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Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval

Elizabeth Groves

University of Rochester

Thesis Defense

February 11th, 2013

Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval

Elizabeth Groves

University of Rochester

Thesis Defense

February 11th, 2013

Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval

Elizabeth Groves

University of Rochester

Thesis Defense

February 11th, 2013

Optical Pulse Propagation

Beer’s Law of Absorption

- 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

Stopped light?

Maybe, but not useful.

We want storage.

Optical Pulse Propagation

Long weak pulses

Short strong pulses

- 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

- 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

Optical Pulse Storage

Long weak pulses

Short strong pulses

- Storage of high-bandwidth pulses is desirable
- Enable higher clock-rates, fast pulse switching
- Nonlinear equations
- Support soliton solutions

- Storaged achieved using Electromagnetically-Induced Transparency (EIT) and related effects
- Linear equations, adiabatic, steady-state conditions

- Nonlinear equations hard!

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

Numerical Methods

Approaches

Approaches

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

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

Problems

Problems

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

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

Solving Nonlinear Evolution Equations (PDEs)

Numerical Methods

Approaches

Approaches

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

Problems

Problems

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

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

Solving Nonlinear Evolution Equations (PDEs)

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

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 waterthat rolled forward with great velocity without change of form or diminution of speed.

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.

Russell’s Wave of Translation

- Experiments showed that the solitary wave speed was proportional to height.
- Data conflicted with contemporary fluid dynamics (by big deals like Newton)

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

1965 Numerical integration by Zabusky & Kruskal

Korteweg-de Vries (KdV) Equation

What are Solitons?

Solitons

Russell’s Wave of Translation

was largely ignored until the 1960s.

Speed is proportional to height

Balanced solitary wave solutions to nonlinear evolution equations

1965 Numerical integration by Zabusky & Kruskal

Korteweg-de Vries (KdV) Equation

What are Solitons?

Solitons

Russell’s Wave of Translation

was largely ignored until the 1960s.

nonlinear superposition

Speed is proportional to height

Collision between two solutions

Minimal energy loss

Both solitary waves recovered

that survive collisions

Balanced solitary wave solutions to nonlinear evolution equations

Solit

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 particles

electrons, muons, ping pongs

Solitary waves

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 particles

electrons, muons, ping pongs

Solitary waves

Alphabet Waves

- Not as unusual as once thought

- May play a role in tsunami and rogue wave formation

Ablowitz & Baldwin

Their

Did On My

Summer Vacation

What I

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

Solitons in Nature

Morning Glory Clouds

Jupiter’s Red Spot

Strait of Gibraltar

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

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

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

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

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

Integrable nonlinear systems can be characterized by the Lax formalism

Lax Form

Analytic Methods

- Inverse Scattering Transform (AKNS Method)
- Zakharov-Shabat Method
- Bäcklund Transformation

Darboux Transformation

Generates soliton solutions!

Integrable nonlinear systems can be characterized by the Lax formalism

Lax Form

Seed solution

Darboux Transformation

New solution

1. Solve linear Lax equations

2. Construct Darboux matrix

Integrable nonlinear systems can be characterized by the Lax formalism

Lax Form

Seed solution

New solution

Darboux parameter determines velocity/height

Solving KdV Equation

First-Order Soliton Solution

Seed solution

1. Solve linear Lax equations

2. Construct Darboux matrix

New solution

Darboux parameters determine velocities/heights

but potentially hard!!

Solving KdV Equation

Second-Order Soliton Solution

Seed solution

1. Solve linear Lax equations

2. Construct Darboux matrix

New solution

Darboux parameters determine velocities/heights

Algebraic Nonlinear Superposition Rule

slower first-order soliton

Solving KdV Equation

Nonlinear Superposition

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

Wavefunction ψ (pure states)

Density matrix ρ (mixed states)

2

1

Short Optical Pulse

Propagation

Long Collection of Atoms

2

dE

Resonant Atoms

Ω

Optical frequency ω

Dipole moment operatord

1

Slowly-varying envelope E

Short Optical Pulse

Propagation

Rabi frequency

dE

Rabi frequency

Ω

Pulse area

1

time

Short Optical Pulse

Propagation

Laser Pulse

Resonant Atoms

Optical frequency ω

Dipole moment operatord

Slowly-varying envelope E

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

Atom-field coupling μ

2

dE

Ω

1

von Neumann’s equation

Maxwell’s slowly-varying envelope equation

Short Optical Pulse

Propagation

Dipole moment operator d

Slowly-varying envelope E

Rabi frequency

Integrable Nonlinear Evolution Equations

Darboux Transformation Method

Zero-Order Soliton Solution

1. Solve linear Lax equations

2. Construct Darboux matrix

First-Order Soliton Solution

2

2

2

2

1

1

1

1

0

First-Order Soliton Solution

McCall-Hahn Self-Induced Transparency (SIT) Pulse

Zero-Order Soliton Solution

Darboux Transformation

First-Order Soliton Solution

2

Temporal pulse width

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

Absorption coefficient

1

0

First-Order Soliton Solution

McCall-Hahn Self-Induced Transparency (SIT) Pulse

Pulse travels at a reduced group velocity in the medium

2

1

Two-Frequency Pulse Propagation in Three-Level Media

Opportunities for interesting dynamics and pulse-pulse control

First-Order Soliton Solution

3

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

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

2

1

Two-Frequency Pulse Propagation in Three-Level Media

Equal atom-field coupling parameters

Control

Signal

3

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

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

2

1

Two-Frequency Pulse Propagation in Three-Level Media

Nonlinear Evolution Equations

Equal atom-field coupling parameters

Same form as two-level equations

Temporally matched pulses

Control

Signal

Darboux Transformation Method

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.

