No Slide Titlewp.optics.arizona.edu/.../2015/01/Complete-Presentation.pdf · 2015. 1. 15. · kk...

transcript

Four-Wave Mixing and Many-Particle Effects in Semiconductors

Rolf BinderCollege of Optical Sciences and Department of Physics

The University of Arizona

A lot of diagrams...

This tutorial is about

using Feynman diagrams and four-wave mixing tostudy many-particle correlations in semiconductors

It is not about

photon echo, holography, phase conjugation,third-harmonic generation, ...

Three theoretical approaches

Fermionic theory

Bosonic theory

Atomic models withmany-body corrections

Acknowledgements

First:

Special thanks toNai H. Kwong

College of Optical SciencesUniversity of Arizona

...and to (past and present) graduate students and postdocsIlya Rumyantsev (now Synopsis)Zhenshan Yang (now Texas A&M Univ.)Ryu Takayama (now Canon)Dan Nguyen (now NP Photonics)Stefan Schumacher (now Heriot-Watt Univ.)Greg RupperBaijie Gu

ERATO, AFOSR, DARPA, JSOP

Introduction Many-particle theory and Green's functionsThird-order optical responseCorrelations beyond third orderFew-level systemsFWM instabilities (time permitting)

Outline:

Weekend experiences withperturbative and non-perturbativetwo-particle correlations

Two-particle interaction

Two-particle correlations: bound states

Two-particle correlation: continuum states

Introduction to semiconductors

GaAs bandstructure

Optical excitationtypically close toΓ

point (k=0)

Parabolic Bandstructure near k=0

conduction band

valence band-------- ---

Ground state:full valence band

-------- -

Optical excitation

-------- -

light absorbed

“hole” in valence band

Optical excitation

Hole = positively charged “particle”

“hole” in valence band

Hole = positively charged “particle”

Excitons

+attractive Coulomb

interaction

real space:

e-h pair

Bound states (Rydberg law) 2b

n gEEn

ε = −

Continuum states

Excitons

real space:

continuum

bulk GaAs

Excitons

Nonlinear excitation: two-exciton states

The concept of four-wave mixing (FWM)

Wave mixing and self diffraction

pkpump

skprobe("signal")

f p sΔ = −k k kGrating with spatial wave vector

Forward four-wave mixing

= + Δ

diffracted light

pkpump

skprobe("signal")

(3)P E E Eχ ∗=

Third-order response:(schematically; integrals over space

and time suppressed)

fie k r sie− k r pie k r pie k rspatial dependence:

2 2 2 2

c t c tε π⎧ ⎫∂ ∂

− ∇ = −⎨ ⎬∂ ∂⎩ ⎭E P

information about excitonic correlations

Strategy in this talk: present P as Feynman diagrams

sk2 p s−k k

sE∗(3)fP

microscopic scattering process

Four-wave mixing

Introduction to many-particle theory

electrons

creation operators

†sa k-

annihilationoperators

†ja k ja k

† †e hband s s s j j j

s jH a a a aε ε= +∑ ∑k k k k k k

† † † † † †, ', ', ' , , ', ', ' , , , , ' ,

allindices

cCoulomb q s s s s j j j j s j j sH V a a a a a a a a a a a a+ − + − + −⎡ ⎤= + +⎣ ⎦∑ k q k q k k k q k q k k k q k q k k

† †, , , ,

allindices

( ) ( )light coupling sj s j sj s jH E t a a E t a aμ μ∗ ∗− − −⎡ ⎤= ⋅ + ⋅⎣ ⎦∑ k k k k

band Coulomb light couplingH H H H −= + +

Fermionic semiconductor Hamiltonian

e and h occupation number operators

electron-hole pair annihilation and creation operators

e-e h-h e-h interaction

† †e hband s s s j j j

s jH a a a aε ε= +∑ ∑k k k k k k

† † † † † †, ', ', ' , , ', ', ' , , , , ' ,

allindices

cCoulomb q s s s s j j j j s j j sH V a a a a a a a a a a a a+ − + − + −⎡ ⎤= + +⎣ ⎦∑ k q k q k k k q k q k k k q k q k k

