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. . . . . . .