Can a laser-pumped laser cavity be laser-cooled by its pumping laser (and pump its pumping laser in return)? Is remote-cooling possible outside a lab? tl;dr: Can one laser "suck" heat from another one (e.g. via Doppler-cooling, but not limited to that) hence remote-cooling it?
I haven't found any non-paywalled literature on the thermodynamics of lasers and it's been a while since I've been to university, but this idea follows me around, so please (gently) point out any misconceptions and let's see if this idea is salvageable implementable in reality:
Preliminaries
Relativistic Doppler effect
If source and receiver of photons move relative to each other with a velocity $\vec v$, the sent frequency $\nu_s$ is rescaled by some factor $D(\vec v)$ for the receiver, i.e.
$\newcommand\labtag[1]{\tag{#1}\label{#1}}$
\begin{align}
  \nu_r = D(\vec v)\nu _s \labtag{Doppler}
\end{align}
where $D$ depends on the relative direction of motion, e.g. for a purely longitudinal approach it's just $D = \sqrt{\frac{1+\beta}{1-\beta}}$ with $\beta = v/c$, while most generally for a source moving at angle $\theta_s$ (from the sender's frame) it is $D = \gamma(1-\beta\cos\theta_s)$ with $\gamma^{-1} = \sqrt{1-\beta^2}$.
Maxwell-Boltzmann / Fermi-Dirac / Bose-Einstein distributions
Details to be added if necessary...
Doppler cooling
Most typically, "laser cooling" refers to Doppler cooling in popular science literature. It basically boils down to emitting photons of an energy slightly below one needed to actually excite a state, but thanks to the temperature-dependent velocity distribution there'll be particles moving towards the source, which can absorb the blue-shifted photons of now fitting energy, thereby reducing its momentum by $\hbar k$. Thanks to spontaneous emission, the excited state relaxes again, but emitting the photon in a random direction, thus statistically not re-increasing the momentum and effectively cooling the medium.
From my understanding, some major reasons why this only works for systems with few particles and low cooling beam intensity are:

*

*with more particles, it becomes more likely that the re-emitted photon is absorbed again by another particle, but now statistically more likely to heat instead of cool

*with more particles, the likelihood of transforming the excitation energy into more heat via e.g. phonons increases

*higher intensity means more photons means it is more likely that the excited state is relaxed via stimulated emission instead, which due to the beam-identical nature of the photon does not contribute to cooling any more

The last point actually led me to my idea:
The idea
In order to actually cool the target medium, the excited state has to be relaxed in a way such that the energy leaves the system in a controlled manner. And what better manner is there than the very resonance a laser provides? If tuned properly, it should emit laser light at a photon energy higher than the one used to cool&pump it, which in turn might even be used to (partially) pump the cooling-laser.
The Back-of-the-envelope "calculation"
For the sake of simplicity, let's assume a sufficiently high temperature that the Maxwell-Boltzmann distribution applies, where the average (radial) velocity is
\begin{align}
  \langle v\rangle = \sqrt{\frac{8 k_BT}{\pi m}}\propto\sqrt T.  \labtag{Boltzmann}
\end{align}
For the average Doppler shift consider all source-frame angles $\theta_s$ equally likely (not entirely sure if that's appropriate; the formula looks less simple for receiver-frame angles - but it doesn't matter too much since the linear relationship between the frequencies remains) to obtain
\begin{align}
  \langle D \rangle &= \frac1{4\pi}\iint D(\theta_s)\, d\phi d\theta_r = \gamma \labtag{Doppler-Average}
\end{align}
which happens to equal the transverse Doppler blueshift at the geometric closest approach. Combining $\eqref{Boltzmann}$ and $\eqref{Doppler-Average}$ into $\eqref{Doppler}$ we obtain an average Doppler shift of
\begin{align}
  \nu_r &= \overbrace{\frac1{\sqrt{1-\frac{8k_b}{\pi m c^2}T}}}^{=:D(T)}\cdot\nu_s
\end{align}
Now let's consider two four-level lasers, the cooling one with index $(C)$ and the "hot" one (to be laser-cooled) with index $(H)$. The cooling laser (probably also cooled to a temperate $T_C$ for stable output, but by conventional means) is pumped with photons of frequency $\nu_{Cp}$ and emits photons of frequency $\nu_{Ce}=:\eta_C\nu_{Cp}$ with $\eta_C<1$ (approximating the efficiency). After the Doppler shift that energy should be equal to the hot laser's pumping frequency $\nu_{Hp}\stackrel!= D(T_H)\nu_{Ce}$ in order to emit (on average) the rest-frame frequency $\nu_{He} = \eta_H\nu_{Hp}$ (also $\eta_H<1$), to which the "hot" cavity should thus be tuned. In the additional case of trying to pump the cooling laser with those very photons, which are of course also Doppler shifted, we obtain the requirement
\begin{align}
  \nu_{Cp} &\stackrel!= D(T_C)\nu_{He} = D(T_C)\eta_H D(T_H)\eta_C \nu_{Cp}
  \\\Rightarrow 1 &\stackrel!= D(T_C)D(T_H)\eta_H\eta_C.  \labtag{Roundtrip}
\end{align}
Since $\eta<1$ but $D(T)\ge1$, that does sound feasible so far. In fact, it even seems possible to use the same kind of medium for both lasers, whereas my gut feeling was requiring two different material specifically tuned to one another, but maybe I've missed something here anyway. Note I haven't considered the losses $1-\eta$ being converted into heat in the worst case, and a proper calculation would have to consider the temperature dependent feasibility of inversion...

So. Is this a truly feasible concept that can be built in reality with real lasers? Aside from potential misconceptions in general I've clearly neglected a proper statistical analysis and thermodynamics in general and would appreciate any input on this. But obviously I don't want an open-ended discussion here; the question is basically just what the title states.
 A: There's a lot of detail in your post. I don't think I'll address that directly, as it is a little confusing exactly what you are proposing. For example, you ask about one laser cooling another which is a funny concept that I'm not sure makes sense. When I think of a laser I think of the the photonic beam emitted by a laser device. Perhaps more precisely you're interested in using one laser to thermally cool the gain medium of another laser?
In any case, it does seem that you are interested in the transfer of energy between a laser and some target medium, and a cavity should be included. The field of physics which studies exactly this kind of thing is the field of cavity optomechanics, which might be considered a sub-field of cavity quantum electrodynamics (cavity QED).
One canonical reference on cavity optomechanics is Aspelmeyer et. al. "Cavity Optomechanics" (2013).
Cavity optomechanics studies the interplay of the photonic field inside of a cavity, represented by photonic amplitude operator $\hat{a}$, and a motional degree of freedom of a quantum harmonic oscillator physically inside of the cavity, represented by position operator $\hat{x}$. The interaction between the cavity and the oscillator is mediated by some type of radiation pressure force which leads to a coupling of the form $g\hat{a}\hat{x}$.
If the cavity is driven to the red/blue of cavity resonance then it is observed that this leads to an extraction/insertion of energy into the harmonic oscillator. So, by detuning the driving laser to the red you can effectively cool the harmonic oscillator. See the chapter on optomechanical cooling in the linked reference.
There are a number of interpretations of this effect. You'll often see comparisons to optical sideband cooling which is a type of laser cooling. But, that said, I believe it is possible to interpret optical sideband as a type of doppler cooling in which either the transition linewidth or the cavity linewidth is narrower than the spacing between vibrational energy levels (i.e. the harmonic oscillator energy splitting).
All of this to say, I think the concept that you are describing is quite similar to cavity optomechanical cooling and I encourage you to look into that field and associated references!