† †, , , ,

allindices

( ) ( )light coupling sj s j sj s jH E t a a E t a aμ μ∗ ∗− − −⎡ ⎤= ⋅ + ⋅⎣ ⎦∑ k k k k

Fermionic semiconductor Hamiltonian

e and h occupation number operators

electron-hole pair annihilation and creation operators

e-e h-h e-h interaction

+k q ' -k q

Show only basic structure:

†bandH a aε=

† †cCoulombH V a a a a=

† †light couplingH E a a E a a∗

− = +

About expectation values and Green's functions

( ) ( ) ( )eh h ep t a t a t= ⟨ ⟩

Expectation values

†( ) ( ) ( )e e ef t a t a t= ⟨ ⟩

†( ) ( ) ( )h h hf t a t a t= ⟨ ⟩

interband polarization

occupation functions

Two-time functions

'( ) ( ) ( ')eh h e t t

p t a t a t=

= ⟨ ⟩

'( ) ( ) ( ')e e e t t

f t a t a t=

= ⟨ ⟩

⟨ ⟩†( ')a t( )a t

Particle propagators

carries information about particle energy

later than 't t

⟨ ⟩†( ')a t ( )a t

Hole propagators

' later than t t

†ˆ( , ') ( ) ( ')nm c n miG t t T a t a t= ⟨ ⟩

→ ∞0t → − ∞

0t → − ∞

†( , ') ( ') ( )nm m niG t t a t a t= −⟨ ⟩if t' later on contour than t:

n m'tt

Schwinger, J. Math. Phys. 2, 407 (1961); Keldysh, Sov. Phys. JETP 20, 235 (1965)

†( , ') ( ) ( ')nm n miG t t a t a t= +⟨ ⟩if t' earlier on contour than t:

→ ∞0t → −∞

0t → −∞

All one-particle Green's functions have "time arrow"

n m†ˆ( , ') ( ) ( ')nm c n miG t t T a t a t= ⟨ ⟩ 'tt

→ ∞0t → −∞

0t → −∞

Usual propagators (non density-type)

†( , ') ( ) ( ')nm n miG t t a t a t= ⟨ ⟩arrow points forward in time

→ ∞0t → −∞

0t → −∞

"Density-type"

†( , ') ( ') ( )nm m niG t t a t a t= ⟨ ⟩arrow points backward in time

A diagram tool box

2, 'k qν −

1,k qν +

1,kν 2, 'kν

1 2 1 2sign( , ) ( ) ( )ci V q t tν ν δ−−

2( )sj

i E tμ ⋅

t{ }, ,e s k{ }, ,h j k−

2( )sj

i E tμ∗ ∗−⋅

t{ }, ,e s k{ }, ,h j k−t 't

(0), ( , ')kiG t tν

{ },kν

2, 'k qν −

1,k qν +

1,kν 2, 'kν

1 2 1 2sign( , ) ( ) ( )ci V q t tν ν δ−−

2( )sj

i E tμ ⋅

t{ }, ,e s k{ }, ,h j k−

2( )sj

i E tμ∗ ∗−⋅

t{ }, ,e s k{ }, ,h j k−

(0), ( , ')kiG t tν

{ },kν

More about Green's functions, propagators and Feynman diagrams

The idea:

†ˆ ( ) ( ')c n mT a t a t⟨ ⟩represent as perturbation seriesvia Feynman diagrams

'ˆ ( ) ( ')c n m t t

T a t a tε= +

⟨ ⟩ obtain expectation valuesfrom equal-time limit

'' '( '')† †

, ,ˆ ˆ( ) ( ') ( ) ( ')

i dt H t

c n m c I n I mT a t a t T e a t a t− ∫

⟨ ⟩ = ⟨ ⟩

0 'H H H= +

Feynman diagrams: expand exponential and factorize:

ixe ix x= + − + ⋅⋅⋅

† † † † † †, , , , , , , , , , , ,I i I j I k I l I i I l I j I k I i I k I j I la a a a a a a a a a a a⟨ ⟩ = ⟨ ⟩⟨ ⟩ − ⟨ ⟩⟨ ⟩

