1
Measuring the information velocity in
fast- and slow-light media
Dan Gauthier and Michael Stenner
Duke University, Department of Physics,
Fitzpatrick Center for Photonics
and Communication Systems
Mark Neifeld
University of Arizona, Electrical and Computer
Engineering, and Optical Sciences Center
Nature 425, 665 (2003)
Institute of Optics, December 10, 2003
Funding from the U.S. National Science Foundation
2
Outline
• Information and optical pulses
• Review of pulse propagation in dispersive media
• How fast does information travel?
• Fast light experiments
• Consequences for the special theory of relativity
• The information velocity
• Measuring the effects of a fast-light medium on the
information velocity
• Measuring the effects of a slow-light medium on the
information velocity
3
Information on Optical Pulses
4
Modern Optical Telecommunication Systems:
Transmitting information encoded on optical fields
http://www.picosecond.com/objects/AN-12.pdf
1
0
1
1
0
RZ data
clock
Where is the information on the waveform?
How fast does it travel?
5
Pulse propagation in dispersive media
"slow-light"
medium
Power (W)
12
tdel= 67.5 ns
10
8
6
Delayed
Vacuum
4
2
0
-200
0
Time (ns)
200
400
40
35
30
25
20
15
10
5
0
Power (W)
dispersive
media
6
12
10
power (W)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
tadv=27.4 ns
8
6
advanced
vacuum
4
2
0
-300
-200
-100
0
100
200
power (W)
"Fast-Light" medium
300
time (ns)
consequences for the special theory of relativity?
7
PULSE PROPAGATION REVIEW:
"Slow" and "Fast" Light
R.W. Boyd and D.J. Gauthier
"Slow and "Fast" Light, in Progress in Optics, Vol. 43,
E. Wolf, Ed. (Elseiver, Amsterdam, 2002),
Ch. 6, pp. 497-530.
8
Propagating Electromagnetic Waves: Phase Velocity
monochromatic plane wave
E
z
E( z, t )  Ae
phase
i ( kz t )
 c. c
  kz  t
Points of constant phase move a
distance Dz in a time Dt
phase velocity
Dz  c
p   
Dt k n
9
Propagating Electromagnetic Waves: Group Velocity
Lowest-order statement
of propagation without
distortion
d
0
d
group velocity
c
c
g 

dn ng
n 
d
10
Propagation "without distortion"
ng
1 dng
2
k ( )  ko  (   o ) 
(   o )  
c
2c d
dng
•
0
d
• pulse bandwidth not too large
"slow" light:
"fast" light:
 g  c (ng  1)
 g  c or  g  0 (ng  1)
Recent experiments on fast and slow light conducted
in the regime of low distortion
11
Pulse Propagation: Slow Light
(Group velocity approximation)
12
Pulse Propagation: Fast Light
(Group velocity approximation)
13
Where is the information?
How fast does it travel?
14
Information Transmission: An Engineering Perspective
Starting from the work of Shannon, we know a lot
about optimizing data rates in noisy channels
No one from the engineering community has
posed the following fundamental question:
What is the speed of information?
That is, how quickly can information be transmitted
between two different locations?
15
Information Transmission: A physics perspective
Interest in the speed of information soon after Einstein's
publication of the special theory of relativity in 1905
Known that optical pulses could have a group velocity
exceeding the speed of light in vacuum (c) when
propagating through dispersive materials
Conference sessions devoted to the topic
Relativity revised: no information can travel faster than c
Faster-than-c information transmission gives rise
to crazy paradoxes (e.g., an effect before its cause)
Garrison et al., Phys. Lett. A 245, 19 - 25 (1998).
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Early Theoretical Studies of Optical "Signals"
A. Sommerfeld, Physik. Z. 8, 841 (1907)
A. Sommerfeld, Ann. Physik. 44, 177 (1914)
L. Brillouin, Ann. Physik. 44, 203 (1914)
L. Brillouin, Wave Propagation and Group Velocity, (Academic, New York, 1960).
Sommerfeld: A "signal" is an electromagnetic wave that
is zero initially.
Luminal information transmission implies that no
electromagnetic disturbance can arrive faster than the
"front" of the wave.
front
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Primary Finding of Sommerfeld
(assumes a Lorentz-model dielectric with a single resonance)
The front travels at c
regardless of the details of the dielectric
Physical interpretation: it takes a finite time for the
polarization of the medium to build up; the first part of the
field passes straight through!
This is an all-orders calculation. The Taylor series expansion
fails to give this result!!!
18
The Sommerfeld and Brillouin Precursors
results of an asymptotic analysis (saddle-point method)
vg has no meaning when vg >c
precursors very small
Sommerfeld:
signal velocity vs depends on detector sensitivity
Brillouin: v  c when v >c
s
g
vs= vg when vg >c
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Fast-Light Experiments
Fast light theory, Gaussian pulses:
C. G. B. Garrett, D. E. McCumber, Phys. Rev. A 1, 305 (1970).
Fast light experiments, resonant absorbers:
S. Chu, S. Wong, Phys. Rev. Lett. 48, 738 (1982).
B. Ségard and B. Macke, Phys. Lett. 109, 213 (1985).
A. M. Akulshin, A. Cimmino, G. I. Opat, Quantum Electron. 32, 567 (2002).
M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, Science 301, 200 (2003)
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Fast-light via a gain doublet
Steingberg and Chiao, PRA 49, 2071 (1994)
(Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000))
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Achieve a gain doublet using stimulated Raman
scattering with a bichromatic pump field
Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000)
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Fast light in a laser driven potassium vapor
dd+
o
AOM
K
vapor
K
vapor
d-
waveform
generator
8
22.3 MHz
egl=1,097
7
gain coefficient, gL
d+
6
large anomalous
dispersion
5
4
3
egl=7.4
2
1
0
190
200
210
220
230
probe frequency (MHz)
240
250
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Some of our toys
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12
10
power (W)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
tadv=27.4 ns
8
6
advanced
vacuum
4
2
0
-300
-200
-100
0
100
200
power (W)
Observation of large pulse advancement
300
time (ns)
tp = 263 ns
A = 10.4%
vg = -0.051c
ng = -19.6
some pulse compression (1.9% higher-order dispersion)
H. Cao, A. Dogariu, L. J. Wang, IEEE J. Sel. Top. Quantum Electron. 9, 52 (2003).
B. Macke, B. Ségard, Eur. Phys. J. D 23, 125 (2003).
large fractional advancement - can distinguish different velocities!
