Band theory of Photonic crystals I have a basic question about the band structure of photonic crystals.
If I have a periodic potential, then Bloch-theory tells me that the bands yield the energy spectrum of the Hamilonian which is described in terms of some quasimomentum $k$ as
$$H(k)\varphi(k)=E(k) \varphi(k),$$ where $E(k)$ are the bands and $\varphi(k)$ the Bloch functions. The Bloch function live here on some fundamental domain of the periodic lattice.
Equivalently one can use the Bloch transform to find (an infinite number of) Wannier states $(\psi_n)$ associated with $\varphi(k).$

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*Question 1: I wonder, can we think of a photon as actually being in a Wannier state or what is the interpretation of the Wannier states?  For electrons I feel this would make more sense, as they can be bound to one lattice site but for photons this seems less plausible to me.


*Question 2: When we say that a band has a certain Chern number, we usually refer to a completely filled band? So in order to fill a band completely in a photonic crystal, wouldn't I need an infinite number of photons? For example one photon in each Wannier state?
Please let me know if you have any questions.
 A: This is a good question.
While there are analogies to electronic crystals, and much of the underlying concepts are the same there are significant differences. Sometimes it might help to look at how photonics crystals are computed numerically to illustrate some of the issues, or look at a special case such as one dimensional case. One big difference is that Photons are polarized and you can have a band structure that looks pretty different if the TM or TE modes are computed.
This arXive paper walks through the one dimensional case of a photonic crystal in detail. It also points out how the Berry phase changes with the band, and how the Wannier function are located (phase change) for different bands. It also points out that spatial the Wannier functions decay exponentially around the defect.
This paper specifically looks at using Wannier functions for defects in photonics crystals and how they can be computed more accurately when using numerical methods. It for example shows

This shows the spatial localization of the photons in the materials. So for an electron in a crystal depending on the details of defect state we wouldn't usually think about it this way. We would think of it more as being a wave trapped in a potential well and would usually be in the lowest available energy state (unless maybe certain types of trapping where it might be metastable for a while). For the photons, I could populate the different modes by changing the frequency, or excite a lot of modes at the same time with a broad spectrum. You can also see the localization can have spatial extent especially for higher order modes.
So to directly answer your first question the Wannier functions are just a convenient basis set that can be used to 'solve' a photonic crystal. If there is a defect, you can use those functions to describe how the electromagnetic modes are spatially localized. How 'localized' depends on the band, or which mode or set of modes you are looking at. You can add as many photons to the mode as you want. If you have sufficient frequency resolution (narrow linewidth laser) you could try to selectively populate the modes.  So in a sense you don't fill the bands like you do with electrons and Fermi-Dirac Statistics, but you can populate the modes.
Taking a look at the Wannier functions, you can see the phase at the center of zone changes. In the one dimensional case  To compute the Chern Number from this paper it appear that you need to understand the Berry Phase and integrate over the Brillouin Zone.

So if I am interpreting this correctly, you don't need to know how 'filled' the band as you state in the electron case.
A: I am not an expert in a photonic crystal, I have learned about it in a couple of lectures and colloquiums. Though, I can comment on the questions posted. Feel free to critique me.

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*By seeing the periodic nature of dielectrics in PC (photonic crystal) analogous to an atomic crystal, it's quite tempting to apply the Bloch theorem in this case too. However, one must be careful and question that is the photon follows the similar Schrodinger's equation analogous to an electron in an ordinary crystal. I did some digging and found this paper Alrefai et al. 2016. It turns out a photon follows a non-linear equation (derived from Maxwell's equation) in a PC. So, it is not as trivial as to just take a complete analogous of Bloch's theorem. The Wannier states show the electron wavefunction "localization" (I have to be very careful to mention the word localization, as many of my colleagues disagree that it is an actual localization) on ions/orbitals of the crystal. There could be a such phenomenon in PC also, but it is now out of my area of expertise.

*Unlike electrons, photons follow Bose-Einstein statistics. This will exclude photons from following the Pauli-exclusion principle. Two electrons can fill one momentum state in an ordinary band and therefore requires a finite number to fill the valence bands, however, that is not the case in photons (it can fill an infinite number in a state). So, I cannot understand what is the meaning of the filled band here. In a sense, yes it would require an infinite number of photons, but it doesn't make sense to ask how to fill a cup with water that has infinite volume.

Hope this helps. There are many such subtle points that prevent a PC from being directly analogous to a crystal.
