The Harper–Hofstadter Hamiltonian and conical diffraction in photonic lattices with grating assisted tunneling

We consider the paraxial propagation of light in a photonic lattice defined by the index of refraction $n={n}_{0}+\delta n(x,y,z)$ ($\delta n\ll {n}_{0}$), where n₀ is the constant background index of refraction, and $\delta n(x,y,z)$ describes small spatial variations, which are slow along the propagation z-axis. The slowly varying amplitude of the electric field $\psi (x,y,z)$ follows the (continuous) Schrödinger equation [26]:

$$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{\partial \psi }{\partial z}=-\displaystyle \frac{1}{2k}{{\rm{\nabla }}}^{2}\psi -\displaystyle \frac{k\delta n}{{n}_{0}}\psi ;\end{eqnarray} \tag{ 1 }$$

here, ${{\rm{\nabla }}}^{2}={\partial }^{2}/\partial {x}^{2}+{\partial }^{2}/\partial {y}^{2}$, and $k=2\pi {n}_{0}/\lambda$, where λ is the wavelength in vacuum. From now on we will refer to $\delta n(x,y,z)$ as the potential or index of refraction. In the simulations we use n₀ = 2.3, corresponding to the systems that were used to implement the optical induction technique [23, 24], and $\lambda =500\;{\rm{nm}}$.

For clarity, let us first present the grating assisted tunneling method in a 1D photonic lattice. If the potential is a periodic lattice, $\delta {n}_{{\rm{L}}}(x)=\delta {n}_{{\rm{L}}}(x+a)$, where a is the lattice constant, and if the lattice is sufficiently 'deep', the propagation of light can be approximated by using a discrete Schrödinger equation [26, 43]:

$$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{{\rm{d}}{\psi }_{m}}{{\rm{d}}z}=-({J}_{x}{\psi }_{m-1}+{J}_{x}{\psi }_{m+1}),\end{eqnarray} \tag{ 2 }$$

where J_x quantifies the tunneling between adjacent waveguides and ${\psi }_{m}(z)$ is the amplitude at the mth lattice site, i.e., waveguide. To create a synthetic magnetic field, here we use the strategy to tune the phase of the tunneling between lattice sites. For this 1D lattice, we seek a scheme which effectively renormalizes J_x to get ${K}_{x}\mathrm{exp}({\rm{i}}{\phi }_{m})$, where ${\phi }_{m}$ denotes the phase for tunneling from site m to site $m+1$. The scheme is illustrated in figure 1. In figure 1(a) we show the potential $\delta n(x,z)$, which can be modeled as a discrete lattice with complex tunneling parameters ${K}_{x}\mathrm{exp}({\rm{i}}{\phi }_{m})$ shown in figure 1(b).

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Let us gradually present this method wherein the index of refraction is varied as in figure 1(a). In figure 1(e), we show the propagation of intensity in a continuous 1D model in the lattice potential $\delta {n}_{{\rm{L}}}(x)=\delta {n}_{{\rm{L}}0}{\mathrm{cos}}^{2}(\pi x/a)$, with $\psi (x,0)=\sqrt{I}{{\rm{e}}}^{-{x}^{2}/{(3a)}^{2}};$ $\delta {n}_{{\rm{L}}0}=4\times {10}^{-4}$, a  =  10 μm. We see the usual diffraction pattern for a spatially broad excitation covering several lattice sites [26]. Next, suppose that we introduce a linear gradient of index of refraction along the x-direction $\delta {n}_{{\rm{T}}}(x)=-\eta x$ in addition to the lattice potential, such that $\delta n=\delta {n}_{{\rm{L}}}(x)+\delta {n}_{{\rm{T}}}(x)$. For a sufficiently large tilt, the tunneling is suppressed. This can be seen from figure 1(f) which shows the propagation of intensity in a tilted potential with $\eta =0.1\;\delta {n}_{{\rm{L}}0}/a;$ the tilt should be smaller than the gap between the first two bands. It should be noted that in the context of continuous periodic lattices with strong tilt, the suppression of tunneling can also be interpreted as Bloch oscillations with negligible amplitude. Finally, let us introduce an additional small grating potential at a small angle θ with respect to the z-axis, $\delta {n}_{{\rm{G}}}(x,z)=\delta {n}_{{\rm{G}}0}{\mathrm{cos}}^{2}(({q}_{x}x-\kappa z)/2)$, such that $\delta n(x,z)=\delta {n}_{{\rm{L}}}(x)+\delta {n}_{{\rm{T}}}(x)+\delta {n}_{{\rm{G}}}(x,z)$. This total potential $\delta n(x,z)$ is illustrated in figure 1(a). The 'frequency' κ is determined by the angle θ of the grating with respect to the z-axis, which is chosen such that $\kappa =\eta {ak}/{n}_{0}$ and the grating forms a z-dependent perturbation resonant with the index offset between neighboring lattice sites $\eta a$ (figure 1(a)). The grating restores the tunneling along the x-axis, hence the term grating assisted tunneling. Restored tunneling is seen in figure 1(g) which shows diffraction for identical initial conditions as in figures 1(e) and (f); the grating parameters are ${q}_{x}=\pi /a$ and $\delta {n}_{{\rm{G}}0}=0.1\delta {n}_{{\rm{L}}0}$.

