The Series Product and Its Application to Quantum Feedforward and Feedback Networks

Abstract

The purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks. These methods build on and generalize existing methods for series (or cascade) connections by allowing for more general interfaces, and by introducing an efficient algebraic tool, the series product. We also introduce another product, which we call the concatenation product, that is useful for assembling and representing systems without necessarily having connections. We show how the concatenation and series products can be used to describe feedforward and feedback networks. A selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology.

Non-Markovian quantum input-output networks

“…For quantum systems this process can be implemented by making explicit measurements, and processing the resulting classical information using a classical system [54][55][56][57][58]. Alternatively a fully quantum version of this process can be realized by coupling the system to a second quantum system without making explicit measurements [9][10][11][12][13][14]35,62]. It is this second method that we consider here, referred to as coherent feedback control.…”

“…[10]). The dynamical equation (43) can be extended readily to n cascade-connected subsystems to obtain the following quantum stochastic differential equatioṅ…”

“…Under the Markovian assumption, the quantum input-output formalism [6] was extended to cascaded systems [7], and has been used to study quantum coherent feedforward and feedback networks [9][10][11][12][13][14]. Markovian quantum input-output networks can be described using two alternative formulations: the Hudson-Parthasarathy formalism in the Schrödinger picture [15]; and the quantum transfer function formalism in the Heisenberg picture [16,17].…”

Principles and applications of quantum control engineering

“…The cascaded system in the instantaneous feedforward limit is in fact equivalent to the single component [19] …”

Sensitivity of intracavity filtering schemes for detecting gravitational waves

“…The sideband fields are coherently fed back to the main interferometer. In some sense, this is one example of coherent feedback that has been extensively discussed in the quantum optics and control communities [8][9][10][11]. Depending on the choices of optical filters, different interferometer responses and sensitivities can be obtained.…”