Photonic quadrupole topological phases

TL;DR

This paper reports the first experimental realization of a photonic quadrupole topological phase in silicon photonics, demonstrating robust corner states in a 2D system, which could lead to advanced on-chip optical devices.

Contribution

It provides the first demonstration of a photonic quadrupole topological phase, showcasing robust corner states and their protection in a 2D silicon photonic system.

Findings

Observation of localized corner states in a 2D photonic system.

Robustness of corner states against certain defects.

Contrast with trivial systems showing no robustness.

Abstract

The topological phases of matter are characterized using the Berry phase, a geometrical phase, associated with the energy-momentum band structure. The quantization of the Berry phase, and the associated wavefunction polarization, manifest themselves as remarkably robust physical observables, such as quantized Hall conductivity and disorder-insensitive photonic transport. Recently, a novel class of topological phases, called higher-order topological phases, were proposed by generalizing the fundamental relationship between the Berry phase and the quantized polarization, from dipole to multipole moments. Here, we demonstrate the first photonic realization of the quantized quadrupole topological phase, using silicon photonics. In this 2nd-order topological phase, the quantization of the bulk quadrupole moment in a two-dimensional system manifests as topologically robust corner states. We unambiguously show the presence of localized corner states and establish their robustness against certain defects. Furthermore, we contrast these topological states against topologically-trivial corner states, in a system without bulk quadrupole moment, and observe no robustness. Our photonic platform could enable the development of robust on-chip classical and quantum optical devices with higher-order topological protection.

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Full text

The Berry phase provides a universal framework which relates the robust quantization of physical observables at the boundaries of a non-interacting system to the topological properties of the bulk. For example, in electronic systems, the Berry phase dictates the presence of quantized zero-dimensional (0D) charges in one-dimensional (1D) insulators Su1979 ; Zak1989 , quantized charge/spin currents along the 1D-edges of a 2D quantum Hall/spin Hall systems Thouless1982 ; Bernevig2006 , and conducting 2D surfaces in 3D topological insulators Liang2011 . In general, this bulk-boundary correspondence relates the quantized charge/currents carried by the $(n-1)$-dimensional boundaries to the quantized electric dipole moments of the $n$-dimensional bulk King-Smith1993 . Recently, this relationship between the charge moments of the bulk and their boundary manifestations has been generalized from electric dipole to multipole moments, leading to higher-order topological phases Benalcazar2017 ; Benalcazar2017b ; Schindler2018 ; Schindler2018b . For example, a quantized quadrupole moment in a 2D system leads to the presence of quantized charges, which are localized at the 0D corners. This is in contrast to the quantized dipole moment which gives rise to 1D edge currents, in a 2D system.

When formulated in terms of the Berry phase, the bulk-boundary correspondence also applies to neutral bosonic systems, that is, the non-trivial topology of the bulk leads to the emergence of robust boundary states. In fact, topologically robust, localized 0D edge states in 1D systems and propagating 1D edge states in 2D systems have been realized using atomic Stuhl2015 ; Atala2013 ; Aidelsburger2015 , photonic Wang2009 ; Hafezi2013 ; Rechtsman2013 ; Kraus2012 ; Cheng2016 ; Ozawa2018 as well as phononic lattices Susstrunk2015 ; Peano2015 ; Yang2015 . Motivated by the search for higher-order topological phases, the 2D quadrupole topological phases with robust 0D corner states were recently observed using microwaves Peterson2018 ; Imhof2018 ; Serra2018arXiv and phononic meta-materials Serra-Garcia2018 ; Xue2018 ; Ni2018 . We report on the first optical realization of the quadrupole topological phase, employing the integrated silicon-photonics platform. Using spectroscopic measurements and direct imaging, we reveal the existence of the localized corner states and show that they are robust against certain fabrication disorders which are ubiquitous in nanophotonic systems. Furthermore, by introducing a quadrupole domain boundary in our system, we show that the observed corner states are not artifacts at the physical corners of the lattice. We note that the corner states can also emerge in systems without a quadrupole moment Noh2018 ; Li2018 . We study corner states associated with zero bulk quadrupole moment and observe that they are not immune to fabrication disorders.