However, the diffraction pattern is drastically changed. In order to interpret it, we point out that, for resonant tunneling where $\kappa =\eta {ak}/{n}_{0}$ and a sufficiently large tilt ($J\ll \eta$), z-averaging over the rapidly oscillating terms shows that the system can be modeled by an effective discrete Schrödinger equation (e.g., see [34] for ultracold atoms):

$$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{{\rm{d}}{\psi }_{m}}{{\rm{d}}z}=-({K}_{x}{{\rm{e}}}^{{\rm{i}}{\phi }_{m-1}}{\psi }_{m-1}+{K}_{x}{{\rm{e}}}^{-{\rm{i}}{\phi }_{m}}{\psi }_{m+1}),\end{eqnarray} \tag{ 3 }$$

where ${\phi }_{m}={\bf{q}}\cdot {{\bf{R}}}_{m}={q}_{x}{ma}$. In figure 1(g) we used ${q}_{x}=\pi /a$, i.e., ${\phi }_{m}=m\pi$. Such a discrete lattice is illustrated in figure 1(c); its dispersion having a 1D Dirac cone at k_x = 0. For a wavepacket that initially excites modes close to k_x = 0, the diffraction in the discrete model (3) yields the so-called (1D) conical diffraction pattern [11], as illustrated in figure 1(h). The initial conditions for propagation in the discrete model correspond to the initial conditions in the continuous system, ${\psi }_{m}(0)=\sqrt{I}{{\rm{e}}}^{-{(m/3)}^{2}}$, and K_x = 0.053 mm⁻¹. Thus, we interpret the diffraction pattern in figure 1(g) as 1D conical diffraction, a signature of the discrete model depicted in figure 1(c). A comparison of the discrete (figure 1(h)) and the realistic continuous model (figure 1(g)), clearly shows that we can use grating assisted tunneling to tune the phases of the tunneling parameters in the discrete Schrödinger equation, thereby realizing synthetic magnetic fields.

Before proceeding to 2D systems, we discuss the amplitude of the tunneling matrix elements K_x as a function of the strength of the grating $\delta {n}_{{\rm{G}}0}$. Figure 1(d) shows K_x (green squares) and J_x (blue circles) versus $\delta {n}_{{\rm{G}}0}$, where J_x corresponds to the potential which includes the lattice and the grating, but no tilt. The amplitudes J_x and K_x are obtained by comparing the diffraction pattern of the discrete with the continuous model, and adjusting J_x and K_x until the two patterns coincide, as in figures 1(g) and (h). Our results in figure 1(d) are in agreement with those in ultracold atoms (e.g., see figure 3(a) in [34]).

The extension of the scheme to 2D lattices is straightforward. We consider a square lattice, $\delta {n}_{{\rm{L}}}=\delta {n}_{{\rm{L}}0}({\mathrm{cos}}^{2}(\pi x/a)+{\mathrm{cos}}^{2}(\pi y/a))$, the tilt in the x-direction, $\delta {n}_{{\rm{T}}}(x)=-\eta x$, and the grating which has the form $\delta {n}_{{\rm{G}}}(x,y,z)=\delta {n}_{{\rm{G}}0}{\mathrm{cos}}^{2}(({q}_{x}x+{q}_{y}y-\kappa z)/2)$. Propagation of light in the total potential $\delta n(x,y,z)=\delta {n}_{{\rm{L}}}(x,y)+\delta {n}_{{\rm{T}}}(x)+\delta {n}_{{\rm{G}}}(x,y,z)$ can be modeled by the discrete Schrödinger equation (the derivation is equivalent to that in [34] for ultracold atoms):

$$\begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}{\psi }_{m,n}}{{\rm{d}}z} & = & -\left({K}_{x}{{\rm{e}}}^{{\rm{i}}{\phi }_{m-1,n}}{\psi }_{m-1,n}+{K}_{x}{{\rm{e}}}^{-{\rm{i}}{\phi }_{m,n}}{\psi }_{m+1,n}\right.\\ & & +\left.{J}_{y}{\psi }_{m,n-1}+{J}_{y}{\psi }_{m,n+1}\right),\end{eqnarray} \tag{ 4 }$$

where ${\phi }_{m,n}={\bf{q}}\cdot {{\bf{R}}}_{m,n}={q}_{x}{ma}+{q}_{y}{na}$. Note that the tunneling along y does not yield a phase because there is no tilt in the y direction; the tunneling amplitude along y depends on the depth of the grating, as illustrated in figure 1(d) with blue circles.

