Discrete, Nonlinear and Disordered Optics

Squeezed states are quantum states of light, which recently proved to play an important role in modern physics due to their connection with quantum information and their employment in gravitational wave detection. The spectral properties of the squeezing operator were discussed in the late 80s. The squeeze operator has a purely continuous spectrum that covers the unitary circle in the complex plane. However, this statement has been recently generalized to include a family of discrete eigenvalues and eigenvectors (Gamow vectors) in a rigged Hilbert space. [1] In this talk, we want to show that squeezed states appear in the development of a classical optical shock phenomenon. A shock wave is a singular solution of a hyperbolic partial differential equation, which is a class of equations that describes a wide variety of wave-like phenomena in physics, ranging from fluido-dynamics to plasmas and Bose-Einstein condensation (see e.g. [2,3]). Recently, it has been proved that shock waves evolution in a nonlinear and highly nonlocal medium can be described through the generalized eigenstates of a reversed harmonic oscillator (RHO), called Gamow states. [4] Here we show that the Gamow vector formalism allows to interpret the squeezing operator as the wave-function propagator of a shock wave. Indeed, we can express the evolution of a Gaussian wave-function in a nonlinear nonlocal medium as a squeezing operator acting on the eigenfunctions of a RHO. Theoretical analysis and numerical simulations reveal that the quadrature (x-p) and (x+p), where x is the position operator and p is the momentum operator, are squeezed during the excitation of a shock wave of a Gaussian light beam both in the temporal and spatial cases. This gives evidence of the presence of squeezed light during shock dynamics. [5] These results open new perspectives on the shock phenomena and links quantum optics and its applications to nonlinear waves in extreme regimes. 1. D. Chruscinski, Phys. Lett. A, 327, 290-295 (2004) 2. W. Wan, Shu Jia and J. W. Fleischer, Nat. Phys. 3, 46 - 51 (2007) 3. M. Conforti, F. Baronio and S. Trillo, Phys. Rev. A 89, 013807 (2014) 4. S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe and C. Conti, Phys. Rev. A, 92, 023801 (2015) 5. M.C. Braidotti, A. Mecozzi, C. Conti, in preparation.

The novel material class of topological insulators exhibits topological protected scatter-free and unidirectional transport along their edges independent of the exact details of the edges. A common way to characterize a two-dimensional topological system is the Chern number of the bands. This number is equal to the difference between the numbers of edge modes entering each band from below and exiting above. However, in periodically driven systems with a time-dependent Hamiltonian the Chern number does not give a full characterization of the topological properties. We implement a waveguide system exhibits chiral edge modes despite the fact that the Chern numbers of all bands are zero. We show the topologically protected edge transport in this fs laser written lattice. Therefore we demonstrate the implementation of such anomalous Floquet topological insulators in the photonic regime. Authors: Lukas J. Maczewsky, Julia M. Zeuner, Stefan Nolte und Alexander Szameit

Following the recent advances in non-Hermitian photonics that were initiated by Parity-Time (PT)-symmetric optics, we will present a new type of waves of constant intensity that exist only in complex media with suitably engineered gain-loss distributions. Several aspects and properties of such waves will be examined in discrete coupled waveguides-cavities (unidirectional invisibility), nonlinear systems (modulation instabilities) and disordered scattering media (perfect transmission).

