Four-Wave Mixing and Many-Particle Effects in Semiconductors
Rolf BinderCollege of Optical Sciences and Department of Physics
The University of Arizona
Copyright © 2009 by 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
Next:
Two-particle interaction
Two-particle correlations: bound states
Two-particle correlation: continuum states
Introduction to semiconductors
Next:
GaAs bandstructure
Optical excitationtypically close toΓ
point (k=0)
ε
k
Parabolic Bandstructure near k=0
conduction band
valence band-------- ---
Ground state:full valence band
ε
k
-------- -
-
Optical excitation
E
-
ε
k
-------- -
-
-
light absorbed
“hole” in valence band
Optical excitation
ε
k
+
-
Hole = positively charged “particle”
“hole” in valence band
ε
k
+
-
Hole = positively charged “particle”
Excitons
ε
k+
-
-
+attractive Coulomb
interaction
real space:
e-h pair
Bound states (Rydberg law) 2b
n gEEn
ε = −
Continuum states
+
-
Excitons
real space:
-
+
1s
2s
-+
continuum
bulk GaAs
Excitons
+
-
+
-
Nonlinear excitation: two-exciton states
The concept of four-wave mixing (FWM)
Next:
Wave mixing and self diffraction
pkpump
skprobe("signal")
f p sΔ = −k k kGrating with spatial wave vector
Forward four-wave mixing
2f p
p s
= + Δ
= −
k k k
k k
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 22
2 2 2 2
4b
c t c tε π⎧ ⎫∂ ∂
− ∇ = −⎨ ⎬∂ ∂⎩ ⎭E P
information about excitonic correlations
Strategy in this talk: present P as Feynman diagrams
pk
pE
pk
sk2 p s−k k
pE
sE∗(3)fP
microscopic scattering process
Four-wave mixing
e2
h1
h2
e1
e1
h1
e2 h2
Introduction to many-particle theory
Next:
electrons
holes
creation operators
ε
k
†sa k-
+
s
j
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
k k
† † † † † †, ', ', ' , , ', ', ' , , , , ' ,
allindices
1 22
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
k k
† † † † † †, ', ', ' , , ', ', ' , , , , ' ,
allindices
1 22
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
k k'
+k q ' -k q
e h
Show only basic structure:
†bandH a aε=
† †cCoulombH V a a a a=
† †light couplingH E a a E a a∗
− = +
band Coulomb light couplingH H H H −= + +
Next:
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
'tt
→ ∞0t → −∞
0t → −∞
"Density-type"
†( , ') ( ') ( )nm m niG t t a t a t= ⟨ ⟩arrow points backward in time
'tt
Next:
A diagram tool box
h e
ε
k
+
-
-
e
-
- -
-
--
---
-
+
++
+
+
+
+
+
+
+
+
-
e
-
-
e e
e e
1t 2t
2, 'k qν −
q
1,k qν +
1,kν 2, 'kν
1 2 1 2sign( , ) ( ) ( )ci V q t tν ν δ−−
1
2( )sj
i E tμ ⋅
t{ }, ,e s k{ }, ,h j k−
1
2( )sj
i E tμ∗ ∗−⋅
t{ }, ,e s k{ }, ,h j k−t 't
(0), ( , ')kiG t tν
{ },kν
1t 2t
2, 'k qν −
q
1,k qν +
1,kν 2, 'kν
1 2 1 2sign( , ) ( ) ( )ci V q t tν ν δ−−
1
2( )sj
i E tμ ⋅
t{ }, ,e s k{ }, ,h j k−
1
2( )sj
i E tμ∗ ∗−⋅
t{ }, ,e s k{ }, ,h j k−
t 't
(0), ( , ')kiG t tν
{ },kν
More about Green's functions, propagators and Feynman diagrams
Next:
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( ) ( ') ( ) ( ')
Ic
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:
2112
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
= ++
+ +
+ + ...
Wick's theorem: sum up all different graphs
+= +
+
+
+
+
+
+
+
+
++
+
All of them???
+= + +
+ +
- --...
+
-
+= + +
+ +
- --...
+=T + +
+= T
describes non-perturbative Coulomb correlation (including possible bound states)
Ladder diagrams
...
