Silicon nanophotonics for scalable quantum coherent feedback networks

Over the last decade, coherent quantum feedback control (CQFC) has emerged as a new interdisciplinary field in the areas of quantum control and quantum engineering, and enjoyed rapid theoretical and experimental advances. In particular, a powerful theoretical framework based on input-output theory has been developed for modeling networks of quantum systems connected by electromagnetic fields [1–6]. Such networks can be designed to operate as autonomous devices for quantum information tasks, e.g., quantum state preparation and stabilization [7–9], as well as ultra-low-power optical processing elements for applications such as optical switching [10, 11]. Recent developments such as the SLH formalism for modular analysis of optical networks [12, 13] and the QHDL language and QNET software tools for specification and simulation of photonic circuits [14] have added important capabilities for efficient and automated design and modeling of CQFC networks.

These advances bring CQFC to the level where it can help design and quantitatively analyze quantum effects in integrated optics systems and, reciprocally, benefit from fabrication capabilities of nanophotonics technology. Indeed, the true potential of CQFC is realized at scales where many optical elements are interconnected as potentially reconfigurable networks [15]. Bulk-optics implementations are impractical for the realization of such complex networks, and a form of integrated optics is necessary. In addition to the obvious size advantage of integrated optics, other benefits include reproducibility and mass production capability, and long-term optical path and phase stability. In particular, CMOS-compatible silicon integrated nanophotonics is seen as a leading platform for constructing large-scale CQFC networks.

In this paper, we present a detailed analysis of the potential of silicon nanophotonics for implementing CQFC networks. We describe the behavior of nanophotonic components relevant for construction of CQFC networks and also discuss the key challenges to silicon nanophotonics implementation of CQFC. Finally, we describe our fabrication and measurement of a simple on-chip CQFC network composed of two coupled cavities and analyze it using the SLH formalism. As many of the general points raised in the paper are revealed in the analysis of this device, this implementation serves as a useful testbed on which to explore the benefits and challenges of silicon nanophotonics realizations of CQFC.

The remainder of the paper is structured as follows. Section 2 presents a brief introduction to the theoretical framework that enables efficient modeling of optical networks composed of modular components connected by quantum fields. In Section 3 we discuss the current capabilities of CMOS-compatible integrated optics platforms, including the common linear and nonlinear components that are available for constructing quantum optical networks. Section 4 explores potential challenges in extending the standard CQFC theory presented in Section 2 to the integrated photonics realm. In Section 5, we present a preliminary on-chip implementation of a network of two coupled cavities, and analyze its properties. Finally, Section 6 concludes with a discussion of potential future directions in integrated optics CQFC.

In this section, we provide a brief summary of the approach to modeling CQFC networks using quantum stochastic calculus, or alternatively input-output theory from quantum optics (for further details, see Refs. [4–6]). The basis of this approach is the decomposition of a quantum optical network into localized components with arbitrary internal degrees of freedom, which are connected via freely propagating unidirectional broadband fields. This allows one to eliminate the explicit description of the fields propagating between components to arrive at an effective description of the network just in terms of the localized degrees of freedom and their couplings. We refer to this modular approach and the associated modeling machinery as the SLH formalism [12, 13].

The key advantage of the SLH formalism is that one can easily construct effective descriptions of arbitrarily connected networks of localized components, each of which is represented by a triple: \(G=(S,L,H)\). Connecting two components in series, parallel, or feedback loop results in another system represented by another SLH triple whose matrices can be derived by simple algebraic rules [12]. As an example, consider connecting two localized systems in series, where the outputs from \(G_{1} = (S_{1}, L_{1}, H_{1})\) are connected to the inputs of \(G_{2} = (S_{2}, L_{2}, H_{2})\), where for simplicity we assume that the number of input ports that \(G_{2}\) has is the same as the number of output ports that \(G_{1}\) has. The resulting system is represented as \(G_{3} = G_{2} \lhd G_{1} = (S_{3}, L_{3}, H_{3})\), where \(S_{3} = S_{2} S_{1}\), \(L_{3} = S_{2} L_{1} + L_{2}\), and \(H_{3} = H_{1} + H_{2} + \operatorname {Im}\{L_{2}^{\dagger}S_{2} L_{1} \}\). See Refs. [5, 12] for more details on the composition rules for the SLH formalism.

We will discuss the validity of these assumptions for integrated optics in the following sections.

