Time-resolved luminescence of excitons in PS is characterized by a stretched exponential lineshape with μs time constants [31]. Analogies with electronic transport in random media allow interpreting the stretched exponential decay of the luminescence as due to the disordered distribution of nanocrystals and of inter-nanocrystal distances, as well as due to the random energy-confining potentials that result from surface passivation. Monte Carlo modeling shows further analogies between the recombination dynamics in PS and diffusion of particles in complex systems: critical scale emerges for cluster percolation and Anderson-like carrier localization [32]. The trap-controlled hopping mechanism of dispersive motion of excitons in a random spatial arrangement of interconnected quantum dots is able to account for the experiments [33]. Recent works point out the relevant role of surface groups in determining PS luminescence [34] and of the excitation conditions in determining the luminescence decay lineshapes [35].

The interplay between the silicon skeleton and the pores is reflected in a porosity dependence of the refractive index (n). An accurate control of the etching parameters (current and duration) allows modulating the effective n of the etched layer and, therefore, the production of PS multilayers (Figure 3A) [36]. In this way, PS microcavities where the emission rate is strongly influenced by the dielectric environment were fabricated [37]. In addition, also free standing dielectric multilayers structures can be fabricated which show in transmission the typical variation in the density of photonic states characteristics of one-dimensional photonic crystals [38]. These developments lead to the engineering of the phase dispersion to form photonic bands with fine features associated with the presence of photonic Bloch modes confined in the finite structure [39]. The group velocity displays wide variations, including peculiar features such as photon slowing down at the band edge and superluminal behavior within the photonic band-gap [40]. Recent progresses show that, by using electrochemical etching and photoacoustic, high-quality multilayers are feasible [41].

Still the use of the analogies between photons and electrons shows that the two are governed by functionally similar equations which reflect their wave-nature. In the one-dimensional case, the electron dynamics is governed by the following Schrödinger equation: ∂ 2 ψ ( x ) ∂ x 2 + 2 m k 2 ℏ 2 ψ ( x ) = 2 m V ( x ) ℏ 2 ψ ( x ) , (1) while photons (i.e., the electric field associated with the photon mode) by the following Helmholtz equations: ∂ 2 E ( x ) ∂ x 2 + ω 2 c 2 E ( x ) = 1 − n 2 ( x ) ω 2 c 2 E ( x ) = ω 2 V e ff ( x ) c 2 E ( x ) , (2)

Therefore, the control of n(x) in a dielectric multilayer allows introducing an effective potential (V _eff (x)) for photons (Figure 3B) and, thus, inducing photon propagation dynamics similar to the ones of electrons. Indeed, by using this analogy we were able to demonstrate photonic Bloch oscillations (Figure 3C) [42], Zener tunneling of photons (Figure 3D) [43], rectification of the transmission though a photonic diode (Figure 3E) [44], and, even, new phenomena such as optical bistability due to capillary condensation [45]. On the other hand, Bloch oscillations and Zener tunneling are not only observed in electronic or photonic systems. Interestingly, similar phenomenologies are observed also in quantum gasses [46], spin waves [47], coupled LC circuits [48], elastic waves [49], etc., which further proves the wave nature origin of the associated physics [50].

Disorder can be also a resource, e.g., in random lasing or in surface enhanced Raman spectroscopy [51]. Interestingly, dielectric multi-layers can be realized as a-periodic one-dimensional quasi-crystals, of the Fibonacci type, to show anomalous photon diffusion behavior. Near the band edges, critically localized states appear which decay weaker than exponentially [52]. On the other hand, purely random one-dimensional structures show exponential decay which reflects the same physics as electronic Anderson localization in amorphous materials (Figure 3E). In addition, interference phenomena in these structures show the building-up of coherent necklace states, observed as modes with a characteristic multiresonance time response and relatively fast decay [53]. The balance between localization and delocalization found in the necklace states can be placed on a more solid ground by modeling the stochastic dissipation in non-hermitian Anderson systems [54, 55]. This effect can also be engineered and exploited in devices such as the “pearl” spectrometer [56].

A first striking optical property of Si-NCs is that, under very specific excitation conditions, they show optical gain (Figure 4) [61]. Since direct electrical excitation of Si-NCs is difficult, the used method was based on the amplified spontaneous emission (ASE) (Figure 4A) [62]. The initial observations were reproduced by using Si-NC formed in different dielectric matrices [63] by CW pump-probe transmission experiments (Figure 4E) and by time-resolved ASE experiments (Figure 4B) [64]. Since a large Stoke shift between absorption and emission is observed, and the gain is spectrally on the high-energy side of the emission (Figures 4A,D) a four-level system was suggested to explain the gain (Figure 4C) [64]. The gain is attributed to the population inversion of interface states (Si-O double bonds) formed at the surface of the Si-NC (Figure 4F) [57]. A very delicate balance between Auger recombination and confined carrier absorption, on one side, and stimulated emission and waveguide losses, on the other side, determines whether loss or gain prevails [65]. This could explain the few late observations of gain [66–68]. Eventhough, claims for lasing in distributed feedback structures appeared [69], the use of Si-NC as an active laser medium seems improbable. Therefore, alternative approaches are now dominating the research and development of a silicon photonic laser, where either quantum dots and other materials are cointegrated in the silicon chip [70–73], or hereogeneous integration and chip bonding of III-V semiconductors are used to make a hybrid III-V laser on silicon [74, 75, 75, 76], or, finally, nonlinear effects are exploited [77].

Having observed luminescence and gain by optical excitation of Si-NC, a strong research effort was paid to get emission from the Si-NC by electrical injection, i.e. to make LEDs. Here, the problem is associated with the fact that the Si-NC are formed in a dielectric matrix (e.g., SRO) which is an insulating material. Therefore, the only chance to get electroluminescence is by tunneling. Two types of tunneling mechanisms are possible, depending on the oxide thickness and on the energy barrier height (Figure 5A):

• Fowler-Nordheim (F-N) tunneling where the carriers are injected in the Si-NCs through a triangular potential barrier caused by the voltage drop across the oxide. In this case, the energetic carrier excites by an inverse Auger process an e-h pair that in turn recombines by emitting a photon. The process is highly inefficient, has an onset at high voltages, and is prone to oxide damage and device failure;

Engineering of the active layer of the LED (Figure 5B) leads to the demonstration of bipolar tunneling [81]. In addition, grading the Si-NC sizes by using large Si-NC near the electrodes, which facilitates injection, and smaller Si-NC in the center of the device, which increases the radiative efficiency (Figure 5E), allows reaching high external quantum efficiency (Figure 5D) [78]. Still record quantum efficiencies in Si-NC LED are achieved by using chemically synthesized or surface-modified Si-NC [82], with record efficiencies of ≃ 10% [83] and luminance of 2060 cd/m² [84].

