On the menu today - photonics.ethz.ch€¦ · 63 Curto et al., Science 329, 930 (2010) Li et al.,...

$: On the menu today - photonics.ethz.ch€¦ · 63 Curto et al., Science 329, 930 (2010) Li et al., J. Biol. Chem. 2000, 275:37048\ Kühn et al., PRL 97, 017402 (2006) ala 839 Quantum$
On the menu today

• Recap: The local density of optical states

• Limits of our theory for calculating decay rates

• Real quantum emitters: beyond two levels

• Resonant energy transfer

• Photon statistics: The second-order correlation function

www.photonics.ethz.ch 1

Previously… photonic structures to control LDOS

• Modulate LDOS on a sub-λ scale

• Rely on resonances of conduction electrons of metal nanoparticles

• Rely on evanescent fields


Optical antennas Micro-cavities

• Modulate LDOS on a λ scale

• Rely on interference of propagating waves

Fermi’s Golden Rule

Local Density of Optical States

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Previously … enhancement – quantum vs. classical

Transition dipole moment is NOT classical dipole moment, but

Classical electromagnetism predicts the decay rate enhancementprovided by a photonic system as compared to a reference system.

Fermi’s Golden Rule gave us: Maxwell’s equations gave us:


Where is the limit of our formalism?

• Emitter decay rate

• Cavity decay rate

• Emitter-cavity coupling rate:


holds in the limit:

The weak-coupling limit


holds in the limit:

• This limit is called “weak-coupling regime”

• Classical model assumes monochromatic source

• Fermi’s Golden Rule assumes weak coupling (perturbation theory)

• Strong-coupling regime: deviation from exponential decay, cyclic energy exchange between emitter and cavity (Rabi oscillations)

Quantum emitters – beyond two levels

• Real emitters (dye molecules, QDs) are no perfect two-level systems

• Electronic states are broadened into bands by vibrational excitations

• Absorption and emission spectra are shifted with respect to each other


S0

S1

grad

gvib

gvib

absorption

emission

That is handy in order to

• Prepare system in excited state (at band edge of S1)

• Separate excitation light from fluorescence with color filter

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gvib >> grad

Resonant energy transfer (RET)


S0

S1

grad

S0

S1

donor acceptor

Resonant energy transfer


S0

S1gDA

S0

S1

donor acceptor

grad



Source@ r0

-+++-

-

Antenna@ rant

α

d

Last time:Rate enhancement provided by an optical antenna

This time:Rate enhancement provided by a second quantum emitter.



Close to source:

Förster radius:

Calculate work done by donor on acceptor



For weak scatterers: Close to source:

Include spectrum of donor:

Förster resonant energy transfer (FRET)

• FRET has a strong distance dependence (R-6 coming from electric dipole near field, compare optical antenna)

• At Förster radius, energy transfer rate equals free space emission rate

• FRET depends on spectral overlap of donor emission and acceptor absorption


Förster resonant energy transfer (FRET)


Robert S. Knox (left) and Theodor Förster (right) preparing for

mechanical energy transfer. Springwater, NY, August 1973.

Resonant energy transfer – example

• FRET is used as a precise ruler for small distances


Li et al., J. Biol. Chem. 2000, 275:37048

Resonant energy transfer – example

• Measure association/dissociation rates of biomolecules


Lakowicz, Principles of Fluorescence Spectroscopy

On the menu today







The Hanbury Brown-Twiss experiment

• Beam of light impinging on a 50/50 beamsplitter (BS)

• Record intensity I(t) in each arm after BS

• Calculate normalized cross correlation between signals I1 and I2


50/50 beamsplitterI1(t)

I2(t)

The second-order correlation function



• Calculate normalized cross correlation between signals I1 and I2



I2(t)

The classical case



• For a classical field I1(t) = I2(t), so g(2) is intensity autocorrelation



I2(t)

Intensity autocorrelation - the classical case



• For a classical field I1(t) = I2(t), so g(2) is intensity autocorrelation

• For long delay times

• correlation at zero delay

• global maximum at zero delay



I2(t)1

t

HW2

Intensity autocorrelation - the coherent case

• Perfectly monochromatic field

• Intensity is therefore



I2(t)1

t

laser

Intensity autocorrelation - the chaotic case

• Collection of sources

• Random phase

• Gaussian distribution of emission frequencies



I2(t)1

t

Intensity autocorrelation – counting photons

• ni(t) is the number of photons on detector i at time t

• Interpret g(2)(t) as the probability of detecting a photon on detector 2 at t= t given that a photon was detected on detector 1 at t=0.



I2(t)

Counting photons – coherent case


• Interpret g(2)(t) as the probability of detecting a photon on detector 2 at t= t given that a photon was detected on detector 1 at t=0

• g(2)(t) = 1 means that photons arrive with Poissonian distribution



I2(t)1

t

Counting photons – chaotic case


• Interpret g(2)(t) as the probability of detecting a photon on detector 2 at t= t given that a photon was detected on detector 1 at t=0

• g(2)(t=0) > 0 means that photons tend to arrive in bunches



I2(t)1

t

Counting photons – a single quantum emitter

• Assume source is a single emitter

• Single emitter can only emit one photon at a time

• If there is a photon on D1 there cannot be a photon on D2 antibunching

• Photon antibunching is at odds with classical electromagnetism

• g(2)(t=0) = 0 is the signature of a single photon source

• What determines the rise time of g(2)(t)?


50/50 beamsplitter

1

I1(t)

I2(t)

t

HW2

Intensity correlation – counting single photons

• How do you know your emitter is a single photon source? For n emitters:

• How does the lifetime show up in the correlation function?Rise time is lifetime in the case of weak pumping.



I2(t)

Beveratos, PhD thesis, Univ. Paris Sud (2002)

Intensity correlation – summary

• Second-order correlation function measures temporal intensity correlation

• Bunching: photons tend to “arrive together”, classically allowed/expected

• Antibunching: photons tend to “arrive alone”, classically forbidden



I2(t)

t

On the menu today







S0

S1

trad

Summary – light matter interaction

Quantum emitters are probes for their electromagnetic environment.


Curto et al., Science 329, 930 (2010)

Li et al., J. Biol. Chem. 2000, 275:37048\

Kühn et al., PRL 97, 017402 (2006)

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Quantum emission can be tailored via the emitter’s electromagnetic environment.

Radiation carries information about• The emitter• The emitter’s environment• The emitter-environment interaction

On the menu today - photonics.ethz.ch€¦ · 63 Curto et al., Science 329, 930 (2010) Li et al.,...

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