Time-dependent Simulations of Electromagnetically Induced Transparency with Intense Ultra-short Pulses Wei-Chih Liu 劉威志 Department of Physics National Taiwan Normal University 2011.12.19@NTHU Outline Introduction to Electromagnetically Induced Transparency (EIT) and time-dependent simulation approach. Single atom response with intense, ultra-short pulses 1D atomic array response with intense, ultrashort pulses with pulse turn-off and turn-on Metamaterials and EIT Electromagnetically Induced Transparency Simulation model E c t Ep t 3 = 3 P 3 2 ,F = 2 , M F = - 2 Coupling field E c t 2 = 3 S 1 2 ,F = 2 , M F = - 2 E p t 1.8 GHz 1-D EM wave and 1-D atomic array Probe field l =589 nm Na atom 1 = 3 S 1 2 ,F = 1 , M F = - 1 Numerical simulation methods The electromagnetic fields are solved by discretizing Maxwell equation and propagating the electromagnetic waves by finite-difference method. 2E 1 2E 1 2P 2 2 2 2 x c t c 0 t 2 With one-directional radiation boundary condition Numerical simulation methods The atomic states and atomic polarization P is simulated by solving time-dependent Schrödinger equation by Runge-Kutta 4th-order method. Using simple cj or density-matrix approach Without rotating wave approximation. No spontaneous emission yet! explicit or implicit method At Resonance - Absorption No coupling field Amplitude -- Probe Field Position (x/λ) EIT – Transparency with coupling field Amplitude -- Probe Field Position (x/λ) EIT from purterbation theory K.-J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991). Energy level shift from simulations coupling field power = 3×104 mW cm-2 Total wave Probe field Scattering field Frequency(ω/ω31) Energy level shift from simulations coupling field power = 3×107 mW cm-2 Total wave Probe field Scattering field Ωc/2 Frequency(ω/ω31) Large Energy level shift - Transparency coupling field power = 1.2×108 mW cm-2 Total wave Probe field Scattering field Frequency(ω/ω31) Mode coupling and energy level shift in EIT 3 2 3 2 3 E c t 2 1 • Ec » Ep 1 Single atom in intense, ultra-short pulses Density-matrix simulation E13 = 0.95 a.u. Decay rate = 2 π / 1000 E12 = 1 a.u. Polarization with various coupling filed intensity coupling field FWHM=256 T/2π probe field FWHM=16 T/2π Ωp=0.01 Polarization with various coupling filed intensity coupling field FWHM=256 T/2π probe field FWHM=16 T/2π Ωc=0.1 Polarization with various coupling filed intensity coupling field FWHM=256 T/2π probe field FWHM=16 T/2π Ωp=1.0 Polarization with various coupling filed intensity coupling field FWHM=256 T/2π probe field FWHM=16 T/2π Ωp=10.0 Time-dependent polarization behavior coupling field FWHM =256 T/2π Ωp=10.0 probe field FWHM =16 T/2π Ωc=0.0 Time-dependent polarization behavior coupling field FWHM =256 T/2π Ωp=10.0 probe field FWHM =16 T/2π Ωc=10.0 Time-dependent polarization behavior coupling field FWHM =256 T/2π Ωp=10.0 probe field FWHM =16 T/2π Ωc=100.0 Time-dependent polarization behavior coupling field FWHM =256 T/2π Ωp=10.0 probe field FWHM =16 T/2π Ωc=400.0 Interaction between light and polarization wave Coupling field turned off by a Gaussian profile profile 1 t 0 profile exp(t 2 / 2 ) t 0 3 E c t 2 1 Coupling field turn-off – = 50 fs -- Polarization between 1-2 level Amplitude -- Probe Field Position (x/λ) Coupling field turn-off – = 20 fs -- Polarization between 1-2 level Amplitude -- Probe Field Position (x/λ) Coupling field turn-off – = 10 fs -- Polarization between 1-2 level Amplitude -- Probe Field Position (x/λ) Coupling field turn-off – = 5 fs -- Polarization between 1-2 level Amplitude -- Probe Field Position (x/λ) Coupling field turn-off – = 1 fs -- Polarization between 