Erbium-doped fiber amplifiers have permitted the development of long-haul optical communication. Therefore, an effort was paid to develop Er-doped waveguide amplifiers to enable on-chip light amplification and, possibly, laser emission [2]. With this respect, the codoping with Er of Si-NCs allows, on one side, to put Er in an environment similar to the one in silica fiber and, on the other side, to exploit the sensitization action of the Er emission by the Si-NC. Sensitization means that the Er ions are indirectly excited by energy transfer from the Si-NC (Figure 5C), which increases by several orders of magnitude the Er excitation efficiency [85]. Er-doped Si-NC waveguide indeed shows signal enhancement but no net amplification of the propagating light (Figure 5F) [86]. Here, careful experimental studies and modeling allow identifying the main limitations to this approach: a low excitable Er fraction through the Si-NC due to the distance-dependent transfer rate [87], cooperative upconversion among the Er ions and confined carrier absorption in the Si-NC [88, 89]. Therefore, most of the introduced Er ions are not contributing to the amplification but provides significant absorption losses that prevent reaching net-optical gain [90]. Time-resolved experiments confirm these observations and elucidate the important role played by Auger recombination in the Si-NC which allows recycling a fraction of the excitons to contribute to the energy transfer [85]. Still, the system can be integrated in complex circuits to realize on-chip electrically pumped optical waveguides that emit at 1,540 nm [91]. Recently, the use of a deep cooling technique to prevent Er clustering has made it possible to achieve high brigthness Er-doped Si LED with superluminescent (stimulated) emission [92]. Alternatively, Er silicate nanocrystals codeposited with Si-NC allow reaching efficient electroluminescence at 1,540 nm in MOS devices [93].

Silicon solar cells have reached records in efficiency and endurance. Still, there are margins of improvement by making a better use of the solar energy. Among other approaches, one enabled by Si-NCs is solar spectrum reshaping [96]. This is based on the modification of the light spectrum, which reaches the active part of the solar cell, to enhance the power conversion efficiency (PCE). There are specifically two regions on which one can act: the low-energy infrared region where silicon does not absorb and the high-energy region where photo-excited e-h pairs suffer by surface recombination. As shown in Figure 6A, solar light from these spectral regions can be moved to the visible either by upconversion (infrared photons are absorbed and e-h pairs recombine emitting higher energy photons [97]) or by down conversion (otherwise named quantum cutting, photons are absorbed at high energy by generating two e-h pairs that recombine at low energy) or by down-shifting (high-energy photons are absorbed, the generated e-h pair thermalizes and recombines by emitting a low-energy photon). Si-NCs allow photoluminescence downshifting [94]. We designed solar cells that are coated by an optimal layer of Si-NCs. In our modeling [98], we considered both the effect of the change of the surface reflectance due to the different effective n of the coating layer as well as the combined action of Si-NCs absorption and emission (Figure 6B). Results show a sizeable increase in the photoresponsivity (Figure 6C). Based on this, state-of-the-art interdigitated back contact (IBC) silicon solar cells were covered by optimized layers of Si-NCs. Experiments show a % increase of the power efficiency mostly due to the increased short circuit current (Figures 6D,E) [95]. Remarkably, the effect was even enhanced when the dielectric environment of the Si-NC was tuned to force the emission to the bulk of the cell with a suitable interposer layer that maximizes the Purcell effect (Figure 6F) [95]. An update review on the use of nanocrystals for downshiting can be found in [99]. On the other hand, Si-NC can be also used in the active layer of a solar cell. Engineering the system allows demonstrating Si/Si-NC tandem cell with 32% PCE [100] or indoor Si-NC hybrid cells with 9.7% PCE [101]. In a different approach, Si-NC are used in luminescent solar concentrators integrated in semi-transparent photovoltaic windows [102].

PS due to its large specific internal surface offers many reaction sites to bind analytes and, thus, PS is a suitable material for a sensor [26]. In addition, different PS properties are influenced by the pore content such as its photoluminescence, refractive index, and conductivity. Specifically, highly luminescent PS microcavities (see Section 2.2), when infiltrated by an organic liquid, show both a refractive index dependence of the microcavity peak position and a low-frequency dielectric constant dependence of the luminescence intensity (Figure 7A). Therefore, PS microcavity sensors have a response that depends on two different analyte properties that allow the discrimination between different organic liquids [104]. Furthermore, PS electrical conductivity is also affected by the ambient. Indeed, a conductometric sensor can be realized by measuring the current flowing through a mesoporous silicon layer. A high sensitivity to the NO₂ pollutant gas at room temperature with limit of detection (LOD) of 12 ppb in dry air and 50 ppb in humid air is observed (Figure 7C) [106]. More interestingly, three independent quantities can be measured in PS, i.e., the electrical conductance, the photoluminescence intensity, and the wavelength of the optical resonance [109]. By monitoring the change of these three parameters as a function of the NO₂ content (0.5–5 ppm), ethanol (300–15,000 ppm), and relative humidity (0–100%), we demonstrated that the analytes affect the measured parameters in a different way (Figure 7B) [109]. In this way, it is possible to distinguish between a pollutant like NO₂ and interfering gases like humidity and ethanol [105]. Despite the semplicity of the PS sensor, its performances are still state-of-the-art. Indeed, typical limit of detection in integrated silicon optical gas sensors is in the few ppb range [110]. Recent progresses in the use of PS as a biosensor are reviewed in [111]. A critical assesment of these is provided by [112].

PS can be also used as a birefringence sensor. Birefringence is the degree of optical anisotropy of a material and depends on the geometrical material parameters as well as on the material properties. Owing to the porous structure of PS, which results from differently oriented and branched pores, its optical anisotropy can be influenced by the etching process and by the pore filling. Therefore, two methods to exploit PS as birefringence sensor have been investigated. In the first, light scattering from a PS layer is used because of the formation of scattering rings whose angles depend on its optical anisotropy [107]. In fact, multiple scattering from a random arrangement of vertical cylinders (the pore walls) generates a scattering cone from an oblique incident beam [113]. The cone aperture depends on the anisotropy of the PS. In the second, we used polarized light transmission through a free standing PS membrane [108]. In particular, heavily doped PS presents very high values of birefringence, as the pore shape is much less branched than in lightly doped PS. This special geometry also produces ring-shaped scattering of light, whose angular maxima are connected to the optical anisotropy of the layer. Filling the pores with different liquids or oxidizing the inner pore structures causes a change to the birefringence which can be measured by the angular position of the scattering rings (Figure 7D) [107]. On the other hand, by etching PS in (110)-oriented silicon wafer, pores grow along the [100] and (010) directions. Therefore, normal incidence light sees a different effective composition, i.e., a different effective n, depending on its polarization. This allows us to use the polarimetric setup of Figure 7E to measure the degree of birefringence of the layer. Filling the pores with different substances allows detecting their presence by measuring the spectral birefringence, as shown in the figure [108]. Recently, it has been demonstrated that this kind of sensors can achieve sensitivity of few ppm when ethanol is probed [114].

