ROBERTO MERLIN University of Michigan
PHOTON-MATTER INTERACTIONS
INTRODUCTION TO LINEAR OPTICS KRAMERS-KRÖNIG RELATIONS DRUDE AND LORENTZ MODELS LATTICE AND CARRIER ABSORPTION PHONON POLARITONS COUPLED LO-PLASMON MODES FANO INTERFERENCES ELECTRONIC EXCITATIONS ELECTRON-PHOTON INTERACTION INTERBAND TRANSITIONS EXCITONS ULTRAFAST SCIENCE NONLINEAR COUPLING MECHANISMS COHERENT AND SQUEEZED STATES
ROBERTO MERLIN University of Michigan
PHOTON-MATTER INTERACTIONS LECTURE 1
A. INTRODUCTION TO LINEAR OPTICS
0 00 0
1 10 0
1 10 0
c t c t
c t c t
∇⋅ = ∇ ⋅ =∇ ⋅ = ∇ ⋅ =
∂ ∂∇× − = ∇× − =
∂ ∂∂ ∂
∇× + = ∇× + =∂ ∂
<<λe Dh B
e Dh H
h Be E
N N=E e
Ne
ND N4πPNe
N N=E D
AIR MEDIUM
HOMOGENIZATION coarse graining, hydrodynamics, elasticity, effective-mass approximation, WKB, etc.
Maxwell’s Equations Continuous Media (no sources) ( )
( )
( ) / 4( ) / 4
i ij j
i ij j
D EB H
= ε ω
= µ ω
= − π= − π
P D EM B H
permittivity permeability
( )ε ω ( )µ ω( ) ( )1 2
2 1 1
4
14
C D
C D
i ic c
ic t c
π ω∇× = + ≡ − ε + ε
ω ∂ ω≡ ε ≡ σ = = − ε
π ∂
H j j E
Dj E E j E
( 1)4iω
σ = − ε −π
Plane Waves ( . )
22
2
~, ( . . 0)
//
i te
i i ck
i i c c
−ωΦ⊥ ∇ = ∇ =× = − ωε ω µε
=× = − ωµ
k r
k E H E Hk H Ek E H
right-handed left-handed
, 0ε µ > , 0ε µ <
1 /
N n i
Z Y −
= + κ = εµ
= = µ ε
refractive index
extinction coefficient
impedance
( ) / / /
0
11
, ~
/2
i n i z c i nz c z c
ZrZ
E H e e e
c
ω + κ ω −ωκ
µ − ε−= =
+ µ + ε
=λ
δ = ωκ =πκ
skin depth
reflection amplitude
reflection coefficient 2 2
22 2
( 1)( 1)nR rn− + κ
= =+ + κ
high reflectivity: n ≈ 0 (metals) κ,n >> 1 (highly absorbing media)
εij(ω) µij(ω) PERMITTIVITY MAGNETIC PERMEABILITY
dielectric function
LOCAL RELATIONSHIP BETWEEN P AND E [if not, εij(ω,k)]
LOCAL RELATIONSHIP BETWEEN M AND H [if not, µij(ω,k)]
• For magnetic materials, µ can be large at low frequen-cies (< THz) with peaks at magnon resonances. It is very close to unity for diamagnets.
• At high frequencies µ ≈1for all materials with the excep-tion of certain metamaterials (metallic compo-sites) or at frequencies in the vicinity of magnetic-dipole transitions.
