Quantum chaos in optical microcavities

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 Prof. J. Wiersig Light matter interaction in semiconductor nanostructures Nonlinear dynamics and charge transport in nanostructures Dr. habil. G. Kasner D. Hommel/A. Rosenauer et al., Bremen Optical properties of microcavities S.W. Cho and Y.D. Park, Seoul Quasicrystals Dipl. Phys. J. Unterhinninghofen D. Heitmann/T. Kipp et al., Hamburg I.R. Fischer et al., Iowa J. Wiersig CIRM 2008 2

Outline 1 Introduction to optical microcavities 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 reflection φ φ φ > φ c φ < φ c Bell labs Light confinement due to TIR tunneling J. Wiersig CIRM 2008 5

Introduction to optical microcavities Microdisk total internal reflection φ φ φ > φ c φ < φ c Bell labs Light confinement due to TIR Whispering-gallery modes Light emission due to tunneling 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 D. Hommel et al. photonic crystal defect cavity 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 Filters... Bell labs 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 h 2 + k 2i ψ(x, y) = 0 and ψ(x, y) = 0 outside, k = q 2mE 2 R. J. Wiersig CIRM 2008 9

Introduction to optical microcavities Wave equation and boundary conditions Quantum billiard: energy eigenstate h 2 + k 2i ψ(x, y) = 0 and ψ(x, y) = 0 outside, k = q 2mE 2 R. Optical microcavity, (TM polarized) mode : E z = Re[ψ(x, y)e iωt ] h 2 + n(x, y) 2 k 2i ψ(x, y) = 0 and continuity of ψ and ψ + outgoing wave conditions, k = ω/c C. lifetime τ = 1 2Im(ω) n(x,y) = n n(x,y) = 1 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 i R, W = V Eigenvalues of H E ± = E 1 + E 2 2 «E1 V H = W E 2 r (E1 E 2 ) ± 2 + VW 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 0Im(E) E ± = E 1 + E 2 2 E i C, W = V : internal coupling r (E1 E 2 ) ± 2 + VW 4 V < V c (weak coupling) V > V c (strong coupling) Re(E) long-lived short-lived Small mixing of eigenstates J. Wiersig CIRM 2008 13

Avoided resonance crossings External coupling 0Im(E) E ± = E 1 + E 2 2 E i C, W V : external coupling r (E1 E 2 ) ± 2 + VW 4 strong coupling Re(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 = ωr/c = kr Re(Ω) 8.24 8.22 8.2 8.18 8.16 8.14 A B C D elliptical microcavity C D E F Im(Ω) 0-0.0001-0.0002-0.0003-0.0004-0.0005 B A D C 0.62 0.64 0.66 0.68 eccentricity E F J. Wiersig CIRM 2008 15

Avoided resonance crossings Avoided crossings despite integrability Re(Ω) A 8.24 8.22 B 8.2 8.18 8.16 8.14 elliptical microcavity C D C D E F A C E B D F Im(Ω) 0-0.0001-0.0002-0.0003-0.0004-0.0005 B A D C 0.62 0.64 0.66 0.68 eccentricity E F Formation of scarlike modes J. Wiersig, PRL 97, 253901 (2006) J. Wiersig CIRM 2008 16

Im(Ω) Avoided resonance crossings Formation of long-lived, scarlike modes Re(Ω) 13.2 13.1 13 12.9 12.8 A B 0 C -0.005 A -0.01 F -0.015 B E -0.02 D 0.72 0.73 0.74 0.75 0.76 0.77 aspect ratio A C C D E E F J. Wiersig, PRL 97, 253901 (2006) Long-lived mode C: Q 23000 (increase by more than one order of magnitude) no diffraction at corners B D F J. Wiersig CIRM 2008 17

Avoided resonance crossings Formation of long-lived, scarlike modes Im(Ω) Re(Ω) 13.2 13.1 13 12.9 12.8 A B 0 C -0.005 A -0.01 F -0.015 B E -0.02 D 0.72 0.73 0.74 0.75 0.76 0.77 aspect ratio C C D E F J. Wiersig, PRL 97, 253901 (2006) Long-lived mode C: Q 23000 (increase by more than one order of magnitude) no diffraction at corners scarlike mode pattern 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 = ωr/c = kr, quality factor Q = Re(Ω) 2 Im(Ω) Re(Ω) 7.05 7.04 7.03 7.02 7.01 7 6e+05 5e+05 long-lived R R 2 d 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

Unidirectional light emission and universal far-field patterns Problem: in the case of multimode lasing we may have modes with different directionality Universal far-field pattern due to unstable manifold H.G.L Schwefel et al., J. Opt. Soc. Am. B 21, 923 (2004) S.-Y. Lee et al., Phys. Rev. A 72, 061801(R) (2005) S.-B. Lee et al., Phys. Rev. A 75, 011802(R) (2007) S. Shinohara and T. Harayama, Phys. Rev. E 75, 036216 (2007) J. Wiersig CIRM 2008 22

