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

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

Information on Optical Pulses

Where is the information on the waveform? How fast does it travel? 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?

Pulse propagation in dispersive media "slow-light" medium

"Fast-Light" medium consequences for the special theory of relativity?

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.

Propagating Electromagnetic Waves: Phase Velocity monochromatic plane wave E z phase velocity phase Points of constant phase move a distance Dz in a time Dt

Propagating Electromagnetic Waves: Group Velocity Lowest-order statement of propagation without distortion group velocity

Propagation "without distortion" pulse bandwidth not too large "slow" light: "fast" light: Recent experiments on fast and slow light conducted in the regime of low distortion

Pulse Propagation: Slow Light (Group velocity approximation)

Pulse Propagation: Fast Light (Group velocity approximation)

Where is the information? How fast does it travel?

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?

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).

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

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!!!

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

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)

Fast-light via a gain doublet Steingberg and Chiao, PRA 49, 2071 (1994) (Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000))

Achieve a gain doublet using stimulated Raman scattering with a bichromatic pump field Wang, Kuzmich, and Dogariu, Nature 406, 277 (2000)

Fast light in a laser driven potassium vapor large anomalous dispersion

Some of our toys

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!

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.

Points of non-analyticity point of non-analyticity P 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"

Speed of points of non-analyticity Spectrum falls off like a power law! 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-analyticity travels at c. Therefore, they associate it with the information velocity.

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!

Measuring the Effects of a Fast-Light Medium on the Information Velocity

Information Velocity: Transmit Symbols information velocity: measure time at which symbols can first be distinguished requested symbols optically generated symbols

Send the symbols through our fast-light medium

Use a matched-filter to determine the bit-error-rate (BER) Detection for information traveling through fast light medium is later even though group velocity vastly exceeds c! Ti Determine detection times a threshold Use large BER to minimize Dt

Origin of slow down? Slower detection time could be due to: change in information velocity vi change in detection latency Dt estimate latency using theory

Estimate information velocity in fast light medium from the model combining experiment and model

Measuring the Effects of a Slow-Light Medium on the Information Velocity

Slow Light via a single amplifying resonance

Slow Light Pulse Propagation

Send the symbols through our slow-light medium vi ~ 60 vg !!

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/

What part of the waveform do you measure? Assumes detection latency is zero.