Trouble understanding phonon polariton

Question

In quantum mechanics, when two states hybridize, a particle can exist in a superposition of those states. Regardless of the specific state it occupies, it remains fundamentally the same particle.

However, in the case of phonon-polaritons or exciton-polaritons, the quasiparticle is described as a hybrid between a photon and a phonon (or an exciton). This seems to imply that the particle is partly light and partly a collective excitation of matter. My confusion lies in the physical interpretation: if we measure such a polariton, it sometimes behaves like a photon (light), and other times like a material excitation (e.g., lattice vibrations). How can a single entity manifest as an electromagnetic wave in one context and as a matter-based excitation in another? How should I understand this dual nature?

-Edit: I would really appreciate a physical picture of this polariton, by physical picture, I mean an exciton can be thought of as a hydrogen-like bounded electron-hole pair, and magnon can be a spin wave. But what would phonon-polariton look like?

Answer 1

The situation is almost completely analogous to the dual-slit experiment at the heart of quantum mechanics (or the Stern-Gerlach one, for that matter).

In many cases, 'quantumness' enters by having different parts of the quantum evolution couple to eigenstates in a different basis.

1. Double slit experiment: $k$-eigenstate is a wave, until $x$-measurement makes it collapse at one of the slits
2. Stern-Gerlach like experiments: spin along one axis is singled out and spins that can be initially in say the superposition $|+>=|1/2>+|-1/2>$ are forced to collapse in either $\pm|1/2>$.
3. Similarly, the polariton is a superposition of a photon and a matter excitation, and this superposition is an eigenstate of the evolution operator. If you perform a measurement, you force it to collapse in either photon or matter excitation (the basis that couples to the measurement apparatus).

Answer 2

I struggled with this concept for a while myself. What helped me was getting a better understanding of Hilbert spaces and abstract algebra generally.

The Hilbert space of a system is usually defined by the span of the eigenvectors of the Hamiltonian that acts on the system. For example, a spin 1/2 system being acted on by the Hamiltonian $\mathscr{H} = -\hbar S_z$ will have a Hilbert space that is spanned by the states $|0\rangle$ and $|1\rangle$. Equivalently, it can be spanned by any 2 orthonormal linear combinations of these vectors.

Often, for a system with Hilbert space $V$ and a system with Hilber space $W$, the combined Hilbert space is the space $V\otimes W$, which is spanned by linear combinations of vectors $|v\rangle \otimes |w\rangle$. This holds for as long as $V$ and $W$ are actually subspaces of the total Hilbert space, and this happens when the Hamiltonian for the total system can be written as a sum of Hamiltonians that act on different spaces i.e. $\mathscr{H} = \mathscr{H}^{(W)}+\mathscr{H}^{(V)}$. However, if there is any coupling between the two systems, or a term that acts on both spaces $\mathscr{U}^{VW}$, then they can no longer be treated as distinct spaces when acted on by the Hamiltonian. This is the mathematical structure underlying the phenomena of "strong coupling", which is necessary for any polaritonic phenomena to occur. Effectively, you must treat the whole space instead of the separate subspaces, and any energy eigenstates of the whole space will be those of new stationary states. These stationary states are "quasiparticles".

In the case of phonon polaritons, there is a strong coupling introduced by the fact that there are overlaps in the energy spectra of optical-frequency phonons (normal modes of lattice vibration) and photons. This introduces a degeneracy, specifically at the 0 momentum point of the optical phonon dispersion. Effectively, the photon will have precisely the right energy and momentum to excite a lattice vibration, and the energy coupling term will create a new localized quasiparticle.

To answer your question regarding what this "looks" like, it would be a transverse localized electromagnetic wave moving through a crystalline lattice, disturbing atoms from their equilibrium positions as it moves. So, it will be an electromagnetic wave that can be seen almost mechanically. I hope this is both helpful and correct. Please let me know if I have said anything in error.

Answer 3

In quantum mechanics, when two states hybridize, a particle can exist in a superposition of those states. Regardless of the specific state it occupies, it remains fundamentally the same particle.

However, in the case of phonon-polaritons or exciton-polaritons, the quasiparticle is described as a hybrid between a photon and a phonon (or an exciton).

A system in QM is described by a single wave function (or density matrix if appropriate), regardless of whether it consists of a single particle or a collection of particles. Sure, if the particles are not interacting, the wave function might decompose into a product of wave particles corresponding to different particles (or a linear combination of such products, like a Slater determinant). However, generally this is not the case, and the eigenstates and the overall state of the system can be viewed as are a superposition of the states of a non-interacting components. In this sense, teh differentce between hybridization for a single particle and for a combination of particles of different nature is just how we call the unhybridized components: whether we name them "eigenstates of particle X" or "particles X, Y, Z..."

Another thing to keep in mind is that excitons and phonons themselves are also hybrids:

• phonons are collective vibrations of atoms in a lattice, i.e., superposition of vibrations of many atoms. Moreover, the forces between atoms are a result of bonds, mediated by electrons - i.e., a composite state in itself.
• electrons in a lattice are not real particles either, but excitations in a crystal consisting of many particles. Indeed, their spectrum (energy bands) is rather different from free electrons. And of course, holes do not even exist without crystal - they are hybridized states as well.
• Likewise, exciton is not an atomic-like complex of a positively and negatively charged particles (although hydrogen model works surprizingly well), but a result of collective motion (hybrid) of many electrons and holes.

Finally, one could try to distinguish between fundamental/elementary particles - i.e., those that are treated as such in QFT, and quasiparticles, which are formed in solid state - conduction electrons, holes, excitons, photons, plasmons, polarons, polaritons, magnons, etc. However, even in QFT the picture is not without ambiguities, since particles transform into each other. E.g., if a proton can decay into a pion and positron, does it mean that the latter two are more fundamental/elementary than the proton itself? Or do the particles just transform into each other, without any of them being more fundamental than the other? - The latter view is a good way of thinking about particles in solid state as well.

Related:
Electrons and holes vs. Electrons and positrons
Is differentiating particle and quasiparticle meaningless?
If a non-interacting particle behaves like an undamped wave, can an interacting particle behave like a damped wave? (this question itself may sound odd, but the answer attempts to provide a self-contained discussion of quasiparticles as poles of a Green's function.)