1-2 level Amplitude -- Probe Field Position (x/λ) Coupling field turn-off – = 1 fs (zoom in) -- Polarization between 1-2 level Amplitude -- Probe Field Position (x/λ) Analyze polarization wave from one atom in the array The polarization between |1> and |2> of one atom in the atomic array under constant coupling field is analyzed.The polarization becomes similar to the envelope of the probe field, while the intensity of the coupling field is large enough 3 E c t 2 1 Atomic Dynamics - Coupling field = 3×107 mW cm-2 -- Polarization between 1-2 level Amplitude -- Probe Field Time (t/T) Atomic Dynamics - Coupling field = 6×107 mW cm-2 -- Polarization between 1-2 level Amplitude -- Probe Field Time (t/T) Atomic Dynamics - Coupling field = 1.2×108 mW cm-2 -- Polarization between 1-2 level Amplitude -- Probe Field Time (t/T) C1*C2e-iw12t component with different coupling light turn-off rate perturbation theory, single atom without atom-atom interaction with atom-atom interaction Coupling field turn-off and on off = 25 period Amplitude - Probe Field -- Polarization between 1-2 level Position (x/λ) Coupling field turn-off and on off = 50 period Amplitude - Probe Field -- Polarization between 1-2 level Position (x/λ) Coupling field turn-off and on off = 75 period Amplitude - Probe Field -- Polarization between 1-2 level Position (x/λ) Coupling field turn-off and on off = 100 period Amplitude - Probe Field -- Polarization between 1-2 level Position (x/λ) ratio Probe pulse reading efficiency vs coupling light turn-off duration atomic density 1×1018cm-3 decay rate Γ3=ω31/20π ratio Probe pulse reading efficiency vs atomic density coupling light turn-off duration τc=τp decay rate Γ3=ω31/20π ratio Probe pulse reading efficiency vs decay rate coupling light turn-off duration τc=τp atomic density 4×1017cm-3 Metamaterial Metamaterials are artificially structured materials that can have profoundly unique electromagnetic or optical properties. - D. R. Smith Metamaterials are artificial materials engineered to have properties that may not be found in nature. Metamaterials usually gain their properties from structure rather than composition, using small inhomogeneities to create effective macroscopic behavior. - Wikipedia Classification of Metamaterials Epsilon-negative (ENG) medium ENG k DNG k Double-negative (DNG) medium Double positive (DPS) medium Re[ ] DPS k Regular Dielectrics Re[ ] MNG k Mu-negative (MNG) medium Realization of DNG Metamaterials 44 R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).2001 Subwavelength Focusing Perfect lens (Pendry, 2000) y = 2d n=1 n=-1 y = -2d 45 Cloaking and Transformation Optics • Is it possible to smoothly bend light around an object? • No backscatter, no shadow = effectively invisible. • Can there really be such an interesting solution still lurking in classical electromagnetics? Pendry et al. [Science, 2006] showed how it can be done. • Key realization: coordinate transformations on electromagnetic fields are completely equivalent to a nonuniform permittivity and permeability. Induced transparency in metamaterials by symmetry breaking Papasimakis and Zheludev, Optics & Photonics News, p22 (Oct 2009) Active metamaterial for losscompensated pulse delays Loss-compensated slow-light device: metamaterial array with EIT-like dispersion placed on a gain substrate (=9.5+035i). At the wavelength of 1.7 µm, it shows single-pass amplification and simultaneously sharp normal dispersion. Metamaterial mimicking EIT N. Papasimakis, et al. Appl. Phys. Lett. 94, 211902 (2009) Acknowledgements Dar-Yeong Ju (朱達勇)at NIU and NTNU Meng-Chang Wu (吳孟昌) (currently at IAMS, AS) Supported by NSC