Quantum chaos in optical microcavities J. Wiersig Institute for Theoretical Physics, Otto-von-Guericke University, Magdeburg Collaborations J. Unterhinninghofen (Magdeburg) M. Hentschel (Dresden) J. Main (Stuttgart) H. Schomerus (Lancaster) Supported by the DFG research group “Scattering Systems with Complex Dynamics”
Theorie der Kondensierten Materie I Light−matter interaction in Nonlinear dynamics and charge semiconductor nanostructures transport in nanostructures Prof. J. Wiersig Dr. habil. G. Kasner D. Hommel/A. Rosenauer et al., Bremen S.W. Cho and Y.D. Park, Seoul Optical properties of Quasicrystals microcavities Dipl.−Phys. J. Unterhinninghofen D. Heitmann/T. Kipp et al., Hamburg I.R. Fischer et al., Iowa J. Wiersig CIRM 2008 2
Theorie der Kondensierten Materie I Light−matter interaction in Nonlinear dynamics and charge semiconductor nanostructures transport in nanostructures Prof. J. Wiersig Dr. habil. G. Kasner D. Hommel/A. Rosenauer et al., Bremen S.W. Cho and Y.D. Park, Seoul Optical properties of Quasicrystals microcavities Dipl.−Phys. J. Unterhinninghofen D. Heitmann/T. Kipp et al., Hamburg I.R. Fischer et al., Iowa J. Wiersig CIRM 2008 2
Outline Introduction to optical microcavities 1 2 Avoided resonance crossings Avoided crossings despite integrability Formation of long-lived, scarlike modes Unidirectional light emission from high- Q modes 3 Unidirectional light emission and universal far-field patterns 4 Fractal Weyl law 5 Summary J. Wiersig CIRM 2008 3
Introduction to optical microcavities J. Wiersig CIRM 2008 4
Introduction to optical microcavities Microdisk total internal φ > φ c reflection φ φ φ < φ c tunneling Bell labs Light confinement due to TIR J. Wiersig CIRM 2008 5
Introduction to optical microcavities Microdisk total internal φ > φ c reflection φ φ φ < φ c tunneling Bell labs Light confinement due to TIR Whispering-gallery modes Light emission due to tunneling High quality factor Q = ωτ Uniform far-field pattern J. Wiersig CIRM 2008 5
Introduction to optical microcavities Types of cavities microdisk microsphere microtorus Bell labs H. Wang et al. V.S. Ilschenko et al. VCSEL−micropillar photonic crystal defect cavity D. Hommel et al. C. Reese et al. J. Wiersig CIRM 2008 6
Introduction to optical microcavities Applications Strong light confinement to a very small volume Individual optical modes Control over light-matter interaction ... Applications Microlasers Single-photon sources Quantum computers Sensors Bell labs Filters ... J. Wiersig CIRM 2008 7
Introduction to optical microcavities Deformed microdisks Directed light emission from deformed disks A. Levi et al. , APL 62, 561 (1993) Open quantum billiard (ray-wave correspondence) sin χ < 1 / n ⇒ no TIR J.U. Nöckel and A.D. Stone, Nature 385, 45 (1997) J. Wiersig CIRM 2008 8
Introduction to optical microcavities Wave equation and boundary conditions Quantum billiard: energy eigenstate ∇ 2 + k 2 i h ψ ( x , y ) = 0 q and ψ ( x , y ) = 0 outside, k = 2 mE � 2 ∈ R . J. Wiersig CIRM 2008 9
Introduction to optical microcavities Wave equation and boundary conditions Quantum billiard: energy eigenstate ∇ 2 + k 2 i h ψ ( x , y ) = 0 q and ψ ( x , y ) = 0 outside, k = 2 mE � 2 ∈ R . Optical microcavity, (TM polarized) mode : E z = Re [ ψ ( x , y ) e − i ω t ] h ∇ 2 + n ( x , y ) 2 k 2 i ψ ( x , y ) = 0 and continuity of ψ and ∇ ψ + outgoing wave conditions, k = ω/ c ∈ C. 1 lifetime τ = − 2Im ( ω ) n(x,y) = 1 n(x,y) = n J. Wiersig CIRM 2008 9
Introduction to optical microcavities Deformed microdisks in experiments A. Levi et al. , APL 62, 561 (1993) C. Gmachl et al. , Science 280 , 1556 (1998) M. Kneissl et al. , APL 84 , 2485 (2004) Improved directionality but Q -factor is strongly reduced Ultimate goal: unidirectional light emission from high- Q modes J. Wiersig CIRM 2008 10
Avoided resonance crossings J. Wiersig CIRM 2008 11
Avoided resonance crossings Avoided level crossings „ E 1 V « H = W E 2 E i ∈ R , W = V ∗ Eigenvalues of H E ± = E 1 + E 2 ( E 1 − E 2 ) 2 r + VW ± 2 4 Vary a parameter ∆ V = 0 V � = 0 E ∆ ∆ Hybridization (mixing) of eigenstates near avoided level crossing J. Wiersig CIRM 2008 12