http://fzu.cz/~dominecf/misc/eps/img/GaAs_big.png
GaAs n
κ
Re(ε) --- Im(ε)
10 THz 103 THz
KRAMERS-KRÖNIG RELATIONS
2 11 2
0( ) 4( ) 11 1( ) 1 ( )d d+∞ +∞
−∞ −∞
′ ′ε ω ε ω −′ ′ε ω = + ω ε ω = − ω′ ′π ω −ω π ωω ω
+−
πσ∫ ∫
metals only
real part
imaginary part
21 22 2 2
12
2
2
( )2 2( ) 1 1 ( )
8 11 ( ) P
d d
d
+∞ +∞
−∞ −∞
+∞
∞
∞
−
ω→
′ ′ε ω ω ′ ′ ′ ′ε ω = + ω − ε ω ω ω′π ω −ω πω
′ ′= − σ ω ω
→
=ω
−ωω
∫ ∫
∫all solids
0
( ) ( ) ( )t
i ij jj
P t F t E t t dt′ ′ ′= −∑∫CAUSALITY: P(t) DEPENDS ON VALUES OF E AT EARLIER TIMES
RELATIONSHIP IS LINEAR AND LOCAL
key assumptions
DRUDE MODEL (METALS)
0( ) ( ) tE t J t J e−γ∝ δ = DELTA PULSE OF ELECTRIC FIELD LEADS TO EXPONENTIALLY DECAYING CURRENT DENSITY
only assumption
2 1
1 2 2
20
2 2 2
411( )
4( )
i t
e m
J t e dti e
m
−
∞− ω
πρε = − ω + γ∝
γ − ω πρ γε = ω ω + γ
∫
1ε
2ε
ω
0
0 1 20.0
0.5
1.0
Refle
ctivi
ty 0γ ≡
P/ω ω
P
↑
ω
P↓ω
LORENTZ MODEL 2
2 0 002 2 2
0
i ti teE eEd x dx ex e x
dt dt M M i
− ω− ω+ γ + ω = − = −
ω −ω − ωγ
ARRAY OF DRIVEN CLASSICAL OSCILLATORS OR INDEPENDENT TWO-LEVEL SYSTEMS (QUANTUM MECHANICS)
22 2 10
4( ) 1 ( )( ) 1 4
x
x
x
P NexNe iPM
E
−
= −π
ε ω = + ω −ω − ωγε ω = + π
LORENTZIAN LORENTZIAN DRUDE
0 0ω ≡
2
1 20
4(0) 1 1NeMπ
ε = + >ω
2 (0) 0ε =
0γ ≡
EXAMPLES
METALS AND DOPED SEMICONDUCTORS
n -Si
Ag
GaAs (semiconductor)
phonons interband transitions
PHOTON-MATTER INTERACTIONS LECTURE 1
A. INTRODUCTION TO LINEAR OPTICS
DAS ENDE
FIN
THE END
ROBERTO MERLIN University of Michigan
PHOTON-MATTER INTERACTIONS LECTURE 1
B. LATTICE & CARRIER ABSORPTION
PHONONS
( )2
IONS 1 21,
, ,...2
jN
j N j
H VM=
= +∑p
r r r
includes direct and electron-mediated interaction
TO LOWEST ORDER, SYSTEM OF COUPLED OSCILLATORS
NORMAL MODES Ω, q PHONON DISPERSION
2
0 , ,, , ,,
s ss s ss
VV V ′ ′′ ′
′ ′
∂= + +
∂ ∂∑ l ll l ll
u uu u
higher-order (anharmonic) terms
s =1, 2
l
,s sG ′ ′l l
, , ,
., ,
( ) ( ),
,
( ) ( )
( )
, 1,2,3
s s s s ss
is s
s ss i si s i i
s i
M G
t e t
M u G u
i i
′ ′ ′ ′′ ′
′′ ′ ′
′ ′
= −
=
= −
′ =
∑
∑
l l l llq l
l 0
u u
u u
q
2,det ( ) 0s ii ss si s iM G′ ′ ′ ′Ω δ δ − =q
n ions/cell – 3n×3n determinant
3n solutions for each value of q (N allowed values) out of the 3n, 3 solutions give (acoustic modes)
0( ) 0
q→Ω →q
3 acoustic branches TA, LA
3(n-1) optical branches
reduction to a single cell
PHONON-PHOTON COUPLING PHOTON MOMENTUM: 2π/λ << π/a0
THE ONLY MODES THAT COUPLE TO ELECTROMAGNETIC RADIATION ARE THOSE AT q≈0 THAT CARRY AN ELECTRIC DIPOLE (MODES THAT TRANSFORM LIKE A VECTOR)
POINT GROUP OF CRYSTAL (32 GROUPS)
4
2
....