Perturbation theory:

full Green's function (propagators)

free particle operators

(0) †, ,

ˆ( , ') ( ) ( ')nm c I n I miG t t T a t a t= ⟨ ⟩t 'tn m

' 0 :H =

+= + +

' 0 :H ≠

+= + +

direct Coulomb interaction exchange Coulomb interaction

Rules and Regulations(short version)

1. Draw all topologically distinct connected diagrams with two external points

2. Sum over all internal indices

3. Attach an additional factor of (-1) for each closed Fermion loop

Complete rules:Thermodynamic equilibrium: Fetter, Walecka, Quantum Theory of Many Particle SystemsOptically excited semiconductors: Kwong, Binder, Phys. Rev. B 61, 8341 (2000)Semiconductors with quantized light: Kwong, Rupper, Binder, Phys. Rev. B 79, 155205 (2009)

Wick's theorem: sum up all different graphs

+ + ...

Wick's theorem: sum up all different graphs

All of them???

+= + +

- --...

+= + +

- --...

+=T + +

describes non-perturbative Coulomb correlation (including possible bound states)

Ladder diagrams

Third-order nonlinear optical response and excitonic diagrams

Dynamics-controlled truncation (DCT)(Axt, Stahl, Z. Phys. B 93, 205 (1994))

Without optical excitation, no density-like propagators:

†( ') ( )I Ia t a t⟨ ⟩

earlier time later time

Green's function approach to DCT: Kwong, Binder, Phys. Rev. B 61, 8341 (2000)

→ ∞0t → −∞

0t → −∞

DCT: "Don't Counterpropagate in Time"

First-order polarization

( ) ( )h ea t a t⟨ ⟩

( , )t ε+ +( , )t +

(1) ( )p t

First-order polarization

( ) ( )h ea t a t⟨ ⟩

( , )t ε+ +( , )t +

(1) ( )p t

sum up ladder diagrams: excitons

In 1s approximation:

center-of-mass momentum

relative e-h momentum

1( , , ) ( ) ( , )eh sp t p tφ=k q k q

optically active

: | ⟩electron spin honot le ation , spin

↓ ↑

↑ ↑ ↓ ↓

optically inactive

↑ ↓

First order excitonic response

↓ ↑

(0)( )xi i p Ep ε γ φ+ + ∗ += − −

In 1s approximation:

cvE+ +≡ d Ewith

The two basic third-order diagrams

direct

electron exchange

h ea a⟨ ⟩ h ea a⟨ ⟩

direct

electron exchange

(3)p E∗

Without Coulomb interaction,this diagram is disconnected

(does not contribute)

The two basic third-order diagrams

e h↓

↑ ↓

PSF 21 (0) 2( )x si i p E A p Ep ε γ φ+ + ∗ + + += − − + | |

p+ p+ ∗

Phase-Space Filling (PSF)

not possible:

not optically activehe

e h↓

↑↓

PSFA p p E+ − +

↓↓ ↑↑and

excitons

exciton-excitoninteraction

(1 of 4 contributions)"direct" "exchange"

2( ) | |dir excHF

ip W W p p= +

↓ ↑ ↓

↑↓↑ ↓

HF 2 (PSF term) | |( )xi i p V p pp ε γ+ + + += − − +

↓ ↑ ↓

↑↓

optically inactive

2cannot be: | |p p+ −Hartree term zero, Fock term only ++

Phase-space filling vs Hartree-Fock in FWM

PSF HF2( )x f s p p s p pfi i p A p p E V p p pp ε γ ∗ ∗= − + +

FWM signal solely due to Coulomb

interaction

PSF=0, only HF contributes

PSF and HF contribute

negative delay time: pump first

homogeneouslybroadened

inhomogeneouslybroadened

negative delay time: pump first

Beyond Hartree-Fock: Excitonic Correlation Functions

Excitonic correlation functions:

