=M
. No sources
{
}
J := (E, H) ∈ L2w (R3 , C6 ) ∇x · εE = 0 ∧ ∇x · µH = 0
3
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
The frequency band picture
M∼
= MZ =
∫
⊕
B
∫ ⊕
=
B
D(MZ ) =
⊔
dk M(k)
(
)
0
+ε−1 (−i∇y + k)×
dk
−µ−1 (−i∇y + k)×
0
(
)
D M(k) ⊂ L2 (B) ⊗ L2w (T3 , C6 )
k∈B
M(k)|G(k) = 0 ⇒ focus on M(k)|J(k)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
The frequency band picture
M∼
= MZ =
∫
⊕
B
∫ ⊕
=
B
dk M(k)
(
)
0
+ε−1 (−i∇y + k)×
dk
−µ−1 (−i∇y + k)×
0
(
)
D M(k) = H1 (T3 , C6 ) ∩ J(k) ⊕ G(k) ⊂ L2w (T3 , C6 )
M(k)|G(k) = 0 ⇒ focus on M(k)|J(k)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
The frequency band picture
M∼
= MZ =
∫
⊕
B
∫ ⊕
=
B
dk M(k)
(
)
0
+ε−1 (−i∇y + k)×
dk
−µ−1 (−i∇y + k)×
0
(
)
D M(k) = H1 (T3 , C6 ) ∩ J(k) ⊕ G(k) ⊂ L2w (T3 , C6 )
M(k)|G(k) = 0 ⇒ focus on M(k)|J(k)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
The frequency band picture
.
Physical bands
.
M(k)φn (k) = ωn (k) φn (k)
Frequency band functions k 7→ ωn (k)
Bloch functions k 7→ φn (k)
both ∈ C, analytic away from band crossings
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
The frequency band picture
.
Physical bands
.
M(k)φn (k) = ωn (k) φn (k)
Frequency band functions k 7→ ωn (k)
Bloch functions k 7→ φn (k)
both ∈ C, analytic away from band crossings
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
The frequency band picture
.
Physical bands
.
M(k)φn (k) = ωn (k) φn (k)
Frequency band functions k 7→ ωn (k)
Bloch functions k 7→ φn (k)
both ∈ C, analytic away from band crossings
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
The frequency band picture
Ω
A+
n4
n3
B+
n2
n1
-Π
Π
k
n-1
n-2
n-3
B-
n-4
A-
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Complex conjugation induces a symmetry
.
Complex conjugation
.
C : L2w (R3 , C6 ) −→ L2w (R3 , C6 ),
(CΨ)(x) = Ψ(x)
CZ : L2 (B) ⊗ L2w (T3 , C6 ) −→ L2 (B) ⊗ L2w (T3 , C6 ),
( Z )
C φ (k) = Cφ(−k) = φ(−k)
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Complex conjugation induces a symmetry
.
Complex conjugation
.
C : L2w (R3 , C6 ) −→ L2w (R3 , C6 ),
(CΨ)(x) = Ψ(x)
CZ : L2 (B) ⊗ L2w (T3 , C6 ) −→ L2 (B) ⊗ L2w (T3 , C6 ),
( Z )
C φ (k) = Cφ(−k) = φ(−k)
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Complex conjugation induces a symmetry
.
Particle-hole-type symmetry
.
}
CεC = ε
=⇒ C M C = −M
CµC = µ
.
⇝ C not a time-reversal symmetry
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Complex conjugation induces a symmetry
.
Particle-hole-type symmetry
.
}
CεC = ε
=⇒ C M(k) C = −M(−k)
CµC = µ
.
⇝ C not a time-reversal symmetry
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Complex conjugation induces a symmetry
.
Symmetrically related bands
.
M(k)φn (k) = ωn (k) φn (k)
⇐⇒
M(k) φn (−k) = −ωn (−k) φn (−k)
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Complex conjugation induces a symmetry
.
Symmetrically related bands
.
M(k)φn (k) = ωn (k) φn (k)
(
)
⇐⇒
(
)
M(k) C φn (k) = −ωn (−k) CZ φn (k)
Z
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Complex conjugation induces a symmetry
Ω
A+
n4
n3
B+
n2
n1
-Π
Π
k
n-1
n-2
n-3
B-
n-4
A-
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Definition (Physical states from a narrow range of frequencies)
.
