Chapter III
Light-matter interaction processes
P.-A. Hervieux
Institut de Physique et Chimie des Matériaux
Strasbourg, France
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P.-A. Hervieux, IPCMS, Strasbourg
Outline
1) Light-matter interaction Hamiltonian
2) Emission and absorption
3) Light scattering
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1) Light-matter interaction Hamiltonian
For one particle:
Many-particle:
(same masse m and same charge q)
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1) Light-matter interaction Hamiltonian
Paramagnetic courant operator:
Density operator:
two choices
: Classical description of the field
: Quantum description of the field
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1) Light-matter interaction Hamiltonian
Using
⇒
with
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2) Emission and absorption
2.1) Emission and absorption rates
Fermi
Golden
Rule
(FGR)
First-order
perturbation theory
Energies of the atomic, molecular, band…states
Density of final states
absorption
emission
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2) Emission and absorption
Note that the states of radiation can also contain any number of photons
in other modes, but here we focus on a process in which a single photon
is emitted or absorbed, so that these other photons are passive spectators
who have no influence on the amplitude of the process.
The hamiltonian H1 is a sum over all the em modes but only one term of
this sum contributes to the amplitude: the one associated to the mode
H1 is a product of an operator acting on the Hilbert space of the field and
an operator acting on the Hilbert space of the material system
field
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matter
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2) Emission and absorption
electromagnetic part
absorption
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2) Emission and absorption
emission
Most notable in these formulas is the presence of factors n and n+1 derived from what
the photon is a boson and are responsible for the emission and absorption of radiation
induced.
The more photons in the initial state, the greater the process
of emission or absorption is likely.
This is the principle of the laser effect.
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2) Emission and absorption
It is also remarkable that the emission of a photon can take place even in
the absence of initial photon (n = 0), it means in the apparent absence of
external stimulation.
Spontaneous emission
It corresponds to the classical radiation of an accelerated charge.
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2) Emission and absorption
2.1.1 Angular distribution of the emitted radiation
using
and
⇒
Emission rate for a given polarization
Emitted power for a given polarization
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2) Emission and absorption
2.1.2 Summation over the polarization states
Generally, we are not interested in the photon polarization state and one
wants to know the probability of emission of a photon of either polarization.
We must therefore sum over the two polarisation states, i.
components of the polarization vector
: Transverse component of the paramagnetic current
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2) Emission and absorption
;
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;
;
13
;
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2) Emission and absorption
Circular polarization states
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2) Emission and absorption
2.2 Electric dipolar radiation (E1)
The photon wavelength being generally much larger than the atomic size
we have
optical photon
atom
Therefore
with
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2) Emission and absorption
(See notes chapter III)
By using
is the non-perturbed hamiltonian
⇒
Electric dipole operator
Electric dipole operator
⇒
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2) Emission and absorption
⇒
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2) Emission and absorption
Hypothesis: the states n and m have a magnetic quantum number = 0
⇒
(z is the quantification axis)
We have used
Total emission rate
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2) Emission and absorption
By replacing D by –ea0 one obtains
is the fine structure constante
⇒
For the hydrogen atom
Thus, the lifetime (relative to the spontaneous emission process) is
much longer than the period of one electron in its orbit.
⇒
Classically we have
with
dimensionless
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quantum world
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Oscillator
strength
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3) Light scattering
Light Scattering process: a photon disappears and another photon appears
During the process of light scattering by an electronic system (atom, molecule,
cluster, solid…, the number of photons remains constant.
It is therefore obvious that the interaction hamiltonian H1 cannot contribute
to first-order in perturbation theory and therefore a calculation at second-order
is necessary.
However, the hamiltonian H2 contributes to first-order. One must generally
take into account these two terms of Hint.
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3) Light scattering
a) Rayleigh scattering
The system cannot be excited and the scattering is purely elastic
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3) Light scattering
b) Raman scattering (inelastic scattering at low energy)
Anti-Stockes
Stockes
- The Stockes processes amplify
the radiation at frequency ω’
- The inverse Stockes processes amplify
the radiation at frequency ω
Inverse process
The system can be excited and the scattering is inelastic
: Ionization potential
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3) Light scattering
Stimulated Raman effect:
(n+1)
Pump
n
The cavity is tuned at frequency ω’ and Nn > Nm
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3) Light scattering
c) Thomson scattering = elastic scattering at high energy
d) Compton scattering = inelastic scattering at high energy
Relativistic regime and
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3) Light scattering
Raman scattering
1) H2 at first-order
Initial state:
Final state:
photons in mode
and 0 photon in mode
photons in mode
and 1 photon in mode
Electronic system: n m
⇒
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3) Light scattering
and
and
is the Fourier transform of the density operator
m
α
First-order
n
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3) Light scattering
Raman scattering
1) H1 at second-order
k: intermediate states
Only two possible intermediate states for the em field
m
α
l
α
second-order
n
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3) Light scattering
Raman scattering
1) H1 at second-order
k: intermediate states
Only two possible intermediate states for the em field
a) no photons
m
α
direct term
l
α
second-order
n
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3) Light scattering
Raman scattering
1) H1 at second-order
k: intermediate states
Only two possible intermediate states for the em field
b) two photons
m
exchange term
l
n
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3) Light scattering
Raman scattering
1) H1 at second-order
direct term
m
l
n
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3) Light scattering
Raman scattering
1) H1 at second-order
exchange term
m
l
n
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3) Light scattering
In the dipole approximation
⇒
If one adds and subtracts
⇒
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3) Light scattering
in
and also adds and subtracts
⇒
(1)
(2)
(3)
⇒
The term (3) cancels out with the first-order contribution of H2
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3) Light scattering
Finally it remains:
Using the FGR one gets:
Kramer’s formula
Fine-structure constant
<< 1
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3) Light scattering
If the system stays in the same state:
then
and
Rayleigh’s formula
The factor
is responsible for the blue color of the sky !
Unpolarized scattering cross-section
Oscillator strength
Due to the atom isotropy one has
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3) Light scattering
⇒
The average over the initial polarizations and the sum over the final ones lead
to
where θ is the angle between
By using
and
electron classical radius
one obtains
surface dimension
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3) Light scattering
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