Our quadrupole topological system is realized using a two-dimensional (2D) lattice of nanophotonic silicon ring resonators (Fig.1, see SI for details on the device fabrication). The unit cell (plaquette) of this lattice consists of four site rings arranged into a square (shaded blue in Fig.1). These site rings are evanescently coupled to their nearest neighbors using a set of auxiliary rings, which we call link rings (shaded green and red). We control the magnitude of the coupling strength between the lattice sites by adjusting the gap between the site-ring and the link-ring waveguides. Furthermore, the resonance frequencies of the link rings are detuned from those of the site rings by introducing an extra path length. This allows us to manipulate the sign (phase) of the coupling between the site rings and introduce a synthetic gauge flux threading each square plaquette Hafezi2011 ; Hafezi2013 . Specifically, when the link ring coupling two site rings along the x-axis is vertically shifted, the photons hopping towards the right travel a slightly longer path compared to those hopping towards the left (Fig.1). This path length difference results in an effective, direction-dependent, hopping phase $\phi$. We choose $\phi=\pi$ such that the shifted link ring (shaded red) introduces a negative coupling. In a unit cell, we arrange the link rings such that $\phi\neq 0$ for only one coupling term, while keeping the amplitude of coupling equal to $\gamma$. This results in a net gauge flux $\pi$ threading the unit cell (Fig.1b). To implement a topological system with quantized quadrupole moment Benalcazar2017 , we realize a 2D array of these unit cells, coupled to their nearest neighbors with coupling strength $\pm\kappa$. Throughout the lattice, we change the sign of the inter-cell coupling $\kappa$, such that a uniform gauge flux $\pi$ penetrates any plaquette, made of $2\times 2$ array of site rings (see Figs.1b,c, and Fig.2a). When $\gamma/\kappa<1$, that is, when the inter-cell coupling is stronger than the intra-cell coupling, the lattice hosts degenerate mid-gap states localized at each of the four corners (Fig.2b,e). These states are characterized by topologically invariant integers and protected against certain disorders, due of the presence of spatial inversion symmetry, and the two mirror symmetries (about $x-$ and $y-$ axis) of the lattice which do not commute as a result of the non-zero gauge flux Benalcazar2017 . In the other case, that is when $\gamma/\kappa>1$, the lattice is topologically trivial and there are no states in the bandgap (Fig.2h).

In our experiment, we fabricate a $5\times 5$ array of these unit cells, as shown in Fig.2a. We design the system to be in the topological regime with the coupling strengths $\gamma$ and $\kappa$ estimated to be $4.8(2)$ GHz and $29.0(8)$ GHz, such that $\gamma/\kappa\approx 0.17$. To probe the presence of corner states in the lattice, we fabricate a waveguide coupler at each of the corners (Fig.2a). This allows us to couple the lattice to a laser and measure the power absorption, as a function of the laser frequency detuning $\left(\delta\nu\right)$ from the ring resonance frequency $\left(\nu\right)$ (see Supplementary Information for further details). Furthermore, we use a microscope objective and an infrared camera to directly image the spatial intensity profile in the lattice, as we sweep the laser frequency. The measured absorption spectra at the four corner sites are shown in Fig.2d. We observe very narrow absorption peaks indicating the presence of four isolated mid-gap states in the lattice (Fig.2b). Figure 2f shows the measured spatial intensity profile in the lattice, integrated over the absorption peaks. In agreement with numerical simulation (Fig.2e), we see the corner modes are remarkably localized at the corner sites. However, contrary to the theoretical prediction (Fig.2b,c), the absorption peaks are shifted with respect to each other, indicating that the corner modes are not degenerate in frequency. This is due the nanofabrication process which introduces very strong disorder in the ring resonance frequencies. The upper bound on the standard deviation in the ring resonance frequencies was measured to be $\approx$ 33 GHz which is very significant compared to the band-gap $2\kappa\approx 58$ GHz. In addition, there is a small disorder in the coupling strengths $\gamma$ and $\kappa$ which was estimated to be $\approx 3\%$ of the mean values, and also a disorder of $0.1$ (radian) in the gauge flux $\phi$. Remarkably, even in presence of such a strong disorder which is comparable to the band-gap, the corner states are still present and very localized which exhibits their topological protection. We note that the “zero energy” in these measurements corresponds to the resonance frequency of a particular longitudinal mode of the ring resonators.

To show the absence of corner states in a trivial system, we analyze the scenario where the inter-cell hopping is much stronger than the intra-cell hopping, that is, $\gamma/\kappa>1$ (Fig.2g). For this system, we swap the two coupling strengths, that is, now $\gamma\approx 29$ GHz and $\kappa\approx 4.8$ GHz, which gives $\gamma/\kappa\approx 5.8$. As such, the lattice is topologically trivial and the mid-gap modes are absent (Fig.2h). Fig.2j shows the measured power absorption spectrum at the four corners and Fig.2l shows the spatial intensity distribution integrated over the absorption bands. In contrast to Fig.2d, the spectra in Fig.2j show two broad absorption bands centered around $\pm\sqrt{2}\gamma$, corresponding to the bulk modes of the lattice Benalcazar2017 . Furthermore, the observed spatial intensity distributions (Fig.2l) confirm that these absorption bands do not correspond to localized modes and are smeared into the bulk of the lattice.