∫ ⊕
(
)
Π0 =
dk 1σrel (k) M(k) so that
B
}
∪ {
1. σ
(k)
=
σ
rel
rel (−k) =
n∈I ωn (k) isolated family of bands
2. source free: ran Π ⊂ ZJ
0
.
. real: CZ Π0 CZ = Π0
3
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
Ω
A+
n4
n3
B+
n2
n1
-Π
Π
k
n-1
n-2
n-3
B-
n-4
A-
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Definition (Physical states from a narrow range of frequencies)
.
∫ ⊕
(
)
Π0 =
dk 1σrel (k) M(k) so that
B
}
∪ {
1. σ
(k)
=
σ
rel
rel (−k) =
n∈I ωn (k) isolated family of bands
2. source free: ran Π ⊂ ZJ
0
.
. real: CZ Π0 CZ = Π0
3
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Definition (Physical states from a narrow range of frequencies)
.
∫ ⊕
(
)
Π0 =
dk 1σrel (k) M(k) so that
B
}
∪ {
1. σ
(k)
=
σ
rel
rel (−k) =
n∈I ωn (k) isolated family of bands
2. source free: ran Π ⊂ ZJ
0
.
. real: CZ Π0 CZ = Π0
3
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Corollary (De Nittis-L. 2013)
.
.Physical states cannot be supported by a single band.
.
Proof.
.
ψRe = Re Z φn =
ψIm = Im Z φn =
(
)
Z
1
2 φn + C φn ,
(
)
Z
1
2i φn − C φn .
.linear combination of Bloch functions to ωn (k) and −ωn (−k).
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Corollary (De Nittis-L. 2013)
.
.Physical states cannot be supported by a single band.
.
Proof.
.
ψRe = Re Z φn =
ψIm = Im Z φn =
(
)
Z
1
2 φn + C φn ,
(
)
Z
1
2i φn − C φn .
.linear combination of Bloch functions to ωn (k) and −ωn (−k).
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Corollary (De Nittis-L. 2013)
. (
)
(
)
c. 1 |ψRe ,Im ⟩⟨ψRe ,Im | = 0 even though c1 |φn ⟩⟨φn | ̸= 0
De Nittis & Gomi (2014) ⇒ No topological effects when w = w
⇝ more on that in talk by De Nittis (Friday 14:00)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Corollary (De Nittis-L. 2013)
. (
)
(
)
c. 1 |ψRe ,Im ⟩⟨ψRe ,Im | = 0 even though c1 |φn ⟩⟨φn | ̸= 0
De Nittis & Gomi (2014) ⇒ No topological effects when w = w
⇝ more on that in talk by De Nittis (Friday 14:00)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Physical states & their geometry
.
Corollary (De Nittis-L. 2013)
. (
)
(
)
c. 1 |ψRe ,Im ⟩⟨ψRe ,Im | = 0 even though c1 |φn ⟩⟨φn | ̸= 0
De Nittis & Gomi (2014) ⇒ No topological effects when w = w
⇝ more on that in talk by De Nittis (Friday 14:00)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
1.
Photonic crystals
2.
Physical states in perturbed photonic crystals
3.
Effective light dynamics
4.
Ray optics equations for physical states
5.
Conclusion
.
.
.
conclusion
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Perturbed photonic crystals
τε−2 (λx)
ε(x)
x [lattice constants]
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Perturbed photonic crystals
.
Perturbation of material constants
.
[lattice spacing]
λ=
≪1
[length scale of modulation]
−2
−2
. ε(x) ⇝ ελ (x) := τε (λx) ε(x), µ(x) ⇝ µλ (x) := τµ (λx) µ(x)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Perturbed photonic crystals
.
Assumption (Slow modulation)
.
τ. ε , τµ ∈ Cb∞ (R3 ), τε , τµ ≥ c > 0
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Adiabatically perturbed Maxwell operator
Mλ = S−2
λ M
( 2
)(
)
τε (λx)
0
0
+ε−1 (x) (−i∇x )×
=
0
τµ2 (λx) −µ−1 (x) (−i∇x )×
0
slow modulation & periodic Maxwell operator
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
existence of physical states
in the presence of perturbations
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Existence of physical states
.
Definition (Physical states: unperturbed)
.
∫ ⊕
(
)
Π0 =
dk 1σrel (k) M(k) so that
B
}
∪ {
1. σ
(k)
=
σ
rel
rel (−k) =
n∈I ωn (k) isolated family of bands
2. source free: ran Π ⊂ ZJ
0
.