Unidirectional light emission and universal far-field patterns Limaçon cavity ρ(φ) = R(1 + ε cos φ) 1 sin χ χ 1/n s ε=0 R leaky region no total internal reflection 0 1/n 1 0 0.2 0.4 0.6 s/s max 0.8 1 0.2 0.4 0.6 s/s max 0.8 1 ρ ε = 0.43 φ sin χ 1 s 1/n 0 1/n 1 J. Wiersig 0 CIRM 2008 23

Unidirectional light emission and universal far-field patterns Unstable manifold Chaotic repeller: set of points in phase space that never visits the leaky region both in forward and backward time evolution Unstable manifold: the set of points that converges to the repeller in backward time evolution (weight according to Fresnel s laws) Subtle differences between TE and TM polarization J. Wiersig CIRM 2008 24

Unidirectional light emission and universal far-field patterns Far-field pattern: ray simulation far-field intensity (arb. units) 1 TE TM 2 1 1s 0 0.5 s/s max 1 2s 1 1 φ FF = 0 2s 2-0.2 -sin χ B -1/n -0.4 sin χ 0 50 100 150 observation angle φ FF in degree Difference between TE and TM modes is due to the Brewster angle 2 1s J. Wiersig CIRM 2008 25

Unidirectional light emission and universal far-field patterns Far-field pattern: mode calculation TE far-field intensity (arb. units) TM Q = 200000 Q = 10 7 0 50 100 150 observation angle φ FF in degree all high-q TE modes show unidirectional emission (universal far-field pattern) J. Wiersig CIRM 2008 26

Unidirectional light emission and universal far-field patterns Husimi representation 1 sin χ 1/n 0 1/n (a) 1 0 0.2 0.4 0.6 0.8 1 s/s max Scarring ensures high Q-factors J. Wiersig CIRM 2008 27

Unidirectional light emission and universal far-field patterns Husimi magnification sin χ 1/n (b) TE 0 1/n 1/n sin χ 0 (c) TM 1/n 0 0.2 0.4 0.6 0.8 1 s/smax Unidirectional emission is due to the unstable manifold Robust: not sensitive to internal mode structure and cavity size works in a broad regime of shape parameter ε and refractive index J. Wiersig CIRM 2008 28

Unidirectional light emission and universal far-field patterns Experiments on the Limaçon cavity Theory: J. Wiersig and M. Hentschel, PRL 100, 033901 (2008) Harayama et al., Kyoto Capasso et al., Harvard Cao et al., Yale J. Wiersig CIRM 2008 29

Fractal Weyl law J. Wiersig CIRM 2008 30

Fractal Weyl law Weyl law for closed systems Density of states ρ(e) = Integrated density of states N(E) = Z E X δ(e E i ) ; E 1 E 2... i=1 ρ(e )de = #{i E i E} split N(E) into a smooth part and a fluctuating part N 8 7 6 5 4 3 2 1 N(E) = N(E) + N fluc (E) E 1 E 2 N(E) E 3... N(E) E J. Wiersig CIRM 2008 31

Fractal Weyl law How to count states in open systems? Re(k) Im(k) J. Wiersig CIRM 2008 32

Fractal Weyl law How to count states in open systems? Re(k) C Im(k) N(k) = {k n : Im(k n) > C, Re(k n) k}. J. Wiersig CIRM 2008 32

Fractal Weyl law How to count states in open systems? Re(k) C Im(k) N(k) = {k n : Im(k n) > C, Re(k n) k}. Conjecture: fractal Weyl law for open chaotic systems J. Sjöstrand, Duke Math. J. 60, 1 (1990), M. Zworski, Invent. Math. 136, 353 (1999) N(k) k α. with non-integer exponent α = D + 1 2 where D is the fractal dimension of the chaotic repeller J. Wiersig CIRM 2008 32

Fractal Weyl law How to count states in open systems? Re(k) C Im(k) N(k) = {k n : Im(k n) > C, Re(k n) k}. Conjecture: fractal Weyl law for open chaotic systems J. Sjöstrand, Duke Math. J. 60, 1 (1990), M. Zworski, Invent. Math. 136, 353 (1999) with non-integer exponent N(k) k α. α = d + 2 2 where d is the fractal dimension of the chaotic repeller in the Poincaré section J. Wiersig CIRM 2008 32

Fractal Weyl law Microstadium refractive index = 1 R refractive index = n L = R χ 2L s J. Wiersig CIRM 2008 33

Fractal Weyl law Microstadium refractive index = 1 R refractive index = n L = R χ 2L s 10µm n = 3.3 (GaAs): weakly open T. Fukushima and T. Harayama, IEEE J. Sel. Top. Quantum Electron., 10, 1039 (2004) n = 1.5 (polymer): strongly open M. Lebenthal et al., Appl. Phys. Lett., 88, 031108 (2006) J. Wiersig CIRM 2008 33