PS optical microcavities show a resonance frequency that depends on the composition of the fluid filling the pore. A similar effect is observed with optical microring resonator (MR) (see Section 5.1). Briefly, MR are formed by a waveguide closed on itself in a loop (of radius R and effective index n _eff) which is side coupled to another waveguide [119]. Measuring the transmission intensity from this side waveguide, transmission dips are observed whenever the light is coupled to the MR. The resonance wavelengths of these transmission dips are mλ _m = 2π n _eff R, where m is an integer. Therefore, if n _eff changes also λ _m changes which allows measuring n _eff: a simple refractometric optical sensor is formed. We have engineered the MR based sensor by studying the delicate balance between the resonance quality factor (Q-factor) and the evanescent field overlap with the surrounding environment [120]. Based on the design, MR were fabricated with SiON as core materials and their bulk sensitivity S = Δλ _m /Δn _eff measured (Figure 8A) [115]. Here, we use the term bulk sensitivity to indicate the sensor response to a bulk change of the cladding material. This differs with respect to the surface sensitivity that refers to a change of n _eff caused by the chemical binding of the analyte on the MR surface. In this case, the surface of the sensor should be functionalized by the formation of active chemical bonds able to capture the specific molecula which one wants to detect. In our case, we were interested to detect low concentration of a toxin in milk because of the danger of its ingestion by humans. Milk is a complex fluid and many biosensors have been developed to detect its different allergens that are based, e.g., on nanoparticles or electrochemical methods [121]. We were specifically targeting the aflatoxyn M1 (AFM1) toxyn that is cancerous if assumed even in small quantities. We developed a specific functionalization protocol based on small strands of DNA (aptamers) or on antibodies (Fab’) (Figure 8E) [116]. Example of the resonance shift due to the flow of a buffer with different concentrations of AFM1 (the sample) is shown in Figure 8B [116]. Notably, the sensorgram (the time response of the sensor) shows the different phases of the arrival of the sample (the initial rapid large shift), the progressive chemical bonding of the analyte on the MR surface (the following slow shift), and the regeneration of the sensor by glicyne flow which recovers the initial resonance position. Measuring the response for different AFM1 concentrations allows determining the sensor sensitivity, while measuring the sensor response to different toxins allows determining the sensor specificity to AFM1 (Figure 8C). In this example, the response of the sensor to Ocratoxin (OTA) is negligible, while it is still responding to as low as a few nM of AFM1 [116]. The proposed sensor is competitive with immune-detection strip in terms of response time (minutes vs hours), while it lacks behind in terms of sensitivity (ppb vs ppt) [122]. However, this example shows the potential interest of silicon photonics sensors that, indeed, have been widely used for various specific targets. Reviews can be found in [20, 123–125].

As the chemical bonding of the analyte to the sensor surface changes n _eff of a waveguide, other interferometer geometries can be used to realize optical sensors. In particular, the work-horse of integrated photonic interferometer is the Mach-Zehnder interferometer (MZI). It has been widely used to realize optical modulators (see the bottom left device in Figure 1) [11]. The MZI transmission T is determined by the interference between the light that propagates along its two arms T = 1 / 2 1 + cos ( Δ ϕ ) , where the phase difference Δ ϕ = 2 π / 2 n e ff , A L A − n e ff , B L B being L _A and L _B the two arm lengths. Therefore, measuring the MZI transmission allows us to determine Δϕ which in turn allows determining n _eff and, hence, sensing the presence of the analyte. Figure 8D shows bulk sensitivity measurements of asymmetric MZI, while Figure 8F shows the calibration of these sensors with respect to the detection of AFM1 in real milk samples [118]. These sensors, when functionalized with Fab’, are able to detect AFM1 in concentrated and filtered milk samples down to a concentration of ≃ 48 pM within a few minutes. MZI-based sensors are extremely interesting for a wide range of applications [125, 126]. For example, they have been also used to detect volatile organic compounds (VOC) at extremly low levels [127].

Si-NC emitters coupled to optical cavities show modified optical properties. In particular, by using optical lithography and etching it is possible to define optical active thin microdisk rich of Si-NC and standing on a pedestal (Figure 9A) [133]. Their emission spectrum is strongly different with respect to the broad emission band of Si-NC. In fact, narrow directional emission lines are observed in the plane parallel to the microdisk (Figure 9A, middle panels). These lines are due to the change in the photonic density of states caused by the optical cavity where whispering gallery modes (WGM) form in a way similar to what Sir. Rayleigh observed for sound waves in the Saint Paul cathedral. WGM spectral positions reflect the disk geometry that can be tuned by using a stressing layer which upward bends the disk in a micro-kylix shape affecting the WGM dispersion and the optical cavity Q-factor [129]. More interestingly, the coupling of the Si-NC with the modified density of photonic modes observed in the WGM microdisk is also reflected in a variation of the Si-NC radiative lifetime. In fact, the radiative lifetime of an emitter is described by the Fermi golden rule that linearly depends on the density of photon modes. Where the photon modal density is large, the radiative lifetime is short and vice versa. This effect named Purcell effect is clearly observed in the Si-NC emission in a microdisk resonator (Figure 9D) [132]. A sizeable reduction of the radiative lifetime is observed whenever the emission wavelength matches a WGM.

Despite this interesting physics, pedestal standing microdisks are difficult photonic components. Careful coupling with nearby tapered fibers or optical objectives are needed to study their properties. A preferred system, which allows also to be inserted into complex photonic circuits, is the one provided by MR coupled to a side waveguide (Figure 9B). The light from the side waveguide (also called bus waveguide) couples with the microresonator due to the overlap between the waveguide mode and the resonator mode which occurs in the coupling region. Here, the physics can be simply described by an optical mode equation based on the temporal coupled mode theory (TCMT) [130]: d α d t = ( i ω 0 − γ − Γ ) α + i 2 Γ E i n [ t ] (3) E o u t [ t ] = E i n [ t ] + i 2 Γ α [ t ] , (4) where α is the electric field amplitude (optical mode) in the resonator, ω ₀ is the resonant angular frequency of the resonator, E _in is the input electric field amplitude, E _out is the output electric field amplitude, and Γ (γ) is the extrinsic (intrinsic) damping rate which are positive real values. Let us note that γ describes the losses in the resonator which are due to intrinsic factors (material absorption, scattering, bending, etc.), while Γ is related to the coupling with the bus waveguide (extrinsic factor). Indeed, a fraction 2 Γ E i n of the waveguide mode is coupled into the resonator. Solution to Eq. 3 yields the transmission spectrum |t _r (ω)|² of the microring which in stationary condition is t r ( ω ) = E o u t / E i n = 1 − 2 Γ i ( ω − ω 0 ) + γ + Γ . (5)

From this solution, the resonator Q-factor is easily derived as Q = ω 0 2 ( Γ + γ ) . (6)

In Figure 9B, both the transmission intensity |t _r (ω)|² and its phase (arg (t _r )) are reported for different values of the damping rates, corresponding to the over-coupling (Γ > γ), critical coupling (Γ = γ), and under-coupling (Γ < γ) regimes. Typical Lorentzian lineshapes are observed for these coupling regimes, with a vanishing transmission dip for the critical coupling case. Also the phase shows a characteristic behavior with a π delay for the over-coupling regime and a zero delay for the under-coupling regime. The reason for these different behaviors can be understood by looking at Eq. 4, which shows that the output electric field results from the interference between the input field and the resonator optical mode. The temporal solution to Eq. 3 allows computing the output electric field as a function of time (Figure 9B, bottom). It is observed that a destructive interference occurs between the optical mode in the resonator and the input electric field in critical coupling conditions, once the resonator is loaded [130].

