A simple method of analyzing quantum feedback circuits is presented. The classical analysis of feedback circuits can be generalized to apply to quantum systems by mapping the field operators of various outputs to other inputs via the standard input-output formalism. Unfortunately, this has led to unphysical results such as the violation of the Heisenberg uncertainty principle for in-loop fields. This paper shows that this general approach can be redeemed by ensuring a self-consistently Hermitian Hamiltonian. The calculations are based on a noncommutative calculus of operator derivatives. A full description of several examples of quantum linear and nonlinear feedback for optical systems is presented.
A simple method of analyzing quantum feedback circuits is presented. The classical analysis of feedback circuits can be generalized to apply to quantum systems by mapping the field operators of various outputs to other inputs via the standard input-output formalism. Unfortunately, this has led to unphysical results such as the violation of the Heisenberg uncertainty principle for in-loop fields. This paper shows that this general approach can be redeemed by ensuring a self-consistently Hermitian Hamiltonian. The calculations are based on a noncommutative calculus of operator derivatives. A full description of several examples of quantum linear and nonlinear feedback for optical systems is presented.
Quantum input-output theory plays a very important role for analyzing the dynamics of quantum systems, especially large-scale quantum networks. As an extension of the input-output formalism of Gardiner and Collet, we develop a new approach based on the quantum version of the Volterra series which can be used to analyze nonlinear quantum input-output dynamics. By this approach, we can ignore the internal dynamics of the quantum input-output system and represent the system dynamics by a series of kernel functions. This approach has the great advantage of modelling weak-nonlinear quantum networks. In our approach, the number of parameters, represented by the kernel functions, used to describe the input-output response of a weak-nonlinear quantum network, increases linearly with the scale of the quantum network, not exponentially as usual. Additionally, our approach can be used to formulate the quantum network with both nonlinear and nonconservative components, e.g., quantum amplifiers, which cannot be modelled by the existing methods, such as the Hudson-Parthasarathy model and the quantum transfer function model. We apply our general method to several examples, including Kerr cavities, optomechanical transducers, and a particular coherent feedback system with a nonlinear component and a quantum amplifier in the feedback loop. This approach provides a powerful way to the modelling and control of nonlinear quantum networks.
Quantum feedback networks have been introduced in quantum optics as a framework for constructing arbitrary networks of quantum mechanical systems connected by unidirectional quantum optical fields, and has allowed for a system theoretic approach to open quantum optics systems. Our aim here is to establish a network theory for quantum transport systems where typically the mediating fields between systems are bidirectional. Mathematically, this leads us to study quantum feedback networks where fields arrive at ports in input-output pairs, making it a special case of the unidirectional theory where inputs and outputs are paired. However, it is conceptually important to develop this theory in the context of quantum transport theory-the resulting theory extends traditional approaches which tend to view the components in quantum transport as scatterers for the various fields, in the process allowing us to consider emission and absorption of field quanta by these components. The quantum feedback network theory is applicable to both Bose and Fermi fields, moreover, it applies to nonlinear dynamics for the component systems. We advance the general theory, but study the case of linear passive quantum components in some detail.
Quantum input-output response analysis is a useful method for modeling the dynamics of complex quantum networks, such as those for communication or quantum control via cascade connections. Non-Markovian effects are expected to be important in networks realized using mesoscopic circuits, but such effects have not yet been studied. Here we extend the Markovian input-output network formalism to non-Markovian networks. The general formalism can be applied to various examples: (i) we show how non-Markovian coherent feedback can reduce the speed of decoherence for an atom in an optical cavity; (ii) we examine the effect of finite-cavity bandwidths in a superconducting circuit.
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 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-known scattering transformation in Schrödinger 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 Riegle’s matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.
Nonlinear quantum optical systems are of paramount relevance for modern quantum technologies, as well as for the study of dissipative phase transitions. Their nonlinear nature makes their theoretical study very challenging and hence they have always served as great motivation to develop new techniques for the analysis of open quantum systems. We apply the recently developed self-consistent projection operator theory to the degenerate optical parametric oscillator to exemplify its general applicability to quantum optical systems. We show that this theory provides an efficient method to calculate the full quantum state of each mode with a high degree of accuracy, even at the critical point. It is equally successful in describing both the stationary limit and the dynamics, including regions of the parameter space where the numerical integration of the full problem is significantly less efficient. We further develop a Gaussian approach consistent with our theory, which yields sensibly better results than the previous Gaussian methods developed for this system, most notably standard linearization techniques.
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 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 Riegle's matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.