User:AZhaoUNM/sandbox/Photonic topological insulator

Photonic topological insulators are artificial electromagnetic materials that support topologically non-trivial states of light. Photonic topological phases are classical electromagnetic wave analogues of electronic topological phases studied in condensed matter physics. Similar to their electronic counterparts, they can provide robust unidirectional channels for light propagation.

The field that studies these phases of light is referred to as topological photonics, even though the working frequency of these electromagnetic topological insulators may fall in other parts of the electromagnetic spectrum such as the microwave range.

Quantum Hall effect in matter
Topological order in solid-state systems has been studied in condensed matter physics since the discovery of the integer quantum Hall effect. In these materials, one models the electrons as a Fermi gas confined to a 2D surface. The application of a strong, perpendicular magnetic field allows one to measure the Hall conductance across the surface. It was discovered experimentally in 1980 that this conductance comes in integer multiples of $$e^2/h$$, with measurement precision up to one part in a billion. This quantized feature appeared highly robust to any sort of material imperfections, composition impurities, and system sizes. Various related phenomena, such as the fractional and anomalous quantum Hall effects, were later observed as well.

These effects are theoretically understood as resulting from the topology of the energy bands in momentum space. Here, we will briefly review key notions of topological invariants in solid-state electronic systems. Consider a single-particle Hamiltonian on the 2D surface which is spatially periodic with period $$\mathbf{a}$$: $$\hat{H}(\mathbf{\hat{r}},\mathbf{\hat{p}}) = \hat{H}(\mathbf{\hat{r}}+\mathbf{a},\mathbf{\hat{p}}).$$Then by Bloch's theorem the eigenstates of this Hamiltonian take the form$$\psi_{n,\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n,\mathbf{k}}(\mathbf{r}),$$where $$u_{n,\mathbf{k}}(\mathbf{r}) = u_{n,\mathbf{k}}(\mathbf{r} + \mathbf{a})$$. One may define the Berry connection for each eigenstate as$$\mathbf{\mathcal{A}}_n(\mathbf{k}) = i \langle u_{n,\mathbf{k}} | \nabla_\mathbf{k} | u_{n,\mathbf{k}} \rangle,$$and subsequently a Berry (geometric) phase$$\gamma_n = \oint d\mathbf{k} \cdot \mathbf{\mathcal{A}}_n(\mathbf{k}),$$which is acquired by a localized electron state (wave packet) moving on the surface along a closed path in momentum space ($$\mathbf{k}$$-space) over the first Brillouin zone (as well as the usual dynamical phase). However, the Berry connection is not invariant to a local $$\mathrm{U}(1)$$ gauge transformation in momentum space, i.e. $$| u_{n,\mathbf{k}} \rangle \mapsto e^{i\chi(\mathbf{k})} | u_{n,\mathbf{k}} \rangle$$. Nonetheless, since the phase factor must be single-valued along the closed path, the Berry phase $$\gamma_n$$ is at least gauge invariant modulo $$2\pi$$. Since the gauge transformation on the connection takes the form $$\mathbf{\mathcal{A}}_n(\mathbf{k}) \mapsto \mathbf{\mathcal{A}}_n(\mathbf{k}) - \nabla_{\mathbf{k}} \chi(\mathbf{k})$$, one may define a gauge invariant Berry curvature as its curl:$$\mathbf{\Omega}_n(\mathbf{k}) = \nabla_\mathbf{k} \times \mathbf{\mathcal{A}}_n(\mathbf{k}).$$This quantity encodes topological information about the $$n$$th eigenstate, hence the $$n$$th energy band of the system. To see this, consider the following. In 2D, this cross product only has one non-zero component $$\Omega_n(\mathbf{k})$$. One may define the (first) Chern number for this system as$$C_n = \frac{1}{2\pi} \int_{\mathrm{BZ}} d^2 k \, \Omega_n (k_x,k_y),$$where $$\mathrm{BZ}$$ is the entire first Brillouin zone. The Chern number is $$0$$ if and only if $$\mathbf{\mathcal{A}}_n$$ can be continuously defined over $$\mathrm{BZ}$$, a direct consequence of Stoke's theorem. Thus if $$C_n \neq 0$$, the Berry connection is not continuously defined over $$\mathbf{k}$$-space. Note that the first Brillouin zone, due to periodic boundary conditions, is topologically equivalent to the torus. This is how topological features may appear even in surfaces of trivial topology.

