Quantum chaos in optical microcavities
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1 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
2 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
3 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
4 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
5 Introduction to optical microcavities J. Wiersig CIRM
6 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
7 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
8 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
9 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
10 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
11 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
12 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
13 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
14 Avoided resonance crossings J. Wiersig CIRM
15 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
16 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
17 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
18 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) A B C D elliptical microcavity C D E F Im(Ω) B A D C eccentricity E F J. Wiersig CIRM
19 Avoided resonance crossings Avoided crossings despite integrability Re(Ω) A B elliptical microcavity C D C D E F A C E B D F Im(Ω) B A D C eccentricity E F Formation of scarlike modes J. Wiersig, PRL 97, (2006) J. Wiersig CIRM
20 Avoided resonance crossings Avoided crossings despite integrability Re(Ω) A B elliptical microcavity C D C D E F A C E B D F Im(Ω) B A D C eccentricity E F Formation of scarlike modes J. Wiersig, PRL 97, (2006) J. Wiersig CIRM
21 Avoided resonance crossings Avoided crossings despite integrability Re(Ω) A B elliptical microcavity C D C D E F A C E B D F Im(Ω) B A D C eccentricity E F Formation of scarlike modes J. Wiersig, PRL 97, (2006) Augmented ray dynamics including the Goos-Hänchen shift J. Unterhinninghofen, J. Wiersig, and M. Hentschel, PRE 78, (2008) J. Wiersig CIRM
22 Im(Ω) Avoided resonance crossings Formation of long-lived, scarlike modes Re(Ω) A B 0 C A F B E D aspect ratio A C C D E E F J. Wiersig, PRL 97, (2006) Long-lived mode C: Q (increase by more than one order of magnitude) no diffraction at corners B D F J. Wiersig CIRM
23 Avoided resonance crossings Formation of long-lived, scarlike modes Im(Ω) Re(Ω) A B 0 C A F B E D aspect ratio C C D E F J. Wiersig, PRL 97, (2006) Long-lived mode C: Q (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
24 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) e+05 5e+05 long-lived R R 2 d Q 4e+05 x1000 3e+05 short-lived d/r Weak internal coupling: weak hybridization of modes J. Wiersig CIRM
25 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 θ Hybridized whispering-gallery mode has Q = and unidirectional emission J. Wiersig CIRM
26 Avoided resonance crossings Unidirectional light emission from high-q modes far-field intensity (arb. units) observation angle θ in degree Small angular divergence and ultra-high Q > 10 8 J. Wiersig CIRM
27 Avoided resonance crossings Unidirectional light emission from high-q modes far-field intensity (arb. units) 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
28 Unidirectional light emission and universal far-field patterns J. Wiersig CIRM
29 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
30 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, (R) (2005) S.-B. Lee et al., Phys. Rev. A 75, (R) (2007) S. Shinohara and T. Harayama, Phys. Rev. E 75, (2007) J. Wiersig CIRM
31 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, (R) (2005) S.-B. Lee et al., Phys. Rev. A 75, (R) (2007) S. Shinohara and T. Harayama, Phys. Rev. E 75, (2007) Our goal: unidirectional emission high Q-factors J. Wiersig and M. Hentschel, PRL 100, (2008) J. Wiersig CIRM
32 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 s/s max s/s max ρ ε = 0.43 φ sin χ 1 s 1/n 0 1/n 1 J. Wiersig 0 CIRM
33 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
34 Unidirectional light emission and universal far-field patterns Far-field pattern: ray simulation far-field intensity (arb. units) 1 TE TM 2 1 1s s/s max 1 2s 1 1 φ FF = 0 2s sin χ B -1/n -0.4 sin χ observation angle φ FF in degree Difference between TE and TM modes is due to the Brewster angle 2 1s J. Wiersig CIRM
