Shaking photons out of a topological material

Over the past decade, there has been great interest in topological effects, with concepts originally developed in the context of electron transport in condensed matter platforms now being extended to optical systems. While topological properties in electronic systems are often linked to the quantization of electric conductivity observed in the integer quantum Hall effect, a direct analog in optics remains elusive. In this paper, we bridge this gap by demonstrating that the response of the Poynting vector (which may be regarded as a photon current) to mechanical acceleration of a medium provides a precise photonic analog of the electric conductivity. It is shown that the photonic conductivity determines the energy irreversibly transferred from periodic mechanical driving of the medium to the electromagnetic field. Furthermore, it is demonstrated that, for nonreciprocal systems enclosed in a cavity, constant acceleration of the system induces a flow of photons along a direction perpendicular to acceleration, analogous to the Hall effect but for light. The spectral density of the photonic conductivity is quantized in the band gaps of the bulk region with the conductivity quantum determined by the gap Chern number.

Figure 1

A closed box filled with an arbitrary material structure is subject to mechanical shaking with frequency $\mathrm{\Omega }$. The light inside the box is originated by thermal and/or quantum fluctuations. The purple arrows represent the equivalent noise sources that generate the thermal-light. The reference system $\left(x,y,z\right)$ is attached to an inertial (laboratory) frame, whereas the reference system $\left(X,Y,Z\right)$ is rigidly attached to the accelerated cavity.

Figure 2

Band structure of the bulk material showing both the positive (green lines) and negative (purple lines) frequency bands and the interband transitions (vertical arrows) that determine the real part of the photonic conductivity. The dispersion of the transverse (longitudinal) plasmons is depicted with solid (dashed) lines. The black arrows (upward transitions) are associated with the generation of light quanta, which are eventually absorbed in the material (extraction of energy from mechanical driving). The light red arrow (downward transition) is associated with the absorption of quanta from the thermal-light field, which is returned to the mechanical driver. The interactions between the positive frequency bands are possible for arbitrarily small oscillation frequencies (${\mathrm{\Omega }}_s>0$), whereas the interactions with positive and negative frequency bands are only feasible for ${\mathrm{\Omega }}_l>2{\omega }_p$.

Figure 3

Photonic conductivity as a function of the normalized mechanical driving frequency $\mathrm{\Omega }/{\omega }_p$ for different temperature values. In the $T=0^{+}$ limit, the real part of the photonic conductivity vanishes exactly for $\mathrm{\Omega }<2{\omega }_p$.

Figure 4

(a) Illustration of a two-dimensional (2D)-type topological cavity subject to constant acceleration $\mathbf{a}$. The bulk region does not support electromagnetic states at the frequency range of interest. For topological systems, the thermal-light spectrum inside the cavity is determined by the number of topological states. Here, it is assumed that the topological gap is associated with a single counterclockwise unidirectional edge state ($\mathcal{C}=-1$). (b) Sketch of the expectation of the Poynting vector lines when the cavity is accelerated along the $+x$ direction. Due to electromagnetic inertia, the thermal energy associated with the perturbed unidirectional edge state tends to be concentrated on the wall $X=-L_x/2$. This originates a photonic Hall effect, with the averaged Poynting vector expectation oriented along $-y$.