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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
[email protected]
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
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