ZOOM Lecture live at 13:30 – light sources at the nanoscaleTAs live at 15:30 [ photonic crystals ]

Till 13:30- Download slides www.koenderink.info/teaching- Q & A

Next session - May 6 – minisymposionFor input: talk to the TA’s. Ilan is your main contact

On May 6: start at 13:00 sharp.

Quantum emittersFermi’s Golden RuleDensity of states

Nanophotonics class UvAFemius Koenderink – f.koenderink@amolf.nl

Motivation - LEDs

SemiconductorsChallenge 1: extraction

TIR limits extractionto ~ 2%

Challenge 2: avoidnon-radiative decay

Osram (2000)

4

Motivation – quantum optics

Suppose Alice has a secret message to communicate to Bob..

Quantum information in 1 photoncan not be eavesdropped

Also: suppose you have two localized qubits. How do you transfer a quantum state from A to B

Possible solution: spin A photon spin B

Single molecules [Moerner & Orrit, ’89]

100 micron

1018 molecules

Keep on diluting

1 molecule can emit about 107 photons per second (1 pW)Observable with a standard [6k€] CCD camera + NA=1.4 objective

Fluorescence from quantum sources

Space• Whereto does the photon go ?• With what polarization ?

Time• How long does it take for the photon to appear ?

Matter• Selection rules – what color comes out?

Light from electron transitions in a quantum object

Energy scale for light 1 to 3 eV

Compare: kBT ~ 25 meV

Vibrations in molecules: 0.1 eV

e- transitions in hydrogen: 13.6 eV [1/n12-1/n2

2]

Band gap in Si: 1.1 eV

Interaction of an atom with light

Consider two states of an atom, with energies and states

Suppose I shine light at frequency w on the system.This gives rise to a time-varying perturbation

Just the first term gives a potential energy

Transition dipole moment

Dipole approximation – a small object k.r<<1

potential

Perturbation theory: transitions are governed by

‘Transition dipole moment’

Matrix element means: selection rules

Typical moleculesLarge conjugated carbon chains

Rhodamines

Pentacene, perylene, teryllene

DBATT

Electronic levels explained by particle in a 1D boxN bond chain: about 2N electrons in a 1D box of length ~ NaGround state: first N levels are completely filledExcited state: one electron goes from level N to level N+1

Quantum dot nanocrystals

TEM/you see single atoms

CdSe (CdTe, PbS, PbSe, CdS)Semiconductor nano-crystalsElectron & hole confined as particlesin a box

II-VI quantum dots in solution: Bawendi & Norris (early ‘90s)

Molecules are not just electronic systems

Thermally populated vibrations, rotations ….

energy scales < electronic transition

Jablonski diagram

S0

S1

Electronic ground state

Electronic excited state

T1Triplet

1. Fluorescence is spin-allowed, nanosecond time scales2. Phosphorescence is spin-forbidden, so very slow

Jablonski diagram

S0

S1

Electronic excited state

Franck-Condon principleElectronic transition is instantaneous compared to the nucleiNuclei rearrange in picoseconds after the e- transitionTransition requires large vibrational wave function overlap

Franck Condon

Absorption & fluorescence probabilities are proportionalto vibrational overlap ‘Franck-Condon factor’

Expect mirror-symmetricemission vs absorptionspectra

Sharp peaks obscured by(1) Ensemble(2) Rotations & collisions

If this is all wavefunctions,.....

why care about nanophotonics?

A. A bare molecule radiates as a dipole

How do you create directivity

B. The rate of emission controls brightness

How do you control rate

Controlling brightness

Radiation resistance – environment sets power to current ratio

The work you need to do keep current j going depends on environment

Radiation resistance

1) Dipole antenna2) Ground plane

(Balanis Antenna Handbook)

RF antenna in front of a mirror

- +

-+

-

+

-

+

The same current radiates a different far field power“Method of image charge”’ - Interference with its mirror image

Single quantum emitter

20

• After one excitation, emits just one quantum of light

• Probabilistic timing of when emission occurs

Laser pulses

Hits ondetector

Hits onAPD 2

Time

S0

S1

Time (ns)

Lounis & Orrit, Single photon sources, Rep. Prog. Phys (2005)

Scanning mirror ‘Drexhage experiment’

• 25mm PS bead covered with 400nm Ag as mirror

• PS bead glued to cleaved fiber, mounted in AFM

• Sideways scanning varies vertical emitter-mirror distance

Experiment first done by B. C. Buchler (2005)

Drexhage experiment

22

Note how: the power is may be always one photon per laser pulsebut the decay rate varies with mirror-geometry

K.H. Drexhage first did this, with ensembles of molecules (1966)

0 40 80t (ns)

10

100

1000

Even

ts

slope

NV-color center in diamond

Understanding Fermi’s Golden Rule

2

2all finalstates

2( )f i f i

f

V E E

Energy conservationMatrix elements:Transition strengthSelection rules

Spontaneous emission of a two-level atom:

Initial state: excited atom + 0 photons.Final state: ground state atom + 1 photon in some photon state

Question: how many states are there for the photon ???

Understanding Fermi’s Golden Rule

2

2all finalstates

2( )f i f i

f

V E E

Energy conservationMatrix elements:Transition strengthSelection rules

Quantum: rates are proportional to number of available final photon states “DOS”

Classical: Density of States = radiation resistance for a source

2

2

0

| | ( )3

if

m w w

How many photon in a L x L x L box of vacuum ?

