Atomic spectra are defined as the spectrum of the electromagnetic radiation emitted or absorbed by an electron during transitions between different energy levels within an atom.
: Number of atoms in per unit volume.
: Number of atoms in per unit volume.
: Rate of spontaneous emission. The probability of transitioning of a single atom from to due to spontaneous emission per unit time.
: Intensity. . Intensity has a dimension of .
: Absorption cross-section, defined such that is the energy absorbed by a single atom when irradiated by intensity with frequency per unit time.
The change in number of the excited atoms (per unit volume) comes from three sources:
- Spontaneous emission: In a unit time, there is a probability of for a single atom to transition to . For atoms, the total transition number is on average.
- Stimulated absorption: According to the definition of absorption cross-section, is the energy absorbed by a single atom per unit time. Since the incident beam has a frequency , each time an atom absorb a photon , they will be excited into (although the energy gap is ) . Therefore, in a unit time, is the total number of atoms being excited into , which generates a term .
- Stimulated emission: Since stimulated absorption and stimulated emission have the equal probability, the mechanism is similar. We have a negative term .
We write down
For steady state, spontaneous emission + stimulated emission = stimulated absorption, . We obtain the steady state equation
When a laser beam with a frequency of and an intensity of is incident on the medium, we have
where is the dissipation (caused by pure absorption) of energy per unit volume per unit time. At the same time, is also the dissipation of energy per unit volume per unit time, where means stimulated emission minuses stimulated absorption, leaving the pure absorption. So
Combining and the steady state equation, we could finally obtain
We have found that for a fixed number of particles and a constant incident light intensity , the energy absorbed by the medium per unit time varies depending on the frequency of the incident light . This is reflected in the dependence of absorption cross-section on ω. Furthermore, we can extract a dimensionless line shape function from it, describing the absorption efficiency of the medium for different frequencies of light. It also means the stimulated emission power spectrum.
- : The probability of a single atom on the ground state.
- : The probability of a single atom on the excited state.
By solving the Optical Bloch Equation, we can obtain
Define a saturation parameter , the density matrix element becomes
When on-resonance and large intensity , of the atoms occupy the excited state, reaches its maximum.
We will now consider the radiation due to a single, isolated atom driven by a monochromatic field. The radiation, viewed as scattered light, is called resonance fluorescence.
The optical Wiener-Хи́нчин theorem writes
where contains . This will become
where is the spectrum of the scattered light with a dimension of , which describes the probability of the fluorescence having a frequency between and . However, is not normalized. The total probability is
This results in
Conclusion: In spite of the fact that atomic spectra are the superposition of waves at different frequencies, the total received power is exactly equivalent to an atom in the excited state emitting a photon with a probability of in a unit time, regardless of whether the external electric field is on-resonant or off-resonant .
It’s worth mentioning that the spectrum of the scattered light is named by Mollow Triplet Spectrum
where is the saturation parameter.
Therefore, by measuring the power of resonance fluorescence, which is actually measuring since , we can draw a graph
To explain the line shape, we recall the expression of :
Define a parameter where evaluates the intensity of the incident beam. The is written
The full width at half maximum (FWHM) is exactly (When we have ). Two mechanism of broadening could be found
- In the weak-field limit, , . This is called natural broadening.
- In the strong-field limit, . The FWHM of the transition is effectively larger due to the strong coupling to the field. This phenomenon is called power broadening of the transition.
Warning: Only self-consistent when on-resonance .
Optical Bloch Equation writes
Naturally . Recall that in classical model
The absorption cross-section now becomes
where . If we define a line shape function
we find . We can obtain the same result taking advantage of energy conservation:
which means, given the same incident intensity and the same population, the efficiency of energy absorption by atoms varies with the changing incident light frequency , and reaches its maximum at resonance .