Quantum Optics in Photonic Crystals Peter Lodahl Dept. of Communications, Optics & Materials (COM) Technical University of Denmark
Acknowledgements AMOLF Institute Amsterdam / University of Twente Ivan Nikolaev, dr. Femius Koenderink, prof. Willem L. Vos. University of Utrecht dr. Floris van Driel and prof. Daniel Vanmaekelbergh. COM DTU Experiments: Theory: Toke Lund-Hansen, Søren Stobbe, Jeppe Johansen dr. Mikael Svalgaard, and prof. Jørn M. Hvam. Peter Kalsen, Philip Trøst Kristensen, Aliaksandra Ivinskaya dr. Andrei Lavrinenko, prof. Jesper Mørk, and prof. Bjarne Tromborg.
Outline Introduction to photonic crystals Bragg diffraction. Photonic bandgap. Propagation of light in a photonic crystal Model of light diffusion due to structural disorder. Experimental results. Emission of light in a photonic crystal Local density of states. Fermi s Golden Rule and beyond. Quantum dots in photonic crystals. Applications. Nanocavities in photonic crystals. Conclusions.
Photonic Crystals λ a Periodic dielectric structure scale a wavelength of light λ Light interacts with the photonic crystal by scattering and subsequent interference from lattice planes low optical absorption. high refractive index contrast.
Optical Bragg Diffraction Interference for a specific set of lattice planes (one k-vector) band of wavelengths is reflected (stop gap). Reflectivity [%] 100 80 60 40 20 λ Δλ 550 0 600 650 700 Wavelength [nm] Frequency Bragg s law (lowest approx., weak PCs) Width of stop gap indicates photonic strength Ψ=Δλ/λ Δω Stop Gap Wave vector λ Β = 2d cosα Vos et al., Phys. Rev B 53, 16231 (1996).
Bandstructure diagram (fcc lattice of air-spheres, ε~12) 3D Photonic Bandgap Frequency Wavevector Busch&John, Phys. Rev. E 58, 3896 (1998) ρ DOS ( ω) = 3 kδ n 1BZ r r d ) ( ω ω ( k ) n Inhibited spontaneous emission in band gap. Enhanced density of states enhanced emission. Bykov, Sov. J. Quant. Electron. 4, 861 (1975); Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
Quantum Dots in Photonic Crystals CdSe quantum dots Ensemble of quantum dots in a 3D photonic crystal TiO 2 photonic crystals Two different effects can occur: 1) Redistribution of the emitted light due (Bragg diffraction) 2) Modified radiative lifetime (density of states)
Redistribution of Light in Photonic Crystals Disorder in PCs leads to multiple scattering of emitted light Mean free path (av. propagation distance): l ~ 20 μm light propagation is diffuse for distances beyond 20 μm. L B : Bragg attenuation length (~ 5-10 μm). α Koenderink & Vos, Phys. Rev. Lett. 91, 213902 (2003). Simplistic picture: The emitted light is diffused in the bulk of the PC. Detected light is scattered close to the sample surface and subsequently modified due to the photonic bandstructure.
Emission Spectra from QDs in Photonic Crystals Angular probability distribution of diffuse light: 1+ R( ω) 3 P( α, ω) = cosα + cosα ω 1 R( ω) 2 [ 1 R( α, )] Reflectivity coefficient Intensity [counts/s] 6000 4000 2000 0 α 14,000 16,000 18,000 20,000 Frequency [cm -1 ] α = 60º α = 20º α = 0º Relative intensity Nikolaev, Lodahl & Vos, Phys. Rev. A 71, 053813 (2005). 3 2 1 0 a = 370 nm (Ref) a = 420 nm a = 500 nm λ = 630 nm 0 10 20 30 40 50 60 Angle α [degrees]
LDOS n 1BZ Local Density of States in Photonic Crystals The emission of light in a photonic crystal is determined by the local density of states: r r r 2 r = 3 r ρ ( ω, ) d k E r ( ) δ Bloch modefunctions n, k ( ω ω ( k )) n Sprik et al., Europhys. Lett. 35, 265 (1996). Vats et al., Phys. Rev. A 65, 43808 (2002). 3 DOS is the unit-cell averaged LDOS: ρdos ( ω) = d rε ( r ) ρldos ( ω, r ) Unit Cell r r r LDOS is strongly modified in a photonic crystal Koenderink et al., Opt. Lett. 30, 3210 (2005).