Avoided resonance crossings Internal coupling E ± = E 1 + E 2 ( E 1 − E 2 ) 2 r + VW ± 2 4 E i ∈ C, W = V ∗ : internal coupling | V | < V c (weak coupling) | V | > V c (strong coupling) Re(E) 0 long-lived Im(E) short-lived ∆ ∆ Small mixing of eigenstates J. Wiersig CIRM 2008 13
Avoided resonance crossings External coupling E ± = E 1 + E 2 ( E 1 − E 2 ) 2 r + VW ± 2 4 E i ∈ C, W � = V ∗ : external coupling strong coupling Re(E) 0 Im(E) ∆ Formation of short- and long-lived modes J. Wiersig CIRM 2008 14
Avoided resonance crossings Avoided crossings despite integrability Boundary element method J. Wiersig, J. Opt. A: Pure Appl. Opt. 5 , 53 (2003) Normalized frequency Ω = ω R / c = kR elliptical microcavity A 8.24 B 8.22 Re( Ω ) C 8.2 C D 8.18 E D 8.16 F 8.14 B 0 D E -0.0001 A Im( Ω ) C -0.0002 -0.0003 -0.0004 F -0.0005 0.62 0.64 0.66 0.68 eccentricity J. Wiersig CIRM 2008 15
Avoided resonance crossings Avoided crossings despite integrability elliptical microcavity A C E A 8.24 B 8.22 Re( Ω ) C 8.2 C D 8.18 E D 8.16 B D F 8.14 F B 0 D E -0.0001 A Im( Ω ) C -0.0002 -0.0003 F -0.0004 -0.0005 0.62 0.64 0.66 0.68 eccentricity Formation of scarlike modes J. Wiersig, PRL 97 , 253901 (2006) J. Wiersig CIRM 2008 16
Avoided resonance crossings Avoided crossings despite integrability elliptical microcavity A C E A 8.24 B 8.22 Re( Ω ) C 8.2 C D 8.18 E D 8.16 B D F 8.14 F B 0 D E -0.0001 A Im( Ω ) C -0.0002 -0.0003 F -0.0004 -0.0005 0.62 0.64 0.66 0.68 eccentricity Formation of scarlike modes J. Wiersig, PRL 97 , 253901 (2006) J. Wiersig CIRM 2008 16
Avoided resonance crossings Avoided crossings despite integrability elliptical microcavity A C E A 8.24 B 8.22 Re( Ω ) C 8.2 C D 8.18 E D 8.16 B D F 8.14 F B 0 D E -0.0001 A Im( Ω ) C -0.0002 -0.0003 F -0.0004 -0.0005 0.62 0.64 0.66 0.68 eccentricity Formation of scarlike modes J. Wiersig, PRL 97 , 253901 (2006) Augmented ray dynamics including the Goos-Hänchen shift J. Unterhinninghofen, J. Wiersig, and M. Hentschel, PRE 78 , 016201 (2008) J. Wiersig CIRM 2008 16
Avoided resonance crossings Formation of long-lived, scarlike modes J. Wiersig, PRL 97 , 253901 (2006) 13.2 A Long-lived mode C : 13.1 B Re( Ω) 13 C Q ≈ 23000 (increase by more D 12.9 E than one order of magnitude) F 12.8 no diffraction at corners 0 C -0.005 A Im( Ω) F -0.01 E B -0.015 D -0.02 0.72 0.73 0.74 0.75 0.76 0.77 aspect ratio A C E B D F J. Wiersig CIRM 2008 17
Avoided resonance crossings Formation of long-lived, scarlike modes J. Wiersig, PRL 97 , 253901 (2006) 13.2 A Long-lived mode C : 13.1 B Re( Ω) 13 C Q ≈ 23000 (increase by more D 12.9 E than one order of magnitude) F 12.8 no diffraction at corners 0 C scarlike mode pattern -0.005 A Im( Ω) F -0.01 E B -0.015 D -0.02 0.72 0.73 0.74 0.75 0.76 0.77 aspect ratio C At the avoided resonance crossing a long-lived, scarlike mode is formed J. Wiersig CIRM 2008 17
Avoided resonance crossings Unidirectional light emission from high- Q modes Normalized frequency Ω = ω R / c = kR , quality factor Q = − Re (Ω) 2 Im (Ω) 7.05 7.04 Re( Ω ) 7.03 7.02 7.01 7 6e+05 long-lived R R 2 d 5e+05 Q 4e+05 x1000 3e+05 short-lived 0.39 0.4 0.41 0.42 0.43 0.44 0.45 d/R Weak internal coupling: weak hybridization of modes J. Wiersig CIRM 2008 18
Avoided resonance crossings Unidirectional light emission from high- Q modes Far-field pattern is dominated by the short-lived component low-Q mode high-Q mode Intensity (arb. units) θ 0 60 120 180 θ Hybridized whispering-gallery mode has Q = 550000 and unidirectional emission J. Wiersig CIRM 2008 19
Avoided resonance crossings Unidirectional light emission from high- Q modes far-field intensity (arb. units) 0 50 100 150 observation angle θ in degree Small angular divergence and ultra-high Q > 10 8 J. Wiersig CIRM 2008 20
Avoided resonance crossings Unidirectional light emission from high- Q modes far-field intensity (arb. units) 0 50 100 150 observation angle θ in degree F. Wilde PhD thesis (2008) Heitmann group, Hamburg Small angular divergence and ultra-high Q > 10 8 J. Wiersig CIRM 2008 20
Unidirectional light emission and universal far-field patterns J. Wiersig CIRM 2008 21
Unidirectional light emission and universal far-field patterns Problem: in the case of multimode lasing we may have modes with different directionality J. Wiersig CIRM 2008 22
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