Observation of robust flat-band localization in driven photonic rhombic lattices

TL;DR

This paper demonstrates that flat-band states in driven photonic rhombic lattices remain localized despite external modulations, due to destructive interference and lattice symmetry, with observed phenomena including Bloch oscillations.

Contribution

It shows the robustness of flat-band localization in photonic lattices under external driving, a novel insight into wave interference and lattice symmetry effects.

Findings

Flat-band states do not diffract under static and high-frequency drivings.

Destructive interference maintains localization despite external modulation.

Observed Bloch oscillations and coherent destruction of tunneling.

Abstract

We demonstrate that a flat-band state in a quasi-one-dimensional rhombic lattice is robust in the presence of external drivings along the lattice axis. The lattice was formed by periodic arrays of evanescently coupled optical waveguides, and the external drivings were realized by modulating the paths of the waveguides. We excited a superposition of flat-band eigenmodes at the input and observed that this state does not diffract in the presence of static as well as high-frequency sinusoidal drivings. This robust localization is due to destructive interference of the analogous wavefunction and is associated with the symmetry in the lattice geometry. We then excited the dispersive bands and observed Bloch oscillations and coherent destruction of tunneling. {\textcopyright} 2017 Optical Society of America.

Equations

$$\displaystyle i\lambdabar\frac{\partial}{\partial z}E(x,z)\!=\!\Big{[}\!-\frac{\lambdabar^{2}}{2n_{0}}\frac{\partial^{2}}{\partial x^{2}}\!-\!\Delta n(x)\!+\!F(z)x\Big{]}E(x,z)$$

$$\displaystyle\big{[}i\partial_{z}+2s\beta\big{]}E^{a}_{s}\!=\!-\kappa\big{(}E^{b}_{s}+E^{b}_{s-1}+E^{c}_{s}+E^{c}_{s-1}\big{)}$$

$$\displaystyle\big{[}i\partial_{z}+(2s+1)\beta\big{]}E^{b}_{s}\!=\!-\kappa\big{(}E^{a}_{s}+E^{a}_{s+1}\big{)}$$

$$\displaystyle\big{[}i\partial_{z}+(2s+1)\beta\big{]}E^{c}_{s}\!=\!-\kappa\big{(}E^{a}_{s}+E^{a}_{s+1}\big{)}$$

Full text

OCIS codes: (130.0130) Integrated optics; (000.1600) Classical and quantum physics; (130.2790) Guidedwaves;(230.7370) Waveguides; (350.3950) Micro-optics.

Nearly a century ago, it was predicted that an electron in a periodic potential and uniform electric field can exhibit periodic Bloch oscillations Bloch (1928); Fukuyama et al. (1973); Dahan et al. (1996); Peschel et al. (1998); Morandotti et al. (1999); Chiodo et al. (2006) due to the formation of Wannier-Stark ladder. An ac electric field, on the other hand, can renormalize the effective tunneling probability Dunlap and Kenkre (1986) causing interesting localization effects such as dynamic localization Dreisow et al. (2008). The dynamics of electrons is also influenced by external magnetic flux, the presence of disorder and particle interactions. It is of great interest to study these fundamental transport phenomena in a lattice geometry supporting non-dispersive (flat) bands; see ref. Tasaki (2008); Huber and Altman (2010); Niţă et al. (2013); Guzmán-Silva et al. (2014); Mukherjee et al. (2015a); Vicencio et al. (2015); taie2015coherent ; Xia et al. (2016); Mukherjee and Thomson (2015); Molina (2015); Nandy and Chakrabarti (2016); Diebel et al. (2016); Weimann et al. (2016); Yang2016 ; Owen2016 for recent works on flat-band lattices. Flat-band eigenmodes are degenerate with infinite effective mass; i.e. a flat-band state has no time evolution. However, in the presence of external electric and magnetic fields, the initial flat-band state of the instantaneous Hamiltonian, $\mathcal{\hat{H}}(t\!=\!0)$, can exhibit complicated dynamics, such as Bloch oscillations associated with Landau-Zener tunneling Cai et al. (2013); Khomeriki and Flach (2016), depending on the lattice geometry and the external drivings. These intriguing phenomena are at the heart of condensed matter physics, and can be investigated in the system of ultracold atoms in optical lattices Bloch et al. (2012) and periodic arrays of coupled optical waveguide (photonic lattices) Longhi (2009); Garanovich et al. (2012). In fact, localized flat-band state was experimentally demonstrated using lattice geometries such as Lieb Mukherjee et al. (2015a); Vicencio et al. (2015); taie2015coherent ; Xia et al. (2016), Kagome Zong et al. (2016), rhombic Mukherjee and Thomson (2015) and stub Baboux et al. (2016).