35 Unidirectional light emission and universal far-field patterns Far-field pattern: mode calculation TE far-field intensity (arb. units) TM Q = Q = observation angle φ FF in degree all high-q TE modes show unidirectional emission (universal far-field pattern) J. Wiersig CIRM
36 Unidirectional light emission and universal far-field patterns Husimi representation 1 sin χ 1/n 0 1/n (a) s/s max Scarring ensures high Q-factors J. Wiersig CIRM
37 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 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
38 Unidirectional light emission and universal far-field patterns Experiments on the Limaçon cavity Theory: J. Wiersig and M. Hentschel, PRL 100, (2008) Harayama et al., Kyoto Capasso et al., Harvard Cao et al., Yale J. Wiersig CIRM
39 Fractal Weyl law J. Wiersig CIRM
40 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 N(E) = N(E) + N fluc (E) E 1 E 2 N(E) E 3... N(E) E J. Wiersig CIRM
41 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 N(E) = N(E) + N fluc (E) E 1 E 2 N(E) E 3... N(E) E Weyl s law for 2D billiard with area A ( 2 /2m = 1) N(E) = A 4π E k 2 J. Wiersig CIRM
42 Fractal Weyl law How to count states in open systems? Re(k) Im(k) J. Wiersig CIRM
43 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
44 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 where D is the fractal dimension of the chaotic repeller J. Wiersig CIRM
45 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 where d is the fractal dimension of the chaotic repeller in the Poincaré section J. Wiersig CIRM
46 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 where d is the fractal dimension of the chaotic repeller in the Poincaré section J. Wiersig CIRM
47 Fractal Weyl law Microstadium refractive index = 1 R refractive index = n L = R χ 2L s J. Wiersig CIRM
48 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, (2006) J. Wiersig CIRM
49 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 J. Wiersig CIRM
50 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 Ω Normalized frequency Ω = kr J. Wiersig CIRM
51 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 harmonic inversion Im(Ω) Ω Normalized frequency Ω = kr Re(Ω) Boundary element method + harmonic inversion statistics of resonances J. Wiersig CIRM
52 Fractal Weyl law Number of modes Probability density Im(Ω) J. Wiersig CIRM
53 Fractal Weyl law Number of modes cutoff C = 0.06 Probability density Im(Ω) ln(n) α ln(ω) J. Wiersig CIRM
54 Fractal Weyl law Number of modes cutoff C = 0.06 Probability density Im(Ω) ln(n) α ln(ω) α [1.96, 2.02] for C [0.03, 0.1] J. Wiersig CIRM
55 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
56 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 (a) 0.73 (b) p p p 1/n s/s max s/s max s/s max 0.41 Chaotic repeller is a fractal with box-counting dimension d 1.68 J. Wiersig CIRM
57 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 (a) 0.73 (b) p p p 1/n s/s max s/s max s/s max 0.41 Chaotic repeller is a fractal with box-counting dimension d 1.68 α = d+2 2 = 1.84, i.e. fractal Weyl law fails! J. Wiersig CIRM
58 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
59 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) α 1.99 information dimension d(1) α 1.96 correlation dimension d(2) α 1.94 d(0) is consistent with fractal Weyl law (α [1.96, 2.02]) J. Wiersig CIRM
60 Fractal Weyl law Low-index stadium n = 1.5 (polymer) Im(Ω) Re(Ω) α [1.68, 1.88] J. Wiersig CIRM
61 Fractal Weyl law Low-index stadium n = 1.5 (polymer) 0 1 p -0.1 Im(Ω) Re(Ω) α [1.68, 1.88] 1/n 0 s/s max d(0) d(1) d(2) The predicted exponent, 1.76 and 1.78, is consistent with the fractal Weyl law J. Wiersig CIRM
62 Fractal Weyl law Low-index stadium n = 1.5 (polymer) 0 1 p -0.1 Im(Ω) Re(Ω) α [1.68, 1.88] 1/n 0 s/s max d(0) d(1) d(2) 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, (2008) J. Wiersig CIRM
63 Lifetime statistics n=3.3, TM stadium RMT P(-Im Ω) Im Ω 40 n=3.3, TE 30 P(-Im Ω) 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
64 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
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