Device can theoretically trap a light 'bit' for an infinite amount of time

Imagine the nanostructure as a bucket and light as water. The goal is to fill the bucket with water and store it there. Light "fills" a material by polarizing the electrons inside the material. However that polarization can and must re-emit light. If there's a 1:1 ratio between the amount of polarization you get in the material for field in the light the reverse process is equally likely, then it turns out the material polarization will always just make light again in the same amount as you add. Ie: there is a hole in the bucket and you cannot fill it.

Some materials polarize non-linearly. Ie: you add 1 drop of water into the bucket and it becomes two drops inside or the reverse. Actually all materials do this but much less than ordinary (linear) polarization. The author's contrived some setup which would exploit this effect to keep the polarization in a nanostructure. Something like (and this is an abuse of analogy): one drop in bucket, it splits, one of those is emmitted and one kept etc. The operating principle of such a device is to control the paths the drops travel on, and separate the directions to enhance the desired processes. Something like a mouse trap.

There are many devices which already exist and maintain states of light given some energy input, for example LASER cavities. Non-linear phenomena are well understood and exploited in all sorts of fascinating optical devices. Non-linear processes are usually weak and rare, so seeing them requires intense light which is expensive to generate and can burn weak materials. Preparing nanostructures which have the desired nonlinearities could be hard or impossible.

For example it's easy to theoretically imagine a crystal that would take the many of infrared photons emitted by the heat of your hand, and upconvert them into fewer visible photons you could see. However any material which could be polarized so much by those low energy photons would not be stable at room temperature so no such thing exists...

Theoretically, the gain element is not necessary. They only need it to compensate for physical imperfections in the device.