Photonic Photonic Photonic Photonic Band GapBand GapBand GapBand GapMaterialsMaterialsMaterialsMaterials

Photonic Photonic Photonic Photonic Photonic Photonic Photonic Photonic Band GapBand GapBand GapBand GapBand GapBand GapBand GapBand GapMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterials

A Sem iconductorfor Light

A Sem iconductorA Sem iconductorfor Lightfor Light

www.physics.www.physics.utorontoutoronto.ca/~john.ca/~john

Photonic Band Gap Materials

– Two Fundamental Optical Principles•• Localization of LightLocalization of Light

– S. John, Phys. Rev. Lett. 53,2169 (1984)– S. John, Phys. Rev. Lett. 58,2486 (1987)

•• Inhibition of Spontaneous EmissionInhibition of Spontaneous Emission– E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987)

! State of research– Since 1995, the number of scientific and engineering

publications per year has been doubling every 18months.

History of Localization and Photonic Band Gaps History of Localization and History of Localization and Photonic Photonic Band GapsBand Gaps

1958: Electronic Localization: All electronic states localizeP.W. Anderson, Phys.Rev. 109, 1492 (1958) "Absence of Diffusion in Certain Random Lattices“1958:1958: Electronic LocalizationElectronic Localization: : AllAll electronic states localize electronic states localizeP.W. Anderson, Phys.Rev. P.W. Anderson, Phys.Rev. 109109, 1492 (1958) ", 1492 (1958) "Absence of Diffusion in Certain Random LatticesAbsence of Diffusion in Certain Random Lattices““

1983: Phonon Localization:Some states are localizedS. John & M.J. StephenPhys.Rev. B28, 6358 (1983)

1983:1983: Phonon LocalizationPhonon Localization::SomeSome states are localized states are localizedS. John & M.J. StephenS. John & M.J. StephenPhys.Rev. Phys.Rev. B28B28, 6358 (1983), 6358 (1983)

1984: Photon Localization:Any states localized?!S. John, PRL 53, 2169 (1984)P.W. Anderson, Phil.Mag. B52,505 (1985)

1984:1984: Photon LocalizationPhoton Localization::AnyAny states localized?! states localized?!S. John, PRL S. John, PRL 5353, 2169 (1984), 2169 (1984)P.W. Anderson, Phil.P.W. Anderson, Phil.MagMag. B52,505 (1985). B52,505 (1985)

ρ(ω)

ω

ω

ρ(ω)

ρ(ω)

ω

""Anderson Transition"Anderson Transition"

ρ(ω)

ω

1985: Weak Localization of Light (Coherent Backscattering) M.P. Von Albada & A. Lagendjik, PRL 55, 2692 (1985) P.E. Wolf & G. Maret, PRL 55, 2696 (1985)

1985:1985: Weak Localization of Light (Coherent Backscattering)Weak Localization of Light (Coherent Backscattering) M.P. Von M.P. Von AlbadaAlbada & A. & A. LagendjikLagendjik, PRL , PRL 5555, 2692 (1985), 2692 (1985) P.E. Wolf & G. P.E. Wolf & G. MaretMaret, PRL, PRL 55 55, 2696 (1985), 2696 (1985)

1987: Prediction of Photonic Band Gap

S. John and E. Yablonovitch

1987:1987: Prediction of Prediction of Photonic Photonic Band GapBand Gap

S. John and E. S. John and E. YablonovitchYablonovitch

1990: Computational Demonstration of PBG (band structure) K.M. Ho, C.T. Chan, C.M. Soukoulis, PRL 65, 3152 (1990) 1990:1990: Computational Demonstration of PBG (band structure)Computational Demonstration of PBG (band structure) K.M. Ho, C.T. Chan, C.M. K.M. Ho, C.T. Chan, C.M. SoukoulisSoukoulis, PRL 65, 3152 (1990), PRL 65, 3152 (1990)

1991: Experimental Demonstration of Microwave Localization and PBG 2D system: S. Shultz et al., Nature 354, 53 (1991) 3D system: A.Z. Genack & N. Garcia, PRL 66, 2064 (1991) E. Yablonovitch, T.J. Gmitter, K.M. Leung, PRL 67, 2295 (1991)

1991:1991: Experimental Demonstration of Microwave LocalizationExperimental Demonstration of Microwave Localization and PBG and PBG 2D system: S. Shultz et al., Nature 354, 53 (1991)2D system: S. Shultz et al., Nature 354, 53 (1991) 3D system: A.Z. 3D system: A.Z. Genack Genack & N. Garcia, PRL 66, 2064 (1991)& N. Garcia, PRL 66, 2064 (1991) E. E. YablonovitchYablonovitch, T.J. , T.J. GmitterGmitter, K.M. Leung, PRL 67, 2295 (1991), K.M. Leung, PRL 67, 2295 (1991)

1990-present: Quantum Electrodynamics in a PBG S. John & J. Wang, PRL 74, 2418 (1990) … T. Quang et al., PRL 79, 5238 (1997)

1990-present:1990-present: Quantum Electrodynamics in a PBGQuantum Electrodynamics in a PBG S. John & J. Wang, PRL 74, 2418 (1990)S. John & J. Wang, PRL 74, 2418 (1990) … … T. T. Quang Quang et al., PRL 79, 5238 (1997)et al., PRL 79, 5238 (1997)

Photon-Atom Bound State

Photon-Atom Photon-Atom Bound StateBound State

1994: Experimental Observation of “Laser Paint” N.M. Lawandy et al., Nature 368, 436 (1994) 1994:1994: Experimental Observation of “Laser Paint”Experimental Observation of “Laser Paint” N.M. N.M. Lawandy Lawandy et al., Nature 368, 436 (1994)et al., Nature 368, 436 (1994)

1995: “Large” Scale 2D PBG Macroporous Silicon U. Gruning, V. Lehman, C.M. Englehardt, Applied Phys. Lett. 66, 3254 (1995) U. Gruning, V. Lehman, S. Ottow, K. Busch, Applied Phys. Lett. 68, 3254 (1996)

1995:1995: “Large” Scale 2D PBG “Large” Scale 2D PBG Macroporous Macroporous SiliconSilicon U. U. GruningGruning, V. Lehman, C.M. , V. Lehman, C.M. EnglehardtEnglehardt,, Applied Phys. Applied Phys. LettLett. 66, 3254 (1995). 66, 3254 (1995) U. U. GruningGruning, V. Lehman, S. , V. Lehman, S. OttowOttow, K. Busch,, K. Busch, Applied Phys. Applied Phys. LettLett. 68, 3254 (1996). 68, 3254 (1996)

1997: Experimental Demonstration of Light Localization D. Wiersma et al., Nature 360, 671 (1997) cover story 1997:1997: Experimental Demonstration of Light LocalizationExperimental Demonstration of Light Localization D. D. Wiersma Wiersma et al., Nature 360, 671 (1997)et al., Nature 360, 671 (1997) cover storycover story

1995-present: “Woodpile”Structures E. Ozbay, Bilkent University S. Noda, Kyoto University S. Lin, Sandia National Lab

19951995-present-present:: ““Woodpile”Woodpile”StructuresStructures E. E. OzbayOzbay, , BilkentBilkent University University S. Noda, Kyoto UniversityS. Noda, Kyoto University S. Lin, S. Lin, Sandia Sandia National LabNational Lab

1998-present: Inverted Opals TiO2, CdSe, Ge, Si, GaP, … 1998-present:1998-present: Inverted OpalsInverted Opals TiOTiO22, , CdSeCdSe, , GeGe, , SiSi, , GaPGaP, …, …

2001: Square Spirals

Si, SiO2

Schrodinger-Maxwell AnalogySchrodingerSchrodinger-Maxwell Analogy-Maxwell Analogy

Can there exist localized (bound) states at energies higher than the highest potential barrier?Can there exist localized (bound) states at energies higher than the highest potential barrier?Can there exist localized (bound) states at energies higher than the highest potential barrier?