What we have described applies to a microresonator side coupled to a bus waveguide by a point coupling geometry (Figure 9C). Another coupling geometry, where a buried waveguide is vertically coupled to the MR, is possible. This vertical coupling has several interesting properties: 1) the two optical components can have different composition and cross-section, 2) the relative position of the bus waveguide allows exciting high order radial modes in a microdisk resonator, 3) the distance between the resonator and the bus waveguide can be controlled at the nm-scale without the need of high-resolution lithography [134]. In addition to these, the vertical coupling shows a new physics associated with the observed oscillatory coupling (Figure 9E) [131]. This last is caused by an extended effective coupling region (named flat zone in Figure 9C). Therefore, the vertical coupling can be modeled as two coupled waveguides of an effective coupling length given by the flat zone. Since oscillation in the power transfer occurs as a function of the effective coupling length, also the transmission oscillates as a function of the vertical distance between the bus waveguide and the resonator. This can be readily measured in experiments (Figure 9E). Further investigations allowed to fully understand the physics of the couplings which leads to an optimization of the system geometry [135].

Let us now consider a MR with clockwise α _CW and counterclockwise α _CCW propagating optical modes (Figure 10A). TCMT shows that d α C C W d t = ( i ω 0 − γ − Γ ) α C C W − β 12 α C W + i 2 Γ E i n (7) d α C W d t = ( i ω 0 − γ − Γ ) α C W − β 21 α C C W (8) where β ₁₂ and β ₂₁ are the complex coupling coefficient between the two modes. These coupled equations show that a continuous energy exchange between the two modes occurs when a mechanism couples them. A possible mechanism is back-reflections due to surface roughness [136]. Equations 7, 8 are written for a forward (F) excitation (i.e. light is input from the left). Extension to a reverse (R) excitation (light input from the right) is straightforward. Solutions to Eqs. 7, 8 for the transmission (t _F and t _R ) and reflection (r _F and r _R ) coefficients are: t F = t R = 1 − 2 Γ ( i Δ ω + γ + Γ ) [ i Δ ω + γ + Γ ] 2 − ( β 12 β 21 ) , (9) r F = 2 Γ β 21 [ i Δ ω + γ + Γ ] 2 − ( β 12 β 21 ) , (10) r R = 2 Γ β 12 [ i Δ ω + γ + Γ ] 2 − ( β 12 β 21 ) , (11) where Δω = ω _inc − ω ₀ is the detuning frequency. If β 12 = − β 21 * we are in the hermitian case and the transmission or reflection coefficients are the same independently of the specific mode excited (i.e. forward or reverse excitations). For a high Q-factor MR, a characteristic balanced doublet appears in the transmission spectrum (Figure 10A, second panel). When the MR has nonsymmetrical scatters on its surface, so that modes propagating in opposite directions interact with the surface defects in a different way, we get β 12 ≠ − β 21 * . We are in the non-Hermitian case and the energy exchange between the two modes is no longer balanced. The transmission doublet is now unbalanced (Figure 10A, third panel) and different reflection coefficients are observed depending on whether the CW or CCW is excited. Note that even in this case, Lorentz optical reciprocity is preserved since the transmission is always the same.

Non-Hermitian physics can be observed in high-Q factor MR due to the excitation of the counter-propagating mode caused by back-scattering of the propagating mode due to the random sidewall roughness [136]. Or it can be observed as a consequence of the coupling between different radial modes in a vertically coupled microdisk through the buried waveguide [137]. In this case, the spatial overlap between the different radial modes governs the interaction and introduces a complex or reactive inter-mode coupling that is manifested by a self-shift of the resonant frequencies. We have described this phenomenon as an analogy to the electronic Lamb-shift term [137]. Notably, the coupling between the narrow first-order mode and the broad second-order mode of the microdisk causes a characteristic Fano lineshape that is characterized by a sharp asymmetry in the response spectrum (Figure 10B, bottom panel). Tuning of the coupling and of the Fano lineshape asymmetry is possible by using nonlinear effects [142]. Clearly, these are only few observations of the phenomena associated to non-Hermitian photonic systems, more studies have been reported in the literature, see e.g., [143–146].

Inspired by the work on coupled miroresonators geometries [147, 148], we have proposed a special resonator geometry where an S-shaped waveguide provides nonreciprocal losses inside the cavity (Figure 10D). Owing to its geometry, we named this resonator a taiji resonator [149]. The taiji MR is a non-Hermitian system and shows direction-dependent properties. Notably in the forward excitation direction, light does not propagate in the S-shaped waveguide and the CW mode is not excited. On the contrary in the reverse excitation direction, light couples to the S-shapes waveguide and the CCW is excited. Therefore, the taiji MR is a prototype system for non-Hermitian physics with, in the ideal case, β ₁₂ = 0 and β ₂₁ ≠ 0. Consequently, the taiji MR behaves as a unidirectional reflector. Note that another consequence of the taiji geometry is a larger power storage in the MR for the reverse configuration than for the forward one. The asymmetric coupling to the S-shaped waveguide allows recirculating the light coupled to it in the reverse case, while it is radiated away in the forward case. This causes a different resonator heating that affects the n _eff and, in turn, causes a nonlinear shift of the transmission resonance [150]. For the same input power, the shift is different for the F and R excitation configuration. In fact, a different transmission is observed (Figure 10D) [139], which means that Lorentz reciprocity is broken in this simple passive and nonmagnetic device. The model here described has to be carefully considered when taiji resonators are arranged in complex topology because interference with other photonic components can alter this simple picture [140]. Taiji MR are used for unidirectional [151] or topological lasers [152, 153] or as building blocks for topological photonic systems [154, 155].

Isolated single MRs show a rich physics that can be used in functional device systems (see e.g., Section 5.4). Coupled MRs show even more interesting properties. Sequence of MR can be coupled side by side in what is called a CROW (coupled resonator optical waveguides) or coupled through side waveguides in what is called a SCISSOR (side-coupled integrated sequences of optical resonators) (Figure 10C). In this last case, in addition to the resonances due to the single resonators (m _R λ _R = 2πn _eff R), the periodicity of the sequence causes the formation of other resonance bands due to a Bragg-like effect (m _B λ _B = 2n _eff L _C , where L _C is the distance between the center of two adjacent rings). In fact, the light dropped by the single optical resonators for specific wavelengths and resonator separation (sequence periods) can interfere constructively in the side waveguide in a way similar to what happens in dielectric Bragg reflectors (DBR). These two resonant conditions are visible as two transmission dips at λ _R and λ _B in the through port (T-port) of the SCISSOR (Figure 10Cb). If λ _B = λ _R , i.e. L _C = πR, then the two resonances spectrally overlap and a coherent feedback occurs. When the two bands are in close spectral proximity and the single resonators have slightly different resonances, then a new sharp resonance peak appears in the T-port transmission (Figure 10Cc). This is due to the coherent coupling of two resonators and, therefore, is named CRIT (coupled resonator induced transparency) [156]. This represents the optical analog of the electromagnetically induced transparency (EIT). The CRIT condition can be engineered in a new SCISSOR geometry which we called tapered SCISSOR since it is made by resonators with slightly increasing radii along the sequence where only two are separated by the positive interference Bragg condition (L _C = πR) and all the other are spaced by a distance L _C ≃ 3/2πR which causes a destructive interference (Figure 10C) [157]. A router device based on a sequence of tapered SCISSOR can therefore be fabricated where the routing wavelength is determined by the CRIT condition (Figure 10E) [138]. Since this is a stringent requirement, the CRIT router is robust with respect to geometrical imperfections and can be easily reconfigured by changing either the single-resonator resonance or the coupling Bragg conditions [158].