In this work, we consider a quasi-one-dimensional photonic rhombic [or diamond, Fig. 1 (a, b)] lattice with three sites ($a$, $b$ and $c$) per unit cell. This lattice geometry has been previously used to theoretically study various interesting effects such as magnetic field induced Aharonov-Bohm caging for both interacting and non-interacting particles Vidal et al. (2000); Creffield and Platero (2010); Longhi (2014), Anderson localization Leykam et al. (2013), conservative and PT-symmetric compactons Yulin and Konotop (2013) and Landau-Zener Bloch oscillations Khomeriki and Flach (2016). In the nearest neighbor tight binding approximation, diagonalizing the Fourier-transformed single particle Hamiltonian of the rhombic chain, one obtains the following dispersion relations Vidal et al. (2000): $\varepsilon^{0,\pm}\!=\!0,\pm 2\kappa\sqrt{1+\cos(kd)}$, where $\kappa$ is the nearest neighbor hopping amplitude (or coupling constant), $d$ is the lattice constant and $k$ is the quasimomentum. The lattice supports three energy bands, two dispersive ($\varepsilon^{\pm}$) and one non-dispersive ($\varepsilon^{0}$) in the middle; see Fig. 1(c). A flat-band state can be excited by initially exciting the $b$ and $c$ sites of a unit cell with equal intensity and opposite phases Mukherjee and Thomson (2015).

Under the influence of a weak analogous static force, we show that a state prepared in the dispersive bands of the rhombic lattice, exhibits Bloch oscillations Dahan et al. (1996); Peschel et al. (1998); Morandotti et al. (1999); Chiodo et al. (2006), however, a flat-band state keeps the compactness of the initial state (as $\partial_{t}\psi_{k}^{0}(t)\!=\!0$; see also R. Khomeriki et al. Khomeriki and Flach (2016)). We then show that an analogous high-frequency sinusoidal force modifies the band structure keeping the flat-band unaffected. Interestingly, the initial flat-band state overlaps only with the degenerate Floquet eigenstates, and hence, remains localized. For a state in the dispersive bands, on the other hand, we observed renormalization of tunneling Holthaus and Hone (1993); Grifoni and Hänggi (1998); Della Valle et al. (2007); Mukherjee et al. (2016), as would be expected.

The propagation of light across a one-dimensional photonic lattice where the axes of all the waveguides are simultaneously and slowly bending along the lattice axis ($x$), is governed by the following scalar-paraxial equation Longhi (2009); Garanovich et al. (2012):

[TABLE]

where $E(x,z)$ is the electric field envelope of the light waves, $n_{0}$ is the average refractive index of the medium, $\lambda=2\pi\lambdabar$ is the free-space wavelength, $\Delta n(x)$ is the transverse refractive index profile, $z$ plays the role of time and $F\!=\!-n_{0}\partial_{z}^{2}x_{0}(z)$. Here $x_{0}(z)$ describes the bending profile of each waveguide and $(x,z)$ is the coordinate system moving with the lattice. Equation (1) is analogous to the Schrödinger equation for a quasiparticle (e.g. an electron) in a one-dimensional lattice driven by an external force, $F$. For weak external fields, neglecting the excitation of higher Bloch bands, the following coupled-mode equations are obtained Longhi (2014); Khomeriki and Flach (2016) for a rhombic lattice

