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Elizabeth Groves University of Rochester

Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval. Elizabeth Groves University of Rochester. Thesis Defense February 11 th , 2013. Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval. Elizabeth Groves University of Rochester. Thesis Defense

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Elizabeth Groves University of Rochester

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  1. Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval Elizabeth Groves University of Rochester Thesis Defense February 11th, 2013

  2. Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval Elizabeth Groves University of Rochester Thesis Defense February 11th, 2013

  3. Soliton Solutions for High-Bandwidth Optical Pulse Storage and Retrieval Elizabeth Groves University of Rochester Thesis Defense February 11th, 2013

  4. Normalvs.Short 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.

  5. Normalvs.Short 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

  6. Normalvs.Short 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.

  7. Analytical Methods 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)

  8. Analytical Methods 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.

  9. Stable solitary wave 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

  10. 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

  11. 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

  12. ons 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

  13. Summarizing Solitons 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

  14. Solitons in Nature 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

  15. (Speculated) 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/

  16. Solving Integrable Equations 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

  17. Solving Integrable Equations Generates soliton solutions! Integrable nonlinear systems can be characterized by the Lax formalism Lax Form Seed solution Darboux Transformation New solution

  18. Solving Integrable Equations 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

  19. 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

  20. 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

  21. faster first-order soliton 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

  22. Dipole moment operator d Wavefunction ψ (pure states) Density matrix ρ (mixed states) 2 1 Short Optical Pulse Propagation Long Collection of Atoms

  23. Slowly-varying envelope E Short Optical Pulse Propagation Optical frequency ω Laser Pulse

  24. Laser Pulse 2 dE Resonant Atoms Ω Optical frequency ω Dipole moment operatord 1 Slowly-varying envelope E Short Optical Pulse Propagation Rabi frequency

  25. 2 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

  26. 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

  27. First-Order Soliton Solution Darboux Transformation Method Zero-Order Soliton Solution 1. Solve linear Lax equations 2. Construct Darboux matrix First-Order Soliton Solution

  28. 1 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

  29. 1 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

  30. 3 2 1 Two-Frequency Pulse Propagation in Three-Level Media Opportunities for interesting dynamics and pulse-pulse control

  31. Nonlinear Evolution Equations 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

  32. 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 Nonlinear Evolution Equations Equal atom-field coupling parameters Same form as two-level equations Temporally matched pulses Control Signal

  33. First-Order Soliton Solution 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

  34. 3 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

  35. 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

  36. Absorption Depths 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?

  37. 1. Solve linear Lax equations 2. Construct Darboux matrix 3 Control Signal 2 1 but potentially hard!! Second-Order Soliton Solution Seed solution New solution

  38. Two first-order soliton solutions Algebraic Nonlinear Superposition Rule Second-order soliton solution Second-Order Soliton Solution

  39. 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

  40. Optical Pulse Storage 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.

  41. Before Collision 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?

  42. 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

  43. 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!

  44. Numerical 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

  45. Control pulse area 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

  46. Control pulse area Numerical Percent Error Numerical Solution High-Bandwidth Optical Pulse Control Step 2: Memory Manipulation Control pulse duration Distance Moved Theoretical Absorption Depths

  47. Original Imprint 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!

  48. Distance Moved 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

  49. Conclusions • 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

  50. Lifetime ~ 26 ns 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

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