Zero-Order Soliton Solution

Darboux Transformation

First-Order Soliton Solution

3

3

3

2

2

2

2

1

1

1

1

First-Order Soliton Solution

Darboux Transformation Method

Zero-Order Soliton Solution

Darboux Transformation

First-Order Soliton Solution

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

3

3

3

Absorption Depths

Absorption Depths

Control

Control

Signal

Signal

2

2

2

1

1

1

Slow SIT pulse

Decoupled pulse

Both pulses active

Important for

finite-length media

First-Order Soliton Solution

Optical Pulse Storage

Absorption Depths

First-Order Soliton Solution

Optical Pulse Storage

Long-lived atomic ground states store pulse information

Interesting, but what else can we do with it?

2. Construct Darboux matrix

3

Control

Signal

2

1

but potentially hard!!

Second-Order Soliton Solution

Seed solution

New solution

Two first-order soliton solutions

Algebraic Nonlinear Superposition Rule

Second-order soliton solution

Second-Order Soliton Solution

Two first-order soliton solutions

Optical Pulse Storage

Memory Manipulation

3

3

Control

Control

Signal

Algebraic Nonlinear Superposition Rule

2

2

1

1

Second-order soliton solution

Signal and control pulse durations

Control pulse duration

Second-Order Soliton Solution

and

Memory Manipulation

3

3

Control

Control

Signal

2

2

1

1

Signal and control pulse durations

Control pulse duration

Second-Order Soliton Solution

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.

After Collision

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

Faster-moving control pulse catching up to the storage solution

Second-Order Soliton Solution

Anticipated Behavior

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

How will the imprint change?

Distance the imprint is moved is given by the phase lag

Second-Order Soliton Solution

Analytic Results

- Wecan choose integration constants cleverlyso 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

Location of original imprint fixed by injected pulse ratios

New imprint location is

Second-Order Soliton Solution

Analytic Results

Relation to finite-length media

Ratio

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!

Control pulse area

Signal pulse area

Pulse ratio

Percent Error

Numerical Solution

High-Bandwidth Optical Pulse Control

Step 1: Pulse Storage

Imprint location

Theoretical

Absorption Depths

Signal pulse area

Pulse ratio

Original Imprint

Numerical Solution

High-Bandwidth Optical Pulse Control

Step 1: Pulse Storage

Imprint location

Theoretical

Numerical

Percent Error

Absorption Depths

Numerical

Percent Error

Numerical Solution

High-Bandwidth Optical Pulse Control

Step 2: Memory Manipulation

Control pulse

duration

Distance Moved

Theoretical

Absorption Depths

Control pulse area

Manipulated Memory

Absorption Depths

Numerical Solution

High-Bandwidth Optical Pulse Control

Step 2: Memory Manipulation

Control pulse

duration

Distance Moved

Theoretical

Numerical

Percent Error

New location

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

New location

well outside medium.

Control pulse area

Theoretical

Numerical Solution

High-Bandwidth Optical Pulse Control

Step 3: Pulse Retrieval

Signal pulse is recovered!

Control pulse

duration

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

Absorption Depths

- 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

F = 3

52P3/2

266.650 MHz

F = 2

156.947 MHz

F = 1

72.2180 MHz

F = 0

Lifetime ~ 26 ns

Lifetime ~ 26 ns

F = 3

F = 3

52P3/2

52P3/2

266.650 MHz

266.650 MHz

384.230 THz

F = 2

F = 2

156.947 MHz

156.947 MHz

F = 1

F = 1

72.2180 MHz

72.2180 MHz

F = 0

F = 0

F = 2

384.230 THz

- 2.56005 GHz

2.56301 GHz

384.230 THz

52S1/2

384.230THz

+ 4.27168 GHz

6.83468 GHz

4.27168 GHz

F = 2

F = 2

2.56301 GHz

2.56301 GHz

F = 1

52S1/2

52S1/2

6.83468 GHz

6.83468 GHz

4.27168 GHz

4.27168 GHz

F = 1

F = 1

87Rb D2 Line Transition

Two-Level Model

Three-Level Model

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

Large bandwidth pulse cannot resolve ground or excited hyperfine states

Pulse bandwidth chosen to resolve ground but not excited hyperfine states

𝜏 < 26 ns

𝜏 < 2 ns

0.15 ns < 𝜏 < 2 ns

Experimental Realizations

Integrable Maxwell- Bloch Equations

Lax Form

Maxwell-Bloch Equations

Traveling Wave Coordinates

Lax Operators

Storage and Retrieval

Absorption Depthsκx

Absorption Depthsκx

Absorption Depthsκx

Two First-Order Soliton Solutions

Linear (but potentially hard!!) Lax equations

Darboux Transformation

Nonlinear Superposition Rule

Hermitian unitary matrix

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

Second-Order Soliton Solution

Asymptotic behavior of the solution

The phase lag

is the only remnant of the collision

Second-Order Soliton Solution

second-order soliton Rabi frequency

second-order soliton Rabi frequency

first-order soliton Rabi frequency

first-order soliton Rabi frequency

Asymptotic behavior of the solution

The phase lag

is the only remnant of the collision

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