In order to demonstrate that the propagation of light in the continuous 2D potential $\delta n(x,y,z)$ is indeed equivalent to the dynamics of equation (4), we compare propagation in the discrete model (4), with that of the continuous equation (1). The lattice parameters are $\delta {n}_{{\rm{L}}0}=4\times {10}^{-4}$, a  =  13 μm; the tilt is given by $\eta =0.1\delta {n}_{{\rm{L}}0}/a;$ the grating is defined by $\delta {n}_{{\rm{G}}0}=0.17\delta {n}_{{\rm{L}}0}$ and ${q}_{x}=-{q}_{y}=\pi /a$, which yields ${\phi }_{m,n}=(m-n)\pi$, and κ is chosen to yield resonant tunneling. The discrete lattice which corresponds to this choice of phases is illustrated in figure 2(a). It has two bands, $\beta =\pm 2\sqrt{{K}_{x}^{2}{\mathrm{sin}}^{2}({k}_{x}a)+{J}_{y}^{2}{\mathrm{cos}}^{2}({k}_{y}a)}$ (β is the propagation constant), touching at two 2D Dirac points at $({k}_{x},{k}_{y})=(0,\pm \pi /2a)$ in the Brillouin zone [44], as depicted in figure 2(b). Suppose that the incoming beam at z  =  0, excites the modes which are in the vicinity of these two Dirac points. Around these points, for a given $\hat{{\bf{k}}}={\bf{k}}/k$, the group velocity, ${{\rm{\nabla }}}_{{\bf{k}}}\beta ({k}_{x},{k}_{y})$, is constant. Thus, the beam will undergo conical diffraction, which has been thoroughly addressed with Dirac points in honeycomb optical lattices [27]. To demonstrate this effect, we consider the propagation of a beam with the initial profile given by $\psi (x,y,0)=\sqrt{I}\mathrm{cos}(y\pi /2a)\mathrm{exp}({-(x/3a)}^{2}-{(y/3a)}^{2})$ in the 2D potential $\delta n(x,y,z)$ (the cosine term ensures that we excite modes close to the two Dirac points). In figure 2(c) we show this beam after propagation for z  =  162 mm. The two concentric rings are a clear evidence of conical diffraction [27, 30]. We also compare this with the propagation in the discrete model (4), with tunneling phases plotted in figure 2(a); K_x = 0.11 mm⁻¹, J_y = 0.14 mm⁻¹. The results of the propagation in the discrete model are shown in figure 2(d). It is evident that for our choice of the tilt and the grating, we have effectively realized the lattice plotted in 2(a). This is in fact a realization of the Harper–Hofstadter Hamiltonian for $\alpha =1/2$, where α is the flux per plaquette in units of the flux quantum [34, 35].

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For the experimental implementation of the scheme, we propose the so-called optical induction technique in photosensitive materials, which can be implemented in photorefractives [22–26]. In these systems, both the lattice $\delta {n}_{{\rm{L}}}(x,y)$ and the grating potential $\delta {n}_{{\rm{G}}}(x,y,z)$ can be obtained in a straightforward fashion by using interference of plane waves in the medium [26]. The lattice constant a, the grating parameter ${\bf{q}}$, and hence the hopping phase ${\phi }_{m,n}={\bf{q}}\cdot {{\bf{R}}}_{m,n}$, are tunable by changing the angle between the interfering beams. This could enable manipulation of the phases of the tunneling matrix elements in real time [23].

The most challenging part of the implementation appears to be creation of the linear tilt potential. To observe the conical diffraction effects, one needs a linear gradient over the ∼20 lattice sites, which implies a total index tilt ${\rm{\Delta }}n=20\eta a\sim 20\times 0.1\delta {n}_{L0}\sim 8\times {10}^{-4}$. In principle, the linear tilt could be achieved by using a spatial light modulator. Another possibility which could create a linear tilt is to use crystals with linear dependence of the index of refraction on temperature, and a temperature gradient across the crystal [46].

However, it should be emphasized that if one uses a superlattice for $\delta {n}_{{\rm{L}}}$, rather than the linear tilt potential, as in [33] for ultracold atomic gases, the grating assisted tunneling scheme proposed here would yield staggered synthetic magnetic fields for photons. The achievement of such staggered fields should be possible with optically induced lattices proposed here.

Before closing, we emphasize that the spatial phase between the periodic lattice potential $\delta {n}_{{\rm{L}}}$ and the grating $\delta {n}_{{\rm{G}}}$ is a relevant parameter. This point has recently been examined in detail in the context of ultracold atoms [45]. To translate it to our system, suppose that in our 1D system, we shift the grating along the x-direction, such that $\delta {n}_{{\rm{G}}}=\delta {n}_{{\rm{G}}0}{\mathrm{cos}}^{2}(({q}_{x}x-\kappa z+\xi )/2)$. This simply shifts the phases in the discrete lattice (shown in figure 1(c)) such that ${\phi }_{m}$ is replaced by ${\phi }_{m}+\xi$, which moves the 1D Dirac point in k_x-space by ${\rm{\Delta }}{k}_{x}=\xi /a$. We have numerically verified that this indeed happens.