[TABLE]

where $\{E^{a}_{s},E^{b}_{s},E^{c}_{s}\}$ are the electric field amplitudes of the light waves at the $a$, $b$ and $c$ sites of the $s$-th unit cell respectively. The analogous static electric field is realized by circularly curving the waveguide axis, $x_{0}^{2}\!=\!R^{2}\!-\!z^{2}$ where $R$ is the radius of curvature, implying almost a linear shift in propagation constant (site energy) along the lattice axis, $\beta\!=\!n_{0}d/(2R\lambdabar)$. Similarly, a sinusoidal ac field is realized by sinusoidally modulating the waveguide paths along the lattice axis. In this case $x_{0}\!=\!A\sin(\omega z)$ and $\beta(z)\!=\!K\sin(\omega z)$, where $A$ and $\omega$ are the amplitude and frequency of the modulation, and $K\!=\!n_{0}A\omega^{2}d/(2\lambdabar)$; see ref. Longhi et al. (2006); Della Valle et al. (2007); Dreisow et al. (2008); Szameit et al. (2009); Longhi and Della Valle (2011); Mukherjee et al. (2015b); Mukherjee et al. (2016) and the references therein for more details on driven photonic lattices. The intensity distribution at the output of the photonic lattice, for a given input excitation, is obtained by numerically solving Eq. (2).

Curved photonic lattices were fabricated inside a borosilicate substrate (Corning Eagle2000) using ultrafast laser inscription technique Davis et al. (1996). The substrate was mounted on $x$-$y$-$z$ translation stages and each waveguide was fabricated by translating the substrate through the focus of the fs laser pulses (350 fs, 500 kHz, 1030 nm). Fabrication parameters were optimized to realize well-confined single-mode waveguides for operation at $780$ nm; see ref. Mukherjee et al. (2015a); Mukherjee and Thomson (2015) for fabrication details. We note that carefully engineered photonic lattices were previously used to experimentally simulate various quantum phenomena including the investigation of localization effects, i.e. the inhibition of transport, caused by disorders Schwartz et al. (2007); Martin et al. (2011), external fields Longhi et al. (2006); Della Valle et al. (2007); Dreisow et al. (2008); Szameit et al. (2009); Longhi and Della Valle (2011); Mukherjee et al. (2015b); Mukherjee et al. (2016) and Kerr nonlinearity Szameit et al. (2006).

It is evident from Eq. (2) that an initial state occupying only $b_{s}$ and $c_{s}$ sites with equal intensity and opposite phases, which excites flat-band eigenmodes, does not have any time evolution in a static field. To demonstrate this, seven circularly curved finite photonic lattices (with propagation lengths 10 mm to 70 mm in steps of 10 mm) were fabricated with 12 unit cells and radius of curvature $R\!=\!1.8$ m. For all the lattices waveguide-to-waveguide separation was $d/\sqrt{2}\!=\!17$ $\mu$m, and the coupling constant was $\kappa\!=\!0.034$ mm*-1*. First, we excited an equal superposition of flat-band eigenmodes by launching the input state $\{E^{b}_{6},E^{c}_{6}\}\!=\!\{1/\sqrt{2},-1/\sqrt{2}\}$ and measured the output intensity distributions at seven different values of propagation distances. As shown in Figs. 2 (a)-(c), we observe that this input state remains localized without significantly tunneling to other lattice sites. However, when the dispersive bands were excited launching an orthogonal mode (we call it equal phase mode, $\{E^{b}_{6},E^{c}_{6}\}\!=\!\{1/\sqrt{2},1/\sqrt{2}\}$), we observe a breathing motion [Fig. 3 and 4 ] of the intensity distribution which is the characteristic of Bloch oscillations. In Fig. 4, the measured variations of the light intensities (red circles: $(|E^{b}_{6}|^{2}+|E^{c}_{6}|^{2})/2$; blue squares: $(|E^{a}_{6}|^{2}+|E^{a}_{7}|^{2})/2$) are plotted as a function of $z$ which are in good agreement with the numerically calculated results. We note that the input states were prepared using a zero-order nulled diffractive optical element; see ref. Mukherjee and Thomson (2015).

