I Polariton, magnon, or something similar

[Demystifier]

In a polarizable medium, one introduces the field
$${\bf D}={\bf E}+{\bf P}$$
where ${\bf E}$ is the electric field and ${\bf P}$ is the polarization. (Units are chosen such that there are no additional factors.) When this field is quantized (second quantization), one gets quanta of ${\bf D}$ which are similar to photons. Are these quanta of ${\bf D}$ called polaritons? If not, then what is the name of those ${\bf D}$-quanta, and where can I find more about them?

Similarly, for magnetic fields we have
$${\bf H}={\bf B}-{\bf M}$$
where ${\bf M}$ is magnetization, etc. Are the quanta of ${\bf H}$ called magnons? If not, then what is the name of those ${\bf H}$-quanta, and where can I find more about them?

 

[TeethWhitener]

Here's Hopfield's original paper coining the term "polariton:"
http://journals.aps.org/pr/pdf/10.1103/PhysRev.112.1555
He quantizes the $\textbf{D}$ field, which yields 2 different creation operators: one for the photon and one for the polariton (Equations 6-8). He goes on to show that many different types of quasiparticles can be classified as polaritons, including excitons and optical phonons.

I don't know too much about magnons, but it seems reasonable to think they're analogous.

 

[Demystifier]

Thanks, that was very useful!

 

[DrDu]

The similarity between electric and magnetic fields is only superficial. In fact, at higher frequencies, like in optics, all effects of the medium can be described in terms of E, D, and B with H=B. The magnetic effects are encoded in a nonlocal dependence of P on E. This is called spatial dispersion and is the usual convention chosen in optics. At lower frequencies, namely radiowaves, it is more common to assume a local dependence of P on E and a local relation between M and B. This is also related to the response of spin degrees of freedom becoming unimportant at optical frequencies.
Magnons are typically the excitations involving spin degrees of freedom.
This is discussed quite in detail in Landau Lifshetz, Electrodynamics of continua.

 

[Demystifier]

In fact, at higher frequencies, like in optics, all effects of the medium can be described in terms of E, D, and B with H=B.

Why would one be only interested in higher frequencies?

 

[DrDu]

Why would one be only interested in higher frequencies?

Did I say so?

 

[DrDu]

What I mean is the following: Both magnons and polaritons are effects that result from the coupling of the electromagnetic field to the mediums current density j.
In the case of magnons, you can express j either in terms of P or M: $j=-\partial P/\partial t$ or $j=\mathrm{rot} M$. So whether they are D-quanta or H-quanta is at best a matter of convention and not a physical distinction.

 

[Demystifier]

So whether they are D-quanta or H-quanta is at best a matter of convention and not a physical distinction.

Is it like saying that whether photon is an E-quanta or B-quanta is a matter of convention?

 

[TeethWhitener]

Is it like saying that whether photon is an E-quanta or B-quanta is a matter of convention?

I bet you could write a Faraday tensor in matter as something like $D^{\mu \nu} = F^{\mu \nu} + a P^{\mu \nu}$ where $F$ is the standard Faraday tensor and $P$ is a polarization/magnetization tensor and just quantize the whole thing. Then polaritons would just be $P$-quanta and everything else (magnons, spinons, phonons, excitons, etc.) would just be a specific instantiation of a polariton.

 

[DrDu]

It is rather a choice of gauge. This is most clear in a 4-vector formulation. The charge density charge current vector $j_\mu$ is related to the polarisation-magnetization vector $\Pi_{\mu \nu}$ via $j_\nu=\partial_\mu \Pi_{\mu \nu}$. This equation actually defines the tensor $\Pi$. But the definition is not unique, as we may add any solution of $\partial_\mu \Pi^{(0)}_{\mu \nu}=0$. A possible choice is $\Pi_{0i}=P_i=-\Pi_{i0}$ and the other components of $\Pi=0$. Then the material currents are described exclusively in terms of polarisation.

 

[Demystifier]

It is rather a choice of gauge. This is most clear in a 4-vector formulation. The charge density charge current vector $j_\mu$ is related to the polarisation-magnetization vector $\Pi_{\mu \nu}$ via $j_\nu=\partial_\mu \Pi_{\mu \nu}$. This equation actually defines the tensor $\Pi$. But the definition is not unique, as we may add any solution of $\partial_\mu \Pi^{(0)}_{\mu \nu}=0$. A possible choice is $\Pi_{0i}=P_i=-\Pi_{i0}$ and the other components of $\Pi=0$. Then the material currents are described exclusively in terms of polarisation.

But $P^{\mu\nu}$ in post #9 is gauge invariant. How is your $\Pi_{\mu \nu}$ related to $P^{\mu\nu}$?

 

[DrDu]

The two are the same. I meant "choice of gauge" in a more general way: While F can be expressed as a derivative of the vector potential A, also j can be expressed as a derivative of the "potential" P. Neither A nor P are completely fixed by specification of F and j, respectively, but there still is some freedom of "gauge".

 

[Demystifier]

The two are the same. I meant "choice of gauge" in a more general way: While F can be expressed as a derivative of the vector potential A, also j can be expressed as a derivative of the "potential" P. Neither A nor P are completely fixed by specification of F and j, respectively, but there still is some freedom of "gauge".

If so, then H and D are also not completely fixed by specification of F and j. Is that what you are saying?

 

[DrDu]

Yes, exactly.