Silicon nanophotonics for scalable quantum coherent feedback networks

Abstract

The emergence of coherent quantum feedback control (CQFC) as a new paradigm for precise manipulation of dynamics of complex quantum systems has led to the development of efficient theoretical modeling and simulation tools and opened avenues for new practical implementations. This work explores the applicability of the integrated silicon photonics platform for implementing scalable CQFC networks. If proven successful, on-chip implementations of these networks would provide scalable and efficient nanophotonic components for autonomous quantum information processing devices and ultra-low-power optical processing systems at telecommunications wavelengths. We analyze the strengths of the silicon photonics platform for CQFC applications and identify the key challenges to both the theoretical formalism and experimental implementations. In particular, we determine specific extensions to the theoretical CQFC framework (which was originally developed with bulk-optics implementations in mind), required to make it fully applicable to modeling of linear and nonlinear integrated optics networks. We also report the results of a preliminary experiment that studied the performance of an in situ controllable silicon nanophotonic network of two coupled cavities and analyze the properties of this device using the CQFC formalism.

Appendix: SLH model for the coupled cavity device

In order to develop the theoretical model of the CCD depicted in Figure 1, we use a schematic representation of the equivalent network shown in Figure 4. Note that the network shown in Figure 4 includes additional fictitious components (a beam splitter and two mirrors) that are used to model losses. This network can be decomposed as a concatenation product [5, 12] of several components in series. Specifically, the SLH triple for the CCD, $G_{\mathrm{CCD}}$, is given by:

$$\begin{aligned} G_{\mathrm{CCD}} =& \bigl[ (G_{p2}\boxplus G_{1}) \lhd G_{\eta} \lhd(G_{\phi}\boxplus G_{1}) \lhd(G_{c2}\boxplus G_{1}) \bigr] \\ &{}\boxplus[ G_{c1} \lhd G_{p1} \lhd G_{w} ] \boxplus G_{p3} \boxplus G_{c3}, \end{aligned}$$

(A1)

where the components are specified by the following SLH triples:

$$\begin{aligned}& G_{p1} = \bigl( 1, \sqrt{\kappa} a, \omega_{p} a^{\dagger}a \bigr), \\& G_{p2} = ( 1, \sqrt{\kappa} a, 0 ), \\& G_{p3} = ( 1, \sqrt{\gamma_{p}} a, 0 ), \\& G_{c1} = \bigl( 1, \sqrt{\kappa} b, \omega_{c} b^{\dagger}b \bigr), \\& G_{c2} = ( 1, \sqrt{\kappa} b, 0 ), \\& G_{c3} = ( 1, \sqrt{\gamma_{c}} b, 0 ), \\& G_{w} = ( 1, \alpha, 0 ), \\& G_{\phi} = \bigl( e^{i\phi}, 0, 0 \bigr), \\& G_{1} = ( 1, 0, 0 ), \\& G_{\eta} = \left( \left[ \textstyle\begin{array}{c@{\quad}c} \sqrt{1-\eta} & \sqrt{\eta} \\ \sqrt{\eta} & \sqrt{1-\eta} \end{array}\displaystyle \right],0,0 \right). \end{aligned}$$

Here, a is the annihilation operator for the fundamental mode of the plant cavity, of frequency $\omega_{p}$, and b is the annihilation operator for the fundamental mode of the controller cavity, of frequency $\omega_{c}$. $G_{p1}$ and $G_{p2}$ represent the “mirrors” of the plant cavity that couple to the bus waveguides. Similarly, $G_{c1}$ and $G_{c2}$ represent the “mirrors” of the controller cavity that couple to the bus waveguides. We assume that all these mirrors have the same leakage rate κ (the leakage rate κ of a cavity mirror relates to its power transmittance T as $\kappa = c T/(2 n_{\mathrm{eff}} \ell)$, where c is the speed of light, $n_{\mathrm{eff}}$ is the refractive index of the cavity medium, and ℓ is the cavity length). $G_{p3}$ and $G_{c3}$ represent the fictitious “mirrors” that leak light and are used to model intrinsic losses in the plant cavity and the controller cavity, respectively; these “mirrors” have leakage rates $\gamma_{p}$ and $\gamma_{c}$, respectively. $G_{w}$ represents the drive field applied to the plant cavity (signal w), which is assumed to be prepared in a coherent state with complex amplitude α. $G_{\phi}$ represents the phase shifter that induces the phase shift ϕ in the feedback field u, and $G_{1}$ represents a simple passthrough. Finally, $G_{\eta }$ represents the fictitious beam-splitter that models the loss in the waveguide linking the controller and plant. The power loss in the waveguide is η, so the $\sqrt{1-\eta}$ portion of the field amplitude is transmitted. We see from this component breakdown that there are seven free parameters in this model: $\omega_{p}$, $\omega _{c}$, κ, $\gamma_{p}$, $\gamma_{c}$, η, and ϕ.