In the presence of a sinusoidal driving, the Hamiltonian that describes the system is periodic in $z$, the analogous time, i.e. $\mathcal{\hat{H}}(z)\!=\!\mathcal{\hat{H}}(z+z_{0})$ where $z_{0}\!=\!2\pi/\omega$. In this situation, using Floquet theory Holthaus and Hone (1993) the quasienergy spectrum is obtained by diagonalizing the evolution operator defined as $\hat{U}\!=\!\mathcal{T}\exp[-i\int_{0}^{z_{0}}\mathcal{\hat{H}}(z^{\prime})dz^{\prime}],$ where $\mathcal{T}$ indicates the time ordering. The Floquet quasienergy spectrum for a high-frequency sinusoidal driving (with $K/\omega\!=\!0.912$) is shown in Fig. 5(a) [open squares]. There are two important aspects to be noted. First, the flat-band is not destroyed by the ac driving and second, an initial flat-band state (i.e. $\{E^{b}_{6},E^{c}_{6}\}\!=\!\{1/\sqrt{2},-1/\sqrt{2}\}$) overlaps only with the degenerate Floquet eigenstates; see the blue circles. Therefore, this input state is expected to remain localized. Figure 5(b) shows the Floquet spectrum as a function of $K/\omega$; irrespective of the value of the $K/\omega$, the initial flat-band state overlaps only with the eigenmodes at zero quasienergies (shown by the blue circles). Note that the dispersive bands of the lattice (pseudo) collapse Holthaus and Hone (1993); Della Valle et al. (2007) when $\mathcal{J}_{0}(K/\omega)\!=\!0$ i.e. $K/\omega\!=\!2.405$ [$\mathcal{J}$ is the Bessel function of the first kind]. In other words, a sinusoidal driving renormalizes the effective coupling constant as $|\kappa_{\text{eff}}/\kappa|\!=\!|\mathcal{J}_{0}(K/\omega)|$.

To observe the effect of sinusoidal driving, we fabricated five sinusoidally modulated rhombic lattices ($70$-mm-long) with the amplitude of modulation $A\!=\!6$ $\mu$m to 14 $\mu$m in steps of 2 $\mu$m, and the spatial frequency $\omega\!=\!1.5708$ mm*-1*. The output intensity distributions for two types of input states are shown in Fig. 6. It was observed that the flat-band modes are not affected by the sinusoidal driving whereas the output intensity distribution for equal-phase mode is determined by $K/\omega$. Note that for Fig. 6(g), the ratio of the field magnitude and the field frequency (i.e. $K/\omega$) is close to the first root of $\mathcal{J}_{0}$ causing coherent destruction of tunneling. The effective coupling constant of a modulated lattice, for a given value of $K$ and $\omega$ was evaluated by simulating a $70$-mm-long straight photonic lattice, and varying the coupling strength to optimally fit the observed output intensity distribution. The estimated values of normalized effective coupling constants, $|\kappa_{\text{eff}}/\kappa|$, are in agreement with the theoretical prediction as shown in Fig. 7(a).

The robust localization of the flat-band modes in the rhombic chain can also be explained intuitively. In the presence of a static or ac driving, the effective propagation constant (or the analogous site energy) of the waveguides in the lattice at a given value of $z$ is determined by the driving strength and its direction Heiblum and Harris (1975); Lenz et al. (1999). If the lattice is driven along the lattice axis, the $b$ and $c$ sites of the s-th unit cell have the same effective propagation constant which is shifted by $\pm\beta$ compared to the $a$ site of the $s$-th and the $(s+1)$-th unit cell respectively, see Fig. 7(b). In this situation, the flat-band input state cannot tunnel to the nearest neighbor sites due to destructive interference as for the undriven case. Note that an extended initial flat-band state in real space will also exhibit this robust localization. On the basis of this argument, one can infer that the symmetry in the lattice geometry is crucial to observe this robust localization in the presence of both static and ac drivings.

In conclusion, we investigated the dynamics of the initial states prepared in the dispersive and non-dispersive bands of the photonic rhombic lattices in the presence of both static and high-frequency sinusoidal drivings along the lattice axis. We experimentally excited flat-band eigenmodes and show that they remain localized in the presence of these drivings. However, when dispersive bands were excited by launching an orthogonal input state, we observed Bloch oscillations and coherent destruction of tunneling respectively. This robust flat-band localization is due to destructive interference of the analogous wavefunction and is associated with the symmetry in the lattice geometry.

Data.$-$ Raw experimental data are available through Heriot-Watt University PURE research data management system data .

Acknowledgments.$-$ S. M. thanks Mr. Alexander Spracklen (UK) and Prof. Nathan Goldman (Belgium) for helpful comments.

Funding.$-$ UK Science and Technology Facilities Council (STFC) ST/N000625/1.