( , ) sin( ) with ( , , )i tE x t Ae l m nL

w k r kStates in an LxLxL box:

l,m,n positive integers

Number of states with |k|between k and k+dk:

3

24( ) 2

8

LN k dk k dk

l,m,n > 0fill one octant

fudge 2 for polarization

2 23 3

2 2 2 3( )

dkN d L d L d

c d c

w ww w w w

w

k

dk

26

Fluorescence decay rates

Fermi’s Rule: Fluorescence rate number of photon states

0 2 4 60

50000

100000

150000

Photo

n s

tate

s p

er

m3, per

Hz

Frequency w (1015

s-1)

Visible light: ~105 photon states per Hz, per m3 of vacuum

Loudon, The Quantum Theory of Light

Example: 3D photonic crystal

27

Air-sphere / Sifcc photonic crystal

1st inverse opal photonic crystal: Wijnhoven & WLV, Science 281 (1998) 802LDOS calculations: Nikolaev, Vos & Koenderink, JOSA-B 5 (2009) 987

Dispersion relation

Stop gap

wave vector k0 π/a

standing wave in n1

standing wave in n2

Freq

ue

ncy

Density of States

Redistribution of states: - photonic band gap - flat bands imply high DOS

Busch & John, Phys. Rev. E (1998)

Observations -2D quantum well

Fujita et al., Science (2005)Two-dimensional: Kyoto [Noda], Stanford [Vuckovic], DTU [ Lodahl], WSI [Finley] ...Three dimension: Lodahl et al. (Nature 2004), Leistikow et al. (PRL ’11)

30

Cavity

Fluorescence in a cavity

0 2 4 60

50000

100000

150000

Photo

n s

tate

s p

er

m3, per

Hz

Frequency w (1015

s-1)

Fermi’s Rule: Fluorescence rate number of photon states

Microcavity: Exactly one extra state per Dw=w/Q in a volume V

Gérard & Gayral, J. Lightw. Technol. (1999)

31

Cavity

Fluorescence in a cavity

0 2 4 60

50000

100000

150000

Photo

n s

tate

s p

er

m3, per

Hz

Frequency w (1015

s-1)

Fermi’s Rule: Fluorescence rate number of photon states

Microcavity: Exactly one extra state per Dw=w/Q in a volume V

Purcell factor

3

2

3

4

QF

V

Gérard & Gayral, J. Lightw. Technol. (1999)

Record high Purcell factor

Akselrod et al.Nat. PhotonicsVol 8, 835 (2014)

Single-crystalAg-cube on Au

8 nm gap (PVP spacer)

Claim:up to 1000-foldEnhancement

50% lost in metal50% appears as light

Local density of states

Consider a molecule / quantum dot / ... - as located at a fixed position- as oriented along a fixed direction

The available modes have to be weighted by how well the dipole orientation and position match to them

DOS: just count

LDOS: local strength

Sprik, v. Tiggelen & Lagendijk, Eur. Phys. Lett. (1996)

State of the art number summary

Microcavities Photonic crystals Plasmonics

Narrowband Dw/w=10-5

Local (mode profile)

Theory: F =103

Data: F=20Single |E|2 dominates

Broadband Dw/w=0.2Global

Theory: F=0 to 20Data: F=0.1 to 10 Many modes count

Broadband Dw/w=0.3Local

Theory: F=104

Data: F=500 to 1000 Problem: loss

Picture: Verhagen Picture: Moerner

F = LDOS / vacuum LDOS - L mean “local”

Why relevant?

1. Outpacing non-radiative decay channels

2. Less timing jitter in a single photon source

3. Brighter source by faster cycling through transition

4. Extracting light via the mode that dominates the LDOS

Why relevant?

1) Nanophotonics to measure quantum efficiency

2) Nanophotonics to improve quantum efficiency

heat/...

Calibration example – single NV center

38

For a mirror the LDOS is exactly knownThe contrast of the oscillation tells you the quantum efficiency

Single emitter quantum-efficiency measurement

Drexhage / Buchler & Sandoghdar/ Barnes / Polman / Frimmer

0 40 80t (ns)

10

100

1000

Eve

nts

slope

AC current - radio, WIFI, GSM… up to 100 GHz frequenciesOptics (200 THz) - no classical AC electronics available

Funneling light into a single beam

Sample: perforated Au film - hexagons of 440 nm pitchSources: dilute fluorophores Atto 640 dye diffusing in H2O

Molecules in the central hole pumped in a confocal microscope

Emission strongly redirected in a narrow beam

Single aperture: 10x brightness enhancement (full NA), pump |E|2

Array: 40x enhancement in forward direction

L. Langguth et al. ACS Nano

Single hole One shell Two shells Three shells

Fourier image kx (up to NA=1.2)

ky

Funneling light into a single beam

Route to quantum

Fermi’s Golden Rule: irreversible decay

Strong coupling QED: regime of reversible interaction“Strong coupling cavity QED” [Haroche, Wineland, 2012]

Conclusions

Absorption <-> stimulated emission Induced by external E

Spontaneous emission without any driving‘stimulated by vacuum fluctuations’

Fermi’s Golden rule

Nanophotonics controls the DOS/LDOS (w)

- How fast and whereto quantum sources emit light- Black body emitters- Any force mediated by ‘vacuum fluctuations’

2

2

0

| | ( )3

if

m w w

44

Fluorescence decay rates

Fermi’s Rule: Fluorescence rate number of photon states

0 2 4 60

50000

100000

150000

Photo

n s

tate

s p

er

m3, per

Hz

Frequency w (1015

s-1)

Visible light: ~105 photon states per Hz, per m3 of vacuum