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

We introduce a grating assisted tunneling scheme for tunable synthetic magnetic fields in photonic lattices, which can be implemented at optical frequencies in optically induced one- and two-dimensional dielectric photonic lattices. We demonstrate a conical diffraction pattern in particular realization of these lattices which possess Dirac points in $k$-space, as a signature of the synthetic magnetic fields. The two-dimensional photonic lattice with grating assisted tunneling constitutes the realization of the Harper-Hofstadter Hamiltonian.


Introduction
Synthetic magnetism for photons is a unique tool for the manipulation and control of light, and for the design of novel topological phases and states in photonics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Topological photonics is a rapidly growing field [18], advancing in parallel to analogous efforts in ultracold atomic gases [19,20], inspired by the development of topological insulators in condensed matter physics [21]. One motivating aspect of topological photonic systems is the existence of unidirectional backscattering immune states [1][2][3][4], which are robust to imperfections, and thus may serve as novel waveguides and for building integrated photonic devices. The first experimental observations of such edge states were in the microwave domain, in magneto-optical photonic crystals [4], theoretically proposed in [1][2][3]. In the optical domain, imaging of topological edge states was reported in Floquet topological insulators, implemented in modulated honeycomb photonic lattices [5], and in the twodimensional array of coupled optical-ring resonators [6]. The strategies for obtaining synthetic magnetic/gauge fields and topological phases for optical photons are closely related to the system at hand. In systems of coupled optical resonators, the strategy is to tune the phase of the tunneling between coupled cavities [6][7][8][9]; for example, by using link resonators of different length [6,7], by time-modulation of the coupling [9] or the resonant frequencies of the cavities [14]. Photonic topological insulators were also proposed in superlattices of metamaterials with strong magneto-electric coupling [15]. In 2D photonic lattices (waveguide arrays), pseudomagnetic fields have been demonstrated by inducing 'strain' in optical graphene [10]. Lattice modulation has also been shown to be a fruitful approach in these systems [11][12][13][14]17]. By modulating 1D photonic lattices along the propagation axis, one can choose the sign of the hopping parameter between neighbouring sites, as was realized in a system with alternating positive and negative couplings and resulted in 1D conical diffraction [11]. Transversely modulated waveguide arrays were proposed for the realization of the nonlinear Harper model [12], whereas topological states were achieved in 1D photonic quasicrystals [13]. Moreover, optical analogues of the so called laser assisted tunneling in atomic systems were suggested for creating synthetic magnetic fields in discrete square lattices [14] and recently experimentally studied in laser-fabricated photonic Wannier-Stark ladders [17].
However, all of the above mentioned photonic lattices are in reality continuous dielectric structures. An important test of the viability of a method considered for producing synthetic magnetism is the comparison of light propagation in a continuous system with the propagation in the corresponding discrete model. Here, we propose the realization of a grating assisted tunneling scheme for tunable synthetic magnetic fields in optically induced one-and two-dimensional dielectric photonic lattices [22][23][24][25][26]. As a signature of the synthetic magnetic fields, we demonstrate conical diffraction patterns in particular realization of these lattices, which possess Dirac points in k-space. We compare the light propagation in these realistic (continuous) systems with the evolution in discrete models representing the Harper-Hofstadter Hamiltonian [31,32], and obtain excellent agreement.
Methods encountered in photonics are sometimes closely related to methods in other branches of physics. For example, synthetic magnetic fields were created by using 'strain' in optical graphene [10], analogously to the scheme with strained real graphene [28]. Furthermore, many of the results obtained with ultracold atoms are viable in photonics, and perhaps even more of them are feasible, because heating by spontaneous emission, which is present in ultracold atomic systems, is absent in photonic systems. The method we consider is analogous to those used in [14,17]; however, instead of using curved laser written waveguides [14,17,29] to create an effective linear tilt of the index of refraction, here we propose to use the optical induction technique and/or a temperature gradient, as elaborated below. All these schemes were inspired by the so called laser assisted tunneling scheme that was implemented in optical lattices with ultracold atoms [33][34][35]. Laser assisted tunneling is based on theoretical proposals in [36,37], subsequently developed to experimentally realize the Harper-Hofstadter Hamiltonian [34,35], and staggered magnetic fields in optical superlattices [33].
Historically, the first synthetic magnetic fields were implemented in rapidly rotating Bose-Einstein condensates (BECs) by using the Coriolis force to mimick the magnetic Lorentz force [38,39], whereas the first implementation based on laser-atom interactions used spatially dependent Raman optical coupling between internal hyperfine atomic states in bulk BECs [40]. The latter scheme used the analogy between the Aharonov-Bohm and the Berry phase [19,20]. Beside the schemes with laser assisted tunneling [33][34][35], methods yielding synthetic magnetism in optical lattices include shaking of the optical lattice [41] and an all-optical scheme which enables flux rectification in optical superlattices [42]. For detailed reviews on synthetic gauge fields and potentials in cold atoms see [19,20].

