Neglecting the linear momentum of photons

It is a common practice to neglect the linear moment of photons of conventional (non-twisted) beams in their interaction with matter. Though this approximation make sense in many situations, in others, it fails to capture important phenomena.

A simple analysis identifies the wave-number k of a monochromatic non-twisted field
\begin{equation}
\mathbf E(\mathbf r,t) = \mathbf E_0 e^{i(\mathbf k \cdot \mathbf r - \omega t)} + c.c.
\end{equation}
with the linear momentum per photon. Using the dispersion relation ω=ck, and energy expression E=ℏω, leads to E=cℏk. In addition, we know that, for photons, the energy relates to the momentum by E=cp. Then, we find the relationship between momentum and wave-number k: p=ℏk.

Consider k from a different perspective. Given that k=2π/λ, fields having a small k, change only appreciable in long distances. Let us put some numbers: (visible) light is in the range λ=400-700 nm. On the other hand, the typical radius of a small semiconductor quantum dot is 5 nm. This implies that, within the quantum dot, a light field changes only very little with kz~0.06 and exp(ikz)~1. Thus, the photon's linear momentum ℏk can be safely neglected, when the dimensions of the system interacting with the electromagnetic field are much smaller than the wavelength. Then, in the case of atoms, molecules and nanostructures the field becomes homogeneous
\begin{equation}
\mathbf E(\mathbf r,t) = \mathbf E_0 e^{-i \omega t} + c.c.
\end{equation}
that is, we take it as a constant in the region of space being considered.

However, for extended systems the approximation of constant field may be inappropriate. Two examples are exciton-polaritons [1] and the so-called photon-drag effect [2]. The photon-drag --most related to the topic of twisted light-- is the generation of an electric current in a rod of material (e.g. p-type germanium) by absorption of light. Photons impinging in the rod produce transitions between states in two valence bands as the figure shows. Despite the fact that each photon transfers a small amount of linear momentum to an electron, the contribution of many photons becomes substantial, and an electric current in the direction of light beam appears. The effect is so well established that one can nowadays buy detector based on it [3].

Figure 1: The photon-drag effect. Light propagating along the z-axis induces
electronic transitions between two valence bands and transfers
a small amount of linear momentum ℏk per photon.

In a future post I will talk about the circular photon-drag effect, that causes circular electric current in a bulk system by illuminating it with twisted light. But the circular photon-drag is not the only new phenomenon that one predicts when the momentum --linear or angular-- of light is taken into account!

References

[1] Haug, H., & Koch, S. W. (2004). Quantum theory of the optical and electronic properties of semiconductors (Vol. 3). Singapore: World scientific. Chapter "Polaritons".
[2] Gibson, A. F., & Walker, A. C. (1971). Sign reversal of the photon drag effect in p type germanium. Journal of Physics C: Solid State Physics, 4(14), 2209.
[3] http://www.hamamatsu.com/resources/pdf/ssd/b749_kird1038e04.pdf