Also CROW can be used as routers. In Figure 10F, we show a design of a reconfigurable interleaver or band router which is based on a two-resonator CROW. Here the idea is to use the CROW as an element of an interferometer where the phase difference (Δϕ) between the CROW inputs is controlled by the power dissipated in a metallic heater [141]. The device shows three output states: through, drop and a 50/50 split state that divides the input equally among the two outputs. Experimental data collected on a four-resonator CROW show that the interferometric switch works as expected (Figure 10F, bottom).

Here, we discussed two devices enabled by coherently coupling MR, actually the related physics is much richer [163, 164]. Other applications are possible such as synchronization of comb frequencies [165] or cavity quantum electrodynamics with trapped atoms [166] or perfect absorption [146].

It is well known that most of the data traffic occurs within data centers. Here one of the most imperative requirements is to handle high bandwidth with low power consumption and data latency. Therefore, reconfigurable optical switches are looked for as the solution for data transport in data centers [8]. The usual MEMS (micro electro-mechanical system) based switches used in telecom networks show limitation due to long switching time, cost, sizes, and resources optimization. Silicon photonics, with electro-optic integration, is currently the most viable alternative. A transponder aggregator (TPA), which distributes selected wavelengths received from the optical node to transponders (devices that convert the electric data flow into optical data flow at a given wavelength, thus connecting the users to the optical network) without any limitation in terms of wavelengths and directions or, in the opposite direction, which combines any wavelengths from transponders to the optical node, is the function of the IRIS (Integrated Reconfigurable sIlicon photonic Switch) optical switch [160].

The concept of the IRIS device is shown in Figure 11A [160]. A matrix of thermally controlled ring resonators are used to switch selected wavelengths. From the 4 network ports (NP) a set of 6 wavelengths are coupled to the IRIS-TPA by using grating couplers (SPGC). Interleavers (INT) separate the even and the odd wavelengths to increase the channel spacing. Then, demultiplexers based on array waveguide grating (AWG) devices [161] separate the different wavelengths along different waveguides that form the various rows of the 48 × 16 switching matrix. Along the rows there are nodes (switching elements (SE) shown as colored dots in Figure 11A) where ring resonators in the add-drop configurations are thermally moved into resonance or out of resonance with respect to the given wavelength. In case the wavelength has to be dropped, a given ring is moved in resonance with this wavelength in such a way that the light is coupled to the ring and dropped to waveguides which form the matrix column. At the end of each column, an interleaver combines the odd and even channels and a grating outputs the wavelength to the selected 8 local ports (LP). The IRIS switch can be also used in the other direction, i.e. a wavelength can be added to the IRIS switch by a specific local port and sent to the desired network port. To this end, the light is coupled to the photonic chip by a grating, then it is routed to the even or odd channel columns and, based to which network port it has to be routed, the switch element at the proper node of the matrix is brought into resonance with this wavelength so that the light is dropped from the column to the row. Then, the wavelengths on the various rows are multiplexed by AWG, the odd and even channels are recombined by the interleaver and via a grating the wavelength is routed to the proper network port. The switch elements are made by two coupled ring resonators (Figure 11G) to get a large channel isolation and a flat top transfer function (Figure 11B). On the top of each ring, a metal heater is fabricated to locally heat the ring and to change the resonance wavelength by controlling the n _eff via the thermo-optic effect. Owing to the small size of the ring, a shift by a free spectral range is achieved within few μs and with few mW which implies order of magnitude improvement with respect to MEMS optical switches. Interleavers and AWG have been designed and fabricated by using novel designs [160]. Specifically, whiskered AWG, where the star coupler surface is designed to trap the reflected light [167], have been used since they are robust to phase errors and show the required performances (Figure 11D) [161]. The various optical components are integrated in a silicon photonic chip that is bonded through metal bumps to a specifically designed ASIC chip (Figure 11H [160]). Electrical currents are flowing between the two chips for setting the desired resonator wavelength (Figure 11F) or to monitor the light signal in the photonic chip by reading integrated photodetectors (PD) [162]. The resulting electric/photonic integrated system has the cross section shown in Figure 11E. Remarkably, a fiber array is also pigtailed to the photonic chip to input and output the optical data (Figure 11H) [159]. In the final packaged device (Figure 11H), 1,398 photonic components controlled by more than 2000 electronic components are integrated in a 65 mm² area. The photonic chip, with a switching matrix of 768 SE, was fabricated with a 193 nm deep-UV lithography in a fully CMOS-compatible process on 8-inch SOI (Silicon on Insulator) wafers, while the electronic chip by the STMicroelectronics’ BCD 0.16 μm technology. The IRIS switch needs to be calibrated before operation since the suitable currents to be injected in the metallic heaters to trim the interleavers or the switch elements have to be determined. This process (Figure 11G) is automatically performed by a purposely designed genetic algorithm [168]. After this, the switch can be operated and the desired wavelength addressed to the desired port (Figure 11C). Remarkably, the presence of on-chip monitor photodiodes allows us to trace the optical signal in different points of the switching matrix and to assess the related device losses. Despite that the switch is able to operate at 25 Gbps without adding extra power penality due to interferring signals and a total power consumption of about 1W, the overall power loss due to the matrix of 22 dB is still too high. By using higher resolution lithography and by integration of semiconductor optical amplifiers (SOA), better performances are achievable [9]. Indeed, these have been demonstrated in a 32 × 32 switch matrix that had only 10 dB of fiber-to-fiber loss and MZIs as switching nodes [169], while improvements due to on-chip SOA are discussed in [170]. A discussion of the state-of-the-art of reconfigurable photonic circuits with an emphasis on the power consumption is reported in [171].

The MR resonance wavelength is determined by its geometrical parameters as well as by n _eff. When the input power is increased, nonlinear effects set-in which change n _eff. Specifically, in the CW case, the high input power induces two-photon absorption (TPA) which generates excess free carriers (with a density ΔN) at a rate (g _TPA ) in the conduction and valence band of silicon waveguide. These free carriers both cause free carrier absorption (with an associated loss η _FCA ) and free carrier dispersion (FCD) as well as they thermalize with a typical free carrier lifetime (τ _FC ) self-heating the material. The resulting increased temperature (ΔT) affects n _eff via the thermo-optic effects [175]. All these are related to the input pump power. Therefore, the linear Eq. 3 is modified by a system of coupled equations of α, ΔN and ΔT [172, 175]: d α d t = ( i ω 0 ( t ) − γ ( t ) − Γ ) α + i 2 Γ E i n [ t ] (12) d Δ N d t = − Δ N τ F C + g T P A | α | 4 , (13) d Δ T d t = − Δ T τ t h + P a b s m c p , (14)

Equation (14) is the Newton’s law, where P _abs is the absorbed power by the material that causes heating, m indicates the mass of the ring, c _p is the silicon-specific heat, and τ _th indicates the thermal decay time due to the heat dissipation with the surrounding medium. The nonlinear effects are given by the change of the resonance frequency ω ₀(t) = ω ₀ + δω _nl (t), where ω ₀ is the MR resonance frequency in the absence of the pump light (cold MR frequency) and δ ω n l = − 1 n 0 d n d T Δ T + σ F C D Δ N , with σ _FCD the free carrier dispersion coefficient. Also the intrinsic loss term γ(t) is affected by the pump power, in fact γ(t) = γ + η _FCA ΔN + η _TPA |α|², where γ is the cold MR intrisic losses and η _TPA the TPA loss coefficient. Therefore, at low input power the behavior of the MR is as the one we described in Section 5.1, while at high input power several new phenomena appear. Indeed by considering the typical times: photon lifetime of the cavity τ _ph = γ ⁻¹ ≃ ps, τ _FC ≃ ns and τ _th ≃ 100ns we note that the dynamical response of a MR is affected by different phenomena depending on the temporal scale of the input signal. In addition, we note that the TO coefficient is negative while σ _FCD is positive; therefore, the two have opposite contributions to ω _nl . Their interplay appears at the output of the resonator as time-instabilities associated to Self-Pulsing (SP), bistability, excitability, or as a nonlinear distortion in the transmitted spectrum.