Nowadays, the integration of a single quantum emitter with different photonic architectures poses a great research challenge. Indeed, this interconnection can be considered as the heart of various cutting edge research topics including cryptography, cavity quantum electrodynamics, and quantum information networks. A common requirement for all these implementations is highly engineered optical photonic platforms that are inherently sensitive to fabrication imperfections. In other words, the term disorder in photonics tends to carry a negative stigma. Nevertheless, it represents also the common denominator of a plethora of phenomena, characterized by a multiple-scattering description. In other words, multiple scattering of photons in disordered dielectric structures offers an alternative route to light confinement. Recent advances in the engineering of light connement in this system, have demonstrated the ability to fully control the spectral properties of an individual photonic mode paving the way for the creation of open transmission channels in strongly scattering media. The optical confinement and the coupling between modes open new perspectives over the control of the light flow in random media, as well as the possibility to build architectures tuned for efficient light-matter interaction. We have manipulated the interplay between order, disorder and correlation to tune and to control the flow of light in complex systems. In details, we have investigated the appearance of quasi-modes, called necklace states, to create a chain of hybridized localized states extended from one end of disordered dielectric membranes to the other. Preliminary results show a profound relations between the mode spatial extent and the reminiscence of the photonic band gap of different random membranes. In other words, the occurrence of pseudogaps in the photon density of states seems to be the fundamental ingredient to design optical-disordered membranes with an high degree for system interconnections.

In recent years, the ever-increasing demand for high-capacity transmission systems has driven remarkable advances in technologies that encode information on an optical signal. Mode-division multiplexing makes use of individual modes supported by an optical waveguide as mutually orthogonal channels. The key requirement in this approach is the capability to selectively populate and extract specific modes. Optical supersymmetry (SUSY) has recently been proposed as a particularly elegant way to resolve this design challenge in a manner that is inherently scalable, and at the same time maintains compatibility with existing multiplexing strategies. Supersymmetric partners of multimode waveguides are characterized by the fact that they share all of their effective indices with the original waveguide. The crucial exception is the fundamental mode, which is absent from the spectrum of the partner waveguide. Here, we demonstrate experimentally how this global phase-matching property can be exploited for efficient mode conversion. Multimode structures and their superpartners are experimentally realized in coupled networks of femtosecond laser-written waveguides, and the corresponding light dynamics are directly observed by means of fluorescence microscopy. We show that SUSY transformations can readily facilitate the removal of the fundamental mode from multimode optical structures. In turn, hierarchical sequences of such SUSY partners naturally implement the conversion between modes of adjacent order. Our experiments illustrate just one of the many possibilities of how SUSY may serve as a building block for integrated mode-division multiplexing arrangements. Supersymmetric notions may enrich and expand integrated photonics by versatile optical components and desirable, yet previously unattainable, functionalities.

Magnetic metamaterials consist of artificial structures whose magnetic response can be tailored to a certain extent. A common realization of such system consists of an array of metallic split-ring resonators (SRRs) coupled inductively. This type of system can feature negative magnetic response in some frequency window, making them attractive for use as a constituent in negative refraction index materials. In the absence of dissipation and nonlinearity, periodic arrays of SRR support the propagation of magnetoinductive waves, and the eigenfrequencies of the system form bands. In this talk, we will show several examples about localized modes, fano resonances, PT-symmetry and flat bands, in the context of magnetic metamaterials. In particular we will focus on: (a) Bulk and surface magnetoinductive breathers in binary metamaterials (b) Defect modes, Fano resonances and Embedded states in Magnetic Metamaterials. (c) Bounded dynamics of finite PT-symmetric magnetoinductive arrays (d) Flat band modes in quasi one dimensional magnetic metamaterials

Wave functions traversing a lattice potential subjected to an external gradient force exhibit a periodic spreading and re-loalization known as Bloch oscillations (BOs). Albeit BOs are deeply rooted into the very foundations of quantum mechanics, all experimental observations of this phenomenon so far have only considered dynamics of one or two particles initially prepared in separable local states. A more general description of the phenomenon should evidently contemplate the evolution of nonlocal states. In our work, we experimentally investigate BOs of two-photon nonlocal N00N states, defined as: $\ket{\psi^{(\theta)}} = \frac{1}{2}[(\hat{a}^{\dagger}_m)^2 + \e^{\i\theta}(\hat{a}^{\dagger}_n)^2]\ket{0}$, where the indices $m$ and $n$ refer to the lattice sites and $\theta$ is a relative phase. \par Using the direct laser writing technique, we have fabricated BO lattices with 16 waveguides following a curved trajectory, designed to investigate the time-evolution of the quantum correlations of N00N states undergoing BOs. Three input states are considered (symmetric: $\theta = 0$, antisymmetric: $\theta=\pi$, and partially symmetric: $\theta\in(0,\pi)$). Importantly, all three states are prepared on-chip by introducing detuned directional couplers, which guarantee ultra-high experimental stability. \par Quantum features of light are probed by two-photon correlations. The spatial correlation function $\Gamma_{k,l}(z)$ is obtained through measurements of the probability of simultaneously detecting one photon exiting from waveguide $k$ and its twin at site $l$ after propagation over $z$. Our results clearly indicate a correlation turning point and verify the theoretically predicted cyclic behavior of the correlation patterns. This may render BO lattices as a quantum simulator capable of tailoring particle statistics.