. real: CZ Π0 CZ = Π0
3
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Existence of physical states
.
Definition (Physical states: perturbed)
.
∫ ⊕
(
)
∼
Πλ =
dk 1σ (k) M(k) + O(λ) so that
B
rel
. σrel (k) = σrel (−k) = ∪ {ωn (k)} isolated family of bands
n∈I
2. source free: “ran Π ⊂ ZJ ” up to O(λ∞ )
λ
λ
3. real: CZ Π CZ = Π + O(λ∞ )
λ
λ
1
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Existence of physical states
.
Definition (Physical states: perturbed)
.
∫ ⊕
(
)
∼
dk 1σ (k) M(k) + O(λ) so that
Πλ =
B
rel
. σrel (k) = σrel (−k) = ∪ {ωn (k)} isolated family of bands
n∈I
2. source free: “ran Π ⊂ ZJ ” up to O(λ∞ )
λ
λ
3. real: CZ Π CZ = Π + O(λ∞ )
λ
λ
1
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Existence of physical states
.
Theorem (De Nittis-L. 2013)
.
Suppose the bands σrel (k) = σrel (−k) are isolated and 0 ̸∈ σrel (0).
Then there exist orthogonal projections
Πλ = Π+,λ + Π−,λ + O(λ∞ )
so that
[
]
Mλ , Π±,λ = O(λ∞ )
.whose range supports physical states.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
1.
Photonic crystals
2.
Physical states in perturbed photonic crystals
3.
Effective light dynamics
4.
Ray optics equations for physical states
5.
Conclusion
.
.
.
conclusion
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
.
Goal
.
Approximate e−it Mλ for physical states from a narrow range of
frequencies
up to O(λn ).
.
Reduction in complexity: only few bands contribute to dynamics
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
.
Goal
.
Approximate e−it Mλ for physical states from a narrow range of
frequencies
up to O(λn ).
.
Reduction in complexity: only few bands contribute to dynamics
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective light dynamics
.
Theorem (De Nittis-L. 2013)
.
Suppose the bands σrel (k) = σrel (−k) are isolated and 0 ̸∈ σrel (0).
Then there exist a unitary Vλ and an effective Maxwell operator
Meff = Vλ−1 Opλ (Meff ) Vλ
which approximates the full light dynamics,
Z
e−itMλ Πλ Re Z = Re Z e−itMeff Πλ Re Z + O(λ∞ ),
Z
e−itMλ Πλ Im Z = Im Z e−itMeff Πλ Im Z + O(λ∞ ),
∞
.and leaves ran Πλ invariant up to O(λ ).
Πλ , Vλ and Meff can be computed explicitly order-by-order in λ
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective light dynamics
.
Theorem (De Nittis-L. 2013)
.
Suppose the bands σrel (k) = σrel (−k) are isolated and 0 ̸∈ σrel (0).
Then there exist a unitary Vλ and an effective Maxwell operator
Meff = Vλ−1 Opλ (Meff ) Vλ
which approximates the full light dynamics,
Z
e−itMλ Πλ Re Z = Re Z e−itMeff Πλ Re Z + O(λ∞ ),
Z
e−itMλ Πλ Im Z = Im Z e−itMeff Πλ Im Z + O(λ∞ ),
∞
.and leaves ran Πλ invariant up to O(λ ).
Πλ , Vλ and Meff can be computed explicitly order-by-order in λ
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective light dynamics
.
Corollary (“Peierl's substitution”, De Nittis-L. 2013)
.
{
}
In case σrel (k) = ω(k), −ω(−k) consists of two topologically trivial
bands, then the symbol to the pseudodifferential operator is
(
)
ω(k)
0
Meff (r, k) = τε (r) τµ (r)
+ O(λ).
0
−ω(−k)
.
⇒ motivates the definition of a Maxwell-Harper operator
⇝ Hofstadter butterfly?
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective light dynamics
.
Corollary (“Peierl's substitution”, De Nittis-L. 2013)
.
{
}
In case σrel (k) = ω(k), −ω(−k) consists of two topologically trivial
bands, then the symbol to the pseudodifferential operator is
(
)
ω(k)
0
Meff (r, k) = τε (r) τµ (r)
+ O(λ).
0
−ω(−k)
.
⇒ motivates the definition of a Maxwell-Harper operator
⇝ Hofstadter butterfly?
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
1.
Photonic crystals
2.
Physical states in perturbed photonic crystals
3.
Effective light dynamics
4.