Self-pulsing is particularly relevant since it transforms a CW input signal in a periodic oscillating output signal. Being simple the description of an isolated MR [175], the physics gets much more interesting in a SCISSOR [172]. Figure 12A shows the different temporal responses of a SCISSOR in the nonlinear regime. By measuring with an IR camera the scattered light from the SCISSOR (which is linearly proportional to |α|²) we can trace the number of rings that are excited by the input light. We note that if two rings are excited a periodic behavior in the T-port output signal is observed (Figure 12A, orange box). If three rings are excited, an a-periodic behavior is observed (Figure 12A, green box). If more rings are excited, a chaotic behavior is observed (Figure 12A, red box). Chaos is demonstrated by showing that the output is extremely sensible to negligible variations of the input [172].

Chaos in MR can be beneficial in many applications, such as achieving two-octave spanning frequency combs [176] or to suppress lasing instabilities [177]. Here, it leads us to consider MR as nonlinear nodes for photonic neural networks (see chap. 10 in [9]). We started from a very simple network, physically realized by a single nonlinear MR [173]. We implemented a reservoir computing scheme by using the nonlinearity of a MR and time multiplexing (Figure 12B). In this type of neural network, the input data are processed by a reservoir whose nodes are interconnected and the strengths of the connections are randomly chosen. In this way, the input is projected in a multidimensional space and the nonlinear dynamics of the reservoir guarantees the network separability. Only the reservoir outputs are trained by varying the weights in a linear combination of the output states of the reservoir. In our implementation, we encoded the input signal in time bins of different light intensity, which are input in the device. This sequence defines the virtual nodes (of number N _v ) of the reservoir. The time bin duration is set of the same order of the nonlinear MR dynamics, which triggers a transient state in the free carrier population. The carrier dynamics defines the connections between the virtual nodes, i.e. the network memory. Specifically, we used a pump-probe technique (Figure 12B): the intensity modulated pump (red) and the CW probe (blue) are injected into the MR. An incoherent transfer of information from the pump to the probe occurs due to FCA and FCD within the resonator, where the different virtual nodes (yellow dots in Figure 12B) interact and process the input data. The probe exits from the D-port of the MR, carrying the result of the computation, and is detected by a fast PD. Virtual nodes are sampled and sent into a linear classifier, trained to perform a specific task. As an example, we classified the different species of the IRIS flower achieving state-of-the-art performances (Figure 12C) [173]. Note that the increase of the input power yields better classification rate up to a power level where self-pulsing is induced.

In [173], the linear combination of the reservoir outputs (perceptron) is done off the chip. Actually, this is also possible in the chip by properly designing the photonic circuit (Figure 12D). Here, it is shown a simple time delay network that integrates a complex perceptron [174]. Gratings are used to couple the light in and out of the chip. A 1 × 4 splitter distributes the input signal (u(t)) to four delay lines, realized by spirals (each spiral adds a delay of Δ _t ). Then, thermal phase modulators allow controlling the relative phases ( w k = e i ϕ k ) of the signals (u _k (t)) along each waveguide. These are then summed by combiners and the resulting signal (∑ _k u _k (t)w _k ) is output via a grating. Finally, a PD performs a nonlinear transformation |⋅|² giving the perceptron output y(t). Therefore, the output at any given time is the coherent weighted sum of the four delayed copies of the input, nonlinearly transformed by the detector. As an example, we show the result of the n-bit delayed XOR task to investigate the node memory and its nonlinear transformation capability. The perceptron has to output the result of the XOR operation between a bit and the nth previous bit (Figure 12E). The 1-bit XOR is accurately performed up to 16 Gbps showing that the network has both nonlinearity and memory. The 2-bit XOR is solved only at high rate due to the fix time delay (Δ _t = 50 ps) which avoids to populate the perceptron at low rates.

More advanced structures have been shown in the literature where both photonic accelerators (devices specialized to perform a single function and which are interfaced with electronic computer) [178, 179] or photonic integrated neural networks [180–183] with extremely high capacity and speed [184, 185] are demonstrated.

Si-NCs have a crystalline core of silicon which shows a Kerr-type nonlinearity (χ ⁽³⁾) that competes with FCD and TO effects. In Si-NC, the variation of the refractive index (Δn) as a function of a pulsed high-field intensity can be modeled by Δn = n _2be |E|² − σ _FCD ΔN, where n _2be is the contribution due to the fast polarization of the electronic clouds while the second term is due to the excess free carriers generated by TPA [194]. Overall, n _2be in Si-NC is one order of magnitude larger than that of bulk Si and three orders of magnitude larger than that of silica [194]. The large value is explained by the cumulative effects of quantum confinement and of the dielectric mismatch between the Si-NC and the dielectric matrix which enhances the field intensity in the high dielectric regions. The high n _2be can be used in MR to achieve all optical switching [186]. In fact, when a Si-NC-based MR is pumped with a high power laser, the resonance frequency of the MR is shifted (Figure 13A). This effect can be used to realize an all-optical switch where a CW probe beam is switched on or off by a pulsed (1ps long pulse, 10 GHz repetition rate) high power beam (Figure 13D). Similarly, the slow TO effects can induce an optical bistable behavior in a MR [191]. Here, an integrated microdisk with embedded Si-NC is vertically coupled to a SiON waveguide. The transmission is recorded as a function of the input power, and depending on the power sweep directions an hysteresis loop is formed (Figure 13G). Optical bistability is only observed when the pump wavelength is on the long wavelength side (red-detuning) of the cold resonance wavelength so that a positive feedback can take place (power is absorbed which increases the temperature which increases n _eff which red-shifts the resonance which increases the absorbed power which ⋯). As a result, above a power threshold the resonance wavelength locks to the pump wavelength (see the spectra in Figure 13G at point 3). The power threshold is different (which explains the hysteresis) if the sweep is performed starting from low or high pump power. Thus, for the same input power in the hysteresis loop, the MR transmission shows two states (points 2 and 4 in Figure 13G). Other applications of Si-NC optical nonlinearities have been reported in [195].

