# Can we change the frequency of a photon in a resonator by changing the refractive index?

First consider the classical case, the simplest optical resonant cavity is a rectangular cavity whose surface is an ideal conductor. The field inside the cavity can be described by Maxwell Equations.

The field is: $$E_x=A_1\cos\left(\frac{m\pi x}{L_1}\right)\sin\left(\frac{n\pi x}{L_2}\right)\sin\left(\frac{p\pi x}{L_3}\right)\cos\omega t$$ $$E_y=A_2\sin\left(\frac{m\pi x}{L_1}\right)\cos\left(\frac{n\pi x}{L_2}\right)\sin\left(\frac{p\pi x}{L_3}\right)\cos\omega t$$ $$E_z=A_3\sin\left(\frac{m\pi x}{L_1}\right)\sin\left(\frac{n\pi x}{L_2}\right)\cos\left(\frac{p\pi x}{L_3}\right)\cos\omega t$$ $$A_1\cdot\frac{m}{L_1}+A_2\cdot\frac{n}{L_2}+A_3\cdot\frac{p}{L_3}=0$$ $$B_x,B_y,B_z\propto\sin\omega t,\ \ \omega=\frac{\pi}{\sqrt{\epsilon\mu}}\sqrt{\left(\frac{m}{L_1}\right)^2+\left(\frac{n}{L_2}\right)^2+\left(\frac{p}{L_3}\right)^2}$$ When $$t=0$$, if we change the refractive index (if we change $$\sqrt{\epsilon\mu}$$), we can easily notice that the field is still an eigenstate of the resonator, we only change its oscillation frequency.

So the question is, can it be put forward into quantum regime? In cavity QED, if the state is a 1-photon state in the cavity, if we change the refractive index inside the cavity, can we change the frequency of the photon?

• A photon doesn't have a frequency. A photon has energy, momentum and angular momentum. A cavity does not even preserve momentum as the field exchanges momentum with the walls. I believe that's also true for angular momentum, probably depending on symmetry. As soon as you put matter inside a cavity you don't even have photons. Now there is a quasi-particle excitation that characterizes the em-field and the matter together. If you want to avoid that, then you have to model individual atoms, but then the notion of a refractive index is lost. Apr 13 at 12:50

The exact answer depends on the actual implementation of this refractive index change - i.e., on the Hamiltonian that describes it.

However, it is worth making some remarks:

• what is changed here is not "photon frequency" by the frequency of photon eigenmodes (equivalent of eigenstates when talking about electrons)
• the perturbation is time dependent. Thus, if we have particular eigenmode excited (i.e., having non-zero number of photons in quantum case), different things may happen:
• if the refractive index changes smoothly/adiabatically, the system will remain in the same eigenstate, i.e., the "photon frequency" would change (see Adiabatic theorem)
• if the perturbation is abrupt (like a sudden jump at $$t=0$$ from one refractive index to another) we will likely have many modes excited
• The only thing that can change the frequency of an electromagnetic wave is a time dependent system. Time-invariant systems with different refractive indices, including systems with jumps, are what we use in most technical optical systems (aka lenses, mirrors, thin films etc.). None of them change photon energy or the frequencies of the spectral components of the light that goes through them. They do change wave vectors/momentum vectors, of course. Apr 13 at 13:18
• @FlatterMann in a more precise form this is stated under the first bullet in my answer. Apr 13 at 13:20
• Maybe I misunderstood your comment. Are you talking about a time dependent change in refractive index? That's hard to come by (it can, of course be done, e.g. by increasing concentrations of a substance in a solvent, chemical reactions etc., but that's a rather unusual system). The phrase "smoothly changing refractive index" is usually meant to mean "spatial change", not temporal change. If that is not what you meant, then your comment is clear. Apologies. Apr 13 at 13:25
• @FlatterMann I think the question clearly speaks about temporal change. How it is done exactly is unclear - as I point in the first sentences of the answer. This is akin to changing suddenly the mass of electron - difficult but not impossible, e.g., if we are talking about effective mass in a crystal. So the problem is not without merit. Apr 13 at 13:29
• In that case I simply misunderstood. I would say the most simple case of a "temporal modulation photon energy changer" is a moving mirror as used in doppler radar. Most of our cars have that built in these days. Apr 13 at 13:31