Spontaneous Emission The decay rate of a dipole emitter is proportional to the projected LDOS in the Wigner-Weisskopf approximation r r πω r r r 2 = ( ) 3 r r γ ( d, ω, ) d k d E r ( ) δ ω ω n k n( k ), 3hε 0 n 1BZ Fermi s Golden Rule Spontaneous emission decay depends on position of emitter Excited state population 1,0 0,8 0,6 0,4 0,2 Position 1 Position 2 0 1 2 3 4 Time
Beyond Fermi s Golden Rule Atomic resonance tuned close to a photonic band edge. Frequency Fractional decay of excited state. Philip Kristensen. See poster for further details. Excited state population Time
Lifetime Measurements CdSe quantum dots Ensemble of quantum dots in a 3D photonic crystal TiO 2 photonic crystals
Quantum Dots in 3D Titania Photonic Crystals Intensity [counts] 1000 100 10 (3) (1) (2) 0 20 40 60 80 Time [ns] (1): a = 340 nm (2): a = 460 nm (3): a = 580 nm Most-frequent decay rate: γ MF = 0.052 ns -1 (inhibited emission) γ MF = 0.081 ns -1 (reference) γ MF = 0.104 ns -1 (enhanced emission) First demonstration that photonic crystals control spontaneous emission (as proposed by Yablonovitch, 1987). Lodahl, van Driel, Nikolaev, Irman, Overgaag, Vanmaekelberg & Vos, Nature 430, 654 (2004). Nikolaev, Lodahl, van Driel, Koenderink & Vos, submitted, arxiv.org > physics/0511133.
Modeling Decay Curves Decay curves are not single-exponential model with a distribution of exponentials φ(γ). I ( t) I φ( γ ) 0 γ t = e dγ Intensity [counts] 10000 1000 χ 2 R =1.1 Log-normal distribution: φ ( γ ) = Ae ln 2 ( γ / γ MF )/ w 2 Residuals 100 4 2 0-2 -4 0 20 40 60 80 Time [ns]
LDOS Variations of Decay Rate Center and width of distribution vary strongly with lattice parameter QDs are distributed over different positions in the PC Nikolaev, Lodahl, van Driel, Koenderink & Vos, submitted, arxiv.org > physics/0511133.
Comparison with theory Variation of center (γ MF ) and width (Δγ) with lattice parameter. γ MF varies 3x Δγ varies 6x Quantitative theory: LDOS Nikolaev, Lodahl, van Driel, Koenderink & Vos, submitted, arxiv.org > physics/0511133.
Spontaneous Emission from Quantum Dots Ideal 2-level excitonic system in a homogeneous medium: γ = 2 e 3πε m 0 2 hc 3 ω 0 Thermal population: supralinear frequency dependence r p e 2 ω Time-resolved emission from CdSe quantum dots in solution Decay Rate T=300 K van Driel, Allan, Delerue, Lodahl, Vos & Vanmaekelbergh, Phys. Rev. Lett. 95, 236804 (2005).
Photonic Crystals versus Cavities High-Q cavity: Enhancement: (Δω/ω) cav 10-4 Inhibition: (Δω/ω) cav 10-3 Gerard et al., PRL 81, 1110 (1998). Bayer et al., PRL 86, 3168 (2001). Photonic crystal: Enhancement and inhibition: (Δω/ω) pc 10-1 >100x bandwidth Large volume, room temperature.
Applications Inhibition: Increase efficiency of miniature lasers, transistors, and solar cells (Yablonovitch 87). Suppress vacuum fluctuations to shield fragile quantum information against decoherence. Enhancement: Enhance efficiency of optical transitions. Diminish photon-jitter noise of single-photon sources.
Quantum Dynamics in a Nanocavity Time Model DOS for a cavity in a photonic crystal with a band gap. Vacuum Rabi oscillations. 1 2 ( e 0 + g 1 ) Entanglement. Excited state population wide cavity narrow cavity
Conclusions Photonic crystals modify both propagation and emission of light. Light propagation is diffused due to structural disorder of the photonic crystal. Emission of light can be accelerated or delayed by employing the LDOS. Nanocavities can be implemented in photonic crystals.