Types of scattering:1. Rayleigh2. Geometric Optics3. Resonance " Microscopic " Macroscopic

Types of scatteringTypes of scattering::1. 1. RayleighRayleigh2. Geometric Optics2. Geometric Optics3. Resonance 3. Resonance "" Microscopic Microscopic "" Macroscopic Macroscopic

S. John, PRL 53, 2169 (1984)S. John, PRL S. John, PRL 5353, 2169 (1984), 2169 (1984)

ε0ε0

εfluctεfluct

2 2

2 2 2

1 1

1 ( )

B EE Bc t c t

BE x Ec t c

ω ε

∂ ∂∇× = − ∇× =∂ ∂

∂∇×∇× = − ⇒∂

! !! ! ! !

!! ! ! !2 2

2 2 2

1 1

1 ( )

B EE Bc t c t

BE x Ec t c

ω ε

∂ ∂∇× = − ∇× =∂ ∂

∂∇×∇× = − ⇒∂

! !! ! ! !

!! ! ! !

Dielectric constantDielectric constantDielectric constant 0( ) ( )fluctx xε ε ε= +0( ) ( )fluctx xε ε ε= +

2 22

02 2( ) ( )fluctE E x E Ec cω ωε ε−∇ + ∇ ∇ − =

! ! ! ! ! !2 22

02 2( ) ( )fluctE E x E Ec cω ωε ε−∇ + ∇ ∇ − =

! ! ! ! ! ! 0 ( )fluct xε ε> −0 ( )fluct xε ε> −

Real, Positive Dielectric ConstantReal, Positive Dielectric ConstantReal, Positive Dielectric Constant

0Extended States

ωω

→ →→ ∞

lscatt

a

l*

ConventionalLocalization

Criterion

ConventionalConventionalLocalizationLocalization

CriterionCriterion

Classical (Elastic)Transport

Mean Free Path

Classical (Elastic)Classical (Elastic)TransportTransport

Mean Free PathMean Free Path

Near Near PhotonicPhotonic Band Gap Band Gap ρ(ω)ρ(ω) << << ρρ00(ω)(ω)

S.John, R.S.John, R.RangarajanRangarajan, Phys.Rev.B. , Phys.Rev.B. 3838, 10101 (1988), 10101 (1988)

Resonance RegimeGeneralized Localization

Criterion

Resonance RegimeResonance RegimeGeneralized LocalizationGeneralized Localization

CriterionCriterion

In the presence of scattering In the presence of scattering resonances resonances the the Photon Density of StatesPhoton Density of States ρ(ω)ρ(ω) is strongly is stronglymodified from the free space value modified from the free space value ρρ00(ω)=ω(ω)=ω22/(π/(π22cc33).).

ll**

λ/2πλ/2πaa

aa Geometric ray opticsGeometric ray optics

ResonanceResonance

RayleighRayleigh

ll**~~λλ44 weak disorderweak disorderstrong disorderstrong disorder

S. JohnS. JohnPRL 53, 2169 (1984)PRL 53, 2169 (1984)

Localization with very weak disorder (l* >> Localization with very weak disorder (l* >> λλ vacuum length) vacuum length)( )22 *( ) 1 cπ ρ ω ≈ ⇒"( )22 *( ) 1 cπ ρ ω ≈ ⇒"

( ) ( )2* phase space 4π× ≈"( ) ( )2* phase space 4π× ≈"

2not necessarily 4 kπ 2not necessarily 4 kπ

* 2Ioffe-Regel: 1πλ

≈"* 2Ioffe-Regel: 1π

λ≈"

Photonic Band Gap (PBG) Formation: A synergetic interplaybetween Microscopic and Macroscopic Resonances

PhotonicPhotonic Band Gap (PBG) Formation: A synergetic interplay Band Gap (PBG) Formation: A synergetic interplaybetween between MicroscopicMicroscopic and and MacroscopicMacroscopic ResonancesResonances

Largest 1-d gap occurs when singlescattering resonance and Braggresonance conditions coincide.

Largest 1-d gap occurs when singleLargest 1-d gap occurs when singlescattering resonance and Braggscattering resonance and Braggresonance conditions coincideresonance conditions coincide..

L= lattice constant a = "sphere radius"L= lattice constant a = "sphere radius"

εε(x)(x)

kkk

ωωω

π/Lππ/L/L 2π/L2π2π/L/L

Macroscopic Bragg ResonanceMacroscopic Bragg ResonanceMacroscopic Bragg Resonance

Microscopic “Mie” ResonanceMicroscopic “Microscopic “MieMie” Resonance” Resonance

mc Lω π=

mc Lω π=

transmission resonancetransmission resonance

maximum reflection maximum reflection λλ /4=2a/4=2a

Refractive index nRefractive index n2

2 (2 )cn c n a

π ω πλω

= ⇒ =2

2 (2 )cn c n a

π ω πλω

= ⇒ =

Illustrative Example: (d=1) Scalar WaveIllustrative Example: (d=1) Scalar Wave2 2

202 2( )fluctE x E E

c cω ωε ε−∇ − =

2 22

02 2( )fluctE x E Ec cω ωε ε−∇ − =

Choose a and L so that microscopic and macroscopic Choose a and L so that microscopic and macroscopic resonances resonances occur at theoccur at thesame frequency: volume filling fraction f = 2a/L = 1/2nsame frequency: volume filling fraction f = 2a/L = 1/2n

PBG Formation (continued)PBG Formation (continued)

Detailed Band Structure Calculation:Detailed Band Structure Calculation:n > 2 Diamond lattice of spheres.n > 2 Diamond lattice of spheres.

K. Ho, C.T. Chan, C.K. Ho, C.T. Chan, C.Soukoulis Soukoulis PRLPRL65, 3152 (1990)65, 3152 (1990)

∆ω∆ω/ω/ω00

1/2n1/2n ff

3D3D

CermetCermet ((nnspheresphere > > nnbackgroundbackground))

•• Low velocity (high index) Low velocity (high index)component is not connected.component is not connected.

••Favoured Favoured by scalar and elasticby scalar and elasticwaveswaves

NetworkNetwork

••Favoured Favoured by EM waves.by EM waves.

Topology of Dielectric MicrostructureTopology of Dielectric Microstructure

Photo-Electrochemical Etching of 2-dand 3-d Silicon Photonic Crystals

Photo-Photo-Electrochemical Etching of 2-dElectrochemical Etching of 2-dand 3-d Siliconand 3-d Silicon PhotonicPhotonic Crystals Crystals

(collaboration with Max-Planck-Institute of Microstructure Physics, Germany)(collaboration with Max-Planck-Institute of Microstructure Physics, Germany)(collaboration with Max-Planck-Institute of Microstructure Physics, Germany)

2-d Si Photonic Crystal(1,0,0) surface

pores etched along (1,0,0) direction

2-d 2-d Si PhotonicSi Photonic Crystal Crystal(1,0,0) surface(1,0,0) surface

pores etched along (1,0,0) directionpores etched along (1,0,0) direction

3-d Si PBG Material(1,1,1) surface

criss-crossing pores along (1,1,3) direction

3-d 3-d SiSi PBG Material PBG Material(1,1,1) surface(1,1,1) surface

crisscriss-crossing pores along (1,1,3) direction-crossing pores along (1,1,3) direction

Micro-Circuitry in a 2-D Photonic Crystal

ε=11.9

ε=4.2

ε=1

r/a=0.48, ri=0.4

Group Velocity Dispersionin 2D Silicon PBGGroup Velocity Dispersionin 2D Silicon PBG

GVD (psec/nm/km)

GroupVelocity

Configurable WDM adddrop filters

Configurable WDM adddrop filters

F1, F2, …

F1

F2

Holey FibreHoley Fibre

Silica Opal TemplatesSilica Opal TemplatesSilica Opal Templates

# Mono-disperse silica spheres, 2-5 % variation in diameter# Self-assembled into fcc lattice# Sintered to induce necking between spheres# Control of infiltration and etching, mechanical and photonic properties

## Mono-disperse silica spheres, 2-5 % variation in diameterMono-disperse silica spheres, 2-5 % variation in diameter## Self-assembled into Self-assembled into fcc fcc latticelattice## Sintered to induce necking between spheresSintered to induce necking between spheres## Control of infiltration and etching, mechanical and photonic propertiesControl of infiltration and etching, mechanical and photonic properties

Si Inverted OpalSi Inverted Opal

Optical Reflection Spectrum in the Γ- L Direction

Density of States for the FCC Latticeair voids in (macroporous) silicon, closed packed

