Measuring the information velocity in fast- and slow-light media
description
Transcript of Measuring the information velocity in fast- and slow-light media
1
Measuring the information velocity infast- 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
Institute of Optics, December 10, 2003Funding from the U.S. National Science Foundation
Nature 425, 665 (2003)
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
http://www.picosecond.com/objects/AN-12.pdf
Modern Optical Telecommunication Systems:Transmitting information encoded on optical fields
RZ data
clock
Where is the information on the waveform?
How fast does it travel?
1 0 1 1 0
5
dispersivemedia
Pulse propagation in dispersive media
"slow-light"medium
Time (ns)
-200 0 200 400
Po
we
r (
W)
0
2
4
6
8
10
12
Po
we
r (
W)
0510152025303540
DelayedVacuum
tdel= 67.5 ns
6
time (ns)
-300 -200 -100 0 100 200 300
pow
er ( W
)
0
2
4
6
8
10
12
pow
er ( W
)
0.00.2
0.40.6
0.81.01.2
1.41.6
advanced vacuum
tadv=27.4 ns
"Fast-Light" medium
consequences for the special theory of relativity?
7
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.
PULSE PROPAGATION REVIEW:"Slow" and "Fast" Light
8
Propagating Electromagnetic Waves: Phase Velocity
monochromatic plane wave
E z t Ae c ci kz t( , ) .( )
phase kz t
E
z
Points of constant phase move adistance z in a time t
phase velocity
p
zt k
cn
9
Propagating Electromagnetic Waves: Group Velocity
Lowest-order statement of propagation withoutdistortion
dd
0
group velocity
g
g
c
ndnd
cn
10
Propagation "without distortion"
k kn
c c
dn
do
g
o
g
o( ) ( ) ( )
12
2
dn
dg
0•
• pulse bandwidth not too large
Recent experiments on fast and slow light conducted in the regime of low distortion
"slow" light:
"fast" light:
g gc n ( )1
g g gc or n 0 1( )
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 lotabout optimizing data rates in noisy channels
No one from the engineering community hasposed the following fundamental question:
What is the speed of information?
That is, how quickly can information be transmittedbetween two different locations?
15
Information Transmission: A physics perspective
Interest in the speed of information soon after Einstein'spublication 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 riseto crazy paradoxes (e.g., an effect before its cause)
Garrison et al., Phys. Lett. A 245, 19 - 25 (1998).
16
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 thatis zero initially.
Luminal information transmission implies that no electromagnetic disturbance can arrive faster than the "front" of the wave.
front
17
The front travels at c
Primary Finding of Sommerfeld
(assumes a Lorentz-model dielectric with a single resonance)
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 expansionfails 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: vs c when vg >c
vs= vg when vg >c
19
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)
Fast-Light Experiments
20
Fast-light via a gain doublet
Steingberg and Chiao, PRA 49, 2071 (1994)(Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000))
21
Achieve a gain doublet using stimulated Raman scattering with a bichromatic pump field
Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000)
22
probe frequency (MHz)
190 200 210 220 230 240 250
ga
in c
oe
ffic
ien
t, g
L
0
1
2
3
4
5
6
7
8
egl=7.4
egl=1,097
22.3 MHz
Fast light in a laser driven potassium vapor
large anomalousdispersion
AOM
o
waveformgenerator
Kvapor
Kvapor
d-
d+
d-
d+
23
Some of our toys
24
time (ns)
-300 -200 -100 0 100 200 300
pow
er ( W
)
0
2
4
6
8
10
12
pow
er ( W
)
0.00.20.40.60.81.01.21.41.6
advanced vacuum
tadv=27.4 ns
Observation of large pulse advancement
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!
25
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.
The Information Velocity
26
Points of non-analyticity
t
Ppoint of non-analyticity
knowledge of the leading part of the pulse cannot be usedto infer knowledge after the point of non-analyticity
new information is available because of the "surprise"
27
Speed of points of non-analyticity
Spectrum falls off like a power law!
k kn
c c
dn
do
g
o
g
o( ) ( ) ( )
12
2
Taylor series
no longer converges even when pulse "bandwidth"(full width at half-maximum) is small! Subtle effect!