The condition for a non-zero Chern number is a breaking of time-reversal symmetry. Consider a time-reversal operator $$\hat \mathcal{T}$$ and parity inversion operator $$\hat \mathcal{P}$$. Here, they both map $$\mathbf{k} \mapsto \mathbf{-k}$$, but $$\hat \mathcal{T}$$ incurs a complex conjugate (not a Hermitian conjugate):

$$(\hat\mathcal{P}\hat\mathcal{T}) \hat{H}(\mathbf{k}) (\hat\mathcal{P}\hat\mathcal{T})^{-1} = \hat{H}(\mathbf{k})^*.$$Suppose the Hamiltonian is initially real-valued, thus is PT-invariant. Any perturbation which breaks this PT symmetry lifts the Dirac point degeneracy in the system, opening an energy gap between band pairs of opposite spins. However, breaking $$\hat \mathcal{T}$$ and $$\hat \mathcal{P}$$ separately are topologically inequivalent. This is because $$\mathbf{\Omega}_n$$ is odd under $$\hat \mathcal{T}$$ but even under $$\hat \mathcal{P}$$:

$$\mathbf{\Omega}_n(\mathbf{k}) = \begin{cases} 0, & \hat\mathcal{P}\hat\mathcal{T} \text{ preserved} \\ -\mathbf{\Omega}_n(-\mathbf{k}), & \text{only } \hat\mathcal{P} \text{ broken} \\ \mathbf{\Omega}_n(-\mathbf{k}), & \text{only } \hat\mathcal{T} \text{ broken}. \end{cases}$$

Integrating over $$\mathrm{BZ}$$ to obtain $$C_n$$, one sees that the integral necessarily vanishes for all cases except when only time-reversal symmetry is broken. Thus we see one manifestation of topological protection: any perturbation to the system which preserves time-reversal symmetry cannot affect the topological states.

Furthermore, one may show that $$C_n$$ must be integer-valued. Thus it is the non-zero Chern number which directly gives rise to the integer quantum Hall effect. In particular, if we ignore interparticle interactions, the Hall conductance is given by $$\sigma = -\frac{e^2}{h}\sum_{n\in\mathrm{occ}} C_n,$$where the sum is taken over the occupied energy bands of the state. For a 2D free electron gas with a strong perpendicular magnetic field, each energy band is a Landau level, all with the same $$C_n = 1$$, so that the Hall conductance is directly proportional to the number of occupied Landau levels.

If one considers the interparticle interactions, generalizations of the Chern number arise which are fractional-valued, leading to the fractional quantum Hall effect. Such states are the consequence of both non-trivial topology and strong correlations, and can exhibit even more exotic properties beyond quantized conductance, such as fractional charge and statistics.

Bulk–edge correspondence
A key feature of topologically non-trivial systems is the fact that the environment in which they live is topologically trivial. The discontinuity between these two topologies gives rise to special states which are localized along the boundary, called edge modes. There are two common types of such modes: helical and chiral edge states. Helical modes arise in pairs due to spin effects; for spinful particles, e.g. possessing up and down states, the direction of the state's spin determines the direction of propagation along the boundary under free evolution. This is in contrast to chiral modes of spinless systems, which enforce unidirectional propagation along the boundary. Hence these chiral modes are responsible for carrying an edge current around the system. The sum of Chern numbers occupied in the bulk is equal to the number of occupied chiral edge states—this relationship between topological invariants in the bulk of a material with phenomena localized on its boundary is called the bulk–edge correspondence.

The unidirectionality of chiral states can be seen by the following. Consider a Hamiltonian with simple charge–magnetic field interaction, ignoring edge effects for now:

$$\hat{H} = \frac{1}{2m}\left( \hat{\mathbf{P}} - \frac{e\hat{\mathbf{A}}}{c} \right)^2 = \frac{1}{2m}\left[ \hat{P}_x^2 + (\hat{P}_y - m\omega_c \hat{X})^2 \right],$$where we have chosen the Landau gauge for the perpendicular magnetic field, $$\mathbf{A} = B x \mathbf{e}_y$$, and $$\omega_c = eB/mc$$ is the cyclotron frequency (we have taken $$\hbar = 1$$ here). This Hamiltonian has well-known eigenstates

$$\psi_{k_y}(x,y) = e^{i k_y y} f_n(x - k_y/m\omega_c),$$

where $$f_n$$ are the 1D simple harmonic oscillator eigenfunctions (Hermite polynomials times Gaussians). From this form, one sees that $$\langle \hat{P}_y \rangle = k_y$$ and $$\langle \hat{X} \rangle = k_y/m\omega_c$$. Then the current operator $$\hat{I}_y = -\frac{e}{m}( \hat{P}_y - m\omega_c \hat{X} )$$ in the y-direction has expectation value

$$\langle \hat{I}_y \rangle = -\frac{e}{m} \left( k_y - m\omega_c \frac{k_y}{m\omega_c} \right) = 0.$$

Hence there is no classical current in the bulk of the material.