Density of States for the FCC LatticeDensity of States for the FCC Latticeair voids in (air voids in (macroporousmacroporous) silicon, closed packed) silicon, closed packed

Kurt Busch & S. John, Phys.Rev. E 58, 3896 (1998)Kurt Busch & S. John, Phys.Rev. E Kurt Busch & S. John, Phys.Rev. E 5858, 3896 (1998), 3896 (1998)

0.00.0 0.10.1 0.20.2 0.30.3 0.40.4 0.50.5 0.60.6 0.70.7 0.80.8 0.90.90.00.0

0.20.2

0.40.4

0.60.6

0.80.8

1.01.0

ωa/2πcωωaa/2/2ππcc

DOS(arbitrary

units)

DOSDOS(arbitrary(arbitrary

units)units)

|E| at Dielectric Band Edge(3-d Si Inverse Opal)

|E| at Dielectric Band Edge|E| at Dielectric Band Edge(3-d (3-d Si Si Inverse Opal)Inverse Opal)

|E| at Air Band Edge(3-d Si Inverse Opal)|E| at Air Band Edge|E| at Air Band Edge(3-d (3-d Si Si Inverse Opal)Inverse Opal)

Liquid Crystal Photonic Band Gap Materials:The Tunable Electromagnetic Vacuum

Liquid Crystal Liquid Crystal PhotonicPhotonic Band Gap Materials: Band Gap Materials:The The TunableTunable Electromagnetic Vacuum Electromagnetic Vacuum

Cross-sectional view through the inverse opal backbone (blue) resulting from incompleteinfiltration of silicon in the air voids of an artificial opal. A tunable PBG is obtained byinfiltrating this backbone with nematic liquid crystal (green) which wets the inner surfaceof each sphere (only one is shown in the figure).

Cross-sectional view through the inverse opal backbone (blue) resulting from incompleteCross-sectional view through the inverse opal backbone (blue) resulting from incompleteinfiltration of silicon in the air voids of an artificial opal. A infiltration of silicon in the air voids of an artificial opal. A tunabletunable PBG is obtained by PBG is obtained byinfiltrating this backbone with infiltrating this backbone with nematicnematic liquid crystal (green) which wets the inner surface liquid crystal (green) which wets the inner surfaceof of eacheach sphere (only one is shown in the figure). sphere (only one is shown in the figure).

Kurt Busch & Sajeev John, PRL 83, 967 (1999)Kurt Busch & Kurt Busch & SajeevSajeev John, PRL John, PRL 8383, 967 (1999), 967 (1999)

Infiltrated Inverted Opal (Si, fSi =24.5%)Infiltrated Inverted Opal (Infiltrated Inverted Opal (SiSi, , ffSiSi =24.5%)=24.5%)Liquid crystal (BEHA): εx=εy=1.96, εz=2.56, fBEHA=36.8%Liquid crystal (BEHA): Liquid crystal (BEHA): εεxx==εεyy=1.96, =1.96, εεzz=2.56, =2.56, ffBEHABEHA=36.8%=36.8%

Total DOS(arbitrary units)

Total DOSTotal DOS(arbitrary units)(arbitrary units)

0.00.00.0 0.10.10.1 0.20.20.2 0.30.30.3 0.40.40.4 0.60.60.6 0.70.70.7 0.80.80.8 0.90.90.90.00.00.0

0.20.20.2

0.40.40.4

0.60.60.6

0.80.80.8

1.01.01.0

ωa/2πcωωaa/2/2ππcc0.50.50.5

Infiltrated Inverted Opal (Infiltrated Inverted Opal (SiSi, , ffSiSi =24.5%)=24.5%)Change in Total Photon Density of StatesChange in Total Photon Density of StatesChange in Total Photon Density of States

0.030.030.03

ϑ = 0ϑϑ = 0 = 0ϑ = π/8ϑϑ = = ππ/8/8ϑ = 5π/16ϑϑ = 5 = 5ππ/16/16

0.760.760.76 0.770.770.77 0.780.780.78 0.790.790.790.000.000.00

0.010.010.01

0.020.020.02

ωa/2πcωωaa/2/2ππcc

Total DOS(arbitrary units)

Total DOSTotal DOS(arbitrary units)(arbitrary units)

Square Spirals StructureSquare Spirals Structure

S. John and O. S. John and O. ToaderToader

Square Spirals (continued)Square Spirals (continued)Direct structureDirect structure Inverted structureInverted structure

15 %15 % 24 %24 %

Density of States forInverted Structure with24% 3-D PBG

SUMMARYLight Localization occurs in carefully engineered dielectricswithout the presence of “classical” turning points

Photonic Band Gap formation is a synergetic interplaybetween microscopic and macroscopic resonances

2-D photonic crystal micro-fabrication is well developed

3-D PBG materials: inverse diamond structure, woodpile structure inverse opals (fcc), square spiral (tetragonal)

Optical Micro-circuitry: Band Gap Engineering Point and Line Defects (sub-gap)

Tunable Photonic Band Gaps

Photonic Photonic Photonic Photonic Band GapBand GapBand GapBand GapMaterialsMaterialsMaterialsMaterials

Photonic Photonic Photonic Photonic Photonic Photonic Photonic Photonic Band GapBand GapBand GapBand GapBand GapBand GapBand GapBand GapMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterialsMaterials

A New Frontier inQuantum and

Nonlinear Optics

A New Frontier inA New Frontier inQuantum andQuantum and

Nonlinear OpticsNonlinear Optics

www.physics.www.physics.utorontoutoronto.ca/~john.ca/~john

ConsequencesConsequencesConsequences

Two Fundamental Optical PrinciplesTwo Fundamental Optical PrinciplesTwo Fundamental Optical Principles

Inhibition of Spontaneous EmissionInhibition of Spontaneous EmissionInhibition of Spontaneous EmissionE. Yablonovitch, Phys.Rev.Lett. 58, 2059 (1987)E. E. YablonovitchYablonovitch, Phys.Rev., Phys.Rev.LettLett.. 58 58, 2059 (1987), 2059 (1987)

Localization of LightLocalization of LightLocalization of LightS. John, Phys.Rev.Lett. 53, 2169 (1984)S. John, Phys.Rev.Lett. 58, 2486 (1987)S. John, Phys.Rev.S. John, Phys.Rev.LettLett.. 53 53, 2169 (1984), 2169 (1984)S. John, Phys.Rev.S. John, Phys.Rev.LettLett. . 5858, 2486 (1987), 2486 (1987)

• Low threshold band-edge lasing without a cavity mode•• Low threshold band-edge lasing without a cavity mode Low threshold band-edge lasing without a cavity mode

• New quantum states of light•• New quantum states of light New quantum states of light

• Low threshold and other anomalous Nonlinear Optical Response•• Low threshold and other anomalous Low threshold and other anomalous NonlinearNonlinear Optical Response Optical Response

• Coherent control: single atom optical memory•• Coherent control: single atom optical memory Coherent control: single atom optical memory

• Optical switching and low threshold All-optical Transistor action•• Optical switching and low threshold All-optical Transistor actionOptical switching and low threshold All-optical Transistor action

• Classical and Quantum Gap Solitons•• Classical and Classical and Quantum Gap Quantum Gap SolitonsSolitons

• Photon-atom bound states•• Photon-atom bound statesPhoton-atom bound states

Quantum Electrodynamics and CollectivePhenomena in a Photonic Band Gap

Quantum Electrodynamics and CollectiveQuantum Electrodynamics and CollectivePhenomena in a Phenomena in a PhotonicPhotonic Band Gap Band Gap

Photon-Atom Bound StatePhoton-Atom Bound StatePhoton-Atom Bound State S. John & J. Wang, PRL 64, 2418 (1990)S. John & J. Wang, PRL S. John & J. Wang, PRL 6464, 2418 (1990), 2418 (1990)

No propagating modes in a PBG, so ordinary spontaneous emission of light is eliminated.But photon can tunnel into classically forbidden gap.No propagating modes in a PBG, so ordinary spontaneous emission of light is eliminated.No propagating modes in a PBG, so ordinary spontaneous emission of light is eliminated.But photon canBut photon can tunnel tunnel into into classically forbidden gapclassically forbidden gap..