The photonic system
We consider the paraxial propagation of light in a photonic lattice defined by the index of refraction d = + ( ) n n n x y z , , 0 (d  n n 0 ), where n 0 is the constant background index of refraction, and d ( ) n x y z , , describes small spatial variations, which are slow along the propagation z-axis. The slowly varying amplitude of the electric field y ( ) x y z , , follows the (continuous) Schrödinger equation [26]: 2 , and p l = k n 2 0 , where λ is the wavelength in vacuum. From now on we will refer to d ( ) n x y z , , as the potential or index of refraction. In the simulations we use n 0 = 2.3, corresponding to the systems that were used to implement the optical induction technique [23,24], and l = 500 nm.

Grating assisted tunneling and conical diffraction
For clarity, let us first present the grating assisted tunneling method in a 1D photonic lattice. If the potential is a periodic lattice, , 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]: where J x quantifies the tunneling between adjacent waveguides and y ( ) z m 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 f ( ) K exp i , 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 . For a sufficiently large tilt, the tunneling is suppressed. This can be seen from 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, . This total potential d ( ) 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 k h = ak n 0 and the grating forms a z-dependent perturbation resonant with the index offset between neighboring lattice sites ha ( 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 p = q a x and d d = n n 0.1 G0 L0 . However, the diffraction pattern is drastically changed. In order to interpret it, we point out that, for resonant tunneling where k h = ak n 0 and a sufficiently large tilt ( h  J ), 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): . 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, y = I 0 e m 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 dn G0 . Figure 1(d) shows K x (green squares) and J x (blue circles) versus dn G0 , 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, d = x a y a cos cos . 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 d ( ) n x y z , , is indeed equivalent to the dynamics of equation ( (β is the propagation constant), touching at two 2D Dirac points at 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 = k k k , the group velocity, b  ( ) k k , x y k , 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 Intensity of a beam, which initially excites modes in the vicinity of the Dirac points, after propagation for z = 162mm. The intensity has two concentric rings corresponding to the conical diffraction pattern. (d) Simulation of the diffraction pattern in the discrete model corresponding to the lattice in (a), which also exhibits conical diffraction (see text for details). initial profile given by y x a y a , , 0 cos 2 exp 3 3 2 2 in the 2D potential d ( ) 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 a = 1 2, where α is the flux per plaquette in units of the flux quantum [34,35].

Proposal for experimental realization
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][23][24][25][26]. In these systems, both the lattice d ( ) n x y , L and the grating potential d ( ) n x y z , , G can be obtained in a straightforward fashion by using interference of plane waves in the medium [26]. The lattice constant a, the grating parameter q, and hence the hopping phase f = · q R m n 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 h d D =~´~´n a n 20 20 0.1 8 10

L0
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 dn 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.

On the phase between the lattice and the grating
Before closing, we emphasize that the spatial phase between the periodic lattice potential dn L and the grating dn 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 d d k x = -+ (( ) ) n n q x z cos 2 x G G 0 2 . This simply shifts the phases in the discrete lattice (shown in figure 1(c)) such that f m is replaced by f x + m , which moves the 1D Dirac point in k x -space by x D = k a x . We have numerically verified that this indeed happens.

Conclusion
In conclusion, we have proposed the grating assisted tunneling scheme for tunable synthetic magnetic fields in optically induced one-and two-dimensional dielectric photonic lattices. As a signature of the synthetic magnetic fields and the Dirac points that these lattices possess in k-space, we have demonstrated conical diffraction patterns (shown in figures 1(c) and 2(a)). The excellent agreement between our results for light propagation in realistic (continuous) systems and the evolution in the corresponding discrete models, clearly confirms that the 2D dielectric photonic lattice with grating assisted tunneling constitutes the realization of the Harper-Hofstadter Hamiltonian. We envision that this approach can open the way for other studies of light propagation in tailored dielectric structures (including nonlinear propagation, solitons, instabilities and the creation of synthetic dimensions [47]) being mapped on intriguing discrete models with complex tunneling matrix elements and synthetic magnetic fields.