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The mathematical theory of quantum feedback networks has recently been developed for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, for the special case of linear dynamical systems Markovian systems with instantaneous feedback connections, that the transfer functions can be deduced and agree with the algebraic rules obtained in the nonlinear case. Using these rules, we derive the the transfer functions for linear quantum systems in series, in cascade, and in feedback arrangements mediated by beam splitter devices.
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Enabled by rapidly developing quantum technologies, it is possible to network quantum systems at a much larger scale in the near future. To deal with non-Markovian dynamics that is prevalent in solid-state devices, we propose a general transfer function based framework for modeling linear quantum networks, in which signal flow graphs are applied to characterize the network topology by flow of quantum signals. We define a noncommutative ring $mathbb{D}$ and use its elements to construct Hamiltonians, transformations and transfer functions for both active and passive systems. The signal flow graph obtained for direct and indirect coherent quantum feedback systems clearly show the feedback loop via bidirectional signal flows. Importantly, the transfer function from input to output field is derived for non-Markovian quantum systems with colored inputs, from which the Markovian input-output relation can be easily obtained as a limiting case. Moreover, the transfer function possesses a symmetry structure that is analogous to the well-know scattering transformation in sd picture. Finally, we show that these transfer functions can be integrated to build complex feedback networks via interconnections, serial products and feedback, which may include either direct or indirect coherent feedback loops, and transfer functions between quantum signal nodes can be calculated by the Riegles matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.
A quantum network is an open system consisting of several component Markovian input-output subsystems interconnected by boson field channels carrying quantum stochastic signals. Generalizing the work of Chebotarev and Gregoratti, we formulate the model description by prescribing a candidate Hamiltonian for the network including details the component systems, the field channels, their interconnections, interactions and any time delays arising from the geometry of the network. (We show that the candidate is a symmetric operator and proceed modulo the proof of self-adjointness.) The model is non-Markovian for finite time delays, but in the limit where these delays vanish we recover a Markov model and thereby deduce the rules for introducing feedback into arbitrary quantum networks. The type of feedback considered includes that mediated by the use of beam splitters. We are therefore able to give a system-theoretic approach to introducing connections between quantum mechanical state-based input-output systems, and give a unifying treatment using non-commutative fractional linear, or Mobius, transformations.
Here we consider the speed at which quantum information can be transferred between the nodes of a linear network. Because such nodes are linear oscillators, this speed is also important in the cooling and state preparation of mechanical oscillators, as well as frequency conversion. We show that if there is no restriction on the size of the linear coupling between two oscillators, then there exist control protocols that will swap their respective states with high fidelity within a time much less than a single oscillation period. Standard gradient search methods fail to find these fast protocols. We were able to do so by augmenting standard search methods with a path-tracing technique, demonstrating that this technique has remarkable power to solve time-optimal control problems, as well as confirming the highly challenging nature of these problems. As a further demonstration of the power of path-tracing, first introduced by Moore-Tibbets et al. [Phys. Rev. A 86, 062309 (2012)], we apply it to the generation of entanglement in a linear network.
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.
This paper concerns the problem of stability for quantum feedback networks. We demonstrate in the context of quantum optics how stability of quantum feedback networks can be guaranteed using only simple gain inequalities for network components and algebraic relationships determined by the network. Quantum feedback networks are shown to be stable if the loop gain is less than one-this is an extension of the famous small gain theorem of classical control theory. We illustrate the simplicity and power of the small gain approach with applications to important problems of robust stability and robust stabilization.
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