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