Coherent-feedback control strategy to suppress spontaneous switching in ultra-low power optical bistability

An optical resonator with intracavity Kerr nonlinearity can exhibit dispersive bistability suitable for all-optical switching. With nanophotonic elements it may be possible to achieve attojoule switching energies, which would be very attractive for ultra-low power operation but potentially problematic because of quantum fluctuation-induced spontaneous switching. In this manuscript I derive a quantum-optical model of two Kerr-nonlinear ring resonators connected in a coherent feedback loop, and show via numerical simulation that a properly designed `controller' cavity can significantly reduce the spontaneous switching rate of a bistable `plant' cavity in a completely embedded and autonomous manner.

Although current nanophotonics research focuses mainly on the design and demonstration of individual optical components, future progress towards technological relevance will surely require the development of nanophotonic circuit theory at a level of sophistication comparable to that of modern electronics. As the performance regime of interest for nanophotonic technologies extends to picosecond switching times and attojoule (few-photon) switching energies, quantum-optical effects will be of great practical significance even if the information processing paradigm remains purely classical (i.e., before the advent of true quantum information technology). Rigorous yet user-friendly theoretical methods (based on new generalizations of classical stochastic systems theory 1 ) for the quantum-optical analysis of photonic circuits have recently been developed 2 , but compatible algorithmic design methods 3,4 are still quite limited in scope. It is thus an opportune moment to begin investigating relatively simple nanophotonic circuit 'motifs' in order to exercise our new analysis methods and to provide guidance for subsequent work on more complex component networks. Given the ubiquity of feedback configurations for noise suppression within microelectronic circuits, it seems natural to focus such preliminary exploration on coherent (optical) feedback motifs for managing quantum fluctuations in ultra-low power nanophotonic circuits.
Here we consider a coherent feedback 3-5 strategy for suppressing spontaneous switching in dispersive optical bistability. Dispersive bistability is of interest as a potential physical basis for the design of ultra-low power nanophotonic switches [6][7][8][9] , but in the attojoule switching regime where the logical states are separated by a small number of photons, quantum fluctuations will induce unwanted spontaneous switching 10-12 that must be accounted for in circuit design. For example, Fig. 1 shows a simple quantum trajectory simulation (performed using the Quantum Optics Toolbox for Matlab 13 ) of the mean intracavity photon number for a Kerr-nonlinear optical resonator, assuming parameters (cavity decay rate κ b = 150, drive detuning ∆ b = 5κ b , and nonlinear coefficient χ B = −∆ b /10 √ 2) that classically would be expected to support dispersive optical bistability with attojoule separation between the logical high and low states. The quantum model 14 , corresponding to the Master Equation clearly predicts spontaneous transitions that would compromise the performance of such a device in a photonic switching context. Here b is the annihilation operator for the intracavity field mode, β is the complex amplitude of a coherent drive field, and κ b3 ≤ κ b is the partial decay rate associated with the input coupler of the resonator. In Fig. 1 we have set β √ κ b3 = 10.4934 √ 50 to achieve approximately equal time-average occupation of the low-and highphoton number states.
In order to motivate our coherent-feedback stabilization strategy for suppression of such 'quantum jumps' we first consider a feedback configuration with a linear static controller. If we assume that the bistable ('plant') cavity has three distinct input-output ports corresponding to the bias input and the feedback-loop input and output, we can model 15 the effects of a simple optical feedback loop with unit gain and total phase shift ϕ (as depicted in the upper left panel of Fig. 2) using the Master Equation (1) Here κ b1,2 are the partial decay rates associated with coupling to the feedback loop; it is assumed that κ b1 + κ b2 + κ b3 = κ b and in what follows we will set them equal. It can be seen that the net effects of the feedback loop are a ϕ-dependent frequency pulling of the effective drive detuning and a ϕ-dependent change in the effective cavity decay rate. Either or both of these effects could potentially be used to suppress spontaneous switching of the bistable cavity if ϕ could be adjusted to a value that stabilizes the low state when the state is low, and to a value that stabilizes the high state when the state is high.
To realize the desired form of nonlinear dynamic controller we consider an auxiliary

The Master Equation for this coherent feedback configuration is given by
where H a is obtained by setting b → a in Eq. It should be noted that within the two-cavity coherent feedback configuration we have considered, and with the key 'structural' parameters (χ b , κ b ) of the plant cavity held fixed, there remains a great deal of room for optimizing the operating conditions (β, ∆ b ) and controller parameters (∆ a , κ a , χ a and ϕ) to achieve potentially superior suppression. Given the rather demanding nature of the numerical computations involved (a total Hilbert space dimension of 625 was used in this work and Master Equation integrations were essential), a brute-force scan of so many degrees of freedom would not seem feasible but it seems likely that a more principled computational optimization approach could be developed.
Recent theoretical investigations-based on classical electromagnetic models-of circuit motifs 16 and optimal pulse shaping for switching applications 18 have offered a glimpse of the great potential for innovative engineering at the signals-and-systems (as opposed to device physics) level in nanophotonics. Here we have attempted to extend this exploration to the quantum optical regime of attojoule switching energy, demonstrating that new theoretical methods 1 can be used to analyze intuitive coherent feedback control schemes in quantitative detail.