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By : Majid Sodagar Supervisor : Dr. Sina Khorasani Faculty : Electrical Engineering Date : Nov. 2008 1

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Literature Review Main Theme Exciton Transfer Matrix Method Matrix Diagonalization Photonic Crystal Cavity Finite Difference Time Domain Quality Factor Photon-Exciton Interaction Time Domain Evolution Energy Splitting Conclusion 2

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Nature, Vol. 445, 2007 = 24.1GHz=100 eV = 8.5GHz=35 eV g = 76 eV Investigating the strong coupling regime using self assembled InAs QD 3 Switzerland, US Q = 13000

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This shows the promising potential of photonic crystal waveguides for efficient single-photon sources. Quantum dots that couple to a photonic crystal waveguide are found to decay up to 27 times faster than uncoupled quantum dots. Phys. Rev. Lett. 101, 113903 (2008) 4 Germany SE DR

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J. Phys.: Condens. Matter 20, 454209 (2008) Discuss the recently discovered non-resonant coupling mechanism between quantum dot emission and cavity mode for large detuning. Spectral dot–cavity detuning is discussed on the basis of shifting either the quantum dot emission via temperature tuning or the cavity mode emission via a thin film deposition technique. 5 Germany

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6 Nature photonics, VOL 2, (2008) Japan fully confined electrons and photons using a combination of three dimensional photonic crystal nanocavities and quantum dots. Important due to polarization issue. Applications : Triggered single-photon sources quantum logic gate for optical fibre-based quantum cryptography communication and quantum repeater systems

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D-Wave 16-bit Q-Computer CQED: All optical quantum information and computation Quantum cryptography Realization of quantum repeaters Single photon sources Qbit realization Strong coupling regime: Fabrication of high-efficiency microcavity LEDs Low-threshold vertical-cavity surface emitting lasers Microsphere lasers Entanglement Weak coupling regime: Modification of the emission diagram Enhancement or inhibition of the SE rate Funneling of SE photons into a single mode Control of the SE process on the single photon level 7

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Disk-Like Quantum Dot Electron-Hole Pair (Exciton) Photonic Crystal Slab (PCS) Methods of Photon Confinement : In plane : Distributed Bragg Reflection Normal : Total Internal Reflection 8

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R 0 =150nm Z 0 =4nm Ga 1-x Al x As GaAs Ga 1-x Al x As X=0.36 V ze = 300 meV V zh = 150 meV Strain for GaAs : xx = yy = -9×10 -4 zz = 8.3×10 -4 a v = -1.116 eV b = -2 eV Pikus-Bir Deformation Potentials Hydrostatic Strain Uniaxial Strain V ≈∞ 9

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EgEg Electron Electron-Hole No Binding Energy Exciton Binding Energy E exciton Exciton types: Frenkel : Localized near single atom Smaller Bohr radius Strong coupling Wannier : Electron holes are far apart in CV and VB Larger Bohr radius Weak coupling 10 λ e λ e

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Envelope Part Bloch Part Wave Function Macroscopic Potential Contributing to Envelope Part Microscopic Potential Contributing to Bloch Part 11 Using real potential in Schrödinger equation makes it unwieldy.

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Electron wave function Influenced by only one band (CB) Simple Schrödinger equation S-Like orbital was taken as Bloch part Hole wave function Influenced by three bands (HH,LH,SO) using 6×6 Luttinger Hamiltonian Combination of P x, P y and P z including spin was taken as Bloch part In contrast to CB, there is a twofold degeneracy in VB besides the closeness of SO 12 Energy 0 Wave Vector Band structure for typical III-V and IV group semiconductor

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Disk like Quantum Dot Thickness << Area For holes In plane treated as single band Bessel Sinusoidal 13

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J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869, (1955). Can be reduced to two 3×3 block diagonal matrix via unitary transformation on basis set. Strain effect can be added directly by updating P, Q, R and S using Pikus-Bir deformation potentials. 14

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15 Solving Methods Plane Wave Expansion (PWE) Transfer Matrix Methods (TMM) Finite Difference (FD) Strain Effect Strain due to lattice mismatch

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Forward- Backward Waves Forward- Backward Waves 1 2 3 a4b4a4b4 a3b3a3b3 a2b2a2b2 a1b1a1b1 B. Chen, M. Lazzouni, Phys. Rev. B,45, 1024, (1992). Total Transfer Matrix: Bounded states Condition: T 22 should has eigenvalue equal to 0 for some E ! 16 Continuity of envelope function probability current

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Zero Crossing points denote hole bound states energies quantized in the z direction Sweeping over system energies so that the coefficients of all non- damping terms vanish. 17 Energy (eV) | (T 22 )| 10Log| (T 22 )| HH1LH1HH2 Applied parameters :

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HH1LH1 18

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HH2 19 Derivative were not conserved

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Big Challenge G.P. Lepage, “VEGAS: An Adaptive Multidimensional Integration Program”, Publication CLNS-80/447, (1980). Using VEGAS Algorithm Using Cluster Computer Using MPI Libraries in C++ Interaction 20 Monte Carlo Non-uniform Stratified Monte Carlo Non-uniform Stratified

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Exciton space basis set consists of 243 vectors, so 59049 matrix elements have been calculated. E g = 1424 meV Exciton wave form: Without interaction With binding energy 21

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22 Periodic structure capable of exhibiting gap in frequency domain PC were used to realize a photon cavity Simulation Tools : Plane Wave Expansion (PWE) Finite Difference Time Domain Frequency Domain Finite Element Time Domain Frequency Domain Multiple Multi pole (MMP) Wannier Functions Method E.Yablonovitch, Phys. Rev. Lett. 67, 2295 - 2298 (1991) First 3D PC realization

