Elizabeth Groves University of Rochester

Elizabeth GrovesUniversity of Rochester

Elizabeth GrovesUniversity of Rochester

Soliton Solutions for High-Bandwidth Optical Pulse

Storage and Retrieval

Thesis Defense February 11th, 2013

Thesis Defense February 11th, 2013

Normal vs. Short Optical Pulse Propagation


Beer’s Law of Absorption

Stopped light? Maybe, but not useful. We want storage.

• Atoms absorb laser pulse energy

• Pulse may be too weak to promote atoms to excited state

• Atoms dephase, return little or no energy to the field

• Laser pulse depleted



Long weak pulses Short strong pulses• Atoms initially absorb laser

pulse energy• Laser pulse drives atoms to

excited state• Atoms don’t have time to

dephase; return energy to the field coherently

• Laser pulse undepleted

• Atoms absorb laser pulse energy

• Pulse may be too weak to promote atoms to excited state

• Atoms dephase, return little or no energy to the field

• Laser pulse depleted

• Storage of high-bandwidth pulses is desirable

• Enable higher clock-rates, fast pulse switching

• Nonlinear equations • Support soliton solutions

• Nonlinear equations hard!

Normal vs. Short Optical Pulse Storage

Normal vs. Short Optical Pulse Storage

Long weak pulses Short strong pulses• Storaged achieved using

Electromagnetically-Induced Transparency (EIT) and related effects

• Linear equations, adiabatic, steady-state conditions

We derived an exact, second-order soliton solution that is a reliable guide for short, high-bandwidth pulse storage and

retrieval.

Solving Nonlinear Evolution Equations (PDEs)

Separation of variables, symmetry arguments, clues from related linear system

Hard!!, idealized conditions, cannot linearly superimpose solutions to find a general solution. Each equation seems to require special treatment.

Analytical Methods

Problems

Approaches

What’s a numerical artifact? Have you really sampled the solution space? How important are the initial conditions you’re using?

Numerical Methods

Finite difference method, method of lines, spectral method (uses Fourier transforms)

Problems

Approaches

Solving Nonlinear Evolution Equations (PDEs)

Hard!!, idealized conditions, cannot linearly superimpose solutions to find a general solution. Each equation seems to require special treatment.

Analytical Methods

Problems

Approaches

What’s a numerical artifact? Have you really sampled the solution space? How important are the initial conditions you’re using?

Numerical Methods

Finite difference method, method of lines, spectral method (uses Fourier transforms)

Problems

Approaches

Certain nonlinear evolution equations can be solved exactly by soliton solutions.

• Experiments showed that the solitary wave speed was proportional to

height.

• Data conflicted with contemporary fluid dynamics (by big deals like

Newton)

What are Solitons?In 1834 John Scott Russell, an engineer, was riding along a canal and observed a horse-drawn boat that

suddenly stopped,

causing a violent agitation, giving rise to a lump of water that rolled forward with great velocity without change of form or diminution of speed.

Stable solitary wave

http://www.bbc.co.uk/devon/content/images/2007/09/19/horse_465x350.jpg

Russell’s Wave of Translation

Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.

http://www.bbc.co.uk/devon/content/images/2007/09/19/horse_465x350.jpg

What are Solitons?

QuickTime™ and aPNG decompressor

are needed to see this picture.

Speed is proportional to height





Balanced solitary wave solutions to nonlinear evolution equations

Solitons

1965 Numerical integration by Zabusky & KruskalKorteweg-de Vries (KdV) Equation

was largely ignored until the 1960s.


Speed is proportional to height

What are Solitons?

Balanced solitary wave solutions to nonlinear evolution equations

Solitons



Collision between two solutions

Minimal energy loss

Both solitary waves recovered

nonlinear superposition

that survive collisions

1965 Numerical integration by Zabusky & KruskalKorteweg-de Vries (KdV) Equation

was largely ignored until the 1960s.