Ray optics equations for physical states
5.
Conclusion
.
.
.
conclusion
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective single-band dynamics
Simplest case:
semiclassical dynamics
aka ray optics
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective single-band dynamics
Ω
A+
n4
n3
B+
n2
n1
-Π
Π
k
n-1
n-2
n-3
B-
n-4
A-
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective single-band dynamics
.
Theorem (De Nittis-L. 2013)
.
For all suitable observables f, the following Egorov-type theorem holds:
(
(
))
Z
Z
2
Π
Π±,λ e+it Mλ Opλ (f ) e−it Mλ − Opλ f ◦ Φ±
±,λ = O(λ )
t
±
Φ
. flow associated to positive/negative frequency band
−
+
Φ+
t (r, k) and Φt (r, k) = Φ−t (r, −k) related by symmetry
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective single-band dynamics
.
Theorem (De Nittis-L. 2013)
.
Φ± is the flow associated to the ray optics equations
(
)( ) (
)
ṙ
∇r M±
0
−id + λ Ω±
sc
rk
=
+id + λ Ω±
+λ Ω±
∇k M±
k̇
sc
kr
kk
±
Ω
. generalized Berry curvature
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective single-band dynamics
.
Theorem (De Nittis-L. 2013)
.
Φ± is the flow associated to the ray optics equations
(
)( ) (
)
ṙ
0
−id + λ Ω±
∇r M±
sc
rk
=
+id + λ Ω±
+λ Ω±
∇k M±
k̇
sc
kr
kk
±
Ω
. generalized Berry curvature
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective single-band dynamics
.
Theorem (De Nittis-L. 2013)
.
Semiclassical Maxwellian
M±
sc = τε τµ ω± +
( ⟨
⟩
⟨
⟩
+ λ i τε φE± , ∇r τµ × φH± − i τµ φH± , ∇r τε × φE± +
⟨
⟩)
{
}
− 2i φ± , S S−2 M(·) − τε τµ ω± S−1 φ±
Symbol
Mλ and analog of Rammal-Wilkinson term
.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Effective single-band dynamics
But: single bands cannot support real states
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
Consider {
twin-band case
}
σrel (k) = ω(k), −ω(−k)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
Writing Πλ = Π+,λ + Π−,λ + O(λ∞ ) leads to equations for ψRe :
t
Z
t
Z
Re Z Πλ e+i λ Mλ Op(f) e−i λ Mλ Πλ Re Z =
= F+ (t) + F− (t) + Fint (t)
Intra-band contribution:
t
Z
Z
t
F± (t) = Re Z Π±,λ e+i λ Mλ Op(f) e−i λ Mλ Π±,λ Re Z
(
)
2
= Π±,λ Opλ f ◦ Φ±
t Π±,λ + O(λ )
⇒ Can be approximated with semiclassical flow Φ±
t
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
Writing Πλ = Π+,λ + Π−,λ + O(λ∞ ) leads to equations for ψRe :
t
Z
t
Z
Re Z Πλ e+i λ Mλ Op(f) e−i λ Mλ Πλ Re Z =
= F+ (t) + F− (t) + Fint (t)
In-band contribution:
t
Z
Z
t
F± (t) = Re Z Π±,λ e+i λ Mλ Op(f) e−i λ Mλ Π±,λ Re Z
(
)
2
= Π±,λ Opλ f ◦ Φ±
t Π±,λ + O(λ )
⇒ Can be approximated with semiclassical flow Φ±
t
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
Writing Πλ = Π+,λ + Π−,λ + O(λ∞ ) leads to equations for ψRe :
t
Z
t
Z
Re Z Πλ e+i λ Mλ Op(f) e−i λ Mλ Πλ Re Z =
= F+ (t) + F− (t) + Fint (t)
In-band contribution:
t
Z
Z
t
F± (t) = Re Z Π±,λ e+i λ Mλ Op(f) e−i λ Mλ Π±,λ Re Z
(
)
2
= Π±,λ Opλ f ◦ Φ±
t Π±,λ + O(λ )
⇒ Can be approximated with semiclassical flow Φ±
t
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
Writing Πλ = Π+,λ + Π−,λ + O(λ∞ ) leads to equations for ψRe :
t
Z
t
Z
Re Z Πλ e+i λ Mλ Op(f) e−i λ Mλ Πλ Re Z =
= F+ (t) + F− (t) + Fint (t)
Band transitions:
t
Z
t
Z
Fint (t) = Re Z Π+,λ e+i λ Mλ Op(f) e−i λ Mλ Π−,λ Re Z
t
Z
t
Z
+ Re Z Π−,λ e+i λ Mλ Op(f) e−i λ Mλ Π+,λ Re Z
= O(λ)
(easy)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
Writing Πλ = Π+,λ + Π−,λ + O(λ∞ ) leads to equations for ψRe :
t
Z
t
Z
Re Z Πλ e+i λ Mλ Op(f) e−i λ Mλ Πλ Re Z =
= F+ (t) + F− (t) + Fint (t)
Band transitions:
t
Z
t
Z
Fint (t) = Re Z Π+,λ e+i λ Mλ Op(f) e−i λ Mλ Π−,λ Re Z
t
Z
t
Z
+ Re Z Π−,λ e+i λ Mλ Op(f) e−i λ Mλ Π+,λ Re Z
= O(λ)
(easy)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
Writing Πλ = Π+,λ + Π−,λ + O(λ∞ ) leads to equations for ψRe :
t
Z
t
Z
Re Z Πλ e+i λ Mλ Op(f) e−i λ Mλ Πλ Re Z =
= F+ (t) + F− (t) + Fint (t)
Band transitions:
t
Z
