Welcome to Gaia! ::

So something has been bugging me. It involves solid state, so I'm hoping Moberticus in particular might have something to say about the following.

We've all heard about the correspondance between classical fields and quantum particles. Of particular interest here would be electromagnetic wave and acoustical wave fields. We can look at the wave equations these objects satisfy and analyze the solutions in Fourier space. We first find we can write the solutions as 3-d Fourier transforms (we still want the time-like coordinate to stay intact), find the solution of these Fourier components by demanding they satisfy the differential equation, then the reality of the fields imposes additional constraints on their Fourier components that allow us to write them as something along the lines of:

a(p)e^ipt+a*(-p)e^-ipt

As the story goes, when we quantize the field and assume the cannonical commutation relationships between the field amplitudes and their canonical momentum (observing, of course, the imposed classical contraints on the fields - lest we lose our canonical relationships). These relationships force a and a* to satisfy the creation and annihilation commutation relationships and an additional analysis of the combined action of a and a* with the Hamiltonian, H, reveals that a is in fact the annihilation operator and a* is the creation operator. The rest is history.

Now all of this was done assuming we've dealing with the linear wave equations. Only in the linear case are we guaranteed that plane wave solutions (the things that get quantized into states that are isomorphic to particle states) are stable solutions to the differential equations. If we toss in a nonlinear term, we're pretty much guaranteed harmonic generation. In this case, we stick in one plane wave and it generates harmonics (wave modes with frequencies that are integer multiples of the original wave).

Now let's try to quantize acoustic waves in highly nonlinear materials and electromagnetic waves governed by Maxwell's nonlinear field equations (arise naturally -> relativistic equations involve the metric, Einstein's field equations say the metric dependson the square of the four-potential, meaning Maxwell's equations depend on the square of the four-potential). We find single phonon/photon states aren't physically realizable - putting in one phonon/photon forces a redistribution of the energy into the other modes.

As I understand the solution to this quandary goes, we still say the field consists of a bunch of phonons/photons. Harmonic generation occurs when two phonons/photons interact, annihilate each other, and produce other phonons/photons with higher and lower frequencies, known as upconversion and downconversion respectively.

My problem with this is: why? It might work mathematically and thinking in terms of this can probably help make life a little nicer, but is this seriously what we think is actually happening physically? It's easier to analyze the motion of the endpoint of a diving board by Fourier decomposing it, in which case the motion looks like a point mass at the end of a superposition of a bunch of different springs. Sure the math works out, and sometimes this might actually help us analyze a problem (hell, I'm 2nd author on a paper where we did just this to obtain an analytical solution of cantilever interaction dynamics in atomic force microscopy methodologies), but the two situations aren't physically the same. So since we will never be able to find the wave fields in a single phonon/photon state, does it really make sense to say it's still comprised of phonons/photons?

Please, discuss or help me sort out my problems with this paradigm.

I'm not having a problem with phonons in general. When the material is linear, the operators for the displacement field admit single linear mode solutions allowing the field state to be in a single phonon state - this is all fine and good.

My problem comes in when we look at nonlinear materials. The field operators can't be written in terms of a single linear mode and no physically realizable field can be in a single phonon state from what I understand (I'm actually have my pen in hand quantizing a simple nonlinear massive scalar field right now just to check). We can have states corresponding to single nonlinear mode excitations where the field operators take the form A=Sum[C(i,B)*Cos(iwt)] (with some sort of spatial dependence of course) where the sum is over i=0 to infinity and B is the nonlinearity parameter for the material. In the case where B->0, C becomes a Kronecker delta and we can write the field amplitude in terms of a single linear mode, meaning we can excite single linear modes when we quantize the system using the tried and true creation operators.

Specifically, my problem arises when we try to interpret nonlinear mode states as a collection of phonon states. Mathematically it works out, but since nonlinear materials can't be in a single phonon state, can we really say the field excitations are truly collections of phonons?

I might be asking that awkwardly or not making the fishy-ness of all this apparent, in which case I apologize and will try again later. For now, I must be off though.

Glad I could help XP

But yeah "behaves as if" is a good disclaimer for these kind of descriptions. And dealing with phonons shouldn't make a situation any more ontologically exotic.