• *Physical* 16, 195

Light confined to an accelerating optical cavity could exhibit a photonic counterpart to the electronic quantum Hall effect.

Place a conductor in a magnetic field and the electric current driven by an applied voltage will not flow in a straight line, but in a direction perpendicular to the electric field – a behavior known as the Hall effect. [1]. Reduce the temperature to the point where the electrons manifest quantum mechanical behavior and the graph gets complicated. Conductivity (defined as the ratio of lateral current to voltage) exhibits discrete jumps as the magnetic field varies – the quantum Hall effect [2]. Because low-temperature electrons and photons obey a similar wave equation [3], should we also expect a quantum Hall effect for light? This question has been bubbling beneath the surface for the past decade, leading to the observation of some aspects of an optical quantum Hall effect. [4, 5]. But the analogy between photons and electrons remains incomplete. Now, a theoretical study by Mário Silveirinha, from the University of Lisbon, Portugal, establishes a new parallel between the physics of electrons and that of photons, by defining the concept of “photon conductivity” that characterizes the flow of light in response to movement of matter [6]. Silveirinha’s innovative view of light-matter interaction could help researchers discover a wide variety of motion-induced wave effects.

The quantum Hall effect can be understood through the lens of topology – that is, by linking Hall conductance with topological invariants known as Chern numbers. In a conductor, atoms form a periodic lattice, in which the behavior of electrons can be fully characterized through a single Brillouin zone—a primitive cell in the reciprocal spatial, or momentum, representation of the lattice. A convenient representation of such a Brillouin zone involves “cutting” it out of momentum space and gluing its edges together to form a closed surface – a torus. [7]. Each electron wavefunction in the conductor is then associated with a point on the surface of this torus. Chern numbers are integers that tell us whether different sets of wave functions “attached” to the torus can be smoothly transformed into each other. [8]. It can be shown that Hall conductivity can be expressed in terms of the product of several physical constants times a sum of these Chern numbers. As these numbers are integers, the formula implies the quantization of conductivity [7].

The topological description shows that the essential ingredients of the quantum Hall effect are not exclusive to electrons in magnetic fields. Chern integers can arise for any type of wave that propagates through a periodic structure with a Brillouin zone that can be mapped onto the surface of a torus. This analogy has been enthusiastically recognized in the last decade, leading to an explosion of research into electromagnetic, acoustic and elastic materials designed with non-zero Chern numbers of the bands that characterize wave propagation. [9]. It is now well established, for example, that, in analogy to the electronic extreme states of a conductor in a magnetic field, non-zero Chern numbers can describe extreme states for classical light waves propagating in systems such as photonic crystals. [4].

However, there are wrinkles. At least two missing ingredients make the connection between electrons and classical waves less than perfect. The first is fundamental: unlike classical electromagnetic waves, electrons are fermions and, therefore, have half-integer spin, obey the Pauli exclusion principle and fill energy bands up to the so-called Fermi level. The second is the absence, in these classical systems, of a quantity that obviously corresponds to a conductance, much less one that assumes quantized values. Can we define a quantity that, like an electric current, flows in response to an applied stimulus?

Silveirinha’s work answers this question for the case of electromagnetic waves. Understanding his approach involves inverting our usual perspective on light-matter interaction. Although we are used to thinking that the electromagnetic field produces acceleration of charge and, therefore, currents, we are much less used to thinking about the opposite: the acceleration of matter can induce a flow of electromagnetic energy. Take radiation pressure, for example. When an electromagnetic wave is reflected from a mirror, the mirror accelerates, leading to a flow of energy from the electromagnetic field to the mirror. We can understand this flow as a current of electromagnetic energy driven by the acceleration of matter – an idea fundamentally related to quantum phenomena such as Unruh and dynamic Casimir effects, in which the acceleration of a mirror in a vacuum leads to the creation of a stream of photons [10]. Taking into account this relationship between the acceleration of the mirror and the flow of electromagnetic energy, Silveirinha develops a mathematical formulation for a “photon conductivity”.

The definition allows you to discover the possibility of direct and unexpected light analogous to the Hall effect. The researcher predicts that such an effect arises in a geometry where light is confined to an optical cavity containing an exotic crystal that is nonreciprocal, meaning it behaves differently in forward and backward propagation (Fig. 1). In this configuration, the mechanical acceleration of the cavity is the counterpart of the electric field within a conductor, while the non-reciprocal medium plays the role of the magnetic field. Silveirinha demonstrates that, under appropriate conditions, there should be a lateral flow of electromagnetic energy – in analogy to the Hall effect for electrons. Furthermore, it demonstrates that the band gap topology of the periodic medium implies that this lateral conductivity is quantized – as in the quantum Hall effect.

For now, experiments to directly test this optical Hall effect are likely to be extremely difficult, as they require non-reciprocal exotic materials and a large, precisely controllable cavity acceleration. And given the intricate mathematical expressions for the derived photon conductivity, it may take some time before the implications of this new view are fully appreciated. But the unifying perspective on wave physics offered by Silveirinha will likely open up new lines of investigation. In particular, the new definition of conductivity may be applicable to broader categories of systems than those envisioned in this work, extending, for example, to acoustic and elastic materials where the presence of topological edge states has long been established.

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