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

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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( )

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Pulse Propagation: Slow Light(Group velocity approximation)

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Pulse Propagation: Fast Light (Group velocity approximation)

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Where is the information?

How fast does it travel?

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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?

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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).

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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 thatis zero initially.

Luminal information transmission implies that no electromagnetic disturbance can arrive faster than the "front" of the wave.

front

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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!!!

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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

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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

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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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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

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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!

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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

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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"

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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.

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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!

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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

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

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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

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-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

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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

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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

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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

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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 !!

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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 thewaveform do you measure?

Assumes detectionlatency is zero.