impurity atomsωv < ω0 < ωc

impurity atomsimpurity atomsωωvv < < ωω00 < < ωωcc

ωcωωcc

ωvωωvv

excitedexcitedexcited

ground stateground stateground statehω0hhωω00

atom innth levelatom inatom innnthth level level

photon inmode λ

photon inphoton inmode mode λλ

Case (i) Atom in VacuumCase (i) Atom in VacuumCase (i) Atom in Vacuum

ωωω

kkk

ckckckCase (ii) Atom in PBGCase (ii) Atom in PBGCase (ii) Atom in PBGωωω

kkkk0kk00

scattering statesscattering statesscattering statesComplex E-planeComplex E-planeComplex E-plane

Resonance Fluorescenceτ = spont. emission timeResonance FluorescenceResonance Fluorescenceτ τ = = spontspont. emission time. emission time

X E1+ih/τX EX E11++ihih//ττ

Complex E-planeComplex E-planeComplex E-plane

Real solution: Photon-atom bound stateReal solution: Photon-atom bound stateReal solution: Photon-atom bound state

XXXhωvhhωωvv hωchhωωcc

( )

1 0Solve Schrodinger equation ; Variational ;n

nn n

H E n nλλ

φ ϕ λ∞ ∞

= =Ψ = Ψ Ψ = +∑ ∑∑ ( )

1 0Solve Schrodinger equation ; Variational ;n

nn n

H E n nλλ

φ ϕ λ∞ ∞

= =Ψ = Ψ Ψ = +∑ ∑∑

Model Hamiltonian for Quantum Optics in a PBG materialModel Hamiltonian for Quantum Optics in a PBG materialModel Hamiltonian for Quantum Optics in a PBG material

ωωω

kkkk0kk00

Resonance Two-level AtomResonance Two-level AtomResonance Two-level Atom

ωAωωAA

bbb

aaa

External Laser FieldExternal Laser FieldExternal Laser Field

Rabi frequency ΩRabi Rabi frequency frequency ΩΩ

1. Rotating Wave Approximation2. Neglect external field for now,3. Simplify by going to a new “rotating frame” with frequency ωA:

1. 1. Rotating Wave ApproximationRotating Wave Approximation2. Neglect external field for now,2. Neglect external field for now,3. Simplify by going to a new “rotating frame” with frequency 3. Simplify by going to a new “rotating frame” with frequency ωωAA::

∆λ = ωλ − ωΑdetuning frequency

∆∆λλ = = ωωλ λ − − ωωΑΑdetuning frequencydetuning frequency

( ) ( )12

L Li t i tA zH a a i g a a i e eω φ ω φ

λ λ λ λ λ λλ λ

ω σ ω σ σ σ σ+ − ++ + − + − + = + + − + Ω − ∑ ∑# # # #( ) ( )1

2L Li t i t

A zH a a i g a a i e eω φ ω φλ λ λ λ λ λ

λ λω σ ω σ σ σ σ+ − ++ + − + − + = + + − + Ω − ∑ ∑# # # #

Unitary OperatorUnitary OperatorUnitary Operator ( ) exp 2A zR t i a a tλ λλ

ω σ + ≡ − +

∑( ) exp 2A zR t i a a tλ λλ

ω σ + ≡ − +

∑DefineDefineDefine ( ) ( )RR t tΨ = Ψ( ) ( )RR t tΨ = Ψ

couplingcouplingcoupling ( )21

02Adg e

Vλ λ λλ

ω µε ω

= ⋅# ! !

#( )21

02Adg e

Vλ λ λλ

ω µε ω

= ⋅# ! !

#

H H a a i g a aλ λ λ λ λ λλ λ

σ σ+ + − + → = ∆ + − ∑ ∑$ # #H H a a i g a aλ λ λ λ λ λλ λ

σ σ+ + − + → = ∆ + − ∑ ∑$ # #

Solve Time-dependent Schrodinger equation projected onto 1-photon sector:Solve Time-dependent Solve Time-dependent Schrodinger Schrodinger equation projected onto 1-photon sector:equation projected onto 1-photon sector:

In free spaceIn free space, we obtain exponential decay., we obtain exponential decay.For For ωω =ck, this is like the integral representation of a =ck, this is like the integral representation of a δδ-function.-function.

In a PBG materialIn a PBG material: S. John & T. : S. John & T. QuangQuang, Phys. Rev. A , Phys. Rev. A 5050, 1764 (1994)., 1764 (1994).

Non Non MarkovianMarkovianRadiative Radiative DynamicsDynamics

PhotonPhotonLocalizationLocalization

bb22(0)=1 (0)=1 $$ atom initially excited atom initially excited

“interaction picture”“interaction picture”“interaction picture”2 1,( ) 2,0 ( ) 1, i tR b t b t e λ

λλ

λ − ∆Ψ = +∑2 1,( ) 2,0 ( ) 1, i tR b t b t e λ

λλ

λ − ∆Ψ = +∑

2 1,

1, 2

( ) ( ) (1)

( ) ( ) (2)

i t

i t

d b t g b t edtd b t g b t edt

λ

λ

λ λλ

λ λ

− ∆

∆

= −

=

∑2 1,

1, 2

( ) ( ) (1)

( ) ( ) (2)

i t

i t

d b t g b t edtd b t g b t edt

λ

λ

λ λλ

λ λ

− ∆

∆

= −

=

∑

Formal solution of (2)Formal solution of (2) 2 20

1, 20

( ) ( ) ( )( ) ( ) 3) (t

itd b t d b G t

dtb t g d b e λτ

λ λ τ τ τ τ τ∆ = − −= ⇒∫ ∫2 20

1, 20

( ) ( ) ( )( ) ( ) 3) (t

itd b t d b G t

dtb t g d b e λτ

λ λ τ τ τ τ τ∆ = − −= ⇒∫ ∫

Memory KernelMemory Kernel ( )2( ) i tG t g e λ τλ

λτ − ∆ −− ≡∑ ( )2( ) i tG t g e λ τ

λλ

τ − ∆ −− ≡∑

22( ) ( ) ( )

2sptspG t t b t e γγ

τ δ τ −− − →% %2

2( ) ( ) ( )2

sptspG t t b t e γγτ δ τ −− − →% %

free spacefree spacespontaneousspontaneousemission rateemission rate

3 221

303

Asp

dc

ωγπε

=#

3 221

303

Asp

dc

ωγπε

=#

larg2 2

( )20

0e t- ( ) exp ( ) ( ) ( ) (

)

t

ck

idk kG t iA k k t i t k eiA t

δ τττ ττ

δ τωω

π − →− − − −−

+ −∫% larg2 2

( )20

0e t- ( ) exp ( ) ( ) ( ) (

)

t

ck

idk kG t iA k k t i t k eiA t

δ τττ ττ

δ τωω

π − →− − − −−

+ −∫%

stationary phaseapproximation

stationary phasestationary phaseapproximationapproximation

ωωω

kkkk0kk00

ωcωωcc

PBG Model Dispersion RelationsPBG Model Dispersion RelationsPBG Model Dispersion Relations

Isotropic modelIsotropic modelIsotropic model

ωωωωcωωcc

ρ(ω)ρ(ω)ρ(ω)

ωωωωcωωcc

ρ(ω)ρ(ω)ρ(ω)

Anisotropic modelAnisotropicAnisotropic model model

Power LawDecaying Memory

Power LawPower LawDecaying MemoryDecaying Memory

For a physical For a physical anisotropic anisotropic dispersion relation, the Band Edge isdispersion relation, the Band Edge isassociated with a single point kassociated with a single point k00 (rather than the sphere |k| = k (rather than the sphere |k| = k00))

Non MarkovianMemory KernelNon Non MarkovianMarkovianMemory KernelMemory Kernel

33(2 )k

V d kπ

→∑ ∫!3

3(2 )k

V d kπ

→∑ ∫!

20( )k c A k kω ω + −%

20( )k c A k kω ω + −%

Laplace transformLaplace Laplace transformtransform ( )3/ 2 2 / 3

1/ 2( )( ) , /

( ) A sp AiG s

s iβ β ω γ ω

δ= − =

−$ ( )

3/ 2 2 / 3

1/ 2( )( ) , /

( ) A sp AiG s

s iβ β ω γ ω

δ= − =

−$

20( )ck A k kω ω + −!