( )sum dir excG W W++ ↔ −

( )( )

dir exc

+ − ↔ −

PSF 21 (0) 2( ) [ ]x si i p A p Ep ε γ φ+ + ∗ + += − − − | |

HF 2V p p+ ++ | |

( ) ( ) ( )p d t G t t p t p t− ∗ +− + −∞

−∞′ ′ ′ ′+ −∫

Phase-space filling

2 ( ) ( ) ( )p d t G t t p t p t+ ∗ + + + +∞

−∞′ ′ ′ ′+ −∫

Hartree-FockCoulomb interaction

Time-retarded two-excitoncorrelations

(incl. biexciton)

Takayama, Kwong, Rumyantsev, Kuwata-Gonokami, Binder, Eur. Phys. J. B 25, 445 (2002)

Same equation for the coherent third order interband polarization:

Dynamics Controlled TruncationAxt , Stahl, Z. Phys. B 93, 195 (1994)

Hubbard operators, force-force correlation functionOestreich, Schoenhammer, Sham, Phys. Rev. B 58,12920 (1998)

Nonequilibrium Green’s functionsKwong, Binder, Phys. Rev. B 61, 8341 (2000)

Cumulant expansionsMeier, Koch, Phys. Rev. B 59, 13202 (1999);Hoyer, Kira, S.W. Koch, Phys. Rev. B 67, 155113 (2003)

see also: Schäfer, Wegener, Semiconductor Optics and Transport Phenomena(Springer, Berlin, 2002)

Degenerate FWM(all fields at frequency ω)

{ }2(1) (1)

2(1) (1)

( ) ( ) ( ) ( ) 2 (2 )

( ) ( ) ( ) (2 )

PSF HFG V G

χ ω χ ω χ ω ω ω

χ ω χ ω χ ω ω

+ + + +

+ − + −

⎡ ⎤ + +⎣ ⎦

⎡ ⎤⎣ ⎦

(1) 1( )x i

χ ωω ε γ− +

(1)( ) 1/ ( )PSFG ω χ ωwith

• Takayama, Kwong, Rumyantsev, Kuwata-Gonokami, Binder, JOSA-B 21, 2164 (2004)• Kwong, Takayama, Rumyantsev, Kuwata-Gonokami, Binder, Phys. Rev. B 64, 045316 (2001)

2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )p s s p pf E E E E Ep ω χ ω ω ω χ ω ω ω ω± + + ∗ + − ∗± ± ±= + ∓ ∓

Takayama, Kwong, Rumyantsev, Kuwata-Gonokami, Binder, Eur. Phys. J. B 25, 445 (2002)

HF: 6.3meVHF: +6.3meV

correlations partly cancel HF

Shift of biexciton =correlation effect

beyond χ(3)

no excitation-induced dephasing (EID) belowtwo-exciton continuum

biexciton resonance below two-exciton

continuum

includes bound two-exciton states (biexciton)

G p p+− + −

only continuum states

G p p++ + +

di a adt

⟨ ⟩

W W Wa a a a a a a a a a a a←⎯→ ←⟨ ⟩ ⟨ ⟩ ⟨ ⟩⎯→ ←⎯→

Hierarchy of correlation functions

† †corra a a aW a a= ⟨⟩ ⟩⟨

h e e h

Figure: Lindberg, Hu, Binder, Koch, Phys. Rev. B 50, 18060 (1994)

Hierarchy of correlation functions

Proven:If initial e-h density zero, one has, exact to order of E,

truncation of hierarchyfactorization to yield closed set of equations of motion

Axt, Stahl, Z. Phys. B 93, 205 (1994)

1 1n ma a a a+ +⟨ ⟩

Some experimental FWM data

Identified biexciton,"local field" (HF),and EID

1.548 1.552 1.556Frequency [eV]

Reflectivity

(x,+,-)

(x,+,+)

(x,y,y)

(x,x,x)

blue: experiment(Gonokami et al., 1997)

red: full theory

green: 2nd Born

• Kwong, Takayama, Rumyantsev, Kuwata-Gonokami, Binder, Phys. Rev. Lett. 87, 27402 (2001)• Kuwata-Gonokami, Inoue, Suzuura, Shirane, Shimano, Phys. Rev. Lett. 79, 1341 (1997)