Next, we demonstrate that the observed corner states are not artifacts due to some defects at the physical corners of the lattice. We break the periodicity of the intra-cell coupling strength such that it creates a domain boundary, with the unperturbed quadrupole domain, this time consisting of an array of $5\times 4$ unit cells (Fig.3a). In this configuration, the corners $C_{2}$ and $C_{3}$ of the quadrupole domain coincide with the physical corners of the lattice whereas, corners $C_{1}$ and $C_{4}$ of the quadrupole domain are shifted from the physical corners, labeled $C^{\prime}_{1}$ and $C^{\prime}_{4}$ in Fig.3a. As before, the waveguide couplers for absorption measurements are located at the physical corners of the lattice. Fig.3b,c show the measured absorption spectra. The spectra at corners $C_{2}$ and $C_{3}$ show a single absorption peak and the associated spatial intensity distribution (Fig.3d) shows localized corner states at $C_{2}$ and $C_{3}$, consistent with our previous observation. However, the spectra measured at corners $C^{\prime}_{1}$ and $C^{\prime}_{4}$ show three absorption peaks. The spatial intensity distribution corresponding to the middle absorption peak around zero frequency detuning, in fact, shows excitation of localized modes at the corners $C_{1}$ and $C_{4}$ of the quadrupole domain (Fig.3d). In contrast, the intensity distribution in the other two sidebands, away from zero detuning, occupies the lattice sites outside the quadrupole domain (Fig.3e). The power absorption in the middle band is lower relative to the sidebands, mainly because the corner states of the quadrupole are very weakly coupled to the physical corner.

To further explore the significance of topological protection of the observed corner modes in the quadrupole system, we compare them with those emerging in a system with zero quadrupole moment. We fabricated a very similar lattice with zero gauge flux, that is, where all the couplings are positive (Fig.4a). The inter-cell and the intra-cell couplings in this lattice still correspond to the topological case (Fig.2a), that is, $\gamma/\kappa<1$. Because of the absence of the gauge flux, the two mirror symmetries (about the $x-$ and the $y-$axis) commute and the net quadrupole moment is zero (see S.I.). Numerical simulation results, in the absence of disorder, show that this system hosts zero-energy modes localized at the corners, similar to those of the quadrupole system (Fig.4b,c,e). However, unlike the quadrupole system, the measured absorption spectrum at the corner shows a multi-mode structure indicating that the corner states are not completely isolated from the bulk states (Fig.4d,f, additional data in the S.I.). The measured spatial intensity distribution confirms that these corner modes indeed couple to the bulk modes of the lattice. This observation suggests that these corner modes, in the zero gauge flux lattice, are susceptible to the fabrication disorder and are not topologically protected. This lack of robustness against disorder is also evident in the energy spectrum of the device, where there is no band-gap at zero energy. Consequently, an on-site potential disorder, which in our experimental setup is the most dominant source of disorder, can easily couple these corner modes to the bulk modes located near zero energy (see supplementary information for more details).

In this work, we have demonstrated an integrated photonic quadrupole topological phase supporting robust corner states immune to disorder in coupling strengths and on-site potential. Our versatile integrated photonics platform opens the route to explore even richer physics in the context of higher-order topological phases, for example, those associated with different choices of gauge flux in the lattice or with asymmetric coupling strengths. Moreover, by integrating active features, e.g., using thermal heaters or electro-optic modulators Mittal2016 , our scheme could allow selective flux insertion or manipulation of the corner or the edge modes of the system and hence, the realization of adiabatic charge pumping in high-order topological systems Benalcazar2017b . Furthermore, the integration of gain/loss in this photonics realization could enable the exploration of active optical devices, such as topological lasers St-Jean2017 ; Bahari2017 ; Bandres2018 , quantum light sources Mittal2018 , and non-Hermitian physics, in the context of higher-order topological physics.

Note: During the submission of this manuscript, we became aware of other related works Ota2018 ; ElHassan2018 which realize corner states of light using generalizations of the SSH model. However, these models lack gauge fields and are not direct realizations of the quadrupole topological phases.

Acknowledgements: This research was supported by AFOSR-MURI FA9550-14-1-0267, Sloan Foundation and the Physics Frontier Center at the Joint Quantum Institute. AP and MG have been supported by the Russian Foundation for Basic Research Grant No. 18-29-20037 and the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS . AP also acknowledges partial support by the Australian Research Council. We thank Hossein Dehghani, Yuri Kivshar and Daniel Leykam for discussions, Evan Yamaguchi and Jonathan Vannucci for experimental help.