Despite that the second-order nonlinearity is negligible in Si, the need of fast electro-optic modulators in silicon photonics motivated a strong research effort to induce second-order nonlinearities in silicon. A simple idea was to break the silicon centrosymmetry by using a stressing layer (see the inset to Figure 13B), i.e. to strain silicon. Therefore, strained silicon waveguides cladded by SiN stressing layers were tested for second harmonic generation (SHG) [187]. Encouraging results were obtained where a significant peak at the double of the pump wavelength was observed (Figure 13B). Micro-Raman measurements revealed the large in-homogeneous strain induced in the silicon waveguide by the stressing layer [196]. Even other geometries were tested with different degrees of stress and chemical composition of the cladding layer [197]. However, some puzzling results were reported. First, no relation between the degree of strain induced by the stressing layer and the SHG efficiency was observed [197]. Second, the electrooptic effects due to the second-order nonlinearity in strained silicon MZI vanish for driving frequency larger than the inverse of τ _FC [198]. Finally, an experiment where strained silicon waveguides are mechanically deformed to increase the strain does not show any influence of the strain on the SHG (Figure 13E) [189]. All these data are explained by the electric field induced second harmonic (EFISH) effect [197]. Charged defects form at the interface between the core silicon waveguide and the stressing layer. These charges induce a static electric field E _DC across the waveguide core. An effective χ ⁽²⁾ is then induced by the combined action of χ ⁽³⁾ and E _DC : χ eff ( 2 ) = 3 χ ( 3 ) E D C which explains the observed SHG. The explanation is confirmed by experiments that remove the interface charges: after UV passivation of the Si/SiN interface states no SHG is observed [189]. Noticeably, if a static E _DC induces a χ eff ( 2 ) , a large nonlinearity should be observed in a waveguide embedded in a p-i-n diode. Indeed, this is the case [192]. p-i-n junctions were fabricated across a silicon rib waveguide. The sign of the p-i-n junctions was periodically reversed to periodically modulate the χ eff ( 2 ) , similarly to the poling technique used in the dielectric optical modulator to achieve quasi-phase matching. As a result a large SHG efficiency dependent on the applied reverse bias was observed (Figure 13H). Similar results have been reported by [199, 200] with a 200 nm wide spectral SHG band demonstrated in [201] by group velocity engineering.

The parametric nonlinear process mostly used in silicon is four wave mixing (FWM) which is based on χ ⁽³⁾ nonlinearities [150]. In FWM, two pump beams mix with a pair of weak beams, called signal and idler beams. The signal is typically the shorter wavelength beam and the idler the longer wavelength beam. FWM fulfills energy conservation and momentum conservation. The former is written as ω p 1 + ω p 2 = ω s + ω i , (16) with ω _j (j = p1, p2, s, i) the frequencies of the pumps, signal, and idler beams. The momentum conservation, also known as phase matching condition, is given by Δ k = k p 1 + k p 2 − k s − k i = 0 , (17) with k _j = n _eff (ω _j )ω _j /c (j = p1, p2, s, i) the wavevectors of the pumps, signal, and idler photons. n _eff is different for each optical mode. The FWM generation efficiency η is given by [150]. η ∝ | f j , q , l , m | 2 s i n c Δ k L 2 π 2 (18) where L is the length of the waveguide and f, _j,q,l,m the overlap integral between the involved modes. Therefore, to get large generation efficiencies, Δk should be almost zero.

As a first technological development, here we consider the application of quantum technology for security. Specifically, we aim to exploit the inherent probabilistic nature of quantum physics to develop photonic systems able to generate sequence of unpredictable bits to be used as seed for cryptography applications. These are called random number generator with the specification that they are based on a quantum effect, i.e. quantum random number generator (QRNG). Reviews of QRNG can be found in [214–216]. State-of-the-art results of a silicon photonics integrated QRNG are reported in [217]. In the following, we review a simple, low-cost, and CMOS-compatible system. The first concept we used is the probabilistic nature of spontaneous emission, by which a light source emits randomly photons whose time of arrival on single-photon detectors (single-photon avalanche photodiode, SPAD) is used to generate random bits (Figure 14A) [209]. To use an all silicon approach, the photon source was a Si-NC LED where, with our setup, we measured a Poisson distribution of the emitted photons [218]. With a poissonian distribution, given a photon flux λ, if we use a time-bin t _w = ln (2)/λ we have an equal probability to detect one photon and no photon. Therefore, we can use the SPAD clicks to generate the random bit sequence. This idea works for short sequences, but fails for long sequences due to unwanted ambient fluctuations that affect the LED emission rate. Thus, we have developed another procedure to generate the random bits which is robust to ambient fluctuations. It is based on a random bit extraction implemented on a dedicated field programmable gate array (FPGA) without any postprocessing of the data. Briefly, the Poisson statistics has the property that if there is only one single arrival in a time interval (0, T), the distribution of the arrival times is uniform throughout the interval. We have therefore built an interval and attributed different hexadecimal symbols to 16 subintervals. In addition, we used non-symbol generating intervals to take into account the dead time and the afterpulsing of the SPAD, and we used cyclic permutations of the symbols to consider for electronic artifacts. The final scheme is shown in Figure 14D. Each time a photon is detected in a subinterval a symbol is generated whose binary code represents the random bits. This procedure turns out to be extremely robust to any external factor that influences the system (see, e.g. Figure 14B where large temperature variations that impact the photon flux generation do not influence the probability to generate bit 1 or 0) [210]. Then, we interfaced Si-NC LEDs and a matrix of SPAD in the silicon photomultiplier (PM) configuration with still high quality random number generation [219]. Finally, we moved to a single-chip QRNG, where the light source is provided by a reverse biased pn junction operating in the avalanche regime within a Si PM interfaced to a SPAD (Figure 14E) [209]. A compact board implements the bit extraction methodology, enables easy of operation, and shows the potential compactness of the QRNG which passes all the different statistical tests for randomness of the generated bit sequence without postprocessing [220].

One of the main advantages of QRNG is the fact that they are based on simple physical principles and do not rely on complex stochastic models to estimate the randomness (entropy) they generate. However, for the QRNG we discussed so far we had to trust the working of the device, i.e., the fact that the random numbers are generated according to the quantum process and not to some other source, such as classical noise. This has raised the issue of the certification of the quantum nature of the process. To this end, we developed a self-testing QRNG based on the chip in Figure 14E [212]. The objective is to model the QRNG with minimal assumption on the devices and compute the expected amount of minimal quantum entropy (H _min) which can be extracted. The methodology is based on a prepare-and-measure approach, where the light source is the preparation stage and the SPAD the measurement one. In other word, we assume to know the source and the properties of the emitted states and we leave completely uncharacterized the detector. Randomness is then generated by an unambiguous state discrimination strategy. When the source generates a state (e.g. one photon or no photon, state x) the SPAD can click or not (Figure 14F). This means that the measurement is not able to discriminate whether a photon was emitted or not. These inconclusive events have a random occurrence (the clicks generate the random bits b). From this, we can compute the conditional probability P (b|x) that allows estimating H _min (Figure 14H). H _min can be as large as 5% (i.e. 5 % of the bits in the random sequence are certified as due to a quantum process) [212].