Fractal Weyl law Numerical scheme Boundary element method J. Wiersig, J. Opt. A: Pure Appl. Opt. 5, 53 (2003) Computing sufficiently many resonances in the complex plane is extremely difficult σ/r 25 20 15 10 0 harmonic inversion Im(Ω) -0.01-0.02-0.03-0.04 5 0 0.5 1 1.5 2 2.5 Ω Normalized frequency Ω = kr -0.05-0.06 0 5 10 15 20 25 Re(Ω) Boundary element method + harmonic inversion statistics of resonances J. Wiersig CIRM 2008 34

Fractal Weyl law Number of modes 60 50 Probability density 40 30 20 10 0 0 0.01 0.02 0.03 0.04 0.05 0.06 -Im(Ω) J. Wiersig CIRM 2008 35

Fractal Weyl law Number of modes 60 50 8 7 cutoff C = 0.06 Probability density 40 30 20 10 0 0 0.01 0.02 0.03 0.04 0.05 0.06 -Im(Ω) ln(n) 6 5 4 3 2 α 1.98 0 0.5 1 1.5 2 2.5 3 ln(ω) J. Wiersig CIRM 2008 35

Fractal Weyl law Number of modes 60 50 8 7 cutoff C = 0.06 Probability density 40 30 20 10 0 0 0.01 0.02 0.03 0.04 0.05 0.06 -Im(Ω) ln(n) 6 5 4 3 2 α 1.98 0 0.5 1 1.5 2 2.5 3 ln(ω) α [1.96, 2.02] for C [0.03, 0.1] J. Wiersig CIRM 2008 35

Fractal Weyl law Chaotic repeller Chaotic repeller: set of points in phase space that never visits the leaky region both in forward and backward time evolution 1 p 1/n 0 s/s max 1 J. Wiersig CIRM 2008 36

Fractal Weyl law Chaotic repeller Chaotic repeller: set of points in phase space that never visits the leaky region both in forward and backward time evolution 1 0.83 (a) 0.73 (b) p p p 1/n 0 0.64 0.719 s/s max 1 0.395 s/s max 0.46 0.4066 s/s max 0.41 Chaotic repeller is a fractal with box-counting dimension d 1.68 J. Wiersig CIRM 2008 36

Fractal Weyl law Chaotic repeller including Fresnel s laws Account for partial escape due to Fresnel s laws real-valued I(s, p) [0, 1] 1 p 1/n 0 s/s max 1 J. Wiersig CIRM 2008 37

Fractal Weyl law Chaotic repeller including Fresnel s laws Account for partial escape due to Fresnel s laws real-valued I(s, p) [0, 1] 1 p 1/n 0 s/s max 1 Multifractal: infinite set of fractal dimensions d(q) with real q box counting dimension d(0) 1.986 α 1.99 information dimension d(1) 1.913 α 1.96 correlation dimension d(2) 1.877 α 1.94 d(0) is consistent with fractal Weyl law (α [1.96, 2.02]) J. Wiersig CIRM 2008 37

Fractal Weyl law Low-index stadium n = 1.5 (polymer) 0-0.1 Im(Ω) -0.2 0 10 20 30 40 50 60 70 Re(Ω) α [1.68, 1.88] J. Wiersig CIRM 2008 38

Fractal Weyl law Low-index stadium n = 1.5 (polymer) 0 1 p -0.1 Im(Ω) -0.2 0 10 20 30 40 50 60 70 Re(Ω) α [1.68, 1.88] 1/n 0 s/s max d(0) 1.512 d(1) d(2) 1.593 1 The predicted exponent, 1.76 and 1.78, is consistent with the fractal Weyl law Conjecture: the fractal Weyl law applies to optical microcavities if the concept of the chaotic repeller is extended by including Fresnel s laws J. Wiersig and J. Main, PRE 77, 036205 (2008) J. Wiersig CIRM 2008 38

Lifetime statistics 80 60 n=3.3, TM stadium RMT P(-Im Ω) 40 20 0 0 0.02 0.04 0.06 0.08 -Im Ω 40 n=3.3, TE 30 P(-Im Ω) 20 10 0 0 0.02 0.04 0.06 0.08 -Im Ω Good agreement with random-matrix theory; see talk by H. Schomerus H. Schomerus, J. Wiersig, and J. Main, submitted (2008) J. Wiersig CIRM 2008 39

Summary Optical microcavities as open quantum billiards Avoided resonance crossings Avoided crossings despite integrability Formation of long-lived, scarlike modes Unidirectional light emission from high-q modes Unidirectional light emission and universal far-field patterns Fractal Weyl law J. Wiersig CIRM 2008 40