Solid-state based realizations of quantum optical networks are possible on several platforms nowadays, including microwave quantum optics on superconducting integrated structures [19–21] and visible/near infrared quantum optics on integrated photonic structures [22]. In this work, we focus on the latter and, in particular, on integrated photonics implementations using silicon and silicon nitride at telecommunications wavelengths centered on 1550 nm. The CMOS compatibility and relative maturity of integrated photonics on these platforms makes them appealing candidates for implementing scalable CQFC networks. See Combes et al. [6] for a review of integrated implementations of quantum optical networks, and a discussion of the issues relevant to superconducting circuit implementations.

Silicon (Si) and silicon nitride nanowire waveguides guide light through total internal reflection, enabled by the high index contrast between the guiding core and the surrounding cladding. Usually the waveguide is designed to guide only a fundamental single mode at a desired wavelength. The shape of the waveguide is fully etched, which naturally fits with CMOS compatible processes. Since the waveguide is rectangular with greater width than height, the profile of the guided mode of light is elliptical. The large index contrast between the core and the cladding allows for waveguide dimensions to be only a fraction of the wavelength (several hundreds of nanometers).

A crucial factor that differentiates transmission in silicon waveguides from transmission in vacuum is scattering of photons due to roughness of waveguide surfaces. This leads to a linear loss mechanism (loss that is independent of light intensity) in waveguides and resonant structures. In the nanowire type single mode waveguides, scattering loss arises due to side-wall inhomogeneities created in the etch process used to define the waveguide. This scattering can be minimized by sophisticated fabrication techniques [23–26], but is an intrinsic non-ideality that cannot be completely mitigated. In the nitride nanowire waveguides, absorption due to unsaturated bonds is another source of propagation loss and can be mitigated by sophisticated passivation, deposition and fabrication techniques, e.g., [27, 28]. Other waveguide types, such as ridge waveguides, can have lower scattering loss although they are intrinsically multi-mode. These ridge modes are usually more tightly confined and hence interact weakly with side-wall roughness. All integrated optical waveguides and resonant structures will have intrinsic losses that cannot be completely mitigated.

As mentioned in the introduction, one of the primary advantages of implementing CQFC networks in integrated photonics is the scalability of this platform. Since the SLH formalism, and accompanying automation tools such as the QHDL language [14] and the QNET simulation package [66], implement the analogue of lumped element analysis for quantum optical networks, they are most powerful for analyzing large-scale modular networks that are difficult to simulate from first principles. Practical realizations of such networks require integrated platforms such as superconducting circuits or silicon photonics. However, originally the SLH formalism was developed primarily with bulk-optics implementations in mind, and needs to be reassessed and possibly modified before application to integrated platforms.

The primary challenges in porting the SLH formalism to integrated photonics stem from the need to capture the range of optical phenomena resulting from electromagnetic field propagation in a nonlinear, dispersive medium. In terms of the assumptions listed at the end of Section 2, assumption 3 is the one that needs further examination, since the propagation medium is no longer vacuum. Specific effects that need to be taken into account are dictated by the material substrate and the wavelength of light used in the nanophotonics platform implementation. The dominant physical phenomena present in silicon and silicon nitride integrated photonics at 1550 nm, and absent in bulk-optics networks, were discussed in Section 3. To reiterate, these are: (i) dispersion, (ii) scattering by the medium, including surface roughness scattering, Raman scattering, and Brillouin scattering, and (iii) two-photon absorption and subsequent free carrier generation and heating in the medium.

In the following, we examine each of the physical effects identified above and assess their impact on the SLH formalism.

In order to study the differences and similarities between bulk-optics and silicon nanophotonics networks, we fabricated one of the simplest CQFC networks: two cavities coupled in a feedback loop; see Figure 1. We refer to this network as the coupled cavity device (CCD), and its implementation was inspired by the disturbance rejection network implemented in bulk optics by Mabuchi in Ref. [75], and previously theoretically analyzed in Ref. [76].

Figure 1 shows the schematic of the CCD, which consists of two thermally controlled ring resonators (denoted as cavity 1 and cavity 2) coupled by Si nanowire bus waveguides with an integrated thermo-optic phase shifter on each bus waveguide. The device has four actual ports (additional fictitious ports can be used to model losses), two at each cavity. A coherent input drive (w) from a widely tunable telecom laser is coupled onto chip from a lensed fiber into port 1. The intensity of the output signal (z) at port 2 is monitored off chip using a power meter. The ports of cavity 2 are unmonitored and have vacuum input. The signal in the output field z is a result of interference between the outputs of both cavities since they are coupled. In Refs. [75, 76], one of the cavities is treated as a plant system and the other as a controller, in which case the signal propagating from cavity 2 to cavity 1 (field u) is viewed as a feedback signal from controller to plant.

