Three-level system interacting with classical noise sources Please can someone help me finding a kind of physical implementation (possible experimental scenario) where a qutrit system interacting with a classical fluctuating noise can be described with the following Hamiltonian :$\mathcal{H}(t) =\varepsilon \mathbb{I}_3+ gC(t)S_{x}$. 
Wherein $\varepsilon$ is the qutrit energy in the absence of noise; $\mathbb{I}_3$ is the 3*3 identity matrix ; $g$ is the coupling strength of the classical noise $C(t)$ with the qutrit system. $S_x$ is the Pauli matrix for spin-1 systems. 
In fact, a similar model has been used for qubits systems ($S_x=\sigma_x$ and $\mathbb{I}_3= \mathbb{I}_2 $) in describing photons propagation and electrons transport in disordered structures. As such, I will like to know to which extent, the model can be suitable for qutrit systems. 
Thanks in advance. 
 A: Right off the bat: the term in $\varepsilon \mathbb I$ does absolutely nothing to the dynamics and it can pretty much just be dropped at the blink of an eye as soon as it's convenient to do so. (More precisely, it only contributes a global phase to the dynamics, and global phases don't matter. If you insist on getting technical, the model maps via a gauge transformation to one with $\varepsilon=0$.)
Once you do that, you get a single term in your hamiltonian,
$$
H = gC(t) S_x,
$$
with no non-commuting operators, so the operator is essentially trivial, and the dynamics can be solved exactly in terms of $C(t)$ in the $S_x$ basis. Moreover, since $H$ doesn't act on the $|0\rangle$ eigenvector of $S_x$, you can drop it out of the dynamics, and you just get a copy if the spin-1/2 system with a bigger $g$.
(Now, if you had a non-commuting self-term, say, $H = \omega_0 S_z+gC(t) S_x,$ then this would be an interesting model. As it stands, I can't really see any real dynamics at all in either the spin-1 or the spin-1/2 versions you've written down.)
As for a physical implementation - sure, just take any spin-1 atom and subject it to a (noisy?) magnetic field. I can't see how you'll get anything interesting out of it, though.
A: As @EmilioPisanty has correctly pointed in their answer, $\epsilon$ can be removed by shifting the energy origin.
The problem with this model is actually not the number of levels, but the ad-hoc nature of noise, which breaks the fluctuation-dissipation theorem. This is why in more complex settings one usually uses a heat bath (e.g., composed of harmonic oscillators) - there exist many mathematical formalisms for doing so, but it is worth perhaps to start with checking Caldeira & Legget approach and look up Bloch equation derivations for two-level system.
If we decide to stick with the model as written in the OP, it is necessary to note that while any two-level system can be written in terms of spin operators and $gC(t)S_x$ is the most general coupling possible (up to a rotation in spin space and neglecting time-dependence of the level energies), this is not the case for a three-level system.