t
Z
Fint (t) = Re Z Π+,λ e+i λ Mλ Op(f) e−i λ Mλ Π−,λ Re Z
t
Z
t
Z
+ Re Z Π−,λ e+i λ Mλ Op(f) e−i λ Mλ Π+,λ Re Z
= O(λ2 )?
(relative phases remain locked)
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
.
Conjecture
.
Fint (t) = O(λ2 ), and thus ray optics dynamics approximate light
dynamics
up to O(λ2 ).
.
⇝ Work in progress
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Semiclassical twin-band dynamics
.
Conjecture
.
Fint (t) = O(λ2 ), and thus ray optics dynamics approximate light
dynamics
up to O(λ2 ).
.
⇝ Work in progress
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
1. Haldane & Raghu (Phys. Rev. A 78, 033834 (2008))
»derivation by analogy«
necessity of slow variation recognized, but small parameter λ
not used
⇝ eom are missing O(λ) terms, only leading-order correct
. Esposito & Gerace (Phys. Rev. A 88, 013853 (2013))
2
derivation of k̇ equation, yields result proposed by Raghu &
Haldane
ṙ equation still missing
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
1. Haldane & Raghu (Phys. Rev. A 78, 033834 (2008))
»derivation by analogy«
necessity of slow variation recognized, but small parameter λ
not used
⇝ eom are missing O(λ) terms, only leading-order correct
. Esposito & Gerace (Phys. Rev. A 88, 013853 (2013))
2
derivation of k̇ equation, yields result proposed by Raghu &
Haldane
ṙ equation still missing
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
1. Haldane & Raghu (Phys. Rev. A 78, 033834 (2008))
»derivation by analogy«
necessity of slow variation recognized, but small parameter λ
not used
⇝ eom are missing O(λ) terms, only leading-order correct
. Esposito & Gerace (Phys. Rev. A 88, 013853 (2013))
2
derivation of k̇ equation, yields result proposed by Raghu &
Haldane
ṙ equation still missing
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
1. Haldane & Raghu (Phys. Rev. A 78, 033834 (2008))
»derivation by analogy«
necessity of slow variation recognized, but small parameter λ
not used
⇝ eom are missing O(λ) terms, only leading-order correct
. Esposito & Gerace (Phys. Rev. A 88, 013853 (2013))
2
derivation of k̇ equation, yields result proposed by Raghu &
Haldane
ṙ equation still missing
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
1. Haldane & Raghu (Phys. Rev. A 78, 033834 (2008))
»derivation by analogy«
necessity of slow variation recognized, but small parameter λ
not used
⇝ eom are missing O(λ) terms, only leading-order correct
. Esposito & Gerace (Phys. Rev. A 88, 013853 (2013))
2
derivation of k̇ equation, yields result proposed by Raghu &
Haldane
ṙ equation still missing
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
3. Onoda, Murakami & Nagaosa (Phys. Rev. E 76, 066610 (2006))
use Sundaram–Niu variational technique + second quantization
semiclassical states Ψ(r, k, z) parametrized by (r, k) ∈ T∗ R3 ,
z ∈ S2 ⇝ find extremals of functional
result is not readily comparable to ours
O(λ) terms eom different, only leading-order correct
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
3. Onoda, Murakami & Nagaosa (Phys. Rev. E 76, 066610 (2006))
use Sundaram–Niu variational technique + second quantization
semiclassical states Ψ(r, k, z) parametrized by (r, k) ∈ T∗ R3 ,
z ∈ S2 ⇝ find extremals of functional
result is not readily comparable to ours
O(λ) terms eom different, only leading-order correct
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Ray optics: Notable previous results
All non-rigorous, single-band results:
3. Onoda, Murakami & Nagaosa (Phys. Rev. E 76, 066610 (2006))
use Sundaram–Niu variational technique + second quantization
semiclassical states Ψ(r, k, z) parametrized by (r, k) ∈ T∗ R3 ,
z ∈ S2 ⇝ find extremals of functional
result is not readily comparable to ours
O(λ) terms eom different, only leading-order correct
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
1.