(pump, probe, signal)

Signature of non-perturbativecontinuum correlations

semiconductor quantum well

DBR (distributed Bragg reflector)

pumpprobefwm

Experiment Theory

microcavity FWM

identified importanceof EID and local field

90o phase shift between FWMand cross correlation

two-color pump(at and below exciton)

1 2( )G ω ω⇒ +

Correlations beyond χ

biexciton

V p p p∗ † †V a a a a a a

[ ]p G p p∗ † †corr

V a a a a a a

V n p† †

corrV a a a a a a

[ ]G n p † †

npV a a a a a a

plus: renormalization of V or Gtriexciton

"incoh. density"

"incoh. dens. assist. trans."

Six Wave Mixing

6wm 3 2p s= −k k k

sE∗sE∗

Six Wave Mixing

6wm 3 2p s= −k k k

[ ]G n p

Experiment

"coherent limit"

"incoherent densities"

"incoherent-densityassisted transitions"

Theory

Experiment Theory

full theorymean field theory

without biexciton

Experiment Theory

beats with inversebiexciton bindingenergy period

See also: Meier, Koch, Phys. Rev. B 59, 13202 (1999)

Experiment Theory

beats with inversebiexciton bindingenergy period

See also: Meier, Koch, Phys. Rev. B 59, 13202 (1999)

e h e h

sE∗pE∗

[ ] [ ]V G p p G p p∗

Electromagnetically-induced transparency (EIT)

pump probe biexcitonω ω ε+ ≈EIT dip at

shifts with increasing pump intensity

1.526 1.528 1.530

probe frequency [eV]

linear spectrumnonlinear, experiment

nonlinear, theory

Phillips, Wang, Rumyantsev, Kwong, Takayama, Binder, Phys. Rev. Lett. 91, 183602 (2003)

1s exciton state

biexciton

groundstate

−probe (signal)

////////////

pump probe biexcitonω ω ε+ ≈EIT dip at

shifts with increasing pump intensity

1.526 1.528 1.530

probe frequency [eV]

linear spectrumnonlinear, experiment

nonlinear, theory

Phillips, Wang, Rumyantsev, Kwong, Takayama, Binder, Phys. Rev. Lett. 91, 183602 (2003)

1s exciton state

biexciton

groundstate

−probe (signal)

////////////

sE −

sp −

pE + pE +

pE∗+ pE∗

excitonrenormalization,"+ −" biexciton ingas of "+" excitons

Electromagnetically-induced transparency (EIT)

Triexciton states?

e h e h

† † † †corr

di a adt

V a a a a a a a a aa ⟨ ⟩

⟨ ⟩

= ⟨ ⟩triexciton

Few-level systems

////////////

1s exciton states

biexcitontwo-exciton continuum

Few level models:

are useful for conceptual analysis of optical nonlinearitiescan be used in "double-sided Feynman diagrams"(e.g. in Li, Zhang, Borca, Cundiff, Phys. Rev. Lett. 96, 057406 (2006))

Answer: yes, at least in χ(3) regime!Question: can they be strictly related to many-particle theory?

see Robert W. Boyd, Nonlinear Optics,(Academic Press, London, 1992)

| ground state⟩

| biexciton⟩

| exciton,hh −⟩| exciton,hh +⟩

two-exciton"continuum"

nα μ+nβ μ+ + nα μ+− nβ μ

− −

+ + + − − −

2n x nε ε δ+

++ = + 2n x nε ε δ+− = + 2n x nε ε δ−

−− = +

( )ij ik kj ik kjk

di H Hdt

ρ ρ ρ= −∑

( ) ( ) ( )* *exc , ,exc ,exc( )n ng n np t t t tα βρ ρ ρ

++ + + += + +∑ ∑

Initial condition: system in ground state

( ) ( )*aP t N p tμ+ + +=

density of few-level 'atoms'

( ) ( ) ( )( ) ( ) ( )