In the presence of strong random scattering the behavior of photons in photonic crystals with degenerate spectra is quite different from Anderson localization of photons in a single band: it creates geometric states rather than confining the photons to an area of the size of the localization length. This type of confinement can be understood as angular localization, where the photons of a local light source can propagate only in certain directions. The directions are determined by the boundary of the spectrum. Thus, the system's properties on the shortest scales determine the behavior of the photon propagation on the largest scales.

Periodically driven quantum systems provide a novel and versatile platform for realizing topological phenomena. Among these are analogs of topological insulators and superconductors, attainable in static systems. However, some of these phenomena are unique to the periodically driven case. I will describe how the interplay between periodic driving, disorder, and interactions gives rise to new steady states exhibiting robust topological phenomena, with no analogues in static systems. Specifically, I will show that disordered two dimensional driven systems admit an “anomalous" phase with chiral edge states that coexist with a fully localized bulk. I will show that the micromotion of particles in this phase gives rise to a quantized time-averaged magnetization density when the system is filled with fermions. Furthermore we find that a quantized current flows around the boundary of any filled region of finite extent. The quantization has a topological origin: we relate the time-averaged magnetization density to a topological winding number characterizing the new phase. Thus, we establish the winding number invariant can be accessed directly in bulk measurements, and propose an experimental protocol to do so using interferometry in cold-atom based realizations. In addition, I will show that the bulk topological invariant can be accessed by measuring the current flowing between two terminals when a large bias is applied.

We consider a quantum particle in a one-dimensional disordered lattice with Anderson localization, in the presence of multi-frequency perturbations of the onsite energies. Using the Floquet representation, we transform the eigenvalue problem into a Wannier-Stark basis. Each frequency component contributes either to a single channel or a multi-channel connectivity along the lattice, depending on the control parameters. The single channel regime is essentially equivalent to the undriven case. The multi-channel driving substantially increases the localization length for slow driving, showing two different scaling regimes of weak and strong driving, yet the localization length stays finite for a finite number of frequency components.

We present experimental and theoretical investigations into noise induced switching between bistable states of an optically pumped semiconductor laser with feedback. In particular we show that even small amounts of noise can destroy a regular pulse train while a much greater amount of noise is required to create it. The statistics of the experimental system are compared to realistic simulations and a good agreement is found. Future applications for creating more robust systems will be discussed.

We demonstrate an experimental measurement of the Berry curvature in an optical network. By coupling two fiber loops of different length, a mesh lattice is generated, which is spanned by a discrete time and position. The dimension of the system is further increased by implementing a temporal driving of the lattice similar to a Thouless pump. However, as our system is topological trivial, a geometrical charge pump is instead used, which relies on the excitation of a spectrally narrow wave packet, which probes the underlying Berry curvature. By tracing the wave packet during its propagation through the lattice with a high accuracy, a lateral displacement is measured, which originates from the anomalous velocity due to the Berry curvature of the driven lattice. Furthermore, the time dependent measurement of the displacement allows for a full reconstruction of the Berry curvature, which we demonstrated for two different temporal drivings. M. Wimmer et al. accepted for publication in Nature Physics.