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23 Stability Condition Minimum Simulation time Spatial Maximum Divisions Maxwell’s Equation Discretized form Yee Cube

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Specifications Fully vectorial in 2D and 3D Dispersive material Dissipative material Various boundary condition were implemented Periodic (Reducing simulation time) Bloch (Virtual physical problems) PML {Mur and SPML} (Finite domain simulations) PEC, PMC Yee algorithm was implemented (Simple V. MD-WDF) Implemented in C++ (Speed) Equipped with harmonic inversion (Extraction efficiency) Subpixel averaging K. S. Yee, IEEE Trans. Antennas Propagat. 14, 302 (1966). 24

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Slab Photonic crystal Triangle air hole lattice x y x z 0 1 0.2 0.1 0.3 0.4 0.5 0.6 0.8 0.7 0.9 Light Cone TE Like Gap Guided modes unguided modes Band folding due to bigger unit cell 12 division per a, d=0.2a, c=0.5a, r=0.4a X J J X KxKx KyKy 25 PEC Applied

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It seams reasonable to choose slab thickness between 0.6a and 0.9a. Maximum gap is achieved with hole radius around 0.35a. 12 division per a, r=0.3a Contour20 Division Per a, d=0.6a Contour Gap Size Versus Hole Radius 1.52.02.53.04.5 Hole Radius (a) 3.54.0 0 0.02 0.04 0.06 0.08 0.10 0.14 0.12 0.20.40.60.81.41.01.21.6 10% 15% 20% 25% 30% 40% 35% 5% 0% Gap Size Versus Slab Thickness Slab thickness (a) 26

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Applying Chirp-ZT on output time date. RED Circles : Cavity modes Green Circles : Due to band edge near zero group velocity 0.200.250.300.350.45 Normalized Frequency 0.40 Response from Broad Band Input 3.5 3.0 2.0 2.5 1.5 0.5 0.0 1.0 Cavity is realized by removing one hole. Applying PML on the top and the bottom of the cavity. The system is stimulated with several broad band dipoles. Log[czt|Amp|] (a.u.) 27

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|E||E y ||E x | |E y ||E x ||E| |E y ||E x ||E| |E y ||E x ||E| |E y ||E x | |E||E y ||E x | 28

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Q is evaluated as 148 Frequency DomainTime Domain S. Guo and S. Albin, Opt. Express 11, 1080-1089 (2003). Evaluation begins as the input vanishes 0.200.250.300.350.450.40 Log[czt|Amp|] (a.u.) 3.5 3.0 2.0 2.5 1.5 1.0 Normalized Frequency E0E0 E1E1 T1T1 T0T0 E y (a.u.) Log[|E y |] 6 4 0 2 -2 -4 -6 1 23465 Time (10 -13 ) 29

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Resonance Frequency and Q are dramatically affected by changing r`. Other modes emerges as r` approaches zero. Freq. Versus r`Q. Versus r` 0.38 0.36 0.32 0.34 0.30 0.28 0.26 0.40.350.300.250.20 Nearest Hole Radius (a) 0.40.350.300.250.20 Nearest Hole Radius (a) 1.2×10 4 1×10 4 6×10 3 8×10 3 4×10 3 2×10 3 0 0.20 0.25 0.30 0.35 Normalized Frequency 0.40 0.45 Modes for r`=0 Output Intensity (a. u.) 30

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31 r = 0.27a r = 0.30ar = 0.33a 21800

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Interaction K. Rewski, R. W. Boyd, Journal of modern optics, Vol. 51, no. 8, 1137–1147, (2004). Minimal Coupling Scheme Ignoring second order term Direct Coupling Scheme Second Quantization form has been exploited in order to represent the Hamiltonian in term of field operator Exciton Field operator: 32

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Approximations Using Rotating Wave Approximation (RWA) Ignoring High Frequency Evolutions Using Dipole Approximation Ignoring Epsilon inhomogeneity Integration Simplification Coupling Coefficients Matrix Element Optical wavelength Excitonic wavelength De Broglie 33

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Exciton Binding Energy Photon Can Results in : Creating Electron-Hole Pair Annihilating Electron-Hole Pair Considering Exciton wave expansion There are only triplet integrals. Summing over all electrons 34

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Can Result in : Changing Electron Position Changing Hole Position Considering Exciton wave expansion There are only triplet integrals. Photon Exciton Binding Energy 35

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36 P Lambropoulos, et al, Rep. Prog. Phys. 63 (2000) 455–503 Hamiltonian is responsible for time evolution Choose only one Photonic and Excitonic State

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37 Frequency Density of States Rabi Oscillation can occur for 2C > c High quality factor Big coupling constant Lorentzian approximation for DOS Time |U(t)| 2 Full Photons Full Exciton Photon-Exciton Combination

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Uncoupled State For Two Eigen frequencies : Detuning Dressed States = 110 eV g = 159 GHz = 2~50 eV Strong Coupling Regime 38 Our system at resonance : g1=0, g2=50, g3=150 GRad

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39 Electronic and hole states were found Excitonic state were evaluated by diagonalization A relatively high quality factor PC cavity were designed and simulated. Coupling coefficient between cavity modes and excitonic states were derived This structure is capable of operating in strong coupling regime Investigating more complicated system such as bi-excitons and more photons Realizing the physical structures Applying the concept in engineering

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End. Thanks for your Patience 40

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