[1 2 ]

i i v p Ep p

p d t G t t p t p t

p d t G t t p t p tγ

ε γ+ + + +

− ++−

+ ∞ + +++−∞

− ∞ + −+−−∞

= − − −

′ ′ ′ ′+ −∫

′ ′ ′ ′+ −∫terms proportional to

PSF 21 (0) 2( ) [ ]x si i p A p Ep ε γ φ+ + ∗ + += − − − | |

H F 2V p p+ ++ | |

( ) ( ) ( )p d t G t t p t p t− ∗ + − + −∞

−∞′ ′ ′ ′+ −∫

Phase-space filling

2 ( ) ( ) ( )p d t G t t p t p t+ ∗ + + + +∞

−∞′ ′ ′ ′+ −∫

Hartree-FockCoulomb interaction

Time-retarded two-excitoncorrelations

(incl. biexciton)

( ) ( ) ( )( )* 2

2 2x n b

i i t tn n n nphen iG t t t t eε δ γβ δ β δ

θ ++ +′− + − −

++⎛ ⎞′ ′− = − − ∑⎜ ⎟⎝ ⎠

( ) ( )( )( )2* x n b

i i t tphenn n n n

iG t t t t eε δ γ

θ α δ α δ′− + − −

+−⎛ ⎞′ ′− = − − ∑⎜ ⎟⎝ ⎠

2phenn nV α δ+− = ∑

phenn nV β δ

+++ = ∑

2 1nα =∑We have set

2112 nv β= − ∑

Identification of few-level parameters

2 2 21

| | | (0 ) | | |1 / | (0 ) |

P S Fs

p h en H F

p h en

μ φ μ

( ) ( )

p h en

G t t G t tN

+ + + +

+ − + −

′ ′− ↔ −

approximate(more 'atomic' levels

yield better agreement)

Kwong, Rumyantsev, Binder, Smirl, Phys. Rev. B 72, 235312 (2005)

| ground state⟩

| biexciton⟩

| exciton,hh −⟩| exciton,hh +⟩

μ+ μ−

nα μ+nβ μ+ + nα μ+− nβ μ

− −

+ + + − − −

Example: 7-level system

-8 -4 0 4 80

a)/πa2 0

Ω−2ε [meV]

a)/πa2 0

Microscopic theory 7-level system

a)/πa2 0

-8 -4 0 4 80

Ω−2ε [meV]

/Na)/π

a2 0 [m

Microscopic theory 7-level system

FWM instabilities

pkpump

skprobe

diffracted light(probe direction)sk

pump pk

pump wave + signal wave

FWM wave

pump wave + FWM wave

signal wave

wave mixing

feedback and possible dynamic instability

HFp p sf

HFp p fs

i V p p pp

=self consistency (positive feedback)

| |λ κ± = ±0teλ=A A

det( ) 0M λ− =

Linear stability analysis:

⎛ ⎞= ⎜ ⎟

⎝ ⎠A 2HF

pV pκ =00i

−⎛ ⎞⎜ ⎟⎝ ⎠

=M=A A with

( ) pi tp pp t p e ω−= ( ) ( )

( ) ( )

i ts s

i tf f

p t p t e

=assume and

Savvidis, Baumberg, Stevenson, Skolnick, Whittaker, and Roberts, Phys. Rev. Lett. 84, 1547 (2000)Huang, Tassone, Yamamoto, Phys. Rev. B 61, R7854 (2000)Ciuti, Schwendimann, Deveaud, Quattropani, Phys. Rev. B 62, R4825 (2000)Stevenson, Astratov, Skolnik, Whittaker, Emam-Ismail, Tartakovskii, Savvidis,Baumberg, Roberts, Phys. Rev. Lett. 85, 3680 (2000)Savasta, DiStefano, Girlanda, Phys. Rev. Lett. 90, 096403 (2003)Savasta, DiStefano, Savona, Langbein, Phys. Rev. Lett. 94,246401 (2005)Klopotowski, Martin, Amo, Vina, Shlykh, Glazo, Malpuech, Kavokin, Andre, Solid State Commun. 139, 511 (2006)Kasprzak, Richard, Kundermann, Baas, Jeambrun, Keeling, Marchetti, Szymanska, Andre, Staehli, Savona, Littlewood, Deveaud, LeSiDang, Nature 443, 409 (2006)