Chiao and Steinberg find point of non-analyticitytravels at c. Therefore, they associate it with theinformation velocity.
28
Detecting points of non-analyticity
Chiao and Steinberg proposal not satisfactory from aninformation-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 t due to noise (classical or quantum). We call this the detection latency.
Detected information travels at less than vi, even in vacuum!
29
Measuring the Effects of a Fast-Light Medium on the Information Velocity
30
Information Velocity: Transmit Symbols
information velocity: measure time at which symbols can first be distinguished
time (ns)
-300 -200 -100 0 100 200 300
optic
al p
ulse
am
plitu
de (
a.u.
)0.0
0.5
1.0
1.5
"1"
"0"
-300 -200 -100 0 100 200 300
wav
efor
m a
mpl
itude
(a.
u.)
0.0
0.5
1.0
1.5
"1"
"0"
time (ns)
-300 -200 -100 0 100 200 300
optic
al p
ulse
am
plitu
de (
a.u.
)
0.0
0.5
1.0
1.5
"1"
"0"
requested symbols optically generated symbols
31
Send the symbolsthrough our fast-lightmedium
time (ns)
-300 -200 -100 0 100 200 300
optic
al p
ulse
am
plitu
de (
a.u.
)
0.0
0.5
1.0
1.5
advanced
vacuum
"1"
"0"
time (ns)
-60 -40 -20 00.6
0.8
1.0
1.2
1.4
1.6
1.8
Y D
ata
0.2
0.4
0.6
0.8
1.0
1.2
vacuum
advanced
A
B
advanced
32
-40 -30 -20 -10 0
BE
R
10-4
10-3
10-2
10-1
100
vacuum
advanced
A
final observation time (ns)
Use a matched-filter to determine the bit-error-rate (BER)
Determine detection times a threshold
Use large BER to minimize t
Detection for informationtraveling through fastlight medium is later eventhough group velocityvastly exceeds c!
Ti
33
final observation time (ns)
0 2 4 6 8 10
BE
R
10-4
10-3
10-2
10-1
100
advanced
vacuum
B
Origin of slow down?
Slower detection time could be due to:• change in information velocity vi
• change in detection latency t
TL L
t tii adv i vac
adv vac FHG
IKJ
, ,
b g
estimate latencyusing theory
34
Estimate information velocity in fast light medium
t t nsadv vac b g12 05. .
i adv c, ( . . ) 0 4 05
from the model
combining experiment and model
35
Measuring the Effects of a Slow-Light Medium on the Information Velocity
36
o-d-462 (MHz)
-4 -2 0 2 4
gain
coe
ffic
ient
, gL
0.0
0.5
1.0
1.5
a
-4 -2 0 2 4
n g
-40
0
40
80
120
b
Slow Light via a single amplifying resonance
AOM
o
d
d
L
Waveformgenerator
Kvapour
37
Time (ns)
-200 0 200 400
Pow
er
(W
)
0
2
4
6
8
10
12
Pow
er
(W
)
0510152025303540
DelayedVacuum
tdel= 67.5 ns
Slow Light Pulse Propagation
38
time (ns)
-40 -30 -20 -10 0 100.6
0.7
0.8
0.9
1.0
1.1
Y D
ata
0.6
0.7
0.8
0.9
1.0
1.1time (ns)
-300 -200 -100 0 100 200 300
optic
al p
ulse
am
plitu
de (
a.u
.)
0.0
0.5
1.0
1.5delayed
vacuum
"1"
"0"
vacuum
delayed
a
b
delayed
Send the symbolsthrough our slow-lightmedium
vi ~ 60 vg !!
39
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/
40
What part of thewaveform do you measure?
Assumes detectionlatency is zero.