Many intriguing phenomena in nature are connected to singularities, ranging from black holes to phase transitions. In optics, the most prominent example of singularities are those appearing in the phase of light as optical vortices or in polarization as points and lines, ranging from sunlight to the light transmitted by birefringent materials. A unique type of singularities common in wave phenomena are caustics, representing singularities in the intensity of a wave field. Caustics appear when multiple rays of light coalesce and form bright focusing features, representing the singular point. They have been first observed in the 70ies and 80ies in natural optics, at rainbows, or as networks of high intensity at the bottom of shallow waters. It is only recently that with the maturity of tailoring and sculpting light the artificial and controlled embedding caustics in light fields led to accelerating Airy and Pearcey beams with unique propagation properties. Like many other types of singularities, caustics have universal properties, and are not restricted to optics. Their structure and diffraction patterns can be described in terms of universal classes, according to the dimensionality of the system. Mathematically, caustics are described in terms of catastrophe theory, a branch of bifurcation theory in nonlinear physic, which studies how the qualitative behavior of dynamical system changes upon small shifts in external control parameters known as bifurcations. Each caustic can be associated with a catastrophe class that dictates its properties. In our contribution, we will exploit the basic class of fold catastrophe leading to Airy beams as well as higher-order caustics to create nonlinear refractive index lattices. Due to the striking features of these caustic light structures including acceleration in the transverse direction, auto-focusing, and form-invariant propagation, we demonstrate photonic lattices with extraordinary wave guiding and splitting behavior. Nonlinear light-matter interaction of complex light with caustic lattices also leads to novel solitary light phenomena.

In previous works [1,2] new bound coherent states construction, based on a Keldysh conjecture, was introduced. We will show in this contribution that these coherent states of [1,2] are formed by bilinesr combination of paraboson states. As was shown in [1] the particular group structure arising from the model leads to new symmetry transformations for the coherent states system. As was shown, the emergent new symmetry transformation is reminiscent of the Bogoliubov ones and was successfully applied to describe an excitonic system showing that is intrinsically related to the stability and its general physical behavior. The group theoretical structure of the model permits to analyze its thermal properties in theoretical frameworks that arise as a consequence of the definition of the squeezed-coherent states as transformed vacua under the automorphism group of the commutation relations, as the thermofield dynamics case given by Umezawa and other similar developments. On the other hand, the idea of a possible application of such coherent state construction to the quantum dot-confined dark exciton in semiconductors has attracted the attention of both experimentalists and theoreticians recently being one of the main questions what happens with the influence of non zero temperature in the case of qubit stability[3]. In this paper we considered the theoretical treatment of the thermal behaviour for dark exciton by mean a new coherent state construction in a quantum field theoretical context. 1) Cirilo-Lombardo, Phys.Part.Nucl.Lett. 11 (2014) 502-505 2) Cirilo-Lombardo, D.J. J Low Temp Phys (2015) 179: 55. doi:10.1007/s10909-014-1236-z 3)I. Schwartz, E. R. Schmidgall, L. Gantz, D. Cogan, E. Bordo, Y. Don, M. Zielinski, and D. Gershoni Phys. Rev. X 5, 011009

For the natural supratransmission phenomenon, there exists a threshold of amplitude above which energy can flow in the lattice. We numerically show that nonlinear band gap transmission is possible with driven amplitude below the threshold using Togueu Motcheyo et al. method’s [Phys. Rev. E 88, 040901(R) (2013)].

X waves are solutions of Maxwell's equations, which exhibit neither diffraction nor dispersion during propagation. Traditionally, X waves are understood as superpositions of zeroth order Bessel beams, therefore carrying no orbital angular momentum (OAM). A generalisation to the case of OAM-carrying X waves, however, has been proposed recently, highlighting new features and possibilities, especially in free-space classical and quantum communications [1]. For the latter case, in particular, a careful analysis of he quantum properties of X waves with OAM is needed. In this work, we present a complete and consistent quantum theory of generalised X waves carrying orbital angular momentum, propagating in nonlinear dispersive media. Our quantisation scheme is based on canonical quantisation in terms of collections of harmonic oscillators solidal with the X wave itself. Within this framework, we show that the resulting quantised pulses are resilient against external perturbation, due to their intrinsic non-diffracting and non-dispersive nature. Moreover, as an example of application of our formalism, we consider explicitly the case of squeezing in $\chi^2$-media. Our findings reveal that the OAM carried by the X wave regulates which quadrature of the field is squeezed, while the X wave velocity (namely, its Bessel cone angle) regulates the amount of squeezing. in particular, we show that there exist an optimal Bessel cone angle, for which the squeezing is maximum. References: [1] M. Ornigotti, C. Conti and A. Szameit, Phys. Rev. Lett. 115, 100401 (2015).