! !%

20( )ck A k kω ω + −!

! !%

large t

(-

2

/ 2

2

)

3

( ) exp ( ) ( )

( )

Aq

i t

dq qG t i

t

i

e

Aq t t

δ ττ

τ τ δ τω

τ

−

− − + −

−→

−∫ !%

large t(

-

2

/ 2

2

)

3

( ) exp ( ) ( )

( )

Aq

i t

dq qG t i

t

i

e

Aq t t

δ ττ

τ τ δ τω

τ

−

− − + −

−→

−∫ !%

δ = ωΑ − ωcδ = δ = ωωΑΑ − − ωωcc

Non Markovian Radiative Dynamics has direct implications on Atomic Line ShapeNon Non Markovian Radiative Markovian Radiative DynamicsDynamics has direct implications on Atomic Line Shape has direct implications on Atomic Line Shape

Define Emission SpectrumDefine Define Emission SpectrumEmission Spectrum

From Equation of Motion,From From Equation of Motion,Equation of Motion,

3-level Atom in Λ-configuration3-level Atom in 3-level Atom in ΛΛ-configuration-configuration

|1>|1>

|2>|2>

|3>|3> ω31 ≅ ω c (band edge)ωω3131 ≅ ≅ ωωcc (band edge) (band edge)

Let δ32 and γ32 be the Lamb shiftand the spontaneous emission ratefor transition ω32 (far from PBG)

Let Let δδ3232 and and γγ3232 be the Lamb shift be the Lamb shiftand the spontaneous emission rateand the spontaneous emission ratefor transition for transition ωω3232 (far from PBG) (far from PBG)

MarkovianMarkovianMarkovian

( )( )2 2 2

0

( ) ( ) (0) . 2 Re ( )Ai tAS dt e b t b c c b iω ωω ω ω

∞− − ∗≡ + = − −∫ $ ( )( )

2 2 20

( ) ( ) (0) . 2 Re ( )Ai tAS dt e b t b c c b iω ωω ω ω

∞− − ∗≡ + = − −∫ $

wherewherewhere 2 20

( ) ( )stb s e b t dt∞

−≡ ∫$2 20

( ) ( )stb s e b t dt∞

−≡ ∫$

convolution theorem convolution theoremconvolution theoremintegration by partsintegration by partsintegration by parts

2 2 2 2 22

1(0) ( ) ( ) ( ) ( )( )

b sb s b s G s b ss G s

− + = − → =+

$ $ $$$2 2 2 2 22

1(0) ( ) ( ) ( ) ( )( )

b sb s b s G s b ss G s

− + = − → =+

$ $ $$$

in freein freespacespace

((WignerWigner--Weisskopf Weisskopf approx.)approx.)

wherewherewhere 2 2Lam2 b

0

1( ) 2

( )

i ts pt

k A

sG s g dt e ie gs i

λλ λ

λ λ

γδ

ω ω

∞− ∆−= = →

+ −+∑ ∑∫$ 2 2

Lam2 b0

1( ) 2

( )

i ts pt

k A

sG s g dt e ie gs i

λλ λ

λ λ

γδ

ω ω

∞− ∆−= = →

+ −+∑ ∑∫$

( )2 3/2

32 32

1( )b si

s is iβ

δ γδ

=

+ + −−

$

( )2 3/2

32 32

1( )b si

s is iβ

δ γδ

=

+ + −−

$

Atomic Lineshapenear Band Edge

Atomic Atomic LineshapeLineshapenear Band Edgenear Band Edge

Atomic Populationon excited states |3>

of 3-level Atom

Atomic PopulationAtomic Populationon excited states |3>on excited states |3>

of 3-level Atomof 3-level Atom

P(t)P(t)P(t)

βtββtt

δ = −βδ = −βδ = −β

γ32=0.0βγγ3232=0.0=0.0ββ

γ32=0.1βγγ3232=0.1=0.1ββ

γ32=0.2βγγ3232=0.2=0.2ββ

γ32=0.5βγγ3232=0.5=0.5ββ0.00.00.0

0.20.20.2

0.40.40.4

0.60.60.6

0.80.80.8

1.01.01.0

000 333 666 999 121212 151515

|1>|1>|1>

|2>|2>|2>

|3>|3>|3>

S(ωλ)S(S(ωωλλ)) δ=ω31−ωcδ=ωδ=ω3131−−ωωcc

δ=2βδ=−0.5β

δ=0

δ=2βδ=2βδ=−0.5βδ=−0.5β

δ=0δ=0

Λ configurationΛΛ configuration configuration

3-level atom3-level atom3-level atom

γ32=βγγ3232=β=β

(ωλ−ω32)/β((ωωλλ−ω−ω3232)/β)/β−6−−66 −4−−44 −2−−22 000 222 444 666

000

111

222

333

444

555

Main conclusionsMain conclusionsMain conclusions

Radiative dynamics near a PBG is different than in ordinary vacuum. It ismuch richer than simply “no spontaneous emission takes place”.Radiative Radiative dynamics near a PBG is different than in ordinary vacuum. It isdynamics near a PBG is different than in ordinary vacuum. It ismuch richer than simply “no spontaneous emission takes place”.much richer than simply “no spontaneous emission takes place”.

(1) Non-Markovian radiative Decay Photon Localization(1) Non-(1) Non-Markovian radiative Markovian radiative Decay Photon LocalizationDecay Photon Localization

(2) Vacuum Rabi Splitting + Fractionalized Steady State Inversion(2) Vacuum (2) Vacuum Rabi Rabi Splitting + Fractionalized Steady State InversionSplitting + Fractionalized Steady State Inversion

(3) Collective Enhancement of Radiative Dynamics Near PBG Edge(3) Collective Enhancement of (3) Collective Enhancement of Radiative Radiative Dynamics Near PBG EdgeDynamics Near PBG Edge

(4) Lasing near a Photonic Band Edge (even though particular superradiance may be hard to observe, similar physics occurs in Laser action)(4) Lasing near a (4) Lasing near a Photonic Photonic Band Edge (even though particular Band Edge (even though particular superradiancesuperradiance may be hard to observe, similar physics occurs in Laser action)may be hard to observe, similar physics occurs in Laser action)

Collective Spontaneous EmissionCollective Spontaneous EmissionCollective Spontaneous Emission

N atomsN atomsN atoms

In ordinary vacuum (Markovian)In ordinary vacuumIn ordinary vacuum ( (MarkovianMarkovian))

Near Photonic Band Edge τ-1~NΦ (non-Markovian)Near Near PhotonicPhotonic Band Edge Band Edge ττ -1-1~N~NΦΦ (non- (non-MarkovianMarkovian))Anomalous exponent Φ determined by band edge singularityAnomalous exponent Anomalous exponent ΦΦ determined by band edge singularity determined by band edge singularity

atomic states of kth atomatomic states of atomic states of kkthth atom atom

Collective Scale Factor τ−1∼ NCollective Scale Factor Collective Scale Factor ττ−−11∼∼ NN ωωω

ρ(ω)ρ(ω)ρ(ω)

ω21ωω2121

ωωω

ρ(ω)ρ(ω)ρ(ω)∼(ω−ωc)−1/2∼(ω−∼(ω− ωωcc))−1/2−1/2

Isotropic PBG: Φ=2/3Isotropic PBG:Isotropic PBG: ΦΦ=2/3=2/3ωωω

ρ(ω)ρ(ω)ρ(ω)

2D PBG: Φ=12D PBG:2D PBG: ΦΦ=1=1

ρ(ω)ρ(ω)ρ(ω)

ωωω

∼(ω−ωc)1/2∼(ω−∼(ω− ωωcc))1/21/2

3D (anisotropic) PBG: Φ=23D (3D (anisotropicanisotropic) PBG:) PBG: ΦΦ=2=2

( ) ( )

1Define ;

Nk k

ij ij ij k kk

J i jσ σ=

= ≡∑ ( ) ( )

1Define ;