Figure 4

A schematic representation of the CCD, used to generate a lumped element model. Each ring resonator is modeled as a triangular cavity with mirrors that couple to two waveguides with leakage rate κ, and one fictitious mirror that represents intrinsic loss with leakage rate $\gamma_{p/c}$. The cavity on the right is the plant and the one of the left is the controller. The driving signal w and the monitored signal z are indicated. The feedback signal u undergoes a phase shift ϕ and power loss η. The overall network has five input ports and five output ports, all of which are indicated; the driving field w enters at input port 3 and the monitored field z exits at output port 1; all input ports except 3 have vacuum fields. Note that there is a re-labeling of the ports with respect to Figure 1 in the main text.

Equation (A1) expresses the power of the SLH formalism; it captures the modular nature of the optical network and also gives us a prescriptive formula for how to model the properties of the entire device by knowing properties of each module. For simple networks like the one considered here, it is possible to model each component and deduce the appropriate product by examination. However, there exist systematic and prescriptive techniques to achieve the same SLH component breakdown for an arbitrary network [5, 12]. Furthermore, the QNET software package [66] enables complete automation of this task, starting from a specification of the network in terms of its physical components.

Carrying out the series and concatenation products in Eq. (A1), yields the SLH triple representing the entire CCD:

$$\begin{aligned} G_{\mathrm{CCD}} =& \left( \left[ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \sqrt{1-\eta}e^{i\phi} & \sqrt{\eta} & 0 & 0 & 0 \\ \sqrt{\eta}e^{i\phi} & \sqrt{1-\eta} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\displaystyle \right], \left[ \textstyle\begin{array}{c} \sqrt{\kappa}( a + \sqrt{1-\eta}e^{i\phi} b) \\ \sqrt{\kappa\eta}e^{i\phi}b \\ \sqrt{\kappa}(a+b) + \alpha\\ \sqrt{\gamma_{p}}a \\ \sqrt{\gamma_{c}}b \end{array}\displaystyle \right],\right. \\ & \omega_{p} a^{\dagger}a + \omega_{c} b^{\dagger}b + \frac {\sqrt{\kappa }}{2i} \bigl(a^{\dagger}\alpha- a \alpha^{*} \bigr) \\ &\left.{}+ \frac{\kappa }{2i} \bigl[\bigl(1-e^{-i\phi}\sqrt{1-\eta} \bigr)a^{\dagger}b - \bigl(1-e^{i\phi }\sqrt{1-\eta} \bigr)ab^{\dagger} \bigr] \vphantom{\left[ \textstyle\begin{array}{c} \sqrt{\kappa}( a + \sqrt{1-\eta}e^{i\phi} b) \\ \sqrt{\kappa\eta}e^{i\phi}b \\ \sqrt{\kappa}(a+b) + \alpha\\ \sqrt{\gamma_{p}}a \\ \sqrt{\gamma_{c}}b \end{array}\displaystyle \right]} \right). \end{aligned}$$

(A2)

The ordering of elements in matrices S and L corresponds to the numbering of input and output ports in Figure 4, i.e., the matrix element $S_{i j}$ is the scattering amplitude from input port i to output port j, and the matrix element $L_{i}$ represents the field at output port i.

Given the SLH triple (A2) for the CCD, one can explicitly see that (i) the Hamiltonian includes a coupling term representing an effective interaction between the plant cavity and the controller cavity, induced by the feedback, and (ii) the signal field z at output port 1 is produced by an interference (i.e., composed of a linear combination) of the two cavity fields: $a + \sqrt{1-\eta} e^{i\phi} b$. The observation made in Refs. [76] and [75] is that by appropriate choice of parameters, especially the phase shift ϕ, this output can be made zero regardless of the driving field w at input port 3.

We can explicitly write output field at port 1 (out1 or z) as

$$\begin{aligned} dB_{\mathrm{out1}} =& L_{1}(t) \,dt + \sum _{l} S_{1l} \,dB^{\mathrm{in}}_{l} \\ =& \sqrt{\kappa}\bigl(a + \sqrt{1-\eta}e^{i\phi} b\bigr) \,dt + \sqrt{1- \eta}e^{i\phi} \,dB_{\mathrm{in1}} + \sqrt{\eta} \,dB_{\mathrm{in2}}. \end{aligned}$$

(A3)

It can be verified that the input-output relationship predicted by this SLH model is identical to that predicted by the transfer function derived in Ref. [75]. This is to be expected since the system is linear and the input is in a coherent state.

More generally, for CQFC networks with non-linear optical components, the SLH formalism can be used to model quantum optical phenomena that have no classical analogues. For example, the SLH model of a network of two coupled OPOs [86] predicts interferences that enable to achieve a degree of control over properties of the output field, such as its squeezing spectrum [8], which would be impossible without the coherent feedback.