A new theoretical framework is developed to study dynamical localization (quantum suppression of classical diffusion) in the context of ultracold atoms in periodically shaken optical lattices. Our method is capable to account for finite-time modulations with different shapes, thus going beyond the paradigmatic kicked rotator.

During this talk, I will discuss the "photonic-lantern" - a remarkable photonic technology that facilitates the efficient coupling of multimode light to single-mode photonic devices. I will also discuss how these devices are being used for a variety of applications, including astronomy and telecommunications.

When waves propagate through weakly scattering but correlated, disordered environments they are randomly focussed into pronounced branch-like structures, a phenomenon referred to as \emph{branched flow} which has been studied in a wide range of isotropic random media. In many natural environments, however, the fluctuations of the random medium typically show pronounced anisotropies. We study the influence of anisotropy on such natural focusing events and find a strong and non-intuitive dependence on the propagation angle which we explain by stochastic ray theory.

Optical photonic lattices attract strong interest as a flexible experimental platform with applications ranging from beam shaping and switching to quantum photonics. Importantly, various array designs were suggested by mapping quantum Hamiltonian dynamics to the optical beam evolution. We suggest a novel concept and formulate a general mathematical procedure to exactly map the system dynamics from arbitrary multi-dimensional lattices to light propagation in a one-dimensional optical waveguide array. We realize this approach experimentally in a fs laser written waveguide lattice, and demonstrate the equivalent mapping of a 2D square lattice to a planar array. We anticipate that our results can open new possibilities for implementation of multi-dimensional effects, including topological phenomena, in various planar integrated platforms including Si photonic chips.

Recent demonstrations of the nonlinear Fourier transform (NFT) based fibre-optic communication systems has shown its potential to overcome some fibre impairments. These systems usually consider the signal to vanish as time tends to infinity. However, working with the periodic signals brings about some benefits which are of avail especially regarding the communication application requirements. To name a few; periodic NFT (PNFT) makes it possible to control the time duration of the signal which makes the encoding procedure of the transmitter more straightforward in comparing with its vanishing signal counterpart. Supporting the idea of inserting cyclic extension to avoid inter-symbol interference, the processing window of the PNFT system can be considerably smaller especially in high rate transmissions. On top of that, one can avoid the sudden drop in the signal power ensued from the necessary condition of the conventional NFT to maintain the vanishing boundaries. In this work we simulate and evaluate the performance of a communication system based on PNFT.

Topologically protected quantum effects can be observed in a periodically driven system where the Hamiltonian is time-periodic. In the limit of low frequency driving (driving frequency ~ hopping amplitude), the standard topological invariants, such as Chern numbers, may not be sufficient to describe the topology of the system. In this situation, chiral edge modes can exist even if the Chern numbers of all the bulk bands are zero. We demonstrate the experimental observation of such "anomalous" topological edge modes in a two-dimensional photonic lattice, where these propagating edge states are shown to coexist with a quasi-localized bulk. Ref. Nat. Commun, 8, 13918 (2017).

