Photon reflection by a quantum mirror: A wave-function approach

We derive from first principles the momentum exchange between a photon and a quantum mirror upon reflection, by considering the boundary conditions imposed by the mirror surface on the photon wave equation. We show that the system generally ends up in an entangled state, unless the mirror position uncertainty is much smaller than the photon wavelength, when the mirror behaves classically. Our treatment leads us directly to the conclusion that the photon momentum has the known value $\hslash \mathbf{k}$. This implies that when the mirror is immersed in a dielectric medium the photon radiation pressure is proportional to the medium refractive index $n$. Our work thus contributes to the longstanding Abraham-Minkowski debate about the momentum of light in a medium. We interpret the result by associating the Minkowski momentum (which is proportional to $n$) with the canonical momentum of light, which appears naturally in quantum formulations.

Figure 1

An electromagnetic field with wave vector $\mathbf{k}$ is reflected by a perfect mirror, resulting in a reflected wave with wave vector ${\mathbf{k}}^{\prime }=\mathbf{k}-2(\mathbf{k}·{\widehat{\mathbf{z}}}_{\mathbf{m}}){\widehat{\mathbf{z}}}_{\mathbf{m}}$.