Third-order nonlinear optical response and excitonic diagrams
Next:
Dynamics-controlled truncation (DCT)(Axt, Stahl, Z. Phys. B 93, 205 (1994))
ε
k
Without optical excitation, no density-like propagators:
†( ') ( )I Ia t a t⟨ ⟩
t '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
e h
( ) ( )h ea t a t⟨ ⟩
( , )t ε+ +( , )t +
(1) ( )p t
E
First-order polarization
e h
( ) ( )h ea t a t⟨ ⟩
( , )t ε+ +( , )t +
(1) ( )p t
E
+-
+
-
e h
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
0q =
~
~
1t
1t2t
2t
ε
k
+k
ε
−
optically active
: | ⟩electron spin honot le ation , spin
↓ ↑
ε
k k
ε
↑ ↑ ↓ ↓
optically inactive
↑ ↓
First order excitonic response
h e
eh
p+
E+
↓ ↑
↓ ↑
(0)( )xi i p Ep ε γ φ+ + ∗ += − −
In 1s approximation:
cvE+ +≡ d Ewith
The two basic third-order diagrams
he
h e
eh
e h
direct
he
e
h e
h
e h
electron exchange
h ea a⟨ ⟩ h ea a⟨ ⟩
he
h e
eh
e h
direct
he
e
h e
h
e h
electron exchange
(3)p E∗
E E
(3)p E∗
E E
Without Coulomb interaction,this diagram is disconnected
(does not contribute)
The two basic third-order diagrams
he
e
h e
h
e h↓
↑
↑
↓
↑ ↓
↑ ↓
PSF 21 (0) 2( )x si i p E A p Ep ε γ φ+ + ∗ + + += − − + | |
p+ p+ ∗
E+ p+
Phase-Space Filling (PSF)
not possible:
not optically activehe
e
h e
h
e h↓
↑
↑
↓
↑↓
↑
↓
PSFA p p E+ − +
+ −
↓↓ ↑↑and
e
h
h
e
e
h
e h
e
h
h
e
eh
e h
excitons
exciton-excitoninteraction
(1 of 4 contributions)"direct" "exchange"
+
- -
+
2( ) | |dir excHF
ip W W p p= +
0q =~
↑
e
h
h
e
e
h
e h
+
+
↓ ↑ ↓
↑↓↑ ↓
+
+
HF 2 (PSF term) | |( )xi i p V p pp ε γ+ + + += − − +
e ↑
e
h
h
e
e
h
h
+
↓ ↑ ↓
↑↓
↑↓
−
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 ∗ ∗= − + +
pE
pp
time
sE
sp
time
pE
ppsE
sp
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++ ↔ −
( )( )
sum
sum
dir exc
dir exc
G W W
W W
+ − ↔ −
+ +
e
h
h
e
eh
e h
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
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++ + +
e h
eh
corr
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+ +⟨ ⟩
n m
Next:
Some experimental FWM data
Identified biexciton,"local field" (HF),and EID
1.548 1.552 1.556Frequency [eV]
DFW
M S
igna
l (ar
b. u
nits
)
0
0
0
0
0.5
0.5
0.5
Reflectivity
1
0.8
0.9
(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)
G
Signature of non-perturbativecontinuum correlations
semiconductor quantum well
DBR
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
(3)
Next:
HF
biexciton
ip =
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
sk
QW
6wm 3 2p s= −k k k
pk
pE
6wmp
pE pE
sE∗sE∗
h e
ehe h
e h
Six Wave Mixing
sk
QW
6wm 3 2p s= −k k k
pkh e
ehe h
e h
[ ]G n p
Experiment
"coherent limit"
"incoherent densities"
"incoherent-densityassisted transitions"
Theory
Experiment Theory
full theorymean field theory
without biexciton
Experiment Theory
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)
h e
eh
e h e h
pE
fwmp
pE pE
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
AB
SO
RB
AN
CE
αL
0
3
I0
2I0
4I0
8I0
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
+pump
−probe (signal)
////////////
pump probe biexcitonω ω ε+ ≈EIT dip at
shifts with increasing pump intensity
1.526 1.528 1.530
AB
SO
RB
AN
CE
αL
0
3
I0
2I0
4I0
8I0
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
+pump
−probe (signal)
////////////
h e
ehe h
e h
sE −
sp −
pE + pE +
pE∗+ pE∗
+
excitonrenormalization,"+ −" biexciton ingas of "+" excitons
Electromagnetically-induced transparency (EIT)
+
--
++
-
Triexciton states?