Broadband coherent light sources are becoming increasingly important for sensing and spectroscopic applications, especially in the mid-infrared and terahertz (THz) spectral regions, where the unique absorption characteristics of a whole host of molecules are located. The desire to miniaturize such light emitters has recently led to spectacular advances, with compact on-chip lasers that cover both of these spectral regions. The long wave-length and small size of the sources result, however, in a strongly diverging laser beam that is difficult to focus on the target that one aims to perform spectroscopy with. In my presentation I will introduce an unconventional solution to this vexing problem, relying on a random laser to produce coherent broadband THz radiation as well as an almost diffraction-limited far-field emission profile [1]. Our quantum cascade random lasers do not require any fine-tuning and thus constitute a promising example of practical device applications for random lasing (see a recent discussion of our work in [2]). References [1] S. Schönhuber, M. Brandstetter, T. Hisch, C. Deutsch, M. Krall, H. Detz, G. Strasser, S. Rotter, and K. Unterrainer "Random lasers for broadband directional emission," Optica 3, 1035 (2016). [2] D. S. Wiersma "Optical physics: Clear directions for random lasers." Nature 539, 360 (2016).

Patterning surfaces with subwavelength spaced metallo-dielectric features (metasurfaces) allows one to generate complex wavefronts by locally controlling the amplitude, phase and polarization of the scattered light.1,2 Recent developments in high performance metalenses and applications to chiral imaging 3 and high resolution ultracompact spectrometers 4 will be discussed, along with spin-to-orbital angular momentum conversion5 and the creation of arbitrary Bessel beams using a single phase plate.6 We will present a method which significantly expands the scope of metasurface polarization optics by enabling the imposition of two independent and arbitrary phase profiles on any pair of orthogonal states of polarization linear, circular, or elliptical relying only on simple, linearly birefringent wave plate elements arranged into metasurfaces.7 Using this approach, we demonstrated chiral holograms characterized by fully independent far fields for each circular polarization and elliptical polarization beam splitters, both in the visible.7 Finally we developed a new approach based on large scale disordered metasurfaces, which combines de-alloyed subwavelength structures at the nanoscale with loss-less, ultra-thin dielectrics coatings.8 By using theory and experiments, we show how sub-wavelength dielectric coatings control a mechanism of resonant light coupling with epsilon-near-zero (ENZ) regions generated in the metallic network, manifesting the formation of saturated structural colors that cover a wide portion of the spectrum. 1. P. Genevet, et al. Optica 4, 139 (2017) 2. N. Yu and F. Capasso, Nature Materials 13, 139 (2014) 3. M. Khorasaninejad et al. Nano Letters 16, 4595 (2016) 4. Alexander Y. Zhu et al. APL Photonics 2, 036103 (2017) 5. Robert C. Devlin et al. Optics Express 25, 377 (2017) 6. Wei Ting Chen et al. Light: Science & Applications (2017) 6 e16259; doi:10.1038/lsa.2016.259 7. J. P. Balthasar Mueller et al. Phys. Rev. Lett. 118, 113901 (2017) 8. H. Galinski, et al. Light: Science & Applications (2017) 6, e16233; doi: 10.1038/lsa.2016.233.

We review our recent work on quantum solitons and quantum-inspired phenomena in nonlinear wave propagation.

The recent development of topological insulator states for photons1–3 has led to an exciting new field of “topological photonics,” which has as a central goal the development of extremely robust photonic devices4. Topological insulator physics has been studied extensively in condensed matter5 and cold atoms6. However, photonic systems offer the possibility of non-Hermitian effects through the engineering of optical gain and loss, which opens an entire new field of physics. In this talk, we will discuss out recent experimental work on non-Hermitian topological phenomena in coupled lattice systems, in particular the first demonstration of non-hermitian PT-symmetric topological interface state. References [1] Haldane, F. D. M. & Raghu, S., Phys. Rev. Lett. 100, 013904 (2008) [2] Wang, Z. et al., Nature 461, 772–775 (2009) [3] Rechtsman, M. C. et al., Nature 496, 196–200 (2013) [4] Hafezi et al., Nature Physics 7, 907-912 (2011) [5] Hasan, M. Z. & Kane, C. L., Rev. Mod. Phys. 82, 3045–3067 (2010) [6] Jotzu, G. et al., Nature 515, 237–240 (2014)