The CCD is controlled using two thermo-optic phase shifters activated by externally applied voltages. One phase shifter, controlled by the voltage \(V_{\mathrm{filter}}\), is used to tune the resonance frequency \(\omega_{c}\) of cavity 2, and the other, controlled by the voltage \(V_{\mathrm{phase}}\), is used to induce a phase shift ϕ in the feedback signal (field u). The variable parameters \(\omega_{c}\) and ϕ are the primary in situ controllable degrees of freedom of the CQFC network implemented by the CCD. During the experiment the probe laser wavelength is swept and the upper phase shifter voltage, \(V_{\mathrm{phase}}\) is varied from 0 to 18 V for fixed controller \(V_{\mathrm{filter}}\) bias. We estimate that the resonators have quality factors in the range \(1\times10^{3} - 4 \times10^{3}\).

The coupled cavities behave in a similar fashion to electromagnetically induced transparency (EIT), where two nearly degenerate resonances can interfere creating sharp null in the transmission spectrum. \(V_{\mathrm{phase}}\) acts as a variable coupling between the two resonant cavities and can be used to shift in wavelength the interference null in the transmission spectrum (cf. Refs. [77, 78]). However, for disturbance rejection we require that the transmission is suppressed at all wavelengths. This can be achieved in the CCD if the linewidth of the controller cavity and the phase shift induced in the feedback signal u can be controlled independently, while the other parameters are fixed [75]. Disturbance rejection means that, with suitable parameter values, the output field z is in the vacuum state regardless of the amplitude and phase of the input field w. Physically, this results from all the input power being routed to the output port \(z'\) due to the interference between the cavities. We are unable to tune the cavity linewidths in situ with this generation of the CCD, however, we will evaluate whether disturbance rejection can be achieved with the parameter values determined at fabrication and the in situ tuning capabilities we do have.

It should be noted that if the CCD is driven by a low-power laser (i.e., the input field w is prepared in a low-intensity coherent state) and the cavities remain in the linear regime (i.e., their Q factors are not too large), then the entire network is linear and thus can be modeled by an equivalent classical transfer function as shown in Refs. [75, 76]. In this case, the input-output relationships predicted by the SLH model and the transfer function agree. The experiment is conducted in this linear regime and therefore the SLH model does not incorporate nonlinearities. In the following, we assess the agreement between this model and experimental data.

Figure 2 shows the spectra of the output field z when the input CW field w is prepared in a coherent state. The input power (after accounting for on-chip coupling loss) is calibrated to be \({\sim}100~\mu\mbox{W}\), and \(V_{\mathrm{filter}}\) is held fixed at 1.4 V. Each subplot shows the output spectrum for a different value of the voltage \(V_{\mathrm{phase}}\), which controls the phase shift ϕ induced on the feedback field u. In each subplot, the experimental spectrum is shown together with the spectrum predicted by the theoretical SLH model (developed in the Appendix) with parameter values selected to achieve the best fit with the experimental data. The parameter fitting was performed independently for each value of \(V_{\mathrm{phase}}\), using a simulated annealing algorithm initialized with a good estimate of the parameter values from knowledge of the chip fabrication details. The fits are seen to be reasonably good, which gives us confidence that the linear model is an accurate representation of the system.

Figure 3 shows the fitted values of the model parameters (\(\lambda_{p}\), \(\lambda_{c}\), \(\gamma_{p}\), \(\gamma_{c}\), ϕ, η, and κ) obtained for each value of \(V_{\mathrm {phase}}\). Note that \(\lambda_{c/p} = \frac{2\pi c}{n_{\mathrm{eff}} \omega_{c/p}}\) with \(n_{\mathrm{eff}}=2.85\) being the effective index of the resonator at 1550 nm. Surprisingly, we see that not only the feedback phase ϕ, but almost all other parameters change with \(V_{\mathrm{phase}}\). This is hardly ideal as it means that the parameters in the system are not independently tunable. In particular, we cannot tune the system to operate in the parameter regime required for disturbance rejection. Physically, the effect of the voltage applied to a phase shifter (where Joule heating changes the material refractive index via the thermo-optic effect) does not seem to be localized to the bus waveguide connecting the cavities; the produced heat also affects the properties of adjacent optical elements, including cavity resonances, cavity-waveguide coupling and cavity losses. This cross-talk effect associated with the physical nature of controls and size characteristics of the nanophotonics platform, is an important issue that distinguishes on-chip implementations of optical networks from their bulk-optics analogues.

