zoom lecture live at 13:30 light sources at the nanoscale
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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
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