Photonic Micro and Nanoresonators Hauptseminar Nanooptics and Nanophotonics IHFG Stuttgart
Overview 2 I. Motivation II. Cavity properties and species III. Physics in coupled systems Cavity QED Strong and weak coupling Spontaneous emission Purcell effect Spontaneous emission control Nonlinear effects IV. Summary
I. Motivation 3 Why microcavities? High electric fields give rise to interesting physics Ideally suited for nonlinear lasing applications Studies of strongly and weakly coupled systems Size of structures Volume of cavity decisive for certain effects Easily integrable on a chip B. Li et al., Single nanoparticle detection using split-mode microcavity Raman lasers, PNAS 111, 14657-14662 (2014) F. Vollmer, Detecting single DNA molecules with optical microcavities, SPIE Newsroom (December 2014)
Cavity 4 Mirrors with reflectivity R Δν Cavity length l How to characterize a cavity? Q = ν ν S. Reitzenstein, AlAs/ GaAs micropillar cavities with quality factors exceeding 150.000, Applied Physics Letters 90, 251109 (2007)
Quality Factor 5 Influences on cavity quality factor 1. Absorption losses 2. Edge-scattering losses 1 Q = 1 + Q intrinsic 1 Q edge scattering + 1 Q absorption
Cavity Species 6 Microdisk Micropillar Microsphere Microcavities Photonic Crystal Microring
Micropillar 7 Cavity Distributed Bragg Reflectors (DBRs) R. Oulton, Quantum dots: Electrifying cavities, Nature Nanotechnology 9, 169-170 (2014) J. Joannapoulos, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 2 nd Edition (2008)
Cavity Species 8 Microdisk Micropillar Microsphere Microcavities Photonic Crystal Microring
Microsphere/Microring/Microdisk 9 Whispering gallery modes Θ c = arcsin n t n e http://groups.jqi.umd.edu/solomon/sites/groups.jqi.umd.edu.solomon/files/images/microdisk _figure_copy.png (20.05.16) http://gaeta.research.engineering.cornell.edu/sites/gaeta.research.engineering.cornell.edu/file s/styles/slide_show/public/slider/microsphere-2.jpg?itok=drdftfxr (20.05.16) J. Leuthold et al., Nonlinear silicon photonics, Nature Photonics 4, 535-544 (2010)
Cavity Species 10 Microdisk Micropillar Microsphere Microcavities Photonic Crystal Microring
Photonic Crystal 11 K. Vahala, Optical Microcavities, Nature 424 (2003) B. Ellis, Ultralow-threshold electrically pumped quantum dot photonic-crystal nanocavity laser, Nature 424 (2003)
Cavity Species 12 Microdisk Micropillar Microsphere Microcavities Photonic Crystal Microring
Overview 13 III. Physics in coupled systems
Cavity Quantum Electrodynamics (cqed) 14 Assumption: 1 photon interacting with a 2-level system e, 0 g, 1 Due to interaction: oscillation of energy with Rabi frequency
Jaynes Cummings Model 15 Hamiltonian of the system H ges = H 0 + H int H ges = ħ 0 σ ee + ħω( a a + 1 2) + ħg( σ ge a + σ eg a) transition photon energy interaction energy energy with vacuum Rabi frequency g = μ eg E ħ
Jaynes Cummings Model 16 Hamiltonian in bare states basis e, n H (n) = ħ nω + ω 0 g n + 1 g n + 1 n + 1 ω ω 0 g, n + 1 diagonalize matrix for new eigenenergies and eigenstates +, n = cos α n 2 g, n + 1 + sin α n 2, n = sin α n 2 g, n + 1 + cos α n 2 e, n e, n E ±,n = ħω n + 1 ± 1 2 ħω n Ω n = Δ 2 + g 2 (n + 1) g n + 1 α n = arctan Δ E ±
Dressed States 17 bare states dressed states
Jaynes Cummings Model for Quantum Dots 18 γ c Eigenenergies stay the same E ±,n = ħω n + 1 ± 1 2 ħω n Rabi frequency takes losses into account γ X Ω = Δ 2 + g 2 γ c γ X 2 For zero detuning 4 Ω = g 2 γ c γ X 2 4 E/ħ
Strong and Weak Coupling 19 coupling characterized by Rabi frequency g strong coupling: E > 0 g > γ c γ x 2 g E micro and nanocavities exhibit large electric fields
Strong Coupling 20 How to realize and measure strong coupling of exciton and photon? 1. Prepare quantum dots with desired spectral transition 2. Fabricate high-q cavity around suited quantum dot Q n, g E can be tuned K.Hennessey et al., Quantum nature of a strongly coupled single quantum dot-cavity system, Nature 445 (2007)
Measurement of Strongly Coupled System cavity exciton 21 1. Anti-crossing 2. Spectral triplet K.Hennessey et al., Quantum nature of a strongly coupled single quantum dot-cavity system, Nature 445 (2007)
Anti-Crossing 22 K.Hennessey et al., Quantum nature of a strongly coupled single quantum dot-cavity system, Nature 445 (2007)
Theory of Anti-Crossing 23 Consider 2-level system: H 0 = E 1 0 0 E 2 with E 1 < E 2 g, n + 1 e, n Eigenstates are degenerate for E 1 = E 2 as ΔE = E 1 E 2 = 0 Levels cross
Theory of Anti-Crossing 24 Introduce pertubation which couples former eigenstates ħg n + 1 H = H 0 + P = E 1 0 0 E 2 + 0 W W 0 = E 1 W W E 2 with E 1 < E 2 New eigenenergies E + = 1 2 E 1 + E 2 + 1 2 (E 1 E 2 ) 2 +4 W 2 E = 1 2 E 1 + E 2 1 2 (E 1 E 2 ) 2 +4 W 2 Coupling lifts degeneracy of eigenstates +, n, n ħω e, n g, n + 1