Summarizing Solitons

Special solutions to nonlinear evolution equations (PDEs) that:

• Are stable solitary waves (pulses/localized excitations)

• Maintain their shape under interaction/collision/nonlinear superposition

Collide like particleselectrons, muons, ping pongs

Solit ons

Solitary waves

Summarizing Solitons

Special solutions to nonlinear evolution equations (PDEs) that:

• Are stable solitary waves (pulses/localized excitations)

• Maintain their shape under interaction/collision/nonlinear superposition

Solitary waves Collide like particleselectrons, muons, ping pongs

Solitons

Solitons in NatureAlphabet Waves

Did On MyWhat IAblowitz & Baldwin

Their

Summer Vacationhttp://www.douglasbaldwin.com/nl-waves.html

• Not as unusual as once thought

• May play a role in tsunami and rogue wave formation

http://www.douglasbaldwin.com/nl-waves.html

(Speculated) Solitons in Nature

Strait of Gibraltar

http://www.lpi.usra.edu/publications/slidesets/oceans/oceanviews/slide_13.html

Jupiter’s Red SpotMorning Glory Clouds

Deep and shallow water waves, plasmas, particle interactions, optical systems, neuroscience, Earth’s magnetosphere...

http://en.wikipedia.org/wiki/File:MorningGloryCloudBurketownFromPlane.jpg

http://www.universetoday.com/15163/jupiters-great-red-spot/

I will use solitons to describe solutions to integrable nonlinear equations generated by the Darboux Tranformation method.

http://www.lpi.usra.edu/publications/slidesets/oceans/oceanviews/slide_13.html

http://en.wikipedia.org/wiki/File:MorningGloryCloudBurketownFromPlane.jpg

Darboux Transformation

Solving Integrable Equations


Lax Form

Analytic Methods

Integrable nonlinear systems can be characterized by the Lax formalism

• Inverse Scattering Transform (AKNS Method)

• Zakharov-Shabat Method• Bäcklund Transformation• •

Lax Form


Seed solution

New solution

Generates soliton solutions!




Seed solution

Lax Form

1. Solve linear Lax equations

2. Construct Darboux matrix




New solution



Solving KdV EquationFirst-Order Soliton Solution

Solving KdV EquationFirst-Order Soliton Solution



Darboux parameter determines velocity/height

Seed solution

New solution



Darboux parameters determine velocities/heights

Seed solution

New solution



but potentially hard!!



Solving KdV EquationSecond-Order Soliton Solution

Solving KdV EquationSecond-Order Soliton Solution

Solving KdV EquationNonlinear Superposition

Solving KdV EquationNonlinear Superposition


are needed to see this picture.faster first-order soliton slower first-order soliton

Algebraic Nonlinear Superposition Rule

Useful for colliding/combining solutions with desirable properties for more complicated systems like short optical pulses



Darboux parameters determine velocities/heights

Short Optical Pulse Propagation


Long Collection of Atoms

1

2Dipole moment operator d Wavefunction ψ (pure states)Density matrix ρ (mixed states)

Laser Pulse

Optical frequency ω



Slowly-varying envelope E

1

2



Laser Pulse



Resonant AtomsDipole moment operator d

Rabi frequency

d EΩ

1

2



time

Rabi frequency

d EΩ

Laser Pulse



Resonant AtomsDipole moment operator d

Pulse area

1

2

Rabi frequency

Short pulses allow us to neglect atomic decay mechanisms and focus on coherent effects

von Neumann’s equation

Integrable Nonlinear Evolution Equations

Dipole moment operator d Slowly-varying envelope E

d EΩ

Maxwell’s slowly-varying envelope equation

Atom-field coupling μ



First-Order Soliton SolutionFirst-Order Soliton Solution

First-Order Soliton Solution

Zero-Order Soliton Solution

Darboux Transformation Method







1

2

1

2

1

2

1

2

0

1

McCall-Hahn Self-Induced Transparency (SIT) Pulse


0

1

McCall-Hahn Self-Induced Transparency (SIT) Pulse

The 2 -area hyperbolic secant pulse shape induces a single Rabi oscillation in each atom

