Optics and Quantum Optics with Semiconductor Nanostructures

Stephan W. Koch Department of Physics, Philipps University, Marburg/Germany

andOptical Sciences Center, University of Arizona, Tucson/AZ

theory: Kira, Hoyer et al., MarburgHader, Moloney et al., Tucson

experiments: Gibbs/Khitrova et al., Tucson,Stolz et al., Rostock

• background: optics with atoms• semiclassical semiconductor optics • semiconductor quantum optics:

“which way” experiments and light – matter entanglement

Overview

Collaborators

Marburg

You do not really understand something unless you can explain it to your grandmother (Albert Einstein)

Marburg

atom

43

2

n=1

From Atoms to Solids …

optical absorption/emission =transitions between atomic levels

atom

unit cell

solid

(2 - 5) * 10-8cm

energy states

bands

43

2

n=1

From Atoms to Solids …

possible energy values of electrons in crystal

intrinsic semiconductor:full valence band(s),empty conduction band

E

kEg

Bandstructure

6

4

2

0

-2

-4

-6

-8

-10

-12

L XΓ

Ene

rgy

(eV

)

Realistic Bandstructure GaAs

effective mass approximation

often: photon momentum typical carrier momentum � perpendicular transitions,

Energy Gap in Semiconductors

gap energy determines frequency and therefore color (wavelength) of absorbed and/or emitted light

1ω� 2ω�

conduction band

valence band

������������� ���������� ������� ����

������������ ��������������

Bandgaps of III-V Alloys (300 K)

5.4 5.6 5.8 6.0 6.2 6.4 6.60.0

0.5

1.0

1.5

2.0

2.5

lattice constant (Angstrom)

Ene

rgy

gap

(eV

)

wav

elen

gth

(mic

ron)

1.00

2.0

AlSb

5.010.0

GaP

GaAs

AlAs

InP

InAs

GaSb

InSb

0.620

0.517

0.775

1.55

Quasi-Two Dimensional Structure

discrete states (z direction) and continuous bands (x-y plane)

TEM picture: quantum well structure

band gap at Γ-point (direct semiconductor)

self organizedquantum dots

Semiconductors as Designer Materials

� quantum well = two-dimensional electronic mobility

� quantum wire = one-dimensional electronic mobility

� quantum dot = no (zero-dimensional) electronic mobility

Interband Light-Matter Interaction:Semiclassical Theory

semiconductor:Bloch basis

classical Maxwell’s wave equation macroscopicoptical polarization

Coulomb interaction of chargecarriers quantum mechanical

many-body problem ofinteracting Fermions

→

Semiconductor Bloch Equations (SBE)

• nonlinearities: phase space fillinging, gap renormalization,Coulomb attraction

• correlation effects: scattering, dephasing, screening, …

field renormalization

energy renormalization

Wannier Excitons

• 2 parabolic bandselectron-hole pairinterband Coulombattraction

• wavefunction

• relative motion(Wannier equation)

• hydrogen atom like solutions, Wannier excitons = quasi atoms (finite lifetime < nanoseconds)

c

v

Coulomb potential

Wannier Excitons

• linear absorption Elliott formula

• linear optics: excitonic resonances• INTERACTION induced resonances, not just transitions between bands

Exciton Saturation

F. Jahnke, M. Kira, and S.W. Koch, Z. Physik B 104, 559 (1997)Born-Markov approximation

Detuning

saturation via excitation induced dephasing (EID)= Coulomb induced destructive interference

between different

Exciton Saturation

F. Jahnke, M. Kira, and S.W. Koch, Z. Physik B 104, 559 (1997)

Detuning

Abs

orpt

ion

• experiment: InGaAs/GaAs QW• Khitrova, Gibbs, Jahnke, Kira, Koch, Rev. Mod. Phys. 71, 1591 (1999)• EID first observed in 4-wave mixing, Wang et al. PRL 71, 1261 (1993)

Lineshape Problem

0.2

0.0

-0.4

-0.2

-0.6 -20 -10 0 10

Detuning

Abs

orpt

ion

[103 /

cm]

full calculation

dephasing rate approximation

•

• gain of two-band bulk material

• nondiagonal scattering contributions → lineshape modification,no absorption below the gap

abso

rptio

n (x

103 /

cm)

abso

rptio

n/ga

in [1

/cm

]

abso

rptio

n/ga

in

Photon Energy (eV)

Current 0-20mA… Density 0.6-3.0x1012cm-2

exp: C. Ellmers et al., theory: A. Girndt et al. MarburgCourtesy of W.W. Chow, P.M. Smowton, P. Blood, A. Grindt, F. Jahnke, and S.W. Koch

8nm InGaAs/AlGaAs 6.8 nm In0.4Ga0.5P/(Al0.5Ga0.5)In0.51P0.49

10nm (9.2nm) InGaAs/AlGaAs

N=1.6, 2.2, 2.5, 3.0*1012/cm2

exp: D. Bossert et al., theory: A. Girndt et al., Marburg

Detuning

Detuning

Optical Gain in Semiconductors: Theory and Experiment

• quantitative understanding of interaction phenomena• strong experiment – theory interactions• predictive capability of theory

CHALLENGES:

• modified photonic environment (nano optics with nano structures)• optimized design for specific applications• nonequilibrium phenomena ….