Nk k

ij ij ij k kk

J i jσ σ=

= ≡∑

( )12 21H a a i g a J J aλ λ λ λ λ λλ λ

+ += ∆ + −∑ ∑# # ( )12 21H a a i g a J J aλ λ λ λ λ λλ λ

+ += ∆ + −∑ ∑# #

Initial State: Single atomic excitationin symmetrical (superradiant) state

Initial State Initial State: Single atomic excitation: Single atomic excitationin symmetrical (in symmetrical (superradiantsuperradiant) state) state

(0) , 1J M JΨ = = −(0) , 1J M JΨ = = −

( )

( )

3 22 11

2 221 12 12 21 3

1 212

J J J J M J

J J J J J J

= − − ≤ ≤

= + +

( )

( )

3 22 11

2 221 12 12 21 3

1 212

J J J J M J

J J J J J J

= − − ≤ ≤

= + +

212

( ) (0) ( ) N tP t t e γ−≡ Ψ Ψ = 212

( ) (0) ( ) N tP t t e γ−≡ Ψ Ψ =

Superradiance and Lasing without a Cavity ModeSuperradiance Superradiance and Lasing without a Cavity Modeand Lasing without a Cavity Mode

Consider an initial state with population inversion and infinitesimal initial polarizationConsider an initial state with population inversion and infinitesimal initial polarizationConsider an initial state with population inversion and infinitesimal initial polarization

Solve Heisenberg Equation of Motion for Collective Atomic OperatorsSolve Solve Heisenberg Heisenberg Equation of Motion for Collective Atomic OperatorsEquation of Motion for Collective Atomic Operators

Non Markovian memory kernel provides feedback forsystem to self-organize into coherent (localized) state.Non Non Markovian Markovian memory kernel provides feedback formemory kernel provides feedback forsystem to self-organize into coherent (localized) state.system to self-organize into coherent (localized) state.

( ) 6

1(0) 1 1 2 ~ 10

N

Nk k

r r r −

=Ψ = + −∏( ) 6

1(0) 1 1 2 ~ 10

N

Nk k

r r r −

=Ψ = + −∏

12 3 120

3 21 120

( ) ( ) ( ) ( )

( ) 2 ( ) ( ) ( ) . .

t

t

d J t d G t J t Jdt

d J t d G t J t J c cdt

τ τ τ

τ τ τ

= −

= − − +

∫

∫

12 3 120

3 21 120

( ) ( ) ( ) ( )

( ) 2 ( ) ( ) ( ) . .

t

t

d J t d G t J t Jdt

d J t d G t J t J c cdt

τ τ τ

τ τ τ

= −

= − − +

∫

∫

Define and perform mean-field factorizationDefine and perform mean-field factorizationDefine and perform mean-field factorization12 3( ) , ( )J J

x t y tN N

≡ ≡12 3( ) , ( )J J

x t y tN N

≡ ≡

0

0

( ) ( ) ( )

2 ( ) ( ) ( ) . .

t

t

dx Ny t d G t xdt

dy Nx t d G t x c cdt

τ τ τ

τ τ τ∗

= −

= − − +

∫

∫

0

0

( ) ( ) ( )

2 ( ) ( ) ( ) . .

t

t

dx Ny t d G t xdt

dy Nx t d G t x c cdt

τ τ τ

τ τ τ∗

= −

= − − +

∫

∫

Localization of Superradiance (Lasing without a Cavity)Localization of Localization of Superradiance Superradiance (Lasing without a Cavity)(Lasing without a Cavity)

Initial State: N atoms with overall population inversion infinitesimal initial polarizationInitial StateInitial State: N atoms with overall population inversion infinitesimal initial polarization: N atoms with overall population inversion infinitesimal initial polarization

Numerical Solution ofHeisenberg Equation of MotionNumerical Solution ofNumerical Solution ofHeisenbergHeisenberg Equation of Motion Equation of Motion S. John & T. Quang, PRL 74, 3419 (1995)S. John & T. S. John & T. QuangQuang, PRL , PRL 7474, 3419 (1995), 3419 (1995)

Spontaneous Symmetry Breaking: macroscopic atomic polarization develops in thesteady state limit t " ϖSpontaneous Symmetry BreakingSpontaneous Symmetry Breaking: macroscopic atomic polarization develops in the: macroscopic atomic polarization develops in thesteady state limit t steady state limit t "" ϖϖ

Peak Intensity of Superradiant Emission ~ NΦ+1=N3 (d=3)Peak Intensity Peak Intensity of of SuperradiantSuperradiant Emission ~ N Emission ~ NΦΦ+1+1=N=N33 (d=3) (d=3)

(i) Collective EmissionDynamics

(i) Collective Emission(i) Collective EmissionDynamicsDynamics

AverageBehaviourAverageAverage

BehaviourBehaviour

Optical pumping + damping effects " Band edge microlaserOptical pumping + damping effects Optical pumping + damping effects "" Band edge Band edge microlasermicrolaser(ii) Fluctuations(ii) Fluctuations(ii) Fluctuations N.Vats & S. John, Phys.Rev. A 58, 468 (1998)N.Vats & S. John, Phys.Rev. A N.Vats & S. John, Phys.Rev. A 5858, 468 (1998), 468 (1998)

2525

NNN

|<J12(t)>|||<<JJ1212(t)(t)>>|| MacroscopicAtomic Polarization

MacroscopicMacroscopicAtomic PolarizationAtomic Polarization

Atomic InversionAtomic InversionAtomic Inversion

000 555 101010 151515 202020−1.0−−1.01.0

−0.5−−0.50.5

0.00.00.0

0.50.50.5

1.01.01.0

βN2/3tββNN2/32/3tt

Atomic polarization amplitudenear anisotropic PBG

Atomic polarization amplitudeAtomic polarization amplitudenear near anisotropicanisotropic PBG PBG

Atomic inversion nearanisotropic PBG

Atomic inversion nearAtomic inversion nearanisotropicanisotropic PBG PBG

<J3(t)><<JJ33(t)(t)>>NNN

r = 10−6r = 10r = 10−6−6δc = 0.1δδcc = 0.1 = 0.1

δc = 0δδcc = 0 = 0

δc = −0.3δδcc = = −−0.30.3

N2β3tNN22ββ33tt

δc = ω21 - ωcδδcc = = ωω2121 - - ωωcc

000 222 444 666 888 101010 121212−1.5−−1.51.5

−1.0−−1.01.0

−0.5−−0.50.5

0.00.00.0

0.50.50.5

r = 10−6r = 10r = 10−6−6

δc = ω21 - ωcδδcc = = ωω2121 - - ωωcc

δc = 0.1δδcc = 0.1 = 0.1

δc = −0.3δδcc = = −−0.30.3

δc = 0δδcc = 0 = 0

121212

N2β3tNN22ββ33tt

000 222 444 666 888 1010100.00.00.0

0.10.10.1

0.20.20.2

0.30.30.3

0.40.40.4

0.50.50.5

0.60.60.6

<J12(t)><<JJ1212(t)(t)>>NNN

Atomic polarization distribution for a system of 100 atoms at an isotropicband edge, subject to quantum fluctuations at early times

Atomic polarization distribution for a system of 100 atoms at an isotropicAtomic polarization distribution for a system of 100 atoms at an isotropicband edge, subject to quantum fluctuations at early timesband edge, subject to quantum fluctuations at early times

δc = 0δδcc = 0 = 0

Re <JRe <J1212>>

5=t

ImIm <

J <

J 1212>>

Steadystate

ImIm <

J <

J 1212>>

11=t

ImIm <

J <

J 1212>>

PBGtt 0=

Re <JRe <J1212>>

ImIm <

J <

J 1212>>

SCH-MQW ACTIVESCH-MQW ACTIVE

n-InP CLADn-InP CLAD

WAFER AWAFER A

WAFER BWAFER B

FUSIONFUSION

p-InP CLADp-InP CLAD

n-InP CLAD withn-InP CLAD withTRIANGULAR-LATTICE STRUCTURETRIANGULAR-LATTICE STRUCTURE

n-InP SUBSTRATEn-InP SUBSTRATE

SURFACE EMITTING REGIONSURFACE EMITTING REGIONELECTRODEELECTRODE

Γ-XΓ-X

Γ-JΓ-J

0.462µm0.462µm

2D Photonic Band Edge Laser2D Photonic Band Edge Band Edge Laser

S. Noda et. al.

Collective Switching and All-Optical Transistor actionin a Doped Photonic Band Gap Material

Collective Switching and All-Optical Transistor actionin a Doped Photonic Band Gap Material

• Waveguide channels carrying“pump” and “probe” laser beamsintersect in a region doped withimpurity atoms.

