Nanophotonics class 4. Density of states

emits a photon

Emission rates are set by Fermi’s Golden Rule
Fermi’s Golden Rule & the number of available photon states (LDOS)
Experiments demonstrating emission rate control via LDOS
Conclusion

eigenstates ψ.
Suppose the system is perturbed, e.g. by incident light.
Perturbing term in hamiltonian:

The coupling can take the atom in initial state ψi to another state ψf
Fermi’s Golden Rule: rate of decay of the initial state ψi

light

Dipole operator

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 ???
(constraint: photon energy = atomic energy level difference)

?

States in an LxLxL box:

l,m,n positive integers

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

l,m,n > 0
fill one octant

fudge 2 for
polarization

As a function of frequency ω (=ck):

Picture from
http://britneyspears.ac

k

dk

vacuum per 1 Hz @ λ=500 nm

n=3.5
Lattice spacing 400 nm

Photonic band gap: no states = no spontaneous emission
Enhanced DOS: faster spontaneous emission according to Fermi G. Rule

feels local mode strength |E|2.
It can only emit into a mode if the mode is not zero at the emitter

DOS: just count states

Local DOS

A

B

Atom at position A can not emit into
cavity mode.

Atom at position B can emit into
cavity mode.

of Europium ions depends
on source position relative to a silver mirror
(λ=612 nm)

Silver mirror

Spacer thickness d

Europium ions

polystyrene spheres [1]
Er ions in 340 nm SiO2 spheres [2]

[1] Schniepp & Sandoghdar, Phys. Rev. Lett 89 (2002)
[2] de Dood, Slooff, Polman, Moroz & van Blaaderen, Phys. Rev. A 64 (2001)

LDOS
normalized to
LDOS in SiO2

matching of sphere
with fluid droplets:
Emitter stays the same
Lifetime change disappears

[1] Schniepp & Sandoghdar, Phys. Rev. Lett 89 (2002)

(no photons but heat)

Measured in experiment

Unknown loss
local chemistry
at source

Fermi’s Golden Rule
LDOS

Measurement technique: vary the nanophotonic configuration
vary LDOS and not the chemistry
Example
Emitter in sphere: index match sphere to vary
Assignment: you can find by varying LDOS