We comment on the behavior of each of the parameters in Figure 3. For the most part, the phase shift ϕ changes predictably and monotonically with \(V_{\mathrm{phase}}\). The power loss in the waveguide, η, also behaves rather regularly: it increases with the applied voltage for \(V_{\mathrm{phase}} \lesssim 5\mbox{ V}\) and then stays approximately constant at higher voltages; this behavior implies that the waveguide becomes lossier when the thermo-optic phase shifter is active, but losses saturate at \(V_{\mathrm{phase}} \gtrsim5\mbox{ V}\). On the other hand, the rate of cavity-waveguide couplings, κ, does not change much for \(V_{\mathrm{phase}} \lesssim5\mbox{ V}\), but decreases when the voltage increases for \(V_{\mathrm{phase}} \gtrsim5\mbox{ V}\); this behavior might indicate that cavity-waveguide interfaces become affected when a higher voltage generates a larger amount of heat. Judging by the changes of cavity resonance wavelengths \(\lambda_{p}\) and \(\lambda_{c}\), the cavity resonances are linearly red-shifted with increasing \(V_{\mathrm {phase}}\), and by roughly the same amount for both cavities. The cavity loss rates behave surprisingly dissimilar for the plant and controller resonators. The plant cavity’s loss rate \(\gamma_{p}\) is constant for \(V_{\mathrm{phase}} \lesssim6\mbox{ V}\) and increases with \(V_{\mathrm {phase}}\) for higher voltages. In contrast, the controller cavity’s loss rate \(\gamma_{c}\) mostly decreases when the applied voltage increases, but its behavior is significantly non-monotonic and least regular of all the parameters. We were unable to find a high quality, monotonic fit to this parameter even after randomizing the initial state for the simulated annealing optimization algorithm.

We also performed experiments where both \(V_{\mathrm{phase}}\) and the resonance frequency of the controller cavity were tuned (via \(V_{\mathrm{filter}}\)), but were also unable to achieve disturbance rejection in this case. We attribute this to the inability to tune the CCD to the disturbance rejection parameter regime due to (i) the cross-talk effect mentioned above inhibiting independent tuning of device parameters, and (ii) the large mismatch between the quality factors of the two cavities due to the much larger linewidth of the controller cavity compared to the plant cavity.

In this work, we analyzed the suitability of integrated silicon photonics to serve as a platform for implementing scalable CQFC networks. In addition to summarizing the strengths of silicon photonics for this application, we also outlined the principal challenges to both the theoretical framework and practical implementations. In particular, Section 4 presented the features of the integrated photonics platform that are not yet taken into account by the SLH formalism, which was largely developed with bulk-optics implementations in mind. We identified a number of extensions to the SLH formalism, which are needed to make it more applicable to linear as well as nonlinear integrated optics networks.

In Section 5, we presented the results of a preliminary experiment that explored an on-chip implementation of a simple CQFC network of two coupled cavities, and analyzed its properties using the SLH formalism. The main lesson that we learned from this analysis is that in situ controls applied to one component of the nanophotonic device significantly affected properties of the rest of the components. This cross-talk is a result of the small size of the device and the physical nature of the controls that utilize Joule heating to manipulate optical properties. Therefore, it is important to understand all the impacts of in situ controls including thermal effects due to the heat transfer and effects on carrier concentration, both of which can change properties of integrated network components. It is also clear that it is advantageous to use in situ control mechanisms that act locally, and have as few side effects, as possible.

The fast progress in the field of nanophotonics technology is likely to generate new advances that will be beneficial to integrated optics implementations of CQFC networks. In particular, recent advances in non-silicon CMOS-compatible platforms utilizing low-loss materials such as silicon nitride and Hydex will likely enable more versatile integrated components, especially for nonlinear optics [79]. For example, the Hydex platform was used to implement an integrated photon pair source [80] employing an above-threshold OPO, a key nonlinear optics element. Another promising development is the use of hydrogenated amorphous silicon (a-Si:H) whose nonlinear optical properties in the telecommunications band (including ultrahigh optical nonlinearity, low nonlinear loss, and reduced impact of free-carrier processes) are superior to those of undoped crystalline silicon [81]. The a-Si:H platform was used to demonstrate integrated photonics implementations of various nonlinear optical processes, including parametric amplification [38], four-wave mixing for low-power optical frequency conversion [82–84], and cross-phase modulation for all-optical switching [85]. The incorporation of new materials into CMOS compatible processes and heterogenous integration into the Si photonics platform will greatly expand the integrated quantum optics toolbox, and enable the construction increasingly complex quantum optical networks.