Selected References:

• Haug/Koch, “Quantum Theory of the Optical and Electronic Properties of Semiconductors” 4th ed., World Scientific Publ. (2004)

• Khitrova et al., Rev. Mod. Phys. 71, 1591 (1999)

Summary of Semiclassical Phenomena

Quantized Light-Matter Interaction

where is proportional to dipole matrix element and mode strength at the QW position

Kira et al., Prog. Quantum. Electron. 23, 189 (1999)

Spontaneous Emission from Quantum Wells

-|q |z+|q |z

+-

recombination in electron-hole system

emission occurs simultaneously to left and right, i.e. with and

no translational invariance perpendicular to QW

no momentum conservation

Non Resonantly Excited Photoluminescence

Experiments

t

QW

BS

AP AP

Path 2Path 1

?

non resonant excitation of QW (weak excitation)

incoherent (random) emission at exciton resonance

different emission directions collected in interferometer setup

measurement combines emission to the left and right directions (less thanone photon in interferometer)

control of phase via delay

Experiments (I)

t

QW

BS

AP AP

Path 2Path 1

QW perpendicular

Experiments (II)

t

QW

BS

AP AP

Path 2Path 1

?QW tilted

Experiments (II)

t

QW

BS

AP AP

Path 2Path 1

QW tilted

Experiments (III)

0 20 40 60 80 100

-2 0 20.0

0.2

0.4

CCD pixels

PL

inte

nsity (

a.u

.)

t (ps)

Contr

astno tilt|

with tilt

single beamintensities

Oct. 2003

clear interferences visible if QW NOT tilted

interferences vanish if QW tilted

Hoyer et al. PRL 93, 067401 (2004)

Summary of Experimental Observations

interferences seen in incoherent (single photon) emission

, but intensity shows interferences

interference shows strong directional sensitivity

Summary of Experimental Observations

interferences seen in incoherent (single photon) emission

, but intensity shows interferences

interference shows strong directional sensitivity

effects predicted in Prog. Quantum. El. 23, 189 (1999)

origin of effects: light-matter entanglement & which-way interferences

Spontaneous Emission from Quantum Wells

-|q |z+|q |z

q||

+-

electron-hole recombination

simultaneous emission in and directions

photon emission with same

recoil momentum transferred to carrier system

CASE A: Emission with same

emission to the left

emission to the right

paths not distinguishable with respect to carrier system (i.e. no entanglement)

Explanation of Interferences (I)

q|| q||

photon

many-body wavefunction with recoil

Explanation of Interferences (II)

q|| q||

BL BR

variable phase

emission intensity IL to the left IR to the right interference

interferometry:

INTERFERENCE can be seen

Explanation of Entanglement (I)

CASE B: Emission with different

emission to the left

emission to the right

paths identified by entanglement

q|| q'||

photons

many-body wavefunction with recoil

Explanation of Entanglement (II)

q|| q'||

emission to the left (BL) und to the right (BR) is combined in detector D = BL +BR

emissions intensity IL to the left IR to the right interference

NO interference pattern due to entanglement

Theory of Entanglement-Interferences

semiconductor luminescence equations PRL 97, 5170 (1997)

photon-assisted correlations

photon correlations

in the presence of Coulomb interaction

QUESTION: WHAT HAPPENS IF WE TAKE MANY QUANTUM WELLS ?

Theory of Entanglement-Interferences

Predictions for n quantum wells with spacing d

perfect interferences for Bragg

no interferences for anti-Bragg (n-even)

0 1 2 3 4 5 6 7 8 9

Number of QWs

0.0

0.5

1.0

/4

/2

Prog. Quantum. El. , 189 (1999)23dI

Entanglement-Interference Experiment (IV)

interferences seen in multiple QW system with λ/2 spacing

interferences vanishin multiple QW system with λ/4 spacing

λ/4 spacing leads to complete randomizing of emission to the left and to the right

confirmation of theoretical predictions

Summary of Entanglement-Interferences

incoherent emission to the left and to the right are entangled with the many-body carrier system

emission to the left and to the right with same q||is not entangled

emission to the left and to the right with sameq|| is entangled

description of entanglement via photon-carrier and photon-photon correlations of the type:

more in: Hoyer et al. PRL 93, 067401 (2004)

• variety of novel quantum optical effects in semiconductors• strong experiment – theory interactions

MANY CHALLENGES:

• optimization and application of non-classical properties (quantum information science, …)

• modified photonic environment (phot. x-tals, …)• role of incoherent excitons, biexcitons, ….

Selected References:

• Haug/Koch, “Quantum Theory of the Optical and Electronic Properties of Semiconductors” 4th ed., World Scientific Publ. (2004)

• Khitrova et al., Rev. Mod. Phys. 71, 1591 (1999)• Kira et al., Prog. Quantum. Electron. 23, 189 (1999)

Summary