•Frequency of atomic resonanceoccurs near an abrupt change(discontinuity) in the local photondensity of states of the host photoniccrystal.

• Waveguide channels carrying“pump” and “probe” laser beamsintersect in a region doped withimpurity atoms.

•Frequency of atomic resonanceoccurs near an abrupt change(discontinuity) in the local photondensity of states of the host photoniccrystal.

44

Coherent Amplification of Weak Probe Beam by Pump LaserCoherent Amplification ofCoherent Amplification of Weak Probe Beam Weak Probe Beam by Pump Laser by Pump Laser

Weak Probe Field: ω, εPWeak Probe Field: Weak Probe Field: ωω, , εεPP

( ) ( )12 212

i t i tL LH e eP Pω ω ω ω

ε σ σ− − −

= +

# ( ) ( )12 212

i t i tL LH e eP Pω ω ω ω

ε σ σ− − −

= +

#

Probe Laser BeamProbe Laser Beam Pump Laser BeamPump Laser Beam

Atomic Excitation by a Coherent Laser FieldAtomic Excitation by a Coherent Laser FieldAtomic Excitation by a Coherent Laser Field

Steady State solution ofEinstein Rate EquationSteady StateSteady State solution of solution ofEinstein Rate EquationEinstein Rate Equation

Vacuum Density of States ρ(ω)=ω2/π2c3Vacuum Density of States Vacuum Density of States ρ(ω)=ωρ(ω)=ω22/π/π22cc33

Einstein picture requiresthat ρ(ω) is smooth onthe scale of Ω so that therate of spontaneousemission is roughly thesame in the Mollowsidebands.

Einstein picture requiresEinstein picture requiresthat that ρ(ω)ρ(ω) is smooth on is smooth onthe scale of the scale of ΩΩ so that the so that therate of spontaneousrate of spontaneousemission is roughly theemission is roughly thesame in the same in the MollowMollowsidebandssidebands..

Fluorescence MollowSpectrum

Fluorescence Fluorescence MollowMollowSpectrumSpectrum

ω0−2Ωωω00−2Ω−2Ω ω0+2Ωωω00+2Ω+2Ωω0ωω00

Mollow Mollow splittingsplitting

Dressed atom picture n=# of photons in laser mode >>1Dressed atom pictureDressed atom picture n=# of photons in laser mode >>1 n=# of photons in laser mode >>1

<2,n-1|Hint|1,n> ~ hε breaks degeneracy of<2,n-1|H<2,n-1|Hintint|1,n> ~ h|1,n> ~ hεε breaks degeneracy of breaks degeneracy of |2,n-1>|2,n-1>|2,n-1>|1,n>|1,n>|1,n>

N2NN22

N1NN11N2+N1=NNN22+N+N11=N=NAverage Incident

Energy density WAverage IncidentAverage IncidentEnergy density Energy density WW

ω=ω0ω=ω=ωω00

hω0hhωω00<2,n-1|<2,n-1|<2,n-1|

<2,n|<2,n|<2,n|

<2,n+1|<2,n+1|<2,n+1|2Ω22ΩΩ

<1,n|<1,n|<1,n|

<1,n+1|<1,n+1|<1,n+1|

<1,n+2|<1,n+2|<1,n+2|

( )2 20ω ω εΩ ≡ − +( )2 20ω ω εΩ ≡ − +

2

( ) 2N WN Wωρ ω

=+#

2

( ) 2N WN Wωρ ω

=+#

0.50.50.5N2/NNN22/N/N

WWW

N two-level atomsN two-level atomsN two-level atoms

Collective Switching and Inversion without Fluctuation: All-OpticalTransistor Effect in a PBG Material

Collective Switching and Inversion without Fluctuation: All-OpticalCollective Switching and Inversion without Fluctuation: All-OpticalTransistor Effect in a PBG MaterialTransistor Effect in a PBG Material

Consider N 2-level atoms (in a colored vacuum) interacting with a Coherent Laser field:Consider N 2-level atoms (in a colored vacuum) interacting with a Coherent Laser field:Consider N 2-level atoms (in a colored vacuum) interacting with a Coherent Laser field:

Dressed State Basis: first diagonalize atom + external field part of HDressed State Basis:Dressed State Basis: first first diagonalize diagonalize atom + external field part of Hatom + external field part of H

( )3 1 12 2112

L Li t i taH J a a H e J e Jω ω

λ λ λλ

ω ω ε −+= + + + +∑# # # ( )3 1 12 2112

L Li t i taH J a a H e J e Jω ω

λ λ λλ

ω ω ε −+= + + + +∑# # # wherewherewhere ( )1 12 21H i g a J J aλ λ λλ

+= −∑# ( )1 12 21H i g a J J aλ λ λλ

+= −∑#

( ) 3 22 111

; N

ij kk

J i j J J J=

= = −∑( ) 3 22 111

; N

ij kk

J i j J J J=

= = −∑Consider the Schrodinger equationConsider the Consider the Schrodinger Schrodinger equationequation dH i

dtΨ = Ψ#

dH idt

Ψ = Ψ#

Define Rotating FrameDefine Rotating FrameDefine Rotating Frame ( ) ( ) ( )Rt R t tΨ = Ψ( ) ( ) ( )Rt R t tΨ = Ψ wherewherewhere 3( ) exp2LJR t i t a aλ λ

λω + = − +

∑3( ) exp

2LJR t i t a aλ λ

λω + = − +

∑

where R Rd dRH i H R HR i Rdt dt

+ +⇒ Ψ = Ψ = −$ $# # where R Rd dRH i H R HR i Rdt dt

+ +⇒ Ψ = Ψ = −$ $# #

/ 2 1 0for single atom

/ 2 0 1a

a

εε

∆ → Ω −∆ −

# #/ 2 1 0

for single atom / 2 0 1

a

a

εε

∆ → Ω −∆ −

# # Define new “Rabi”frequency

Define new “Define new “RabiRabi””frequencyfrequency

andchoose

andandchoosechoose

( )22 / 2aεΩ = + ∆( )22 / 2aεΩ = + ∆

22 2

sgn( )1sin 12 1 4 /

a

a

ϑε

∆ = − + ∆

22 2

sgn( )1sin 12 1 4 /

a

a

ϑε

∆ = − + ∆

Introduce Unitary Transf.(Dressed States)

Introduce Unitary Introduce Unitary TransfTransf..(Dressed States)(Dressed States)

02πϑ≤ ≤02πϑ≤ ≤

1 cos 1 sin 2

2 sin 1 cos 2

ϑ ϑ

ϑ ϑ

= −

= +

$

$

1 cos 1 sin 2

2 sin 1 cos 2

ϑ ϑ

ϑ ϑ

= −

= +

$

$

Define Dressed Collective Atomic OperatorsDefine Define Dressed Collective Atomic OperatorsDressed Collective Atomic Operators

We wish to describe the response of this system of 2-level atoms in a Statistical Senseincluding the effects of dipole dephasing interactions with environment and other dampingWe wish to describe the response of this system of 2-level atoms in a We wish to describe the response of this system of 2-level atoms in a Statistical SenseStatistical Senseincluding the effects of including the effects of dipole dipole dephasingdephasing interactions with environment and other interactions with environment and other dampingdamping

pψ = probability that system is in state | ψ >ppψψ = probability that system is in state | = probability that system is in state | ψψ > >

( ) 3 22 111

N

ijk k

R i j R R R=

= = −∑ $ $( ) 3 22 111

N

ijk k

R i j R R R=

= = −∑ $ $

It is easily verified thatIt is easily verified thatIt is easily verified that

( ) ( )

2 212 3 21 12 21 12

2 23 3 21 12

sin cos sin cos ;

cos sin 2sin cos

TJ R R R J J

J R R R

ϑ ϑ ϑ ϑ

ϑ ϑ ϑ ϑ

= − + =

= − − +( ) ( )