h
v
d
d
OCTD
DIAMOND
ZINC-BLENDE
CHARACTER TABLES IRREDUCIBLE ‘IR-ACTIVE’
REPRESENTATIONS (x,y,z)
A GIVEN CRYSTAL MAY OR MAY NOT HAVE PHONONS WHICH TRANSFORM LIKE
THE IR-ACTIVE REPRESENTATIONS
IF CRYSTAL HAS INVERSION SYMMETRY, ONLY ODD MODES CAN
COUPLE TO LIGHT
MICROSCOPIC MODEL (CLASSICAL)
−u
+u
CUBIC CRYSTAL WITH TWO ATOMS/CELL
( )D Z e∗ + −δ = −p u u
Szigeti effective charge
induced electric dipole
1 ( )C
M M
M M v
+ −
+ −
+ −+
= −W u u
2 2 220
12 4L e
V V VC R
V Z eH dV dV dVv M
∗
= +Ω − ⋅ + + ⋅ π∫ ∫ ∫wP W W E E P E
normal mode coordinate 3-fold degenerate
volume of crystal
same for each cell
local electric
field
energy density electronic polarization
(induced by displacement of ion’s electronic cloud)
( )e e e L+ −= α + αP E
( )
220
C R
e e
CC R
d Z edt v M
Z evv M
∗
+ −∗
= −Ω +
α + α= +
W W E
P W E
polarization
MICROSCOPIC MODEL (CLASSICAL)
short range
long-range
ionic
electronic
( )( )
( )( )( )
2 2
2 20
2 22LO0 0
2 2 2 20 TO
4( ) 1 4
( )1
e e
C C R
Z ev v M
+ − ∗
∞∞ ∞
α + α πε Ω = + π −
Ω −Ω
Ω −Ωε − ε Ω= + ε − = ε
Ω −Ω Ω −Ω
4= + πD E P
( )
( )( )
2 20
2 2
2 20
1
C R
e e
C C R
Z ev M
Z ev v M
∗
+ − ∗
− Ω −Ω =
α + α = −
Ω −Ω
W E
P E
LORENTZIAN
-2
0
2
4
ε0 ε∞
ΩTO
ΩLO
Ω
ε
LONGITUDINAL EXCITATIONS PHYSICAL MEANING OF ΩLO:
LONGITUDINAL SOLUTIONS FOR ε = 0
COULOMB-LIKE BUT TIME VARYING.
0 0∇ ⋅ = ε ⋅ =D k D
( )24 0= − π ∇ =E P E
LO MODES CARRY A LONGITUDINAL FIELD (PARALLEL TO DISPLACEMENT). STRONG INTERACTION WITH CARRIERS (FRÖHLICH INTERACTION)
33 2 2
.
4ii d q
r qe=
π ∫ q rr q
ΩLO ΩTO
k
c k∞
Ω =ε
0
c kΩ =ε
LONGITUDINAL SOLUTION
TRANSVERSE SOLUTIONS
( )c k
nΩ =
Ω
POLARITONS: COUPLED PHOTON-PHONON MODES
2 22 2 22 2 TO
2 2LO
( )( ) ( )c c kk
∞
Ω −ΩΩ = =
ε Ω ε Ω −Ω
NO SPONTANEOUS DECAY γ: ANHARMONICITY (PHONON-PHONON COUPLING)
TRANSVERSE EXCITATIONS
TWO-PHONON ABSORPTION
( ) ( ) ( )0
,( ) ...i j j
i jji jj
C u u ′′≡ −
′
= α + +∑ ∑q q qq
P u q
first-order
constants branch labels
FIRST-ORDER: LIGHT COUPLES TO ALL INFRARED-ACTIVE MODES AT q=0.
SECOND ODER: LIGHT COUPLES TO PAIRS OF MODES THROUGHOUT THE BRILLOUIN ZONE
SINCE EVERY CELL UNDERGOES SAME DISPLACEMENTS, q1+q2=0 SPECTRUM USUALLY DOMINATED BY OVERTONES (WEIGHTED DENSITY OF STATES: NON-ANALYTICAL VAN-HOVE SINGULARITIES)
2Ω
ACOUSTIC MODES
OPTICAL MODES WHY IS H2O BLUE?