Measurement of Strongly Coupled System cavity exciton 25 1. Anti-crossing 2. Spectral triplet K.Hennessey et al., Quantum nature of a strongly coupled single quantum dot-cavity system, Nature 445 (2007)
Spectral Triplet 26 Four transitions with three different energies Indication for strong coupling K.Hennessey et al., Quantum nature of a strongly coupled single quantum dot-cavity system, Nature 445 (2007)
Strong and Weak Coupling 27 coupling characterized by Rabi frequency g strong coupling: E > 0 g > γ c γ x 2 g E micro and nanocavities exhibit large electric fields weak coupling: E = 0 g < γ c γ x 2 no energy splitting observed
Weak Coupling 28 Eigenstates cross Indicates weak coupling P. Michler, Single Semiconductor Quantum Dots, Springer Verlag (2009)
Overview 29 III. Physics in coupled systems
Spontaneous Emission (SE) 30 Vacuum flactuations cause excited state to decay Describe SE rate with Fermi s golden rule density of final states Γ i f = 2π ħ f H i 2 ρ f SE rate proportional to density of final states
Purcell Effect 31 Cavity changes density of modes for 2-level system to decay into Γ i f ρ f Changed probability for SE SE rate changed
Purcell Effect 32 SE enhancement (Purcell factor) F = modified cavity enhanced SE rate SE rate into free space = Γ Γ 0 = 3 4π λ cav n 3 Q V Figure of merit for Purcell effect Lifetime τ = 1 Γ If If ρ f > ρ 0 F > 1 τ < τ 0 SE enhancement ρ f < ρ 0 F < 1 τ > τ 0 SE suppression
Spontaneous Emission Enhancement/Suppression 33 D. Englund et al., Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal, Phys. Rev. Let. 95 (2005)
Conclusions for Strong and Weak Coupling 34 Strong coupling Weak coupling Overcomes dissipation in the system Exhibits large electric fields Enables studies and on-chip realization of nonlinear effects at low intensities Supports Purcell effect Allows control of SE rate SE enhancement for bright single photon sources Increased SE rate improves indistinguishability of photons of single photon sources SE suppression useful for photonic devices https://www.sciencedaily.com/releases/2016/03/160307113945.htm (1.06.2016)
Overview 35 III. Physics in coupled systems
Temporal Coupled-Mode Theory 36 Describes how amplitude of electric field in coupled resonator evolves cavity da dt = iω 0a 1 τ a +κs +
Temporal Coupled-Mode Theory 37 da dt = iω 0a 1 τ a +κs + assume s + e iωt a = κs + i ω ω 0 + 1 τ Energy in resonator E = a 2
Mode Coupling in Si Photonic Crystal 38 T. Uesugi et al., Investigation of optical nonlinearities in an ultrahigh-q Si nanocavity in a two-dimensional photonic crystal slab, Opt. Expr. 14, 377-386 (2006)
Measurement for Higher Pump Powers 39 3 rd nonlinear effects cause asymmetric response 1. Two photon absorption Free-carrier generation Thermo-optic effect 2. Kerr effect T. Uesugi et al., Investigation of optical nonlinearities in an ultrahigh-q Si nanocavity in a two-dimensional photonic crystal slab, Opt. Expr. 14, 377-386 (2006)
Modeling Influence of Nonlinear Effects 40 Employ coupled-mode theory a = κs + i ω ω 0 + 1 τ Modify equation for this case a = 1 2τ e S 1 with total decay rate i ω ω 0 + 1 1 2τ total = 1 + 1 + 1 + 1 τ total τ v τ e τ FCA τ TPA and shifted resonance frequency ω 0 = 2πc (λ 0 + Δλ free + Δλ thermal + Δλ Kerr ) T. Uesugi et al., Investigation of optical nonlinearities in an ultrahigh-q Si nanocavity in a two-dimensional photonic crystal slab, Opt. Expr. 14, 377-386 (2006)
Infuences of Nonlinearities on Refractive Index 41 Refractive index change due to free carriers Additional free carriers modify permittivity ε ω = ε ω ω P 2 (only real part) ω 2 ω 2 P = e2 N with ε 0 m n(ω) Re ε ω = ε ω P 2 ω 2 1 2 n 0 ω P 2 2nω 2 = n 0 + Δn free Δn free = ω P 2 2nω 2
Infuences of Nonlinearities on Refractive Index 42 Refractive index change due to thermo-optical effect Δn thermal = n T ΔT ΔT a 2 1 τ FCA + 1 τ TPA Refractive index change due to Kerr effect Δn Kerr = A c a 2 n V Kerr Δn total A c n V Kerr a 2 + n T a 2 1 + 1 ω 2 P τ FCA τ TPA 2nω 2
Refractive Index Change and Radiation Efficiency 43 T. Uesugi et al., Investigation of optical nonlinearities in an ultrahigh-q Si nanocavity in a two-dimensional photonic crystal slab, Opt. Expr. 14, 377-386 (2006)
Conclusion 44 Ultrahigh-Q resonator changes cavity response Nonlinear effects change refractive index of material Shifts resonance frequency Improves sensitivity for sensing applications Nonlinear effects are observed for quite low pump powers Important for applications like higher harmonic generation
Summary 45 Cavities Characterization of cavities Types of cavities cqed Jaynes Cummings Model Strong coupling Weak coupling
Summary 46 Weak coupling Purcell effect Spontaneous emission enhancement and suppression Nonlinear effects in ultrahigh-q resonators Nonlinear effects affect cavity response Strong nonlinear effects at low pump powers