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The Information Velocity
No working definition of the information velocity
The information theory community has not considered
this problem
An interesting proposal can be found in
R. Y. Chiao, A.M. Steinberg, in Progress in Optics
XXXVII, Wolf, E., Ed. (Elsevier Science, Amsterdam,
1997), p. 345.
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Points of non-analyticity
P
point of non-analyticity
t
knowledge of the leading part of the pulse cannot be used
to infer knowledge after the point of non-analyticity
new information is available because of the "surprise"
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Speed of points of non-analyticity
Spectrum falls off like a power law!
Taylor series
ng
1 dng
k ( )  ko  (   o ) 
(   o ) 2  
c
2c d
no longer converges even when pulse "bandwidth"
(full width at half-maximum) is small! Subtle effect!
Chiao and Steinberg find point of non-analyticity
travels at c. Therefore, they associate it with the
information velocity.
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Detecting points of non-analyticity
Chiao and Steinberg proposal not satisfactory from an
information-theory point of view: A point has no energy!
transmitter
receiver
Point of non-analyticity travels at vi = c (Chiao & Steinberg)
Detection occurs later by an amount Dt due to noise
(classical or quantum). We call this the detection latency.
Detected information travels at less than vi, even in vacuum!
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Measuring the Effects of a Fast-Light
Medium on the Information Velocity
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Information Velocity: Transmit Symbols
information velocity: measure time at which symbols can first be distinguished
optically generated symbols
optical pulse amplitude (a.u.)
optical
pulseamplitude
amplitude(a.u.)
(a.u.)
waveform
requested symbols
1.5
1.5
"1"
"1"
1.0
1.0
0.5
0.5
0.0
0.0
-300
-300
"0"
"0"
-200
-200
-100
-100
00
time (ns)
100
100
200
200
300
300
1.5
"1"
1.0
0.5
"0"
0.0
-300
-200
-100
0
time (ns)
100
200
300
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A
1.5
"1"
Y Data
optical pulse amplitude (a.u.)
1.0
0.5
advanced
advanced
"0"
vacuum
0.0
-300
-200
-100
0
100
200
time (ns)
1.2
Send the symbols
through our fast-light
medium
B
300
1.8
advanced
1.6
1.0
1.4
0.8
1.2
0.6
1.0
0.4
0.2
0.8
vacuum
0.6
-60
-40
-20
time (ns)
0
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Use a matched-filter to determine the bit-error-rate (BER)
100
advanced
Detection for information
traveling through fast
light medium is later even
though group velocity
vastly exceeds c!
BER
10-1
Ti
10-2
vacuum
10-3
A
10-4
-40
-30
-20
-10
final observation time (ns)
Determine detection times a threshold
Use large BER to minimize Dt
0
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Origin of slow down?
Slower detection time could be due to:
• change in information velocity vi
• change in detection latency Dt
F
L
T G
H
i

i ,adv
L
 i ,vac
I  bDt
J
K
adv
 Dt vac
g
100
advanced
10-1
BER
estimate latency
using theory
10-2
vacuum
10-3
B
10-4
0
2
4
6
final observation time (ns)
8
10
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Estimate information velocity in fast light medium
from the model
bDt
adv
g
 Dtvac  12
.  0.5 ns
combining experiment and model
 i ,adv  (0.4  0.5)c
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Measuring the Effects of a Slow-Light
Medium on the Information Velocity
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Slow Light via a single amplifying resonance
AOM
o
d
L
K
vapour
Waveform
generator
d
b
1.5
120
1.0
80
ng
gain coefficient, gL
a
0.5
40
0
0.0
-40
-4 -2 0
2
4
-4 -2 0
o-d-462 (MHz)
2
4
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Power (W)
12
tdel= 67.5 ns
10
8
6
Delayed
Vacuum
4
2
0
-200
0
Time (ns)
200
400
40
35
30
25
20
15
10
5
0
Power (W)
Slow Light Pulse Propagation
38
a
1.5
Send the symbols
through our slow-light
medium
"1"
delayed
Y Data
optical pulse amplitude (a.u.)
1.0
vacuum
0.5
"0"
delayed
0.0
-300
1.1
-200
-100
0
100
200
time (ns)
b
1.0
300
1.1
vacuum
1.0
0.9
0.9
0.8
0.8
0.7
0.6
-40
delayed
0.7
0.6
-30
-20
-10
time (ns)
0
10
vi ~ 60 vg !!
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Summary
• Investigate fast-light (slow-light) pulse propagation
with large pulse advancement (delay)
• Transmit symbols to measure information velocity
• Estimate vi ~ c
• Consistent with special theory of relativity
• Special theory of relativity may only be
an approximation?
http://www.phy.duke.edu/research/photon/qelectron/proj/infv/
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What part of the
waveform do you
measure?
Assumes detection
latency is zero.