Pulse travels at a reduced group velocity in the medium

Absorption coefficient

Temporal pulse width

2

1



Two-Frequency Pulse Propagation in Three-Level

Media


Media

Opportunities for interesting dynamics and pulse-pulse control1

3

2


Media


MediaNonlinear Evolution Equations

Equal atom-field coupling parameters


Q-Han Park , H. J. Shin (PRA 1998)

B. D. Clader , J. H. Eberly (PRA 2007, 2008)

1

3

2

Signal Control


Media


Media

1

3

2

Nonlinear Evolution Equations

Equal atom-field coupling parameters


Q-Han Park , H. J. Shin (PRA 1998)

B. D. Clader , J. H. Eberly (PRA 2007, 2008)

Same form as two-level equations

Temporally matched pulses

Signal Control






Warning! Soliton solutions are labelled by the number of applications of the Darboux transformation. Order corresponds to maximum number of solitary waves of a particular frequency.






1

3

21

3

21

3

21

3

2

First-Order Soliton SolutionOptical Pulse Storage


€

Ω13

€

Ω23

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ρ11

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ρ12

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ρ22€

ρ13

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ρ33

€

ρ23

QuickTime™ and aPhoto - JPEG decompressor




Absorption Depths

Absorption Depths

1

3

2

Slow SIT pulse1

3

2Decoupled pulse

1

3

2

Both pulses active

SignalControl

SignalControl

Ratio of pulses at any x is given by a simple relationship

Important for finite-length media

€

Ω13

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Ω23

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ρ11

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ρ12

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ρ22€

ρ13

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ρ23

Absorption Depths

Absorption Depths

Long-lived atomic ground states store pulse information



Interesting, but what else can we do with it?

Second-Order Soliton Solution


Seed solution

New solutionbut potentially hard!!



1

3

2

Signal Control




Second-order soliton solution

Two first-order soliton solutions




Second-order soliton solution

Two first-order soliton solutions

1

3

2

Signal Control

Signal and control pulse durations

Optical Pulse Storage Memory Manipulation

1

3

2

Control

Control pulse duration

and



1

3

2

Signal Control

Signal and control pulse durations

Optical Pulse Storage Memory Manipulation

1

3

2

Control


and

Warning! The concept of collision is much more complicated than it was for the KdV equation. We should think carefully about when we

want the faster-moving control pulse to catch up with the slower storage solution

If we are clever, we can arrange for the signal pulse to be stored before the faster-moving control pulse catches up.



Anticipated Behavior


Anticipated Behavior

After CollisionAfter CollisionFaster-moving control pulse moving ahead of the pulse storage solution

Faster-moving control pulse moving ahead of the pulse storage solution

Before CollisionBefore CollisionFaster-moving control pulse catching up to the storage solution

Faster-moving control pulse catching up to the storage solution

How will the imprint change?

How will the imprint change?



• We can choose integration constants cleverly so the signal pulse is stored before the new control pulse arrives/collides

• Faster-moving control pulse hits the stored signal pulse imprint and recovers the stored signal pulse

• Recovered signal pulse soon re-imprinted at a new location

• Distance the imprint is moved is given by the phase lag


Analytic Results


Analytic Results

Relation to finite-length mediaRelation to finite-length media

Location of original imprint fixed by injected pulse ratios

Distance the imprint is moved is given by the phase lag

Ratio

New imprint location is

Warning! If these guides are unreliable, we may push the imprint too close to the edge of the medium – recovering part of the signal pulse before we are ready for it!


