lasercool.md

title	permalink
Lasercold	/Lasercold

Kinda part of my Theory of Everything - Some Novel Approach Including Love ❤️ – Math on Demand Edition

Super-simple overview

One photon, one two-level system

Resonance at rest

.
.                ------- E_1
.      w_p          A
.     ~~~>          |
.                   |--w_0
.                   V
.                ---x--- E_0

One of the most simple interactions between light and matter (aside from the fundamental QED Feynman graph) is that between a single photon of energy $\hbar\omega_p$ (and frequency $\omega_p$) and a two-level system with a resonance frequency $\omega_0 = (E_1-E_0)/\hbar$. If those frequencies match, the photon can get absorbed to make the system excited.

.
.                ---x--- E_1
.                   A
.                   |
.                   |--w_0 = w_p
.                   V
.                ---o--- E_0

The excitement won't last forever though and due to a process know as spontaneous emission the original photon can get emitted again in a random direction away, leaving the system again in its original, un-excited state.

.
.                ------- E_1
.                   A          w_p
.                   |         ~~~> (random direction away)
.                   |--w_0
.                   V
.                ---x--- E_0

But in reality, there are several assumptions not met exactly in the sentence "One photon interacts with one two-level system at rest at exact resonance", and we'll go through each semantic block in this:

• There is more than "one photon", leading to stimulated emission as described in Multiple photons, one two-level system
• The photon could not interact with the system at all, though that's trivial and thus boring
• There is more than "one two-level system", leading to statistical effects, especially temperature
• There are more than "two-level"s for the system, which is actually important for a laser
• The system is not exactly "at rest" (with respect to the observer's laboratory system), see relativistic Doppler shift in Resonance in motion
• The interaction doesn't happen "at exact resonance", i.e. the photon energy does not exactly match, see Rabi oscillations
• The interaction could be between something else than a photon and a system to be precise, but that is out of scope for this document

Resonance in motion

As described in the Wikipedia article on the relativistc Doppler shift (named after physicist Christian Doppler, though his description is only valid for non-relativistic velocities), if the light source and the system are in motion relative to each other, a frequency shift depending on the relative velocity occurs. Light is "blue-shifted" (or more precisely, perceived as having a higher frequency than at relative rest) when the light source and the system are moving towards each other and "red-shifted" (having lowered frequency).

The energetic symmetry of resonance at rest is no longer guaranteed due to the random direction the photon can get emitted, thus leading to a change of the system's velocity and thus energy. In a simplified way this can be seen as an initial description of laser cooling if a red-shifted photon gets absorbed, since it is more likely to get re-emitted in any other direction which is less red-shifted. Conversely, leaser heating can also be described that way by re-emission of an originally blue-shifted photon.

Multiple photons, one two-level system

.
.                ---x--- E_1               -------
.      w_p          A                                  2 identical photons
.     ~~~>          |           ------->               ~~~>
.                   |--w_0                             ~~~>
.                   V
.                ---o--- E_0               ---x---

If a photon interacts with an already excited system, it can cause an identical photon to be emitted via stimulated emission instead of the random spontaneous emission mentioned before. In this case, the system will be back to its original velocity, only at a slightly shifted trajectory due to the temporary change of velocity during the excited phase.

This effect can counter the laser cooling/heating described in Resonance in motion. This is a simple explanation why laser cooling only works at a limited speed, since the next photon should only interact once spontaneous emission has sufficiently likely occurred. Also note how the system changes velocity and thus its relative resonance frequency once a first photon has been absorbed, so off-resonance effects need to be considered as well.

While stimulated emission may sound detrimental to the idea of laser cooling at first, the effect is crucial for the actual function of a laser itself -- however, due to the random nature of spontaneous emission, a two-level system cannot lase on its own since lasing requires a majority of excited systems to resonate with. It is the very effect described in Multiple levels later on that is required for lasers to work at all as well as the foundation for the idea of cooling described in this document.

Multiple (two-level?) systems

Multiple levels

Off-resonance