Photonic crystals
2.
Physical states in perturbed photonic crystals
3.
Effective light dynamics
4.
Ray optics equations for physical states
5.
Conclusion
.
.
.
conclusion
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Recap
. Maxwell ≈ quantum? Question of dynamics!
2. ⇝ Rewrite Maxwell equations as Schrödinger equation
1
. Effective light dynamics: bona fide multiband problem
4. Complex conjugation: particle hole-type symmetry, not
3
time-reversal symmetry
.5 In non-gyroscopic PhCs: frequency bands and Bloch functions
come in conjugate pairs
6. For ray optics: control of band transition term
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Recap
. Maxwell ≈ quantum? Question of dynamics!
2. ⇝ Rewrite Maxwell equations as Schrödinger equation
1
. Effective light dynamics: bona fide multiband problem
4. Complex conjugation: particle hole-type symmetry, not
3
time-reversal symmetry
.5 In non-gyroscopic PhCs: frequency bands and Bloch functions
come in conjugate pairs
6. For ray optics: control of band transition term
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Recap
. Maxwell ≈ quantum? Question of dynamics!
2. ⇝ Rewrite Maxwell equations as Schrödinger equation
1
. Effective light dynamics: bona fide multiband problem
4. Complex conjugation: particle hole-type symmetry, not
3
time-reversal symmetry
.5 In non-gyroscopic PhCs: frequency bands and Bloch functions
come in conjugate pairs
6. For ray optics: control of band transition term
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Recap
. Maxwell ≈ quantum? Question of dynamics!
2. ⇝ Rewrite Maxwell equations as Schrödinger equation
1
. Effective light dynamics: bona fide multiband problem
4. Complex conjugation: particle hole-type symmetry, not
3
time-reversal symmetry
.5 In non-gyroscopic PhCs: frequency bands and Bloch functions
come in conjugate pairs
6. For ray optics: control of band transition term
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
Thank You for your attention!
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
References
M. S. Birman and M. Z. Solomyak. L2 -Theory of the Maxwell
operator in arbitrary domains. Uspekhi Mat. Nauk 42.6, 1987,
pp. 61--76.
A. Figotin and P. Kuchment. Band-gap Structure of Spectra of
Periodic Dielectric and Acoustic Media. II. 2D Photonic Crystals.
SIAM J. Appl. Math. 56.6, 1996, pp. 1561--1620.
J. D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D. Maede.
Photonic Crystals. Molding the Flow of Light. Princeton
University Press, 2008.
P. Kuchment. Mathematical Modelling in Optical Science.
Chapter 7. The Mathematics of Photonic Crystals. pp. 207--272,
Society for Industrial and Applied Mathematics, 2001.
.
.
.
.
.
.
unperturbed
perturbed
effective dynamics
ray optics
conclusion
References
P. Kuchment and S. Levendorskiî. On the Structure of Spectra of
Periodic Elliptic Operators. Transactions of the American
Mathematical Society 354.2, 2001, pp. 537–569.
M. Onoda, S. Murakami and N. Nagaosa. Geometrical aspects in
optical wave-packet dynamics. Phys. Rev. E. 74, 066610, 2010.
G. Panati, H. Spohn, and S. Teufel. Space Adiabatic Perturbation
Theory. Adv. Theor. Math. Phys. 7, 2003, pp. 145--204. arXiv:
0201055 (math-ph).
S. Raghu and F. D. M. Haldane. Analogs of quantum-Hall-effect
edge states in photonic crystals. Phys. Rev. A 78, 033834, 2008.
.
.
.
.
.
.