One of the most intriguing characteristics of quantum mechanics is the fact that two particles might show one degree of freedom quantum correlated (entangled) even at large distances (nonlocality). More intriguing is the fact that the results of the measurement of a state depend on the measurement (contextuality). Entanglement of two degrees of freedom of a same particle (single-particle entanglement, SPE) does not rely on the nonlocality but relates to the non-contextuality nature of quantum physics [221]. We have worked on the use of SPE as a resource for quantum information due to the fact that this kind of entanglement is easy to be produced and is robust to decoherence. In addition, SPE is not shared nonlocally by two space-like separated photons which means that there is no need of time correlation between different particles. Finally, generation of SPE needs less resources because it relays on single photons generated by any light sources (either an attenuated laser, a LED, or a lamp) [213]. We considered two degrees of freedom (momentum, i.e. direction, and polarization) of a photon. Given a photon with a given direction and polarization (state 0 , V ), the SPE state is generated by a simple method that uses only linear optical components (Figure 14C, generation). A beam splitter (BS) generates the state 1 / 2 ( 0 + i 1 ) V , i.e., the photon is now in a superposition of a reflected and a transmitted state. Then, a half-wave-plate on the BS transmission arm rotates the polarization to horizontal, i.e. the SPE entangled state is formed. Finally, at the end of the generation stage we have a photon in a SPE Bell state Ψ + of the form: Ψ + = 1 2 ( 0 V + 1 H ) . (21)

The usual method to test the entanglement is to verify whether the state violates the Bell inequality. To this aim, we projected the SPE state on all the different combinations of the momentum and polarization (Figure 14C, rotation by angles ϕ, ϕ′, θ, θ′) and detected the different occurrence probabilities (Figure 14C, measurement) from which the correlation coefficients and the violation can be measured. In particular, we test the Clauser, Horne, Shimony, and Holt (CHSH) inequality |S (ϕ, ϕ′, θ, θ′) ≤ 2| and found a significant violation with all the different light sources used (Figure 14G) [213]. We finally implemented this source of SPE as a self-testing QRNG. In fact, the specific SPAD that clicks for a set of rotation angles in the CHSH inequality maximum violation condition is a random variable as a function of time (Figure 14C, bottom table). Careful analysis of the experimental setup and of the generation protocol [222] allows us to estimate H _min with different degrees of trust on the experiment. Values that span from 2.5% up to 30.1% are observed for different levels of description of the setup (Figure 14I) [211]. Moreover, it has been reported that SPE is a resource for other kind of quantum information applications, for example quantum key distribution [223, 224], quantum teleportation [225], state swapping [226], entanglement concentration [227], and single-photon steering [228, 229].

In photonic quantum computing and quantum measurements, single photons are a relevant resource. Ideal single-photon sources emit deterministically only one photon at a time with unit probability. Quantum dots or localized color centers are used to produce such sources with limited performances in terms of indistinguishability. An alternative, which is probabilistic in nature, is to use heralded single-photon sources [202]. Here, a source of time-correlated photons is used where one photon of the pair is detected to herald the other photon. To generate the pair, nonlinear optics is used. Figure 15A shows an example of a heralded single-photon source based on the spontaneous intermodal FWM process (SFWM) [230]. A pulsed pump laser is coupled into a monomode silicon waveguide. Then, a directional coupler (DC) splits the pump beam equally. One part enters a multimode waveguide (MMWG) and excites its fundamental mode, while the other part is coupled to the multimode waveguide by using an asymmetrical direction coupler (ADC) engineered to excite the second-order mode of the MMWG. The MMWG is engineered so that an intermodal phase-matching of the 1221 configuration is achieved. In this way, the pump photons in the TE0 and TE1 modes spontaneously generate idler photons on TE0 and signal photons on the TE1 mode of the MMWG. Then, another ADC is used to couple the signal photon to the TE0 mode of the output waveguide. The chip thus generates pairs of correlated photons at different wavelengths that are output by different waveguides. Detecting the idler photon heralds the signal photons. To verify that the source is indeed a single-photon source, a triggered Hanbury–Brown and Twiss (HBT) interferometer is used which allows us to measure the antibunching characteristics of the signal photons [202]. In fact, single photons are never detected contemporaneously at the two output ports of the BS (coincidence measurements between the two detectors yield 0 for zero time delay). To this end, a BS is placed on the signal path and a three-photon coincidence measurement is performed [230] to measure the heralded second-order coincidence function g H ( 2 ) ( 0 ) . The observation of low values of g H ( 2 ) ( 0 ) (Figure 15B) demonstrates the quality of the source. Let us note that the generated idler and signal photons have a large spectral difference, the pump is at 1,550 nm, the idler photon at 1,260 nm, and the signal photon at 2,014 nm. Therefore to detect the signal photon, no standard telecom single-photon detector can be used. To this scope, we have developed a nonlinear method to spectrally translate MIR photons in the visible where silicon SPAD can be accessed [231]. This is based on an intracavity upconversion process implemented in a compact optical system (Figure 15D). A Nd:YVO₄ rod is placed into a laser cavity together with a nonlinear crystal in such a way that the nonlinear crystal is heavily pumped (up to 100 W). MIR photons enter the cavity and are upconverted in the nonlinear crystals to VIS photons. At the output port, filters remove the pump and other undesired noisy photons. With this setup conversion efficiency as high as 10% is achieved while the statistical properties of the photons are maintained [231]. Therefore, it allows performing single photon and coincidence measurements with a room temperature system. Quite interestingly, the dependence of the intermodal SFWM allows a large tuning of the signal-photon wavelength by a small tuning of the pump photon wavelength (Figure 15C) which opens the use of this source to MIR quantum spectroscopy [230]. Other works show the interest in the MIR region for quantum optics applications, e.g. imaging [233] or sensing [234] or space communication [235], and the need of advancements [236–238].

Another important property of intermodal SFWM is the narrow, spectrally displaced, and discrete generation bands (Figure 13C). This is important to achieve heralded single-photon sources of high purity [202]. In fact, in single-mode waveguides photon pairs are generated by SFWM broadly about the pump with a high degree of spectral anticorrelation which should be removed by using narrow band-pass filters to increase the single-photon purity at the expense of the heralding efficiency. Alternatively, by engineering intermodal SFWM in spiralled MMWG, it is possible to generate photons in discrete pump sidebands with a width that matches the pump pulse bandwidth. In addition, to take into account residual correlations due to the spatial wake-off of the two pump photons, a temporal delay can be added as illustrated in Figure 15G [232]. The chip allows also separating the signal and idler photons in two spatial modes via mode converters. Figure 15E reports the measured biphoton joint spectral intensity that shows no correlation between the idler and signal photons (purity of 0.9904 ± 0.0006) when a delay of 1.46 ps is used. Another important property of the source is the fact that it is integrated in silicon photonics, which means that more sources can be easily fabbricated in the very same chip. In this case, it is important that the generated single pure photons are indistinguishable and that the system does not add unwanted multiphoton noise. The performed experiment is an on-chip 4-photon heralded Hong-Ou-Mandel interference experiment (see the inset of Figure 15F) by which the idler photons generated by two sources herald the quantum interfering signal photons. The measured visibility was as high as 0.96 (2) (Figure 15F) which is a consequence of almost perfect mutual indistinguishability of 0.987(2) [232]. These results pave the way for a massive scaling of photonic quantum information processing [239–241] or for chip-based quantum communications [242], enabled by nonlinear integrated quantum sources [243].