4-5 PHONON ABSORPTION O-H BOND
NEW LONGITUDINAL MODES
PHONON
TOTAL
CARRIER
LO
P
00
0
ε = ω
ε =ε = ω
COUPLED LO-PLASMON MODES
1.0 1.2 1.4 1.6 1.8 2.0
-100
-50
0
1.30 1.35 1.40 1.45 1.500.0
0.2
0.4
0.6
0.8
1.0
1.2
TOTALε
R
INFRARED-ACTIVE MODES IN METALS AND DOPED SEMICONDUCTORS
PHONONS ALONE: PHONON
2 2LO
2 2TO
∞
ω −ωε = ε
ω −ω
CARRIERS ALONE: CARRIER
2P21 ω
ε = −ω
TOTAL PHONON CARRIER
2 2 2LO P
2 2 2TO
1
∞
ε = ε + ε −
ω −ω ω= ε − ω −ω ω
TOTAL PHONON CARRIERχ = χ + χRANDOM-PHASE
APPROXIMATION:
TRANSPARENT (ε=1)
2 224 2 2 P TOP
LO 0∞ ∞
ω ωωω − +ω ω + = ε ε
EXPERIMENTAL DATA (PHONONS)
(KxRb1-x)2SeO4 Soft Optical Modes, Phase Transitions and Internal Modes (molecular crystal)
Infrared Reflectivity
SeO4
soft modes: phase transitions
perovskite BaTiO3
PbS (rock salt)
huge TO-LO splitting
GaAs
3×1018 cm-3
8×1017 cm-3
2×1017 cm-3
Reflectivity Two-Phonon Absorption
InN (wurtzite) heavily doped
PHONON DIPS
Semicond. Sci. Technol. 21, 544-549 (2006)
Density of States
Two-Phonon Absorption
dα/dE Diamond (IR inactive)
Germanium (IR inactive)
FANO-TYPE INTERFERENCES
DISCRETE STATE COUPLED TO CONTINUUM
Barker and Hopfield, Phys. Rev. 135, A1732 (1964)
FANO INTERFERENCES (CLASSICAL MODEL)
( )
0
0
( )
( )A A A B A A A
B B B A B B B
A B B
i
t
t
i
A
u k u u u Z e
u k u u u Z e
P e Z u Z
e
E
E
u
e−
−
ω
ω
+ + κ − κ + Γ = ×
+ + κ − κ + Γ = ×
= +
external electric field
2
, , ,
0
2
2
2 22
2 0
( )
2
A B A B A B
i
B B AA
A B
A A BB
A B
t
B B BA A A
BA
i t
k i
Z Zue
Z ZuE e
Z Z Z ZP e E e
− ω
− ω
Σ = −ω + + κ − Γ
κ + Σ = Σ Σ − κ × κ + Σ = Σ Σ − κ
Σ +Σ + κ= ×Σ Σ − κ
2 22
2 2 0
2 2 2 2
2 2( ) 4 1
iA B
A A B B
A B
A A B
t
B
Z ZP ek i k i
Z e Z ek i k
E e
i
− ω = + × −ω + − Γ −ω + − Γ
ε ω = π + + −ω + − Γ −ω + − Γ
0 2 40
2
4
no coupling (κ=0)
Im( )ε
FANO INTERFERENCES (CLASSICAL MODEL) 2 2
22
2 20
02
( )
A B B A B A
A B
A
i t
B
EZ Z Z ZP e
i
e− ωΣ + Σ + κ= ×
Σ Σ − κ
Σ ≈ − ωΓΣ ≈ ω −ω
overdamped (continuum)
22 20
2 2 2 2 2 40
( )Im( )
( )A BZ Z ω −ω + κ ε = ωΓ
ω Γ ω −ω + κ
coupling-induced transparency
2 2
0 ( )/B AZ Z↑
ω =ω + κ
Im( )ε
minimum of Im(ε) for w1=w2
1 1
2 2
cos sinsin cos
w uw u
θ θ = − θ θ
Fano lineshape 2
2
( )1q + δ+ δ
Phys. Rev. Lett. 76, 784
H2O
Phys. Rev. Lett. 33, 1372
Ni (EELS)
p-Si (Raman)
Phys. Rev. B 9, 4344
Phys. Rev. Lett. 97, 023603
Rb vapor (EIT)
JOSA B 26, 813
WG Modes
Coupled Modes: Interference-Induced Transparency (no losses)
DATA (FANO INTERFERENCES AND EIT)
PHOTON-MATTER INTERACTIONS LECTURE 1
B. LATTICE & CARRIER ABSORPTION
DAS ENDE
FIN
THE END