Analytic Results


Analytic Results

QuickTime™ and aGIF decompressor


Absorption Depths

€

Ω13

€

Ω23

€

ρ11

€

ρ12

€

ρ22

€

ρ13

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ρ23

Numerical SolutionNumerical SolutionHigh-Bandwidth Optical Pulse Control

Step 1: Pulse Storage

Signal pulse area

Control pulse area

Pulse ratio

Theoretical

Imprint location

Numerical

Percent Error

Original Imprint

Absorption Depths

€

Ω13

€

Ω23

€

ρ11

€

ρ12

€

ρ22

€

ρ13

€

ρ33

€

ρ23


Theoretical

Imprint location

Step 1: Pulse Storage

Signal pulse area

Control pulse area

Pulse ratio

Numerical

Percent Error



Absorption Depths

€

Ω13

€

Ω23

€

ρ11

€

ρ12

€

ρ22

€

ρ13

€

ρ33

€

ρ23


Step 2: Memory Manipulation

Control pulse area


Theoretical

Distance Moved

Numerical

Percent Error


Step 2: Memory Manipulation

Control pulse area


Theoretical

Distance Moved

Numerical

Percent Error €

ρ11

€

ρ12

€

ρ22

€

ρ11

€

ρ12

€

ρ22

Original Imprint

Manipulated Memory

Absorption Depths

Our second-order soliton solution gives us remarkably tight control of the imprint!

New location



Absorption Depths

€

Ω13

€

Ω23

€

ρ11

€

ρ12

€

ρ22

€

ρ13

€

ρ33

€

ρ23


Step 3: Pulse Retrieval

Choose control pulse width so that the new storage location is outside the boundary of the medium

Control pulse area


TheoreticalDistance Moved

New location well outside medium.

Signal pulse is recovered!

ConclusionsConclusions

• Demonstrated control possibilities to convert optical information into atomic excitation and back again, on demand, without adiabatic or quasi-steady state conditions

• Focused on broadband pulses, enabling faster pulse-switching and higher clock-rates

• Combined numerical and analytical methods to develop a novel three-step procedure to store, move, and retrieve a signal field with high-fidelity

• Our new, second-order soliton solution indicates how to control the imprint location by adjusting injected pulse ratios and temporal durations

• Numerical studies indicate the general procedure works even for non-idealized input conditions, including pulse areas and shape

Experimental RealizationsExperimental Realizations

Focus on coherent effects by using laser pulses shorter than excited-state lifetime

384.230 THz

52S1/2

52P3/2

F = 1

F = 0

F = 2

F = 3

2.56301 GHz

F = 1

F = 2

4.27168 GHz

6.83468 GHz

Lifetime ~ 26 ns

72.2180 MHz

156.947 MHz

266.650 MHz

87Rb D2 Line Transition

52S1/2

52P3/2

F = 1

F = 0

F = 2

F = 3

2.56301 GHz

F = 1

F = 2

4.27168 GHz

6.83468 GHz

Lifetime ~ 26 ns

72.2180 MHz

156.947 MHz

266.650 MHz

384.230THz + 4.27168 GHz

384.230 THz - 2.56005 GHz

Three-Level Model

Pulse bandwidth chosen to resolve ground but not excited hyperfine states

Two-Level Model

52S1/2

52P3/2

F = 1

F = 0

F = 2

F = 3

2.56301 GHz

F = 1

F = 2

4.27168 GHz

6.83468 GHz

Lifetime ~ 26 ns

72.2180 MHz

156.947 MHz

266.650 MHz

Large bandwidth pulse cannot resolve ground or excited hyperfine states

384.230 THz

Traveling Wave Coordinates

Lax Form

Integrable Maxwell- Bloch Equations

Integrable Maxwell- Bloch Equations

Maxwell-Bloch Equations

Lax Operators



Absorption Depths κx€

Ω13

€

Ω23





€

ρ11

€

ρ12

€

ρ22

€

Ω13

€

Ω23

€

ρ11

€

ρ12

€

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€

Ω13

€

Ω23

Absorption Depths κx

Absorption Depths κx





Two First-Order Soliton Solutions

Hermitian unitary matrix

Linear (but potentially hard!!) Lax equations

Nonlinear Superposition Rule

combines two first-order soliton solutions no integration required!!



second-order soliton Rabi frequency



The phase lag

Asymptotic behavior of the solution

is the only remnant of the collision



first-order soliton Rabi frequency

second-order soliton Rabi frequency

Asymptotic behavior of the solution

first-order soliton Rabi frequencyQuickTime™ and aPNG decompressor


The phase lag is the only remnant of the collision

EITEIT

Elizabeth Groves University of Rochester

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