However, now consider edge effects. Qualitatively, at the boundary there is a confining pseudo-potential which keeps the electrons from "falling off" the material. The effect of this potential is that it deforms the eigenstates near the edge, so that the wavefunction is weighted more toward the center. Consider an edge state localized on the left edge of the surface: the deformation will result in $$\langle \hat{X} \rangle > k_y/m\omega_c$$, hence

$$\langle \hat{I}_y \rangle > 0.$$Thus there is a current carried by these chiral edge states, indicating that they propagate unidirectionally along a given boundary.

Coherent pumping of classical light
At the first level of approximation, one may consider classical states of light injected into topologically interesting matter. Assuming no magnetoelectric coupling, Maxwell's equations for the electric field with frequency $$\omega$$ can be written as $$\nabla_{\mathbf{r}} \times \left[ \frac{1}{\mu(\mathbf{r})}\nabla_{\mathbf{r}} \times \mathbf{E}(\mathbf{r})\right] = \omega^2 \epsilon(\mathbf{r}) \mathbf{E}(\mathbf{r}),$$ where $$\epsilon(\mathbf{r})$$ and $$\mu(\mathbf{r})$$ are the (spatially periodic) permeability and permittivity of the 2D crystalline material. For crystal momenta $$\mathbf{k}$$ and energy bands indexed by $$n$$, denote the modes of the electric field (coupled to the matter) by $$\mathbf{E}_{n,\mathbf{k}}(\mathbf{r})$$. Define a scalar product between fields by$$( \mathbf{E}_1 \mid \mathbf{E}_2 ) = \sum_{\mu,\nu \in \{ x,y,z \}^2 } \int d^2r \, E_1^\mu(\mathbf{r})^{*} \epsilon_{\mu\nu}(\mathbf{r}) E_2^\nu(\mathbf{r}),$$where we have generalized the permittivity to its tensor form, $$\epsilon_{\mu\nu}(\mathbf{r})$$. Then the Berry connection for the electric field modes is

$$\mathbf{\mathcal{A}}_n(\mathbf{k}) = i ( \mathbf{E}_{n,\mathbf{k}} \mid \nabla_{\mathbf{k}} \mathbf{E}_{n,\mathbf{k}} ) = i \sum_{\mu,\nu \in \{ x,y,z \}^2 } \int d^2r \, E_{n,\mathbf{k}}^\mu(\mathbf{r})^{*} \epsilon_{\mu\nu}(\mathbf{r}) E_{n,\mathbf{k}}^\nu(\mathbf{r}). $$The Berry curvature and Chern number follow analogously as before, and they display the usual behavior based on the topology of momentum space. Although these are classical states of light, they exhibit features analogous to the quantum electron states discussed above because Maxwell's equations, in the appropriate limits, reduce to Schrödinger-like equations under these conditions. In particular, the bulk–edge correspondence holds, and so robust chiral edge states of light propagate unidirectionally along the boundary of the surface. Besides trivial toy models, however, the solutions for these states must be numerically computed.

Quantized field formalism
We may promote the electric field to its quantized operator form:

$$\hat\mathbf{E}(\mathbf{r}) = \sum_j \left( \mathbf{E}_j(\mathbf{r}) \hat{a}_j + \mathbf{E}_j(\mathbf{r})^* \hat{a}_j^\dagger \right),$$where $$\hat{a}_j,\,\hat{a}_j^\dagger$$ are creation and annihilation operators for a basis of localized photon modes satisfying the canonical boson commutation relations, and $$\mathbf{E}_j(\mathbf{r})$$ are suitably normalized mode profiles, corresponding to the classical eigenmodes of Maxwell's equations with frequencies $$\omega_j$$. The simplest model one may consider is the field Hamiltonian with a hopping term between modes:

$$\hat{H}_0 = \sum_j \hbar\omega_j \left( \hat{a}_j^\dagger \hat{a}_j + \frac{1}{2} \right) - \sum_{j,k} J_{jk} \hat{a}_j^\dagger \hat{a}_k.$$An important consideration of photonic systems, which is not a major concern in the solid-state case, is the radiation of photon states into the environment. To capture this, define a continuum of radiative modes $$\hat{A}_\eta$$ coupled to the field modes by a coupling strength $$g_{j\eta}$$:

$$\hat{H} = \hat{H}_0 + \int d\eta \, \hbar \omega_\eta \hat{A}_\eta^\dagger \hat{A}_\eta - \sum_j \int d\eta \left( \hbar g_{j\eta} \hat{A}_\eta^\dagger\hat{a}_j + \text{h.c.} \right). $$Assuming each field mode couples independently to its own set of radiative modes, the Heisenberg equation of motion for each field mode reads

$$i \frac{d\hat{a}_j}{dt} = \omega_j\hat{a}_j - \sum_k J_{jk}\hat{a}_k - \frac{i\Gamma_j}{2}\hat{a}_j + \hat{A}_j^{\mathrm{in}},$$where the radiative damping rate is $$\Gamma_j = 2\pi|g_{j\eta}|^2/|d\omega_\eta/d\eta|$$ and the bosonic input operator is $$\hat{A}_j^{\mathrm{in}} = -\int d\eta \, g_{j\eta}^* \hat{A}_\eta$$.