h e
eh
e h e h
† † † †corr
corr
di a adt
V a a a a a a a a aa ⟨ ⟩
⟨ ⟩
= ⟨ ⟩triexciton
Few-level systems
Next:
////////////
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 +⟩
μ+
two-exciton"continuum"
μ−
nα μ+nβ μ+ + nα μ+− nβ μ
− −
+ + + − − −
xε
2 xε
0
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'
( )
( ) ( ) ( )( ) ( ) ( )
2
2
2
*
*
's
[1 2 ]
2
x
phen
phen
phen
phen
i i v p Ep p
V p p
V p 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+− = ∑
212
phenn nV β δ
+++ = ∑
2 1nα =∑We have set
2112 nv β= − ∑
Identification of few-level parameters
2 2 21
1
| | | (0 ) | | |1 / | (0 ) |
1
1 0
a s
P S Fs
a
p h en H F
a
p h en
a
N
v AN
V VN
VN
μ φ μ
φ
+ +
+ −
↔
↔
↔
↔
( ) ( )
( ) ( )
1
1
p h en
a
p h en
a
G t t G t tN
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
− −
+ + + − − −
xε
2 xε
0
Example: 7-level system
+ −
+ −
-8 -4 0 4 80
2
4
6
8
-Im(G
phen
+-/N
a)/πa2 0
[meV
]
Ω−2ε [meV]
-4
-2
0
2
4
R
e(G
phen
+-/N
a)/πa2 0
[meV
]
Microscopic theory 7-level system
+ +
+ +
+ −
+ −
-4
-2
0
2
4
Re(
Gph
en+
+/N
a)/πa2 0
[meV
]
-8 -4 0 4 80
2
4
6
8
Ω−2ε [meV]
-Im(G
phen
+ +
/Na)/π
a2 0 [m
eV]
Microscopic theory 7-level system
FWM instabilities
Next:
pkpump
skprobe
fkFWM
diffracted light(probe direction)sk
fkFWM
pump pk
pump wave + signal wave
FWM wave
pump wave + FWM wave
signal wave
wave mixing
wave mixing
feedback and possible dynamic instability
HFp p sf
HFp p fs
i V p p pp
i V p p pp
∗
∗
=
=self consistency (positive feedback)
| |λ κ± = ±0teλ=A A
det( ) 0M λ− =
Linear stability analysis:
Time
teλ+
s
f
pp∗
⎛ ⎞= ⎜ ⎟
⎝ ⎠A 2HF
pV pκ =00i
Mi
κκ
−⎛ ⎞⎜ ⎟⎝ ⎠
=M=A A with
( ) pi tp pp t p e ω−= ( ) ( )
( ) ( )
p
p
i ts s
i tf f
p t p t e
p t p t e
ω
ω
−
−
=
=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
-5
0
5
cavity
k (106m-1)
frequ
ency
(meV
)
εxk
DBR DBR
QW
2 2
2 2 2cav /
zc k k
c k c
ω
ω
= +
= +
2 2
0 2x xk
kM
ε ε= +
//////
//////
//////
//////
//////
//////
ϑ
totk
zk
k
cavzck ω=
totc kω =
z
-5 0 5
-5
0
5
cavity
UPB
k (106m-1)
frequ
ency
(meV
)
εxk
LPB
DBR DBR
QW
//////
//////
//////
//////
//////
//////
ϑ
totk
zk
k
cavzck ω=
totc kω =
upper polariton branch
lower polariton branch
Coupled modes:
z
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
-5
0
5
cavity
UPB
k (106m-1)
frequ
ency
(meV
)
εxk
LPB
2 1 0 -1
-10
0
10
Τ (a20Eb)
Ω-2
εx 0 (meV
)
ReIm
instability threshold low at "magic angle"
in-plane wave vector(1/μm)
-4 0 4
-8
-4
0
4
8
frequ
ency
(meV
)
ϑangle of incidence0
DBR DBR
QW
ϑ
pump
• 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)
pump
DBR DBR
QW
pump
switch
in-plane wave vector(1/μm)
-4 0 4
-8
-4
0
4
8
frequ
ency
(meV
)
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
Last:
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