2 212 3 21 12 21 12

2 23 3 21 12

sin cos sin cos ;

cos sin 2sin cos

TJ R R R J J

J R R R

ϑ ϑ ϑ ϑ

ϑ ϑ ϑ ϑ

= − + =

= − − +

Hamiltonian in Dressed State BasisHamiltonian in Dressed State BasisHamiltonian in Dressed State Basis 0 1 0 3 where H H H H R a aλ λ λλ

+= + = Ω + ∆∑# #0 1 0 3 where H H H H R a aλ λ λλ

+= + = Ω + ∆∑# #

Interaction PictureInteraction PictureInteraction Picture 0 0

1( )i iH t H t

H t e H e−

≡ # #$ 0 0

1( )i iH t H t

H t e H e−

≡ # #$

H(t) is the same as H1 except with all operators replaced by interaction picture operatorsas defined byH(t) is the same as HH(t) is the same as H11 except with all operators replaced by interaction picture operators except with all operators replaced by interaction picture operatorsas defined byas defined by

~~

[ ]0 ,II

dA i H Adt

=#[ ]0 ,I

IdA i H Adt

=#

21 21

e.g. ( ) (0)exp

( ) (0)exp 2

a t a i t

R t R i tλ λ λ+ += ∆

= Ω

$ $

$ $

21 21

e.g. ( ) (0)exp

( ) (0)exp 2

a t a i t

R t R i tλ λ λ+ += ∆

= Ω

$ $

$ $

later we will specify these to beinteraction picture state vectorslater we will specify these to belater we will specify these to beinteraction picture state vectorsinteraction picture state vectors

Consider Density OperatorConsiderConsider Density Operator Density Operator ( ) ( ) ( )X t p t tψψ

ψ ψ≡ ∑( ) ( ) ( )X t p t tψψ

ψ ψ≡ ∑

Transistor Action in a Doped PBG MaterialTransistor Action in a Doped PBG MaterialTransistor Action in a Doped PBG Material

Spontaneous Emission Rates" γ+ for ω0 > ωc" γ− for ω0 < ωcDipole dephasing rate γP due to phonons

Spontaneous Emission RatesSpontaneous Emission Rates"" γγ++ for for ωω00 > > ωωcc"" γ γ−− for for ωω00 < < ωωccDipole Dipole dephasing dephasing rate rate γγPP due to phononsdue to phonons

Collective Switching from aPassive to Active Medium

Collective Switching from aCollective Switching from aPassive to Active MediumPassive to Active Medium

S. John & T. Quang, PRL 78, 1888 (1997)S. John & T. S. John & T. QuangQuang, PRL , PRL 7878, 1888 (1997), 1888 (1997)

PopulationInversionPopulationPopulationInversionInversion

ThresholdThresholdThreshold

PassivePassivePassive

ActiveActiveActive

0.50.50.5

1.01.01.0

Laser IntensityLaser IntensityLaser Intensity

βββVacuum Rabi splittingPhoton localizationVacuum Vacuum RabiRabi splitting splittingPhoton localizationPhoton localization

For true photonic band edge γ− =0 " Non-Markovian regimeFor true For true photonic photonic band edge band edge γγ−− =0 =0 "" Non-Non-MarkovianMarkovian regime regime

External Laser Field + N atoms " Mollow Splitting in a “Colored” VacuumExternal Laser FieldExternal Laser Field + + N atomsN atoms "" Mollow Mollow Splitting in a Splitting in a ““ColoredColored”” Vacuum Vacuum

∆=ωL−ω0>0Ω >> β

∆=∆=ωωLL−ω−ω00>>00Ω >> βΩ >> β

γ+γγ++γ−γγ−−

atomatomatom

hω0hhωω00

ρ(ω)ρ(ω)ρ(ω)

ωωωωcωωcc

ω0ωω00Densityof StatesDensityDensityof Statesof States

49

Optical Mode Density and Relevant Frequency ScalesOptical Mode Density and Relevant Frequency ScalesOptical Mode Density and Relevant Frequency Scales

ωωω

ρ(ω)ρρ((ω)ω)

ωωLL−2−2ΩΩ

ωωCC ωωAA

ωωLL

ωωLL+2+2ΩΩ

ωC the point of the DOS discontinuityωA, ωL the laser and the resonant atomic frequency Ω the generalized Rabi frequency

ωωC C the point of the DOS discontinuitythe point of the DOS discontinuityωωAA, , ωωLL the laser and the resonant atomic frequency the laser and the resonant atomic frequency ΩΩ the generalized the generalized Rabi Rabi frequency frequency

∆AL= ωA- ωL < 0∆∆ALAL== ωωAA- - ωωL L < 0< 0

Atomic Switching in Photonic Crystals

Collective Atomic Effectin the Markov Approximation

Single Atom Non-MarkovianSwitching near an Anisotropic Photonic Band Edge

Fluctuations inAtomic Inversion:

Fluctuations inFluctuations inAtomic Inversion:Atomic Inversion:

Average InversionAverage InversionAverage Inversion

Atomic population per atom on thebare excited state <J22>/N, as afunction of ε/∆ for ∆=−1, N=5000,γP/γ+=0.5 and γ−/γ+=0.3,0.4,0.5.

Atomic population per atom on theAtomic population per atom on thebare excited state <Jbare excited state <J2222>/N, as a>/N, as afunction of function of ε/∆ε/∆ for for ∆=−1∆=−1, N=5000,, N=5000,γγPP/γ/γ++==0.5 and 0.5 and γγ−−/γ/γ++==0.3,0.4,0.5.0.3,0.4,0.5.

q-Mandel parameter Qb as a functionof ε/∆ for ∆=−1, N=5000, γ−/γ+=0.001and γP/γ+=0.01,0.1,0.5. Inset shows anexpanded view of the same curves inthe regime of sub-Poissonian statisticsof the excited atoms.

q-Mandel parameter q-Mandel parameter QQbb as a function as a functionof of ε/∆ε/∆ for for ∆=−1∆=−1, N=5000, , N=5000, γγ−−/γ/γ++==0.0010.001and and γγPP/γ/γ++==0.01,0.1,0.5. Inset shows an0.01,0.1,0.5. Inset shows anexpanded view of the same curves inexpanded view of the same curves inthe regime of sub-the regime of sub-Poissonian Poissonian statisticsstatisticsof the excited atoms.of the excited atoms.

<J22>/N<J22>/N

ε/∆ε/∆

γ−/γ+=0.5,0.4,0.3γγ−−/γ/γ++==0.50.5,,0.40.4,,0.30.3

11 22 33 440.20.2

0.30.3

0.40.4

0.50.5

0.60.6

0.70.7

0.10.1 0.20.2 0.30.3 0.40.400

100100

200200

300300

400400

500500QbQb

ε/∆ε/∆

γP/γ+=0.5,0.1,0.01γγPP/γ/γ++==0.50.5,,0.10.1,,0.010.01

0.150.15 0.250.25 0.350.3500

55

101022

22 22

22b

J JQ

J

−=

2222 22

22b

J JQ

J

−=

Photonic Crystal Weak Probe Absorption Spectrum

PhotonicPhotonic Crystal Crystal Weak Probe Absorption Spectrum Weak Probe Absorption Spectrum

thε ε<< thε ε<<thε ε≤ thε ε≤

thε ε>> thε ε>>thε ε≥ thε ε≥

If (RED) pump I(t)> I, (BLUE) probe is amplifiedIf (RED) pump I(t) <I, (BLUE) probe is absorbed

APPLICATIONS:Optical Micro-TransistorOptical Wavelength ConverterAll-Optical Packet Switch

APPLICATIONS:Optical Micro-TransistorOptical Wavelength ConverterAll-Optical Packet Switch

SUMMARY

Light Localization " Non-Markovian Radiative Dynamics Photon-Atom Bound State

Fractionalized Steady State Inversion and Vacuum Rabi Splitting

Collective Time Scale Factors near a Photonic Band Edge

Band Edge Lasing (lasing without a conventional cavity)

Inversion of a Two-Level System by Coherent, Resonant Pumping

Collective Atomic Switching " All-Optical Micro-Transistor Optical Wavelength Converter Sub-Poissonian Statistics