Stimulated polariton scattering in semiconductor microcavities

Savvidis, Baumberg, Stevenson, Skolnick, Whittaker, and Roberts, Phys. Rev. Lett. 84, 1547 (2000)Huang, Tassone, Yamamoto, Phys. Rev. B 61, R7854 (2000)Ciuti, Schwendimann, Deveaud, Quattropani, Phys. Rev. B 62, R4825 (2000)Stevenson, Astratov, Skolnik, Whittaker, Emam-Ismail, Tartakovskii, Savvidis,Baumberg, Roberts, Phys. Rev. Lett. 85, 3680 (2000)Savasta, DiStefano, Girlanda, Phys. Rev. Lett. 90, 096403 (2003)Savasta, DiStefano, Savona, Langbein, Phys. Rev. Lett. 94,246401 (2005)Klopotowski, Martin, Amo, Vina, Shlykh, Glazo, Malpuech, Kavokin, Andre, Solid State Commun. 139, 511 (2006)Kasprzak, Richard, Kundermann, Baas, Jeambrun, Keeling, Marchetti, Szymanska, Andre, Staehli, Savona, Littlewood, Deveaud, LeSiDang, Nature 443, 409 (2006)

Stimulated polariton scattering in semiconductor microcavities

The publications in 2000 spurred major activities in bosonic aspects of excitons.In this tutorial, only FWM aspects arecovered, not the bosonic aspects.

-5 0 5

cavity

k (106m-1)

DBR DBR

2 2 2cav /

zc k k

0 2x xk

ε ε= +

//////

cavzck ω=

totc kω =

-5 0 5

cavity

k (106m-1)

DBR DBR

//////

cavzck ω=

totc kω =

upper polariton branch

lower polariton branch

Coupled modes:

On LPB, two-exciton correlation dominated by Hartree-FockSmall excitation-induced dephasing at LPB facilitates instability(in contrast to single quantum well(*) )

H FV G + ++

• Savasta, DiStefano, Girlanda, Phys. Rev. Lett. 90, 096403 (2003)• Schumacher, Kwong, Binder, Phys. Rev. B 76, 245324 (2007)• (*)Schumacher, Kwong, Binder, Europhys. Lett. 81, 27003 (2008)• (*)Schumacher, Kwong, Binder, Smirl, Phys. Stat. Sol. (b) 246, 307 (2009)

neglecting polaritoneffects in exciton-excitonscattering

-5 0 5

cavity

k (106m-1)

2 1 0 -1

Τ (a20Eb)

εx 0 (meV

instability threshold low at "magic angle"

in-plane wave vector(1/μm)

-4 0 4

ϑangle of incidence0

DBR DBR

• Savvidis, Baumberg, Stevenson, Skolnick, Whittaker, and Roberts, Phys. Rev. Lett. 84, 1547 (2000)• Huang, Tassone, Yamamoto, Phys. Rev. B 61, R7854 (2000)• Ciuti, Schwendimann, Deveaud, Quattropani, Phys. Rev. B 62, R4825 (2000)

sk pk fk

Low-intensity directional manipulationsemiconductor proposal for low-intensity switch demonstrated in

Dawes, Illing, Clark, Gauthier, Science 308, 672 (2005)

DBR DBR

switch

in-plane wave vector(1/μm)

-4 0 4

ϑangle of incidence0

Schumacher, Kwong, Binder, Smirl, Phys. Stat. Sol. RRL 3, 10 (2009)Dawes, Gauthier, Schumacher, Kwong, Binder, Smirl, Laser & Photon. Rev. (2009)

Conclusion

During the last 20 years, FWM techniques developed byD.S. Chemla and others have given us deep insightinto many-particle processes in optically excited semiconductors, including the observation of excitoniccorrelation effects.

The physical processes underlying these effects canbe visualized (and analyzed) with the help of Feynmandiagrams.

www.optics.arizona.edu/binder

This tutorial talk is available at

No Slide Titlewp.optics.arizona.edu/.../2015/01/Complete-Presentation.pdf · 2015. 1. 15. · kk...

Documents