# Evanescent waves and photon tunneling

Context

I am reading about near-field heat transfer. Generally this phenomena is describes using (classical) Maxwell equations. In vacuum heat transfer from a body A to a body B such that $T_A > T_B$ happens through radiation. If the distance between these bodies is smaller than the wavelength of the blackbody radiation of A (more specifically smaller than the wavelength given by Wien's displacement law) heat transfer is enhanced by several orders of magnitudes.

Electromagnetic waves have to cross two interfaces before being absorbed in body B. Namely the A-Vacuum (A-V) interface and the Vacuum-B (V-B) interface. For some materials you can get total reflection at the A-V interface. In this situation an evanescent wave. The latter travels parallel to the A-V interface and decay exponentially perpendicularly with respect the A-V interface. The Poynting vector is zero in the perpendicular direction. If now the V-B interface is close enough (and parallel to A-V) to the evanescent wave, an EM wave can be observed in B with non-zero energy density. The energy passed through the vacuum gap even though the Poynting vector is zero (perpendicularly to A-V).

Question

The mathematics is extremely similar to the tunnelling of, for instance, electrons in quantum mechanics. In some books/paper I saw people referring to this phenomenon as photon tunnelling. Is this energy transfer completely described with classical equations photon tunnelling? Is then tunnelling related to wave mechanics and NOT strictly to quantum mechanics?

I think you are probably getting a bit too worried about words and their meaning and are possibly trying to ascribe more precision to natural English words than they can give you without further precise description in mathematical language.

As you witness, the mathematics describing photon tunnelling and evanescent waves is exactly the same as that describing electron tunnelling into classically forbidden regions. Maxwell's equations are both classical equations and can be interpreted as the propagation equation for a one-photon state. So both phenomena - photon and electron tunnelling - are equally quantum mechanics and the mechanics of waves. The two aren't mutually exclusive: the propagation equations in quantum mechanics naturally lead to D'Alembert's and like equations. If you interpret Maxwell's equations for a tunnelling wave into a system of dielectric layers as the propagation equation for a lone photon, the energy density as a function of position for the properly normalized solution is interpreted as the probability to destructively detect the photon at the point in question with a detector when one photon states propagate into the layers separately and the Poynting vector becomes the flux of this probability.

Question from OP

I fully agree with what you say. Maxwell <-> photon tunnelling and Shrodinger <-> electron tunnelling are the "same" thing. But Shrodinger equation leads to quantised states (example hydrogen atom), hence it is part of quantum mechanics. Can Maxwell equations lead to quantised states as Shrodinger does? What is puzzling to me, I think, is the following: Why is Shrodinger description of particles (wave nature) considered quantum mechanics and Maxwell description is not?

Certainly Maxwell's equations lead to bound states. Look at the bound states of an optical fiber, which are shift-invariant eigenfunctions of the form $\Psi(x,\,y)\,e^{i\,\beta\,z}$, where the z direction is along the optical axis of the fiber, and where the propagation constant $\beta$ lies between the core and cladding wavenumbers. This is the discrete spectrum of the relevant Sturm-Liuoville system. As a quantum mechanical description when there is one photon in the mode system at a time, the propagation equation is actually the propagation for a pseudo particle called various things - polariton is probably the most apposite to an optical fiber propagation. The pseudo particle is a quantum superposition of pure EM field one-photon states and excited matter states in the fiber's material.

You ask why this kind of thing isn't called "quantum mechanics". Well it most certainly is part of quantum mechanics and the reason it isn't often referred to as such is probably historical. There is no nonrelativistic description of the photon - Maxwell's Equations are fully Lorentz-covariant - in contrast with the atomic electron Schrödinger equation which describes a nonrelativistic approximation. Such approximations admit position co-ordinates where the wavefunction can be loosely interpreted as definining, through its magnitude, the probability of "finding" an electron at a given position. This kind of thing isn't possible for the relativistic photon, or, for that matter, the relativistic electron described by the Dirac equation (note that Maxwell's equations can indeed be written as a Schrödinger equation and also that Maxwell's equations are equivalent to the Dirac equation for a massless particle). See my answer here and also here and here for further details. The question of photonic wave functions is addressed in detail in the works of Iwo Bialynicki-Birula, for example, cited in my answers.

• I fully agree with what you say. Maxwell <-> photon tunnelling and Shrodinger <-> electron tunnelling are the "same" thing. But Shrodinger equation leads to quantised states (example hydrogen atom), hence it is part of quantum mechanics. Can Maxwell equations lead to quantised states as Shrodinger does? What is puzzling to me, I think, is the following: Why is Shrodinger description of particles (wave nature) considered quantum mechanics and Maxwell description is not? – Worldsheep Aug 26 '16 at 10:26
• @Worldsheep See my updates – WetSavannaAnimal Aug 26 '16 at 11:02