# Can we define potentials for non-conservative force?

I have just started out on Quantum Cavity Optomechanics from this EdX course , and have learnt that radiation pressure force is non-conservative. But, while dealing with the optical spring effect we define the induced spring constant by the radiation force as $$k_{\text{field}}=-\dfrac{dF}{dx}$$

or, says the instructor, looking at this equation from the potential point of view $$F_{\text{radiation}}=-\dfrac{dV_{\text{optical}}}{dx}$$

I don't get it. If it is possible to define this function, why can we say that the force is non-conservative?

• I don't know enough to give a definitive answer, but could it be that for the special case of restricting to 1 dimension, and for a certain region in the experimental setup, the radiation pressure can BE MODELLED by a potential? – Quantumwhisp Mar 25 at 11:22
• Yes, that was my first guess. But the story does not end here. In optomechanical systems, the work done in a closed cycle by the radiation pressure force can be positive or negative. – LoneAcademic Mar 25 at 11:39

I got the answer to this. This is rather confusing at first, but is pretty simple. For one dimensional optomechanical systems, the radiation pressure force can be written as- $$\textbf F(x)=f(x) \hat i$$ which obviously has a zero curl, and hence is conservative for those cases. But the work done by such forces in a cycle can still be non zero, because of the finite cavity decay rate $$\kappa$$.
If we are moving infinitely slowly, the number of photons in the cavity get enough time to follow the functional form $$f(x)$$ given above and be conservative, but for fast processes, photons take a finite amount of time to catch up with the functional form above, and hence based on how that catching up is related to the system, can result in positive or negative work. And this is the basic principle of optomechanical cooling and heating. $$\oint F\ dx < 0$$ implies cooling and $$\oint F\ dx>0$$ implies heating. 