The most relevant case, however, is when we have a coherent source, such as a laser, in which case we can consider the expectation value of the equation of motion:

$$i \frac{d\alpha_j}{dt} = \omega_j\alpha_j - \sum_k J_{jk}\alpha_k - \frac{i\Gamma_j}{2}\alpha_j + \langle \hat{A}_j^{\mathrm{in}} \rangle,$$where we assume a coherent state $$| \alpha_j \rangle$$ in the mode. The source term $$\langle \hat{A}_j^{\mathrm{in}} \rangle$$ corresponds to the classical amplitude of the incident field, and may in general be time-dependent.

Experimental implementations
The general theory behind topological photonics is very broad, essentially encompassing any topologically non-trivial state of light (quantum or classical). Here, we outline some major developments in observing such states in real, physical systems.

Gyromagnetic photonic crystals
In 2008, Wang et al. demonstrated electromagnetic chiral edge states in a 2D square lattice photonic crystal. The crystal is physically constructed by placing a vertical array of gyromagnetic ferrite rods between two horizontal metallic plates. The sides are then confined by non-magnetic metallic slabs to minimize radiative loss. A variable length scatterer is introduced along one of these edges to studying the effects of photon scattering. This allows the system to mimic transverse magnetic (TM) field mode in 2D. The system is visualized in the figure to the right, (a).

A uniform, static magnetic field of 0.2 T is applied, which breaks the time-reversal symmetry in the system. This forms the TM bands seen in the figure, (b). Each TM band corresponds to a Chern number, with robust gaps due to symmetry breaking. Note that electromagetic modes of particular frequencies do not support chiral edge states for any wavenumber, e.g. around 3 GHz. The experiment of Wang et al. observed that, within modes that do support edge states (e.g. around 4.5 GHz), the backscattering transmission rate is highly suppressed, while the forward propagation remains fairly strong (see the figure, (d)). Outside of this frequency band, however, there is no discrepancy between backward and forward propagation, since those states are topologically trivial, hence non-chiral.

Figure, (c) depicts the effects of a large obtrusion along the edge of the crystal. These numerical simulations demonstrate the robustness of chiral edge states to such a scattering object, as one sees that the propagation remains unidirectional. Although not pictured in the figure, Wang et al. observed essentially no change to the backscattering rate when this scatterer was introduced, regardless of the obtrusion's length.

Direct observation of edge states
The experiment of Wang et al. provided evidence for chiral edge states by probing the unidirectional propagation of EM power. In 2012 and 2013, Kraus et al. and Hafezi et al. demonstrated direct imaging of these photonic edge states in 1D and 2D optics, respectively.

In 1D, Kraus et al. used a simple setup of injecting coherent light through an Aubry–André photonic quasicrystal. The quasicrystal features an array of 100 waveguides across which the photons may tunnel. The input light is focused into one waveguide port, and the output light is imaged to see through which of the other waveguides it tunnels through. As seen in the figure to the left, (c), the intensity spread is wide when injected into the center or rightmost sites, but remains highly localized at the left boundary. They define this leftmost site as their edge mode, as there is no backscattering (i.e. tunneling to the right sites). The topological feature of the system is attributed to the exotic nature of the Aubry–André quasicrystal model.

In 2D, Hafezi et al. implemented a much more elaborate setup, consisting of a 10 x 10 lattice of plaquettes. Each plaquette is made of 4 optical resonators "on-site", and 4 "link" resonators which couple the site resonators together. An asymmetry is introduced by making half the link resonators have a longer effective length. Thus the Hamiltonian describing each plaquette consists of hopping terms, but with a relative phase difference between terms depending on whether their coupling link resonator is the longer variant or not. This accrued phase factor is the source of the topologically non-trivial energy band structure.

With numerical simulations and experimental data, they observe these edge states by measuring the intensity of light scattered from the lattice after injecting coherent light into particular plaquettes. When states of light are injected into the bulk of the lattice, they propagate throughout the other plaquettes of the lattice, much like the spread of light through the 1D waveguide sites. However, when an edge plaquette is excited, the state remains tightly localized in that state. In particular, they observed that the intensity remains peaked at the boundary plaquettes.