Classical feedback control and system theory are playing an important role in modelling, controlling and analysing complex devices in many branches of engineering. Recent developments like quantum computers and miniaturisation of existing applications and devices are increasing the importance of the ability to control systems with quantum effects. Efforts have been made recently to extent the simplicity and power of the language of classical control theory to quantum mechanical systems. Within this framework of “Quantum Feedback Networks” we are investigating two problems. The first problem concerns the enhancement of squeezed states. It has been observed that the squeezing effect of squeezing devices can be enhancement by measurement based feedback techniques or use of optical cavities. We are investigating the possibility of feedback enhanced squeezing using coherent feedback control. Considered is a static ideal squeezing devices interacting with a single mode cavity undergoing coherent feedback using a beam splitter. We show that the overall squeezing of the output depends on the beam-splitter’s reflectivity and that we are thus able to enhance the squeezing by choosing an appropriate configuration of the beam-splitter. In the second part we investigate the question of compatibility of a rigorous approach to the adiabatic elimination of some degrees of freedom of a quantum mechanical systems and instantaneous feed-forward and feedback limits for quantum mechanical networks. The commutativity of both limits is not obvious but frequently assumed in quantum optics. We show that both limit procedures are instances of Schur complements and prove the commutativity of both limits by generalising a statement about successive Schur complements.
Classical feedback control and system theory are playing an important role in modelling, controlling and analysing complex devices in many branches of engineering. Recent developments like quantum computers and miniaturisation of existing applications and devices are increasing the importance of the ability to control systems with quantum effects. Efforts have been made recently to extent the simplicity and power of the language of classical control theory to quantum mechanical systems. Within this framework of “Quantum Feedback Networks” we are investigating two problems. The first problem concerns the enhancement of squeezed states. It has been observed that the squeezing effect of squeezing devices can be enhancement by measurement based feedback techniques or use of optical cavities. We are investigating the possibility of feedback enhanced squeezing using coherent feedback control. Considered is a static ideal squeezing devices interacting with a single mode cavity undergoing coherent feedback using a beam splitter. We show that the overall squeezing of the output depends on the beam-splitter’s reflectivity and that we are thus able to enhance the squeezing by choosing an appropriate configuration of the beam-splitter. In the second part we investigate the question of compatibility of a rigorous approach to the adiabatic elimination of some degrees of freedom of a quantum mechanical systems and instantaneous feed-forward and feedback limits for quantum mechanical networks. The commutativity of both limits is not obvious but frequently assumed in quantum optics. We show that both limit procedures are instances of Schur complements and prove the commutativity of both limits by generalising a statement about successive Schur complements.
A coherent feedback scheme is used to enhance the degree of squeezing of the output field in a cavity optomechanical system. In the feedback loop, a beam splitter (BS) plays the roles of both a feedback controller and an input-output port. To realize effective enhancement, the output quadrature should take the same form as the input quadrature, and the system should operate at the deamplification situation in the meantime. This can be realized by choosing an appropriate frequency-dependent phase angle for the generalized quadrature. Additionally, both the transmissivity of the BS and the phase factor induced by time delays in the loop affect optical squeezing. For the fixed frequency, the optimal values of transmissivity and phase factor can be used to achieve the enhanced optical squeezing. The effect of optical losses on squeezing is also discussed. Optical squeezing is degraded by the introduced vacuum noise owing to the inefficient transmission in the loop. We show that the enhancement of squeezing is achievable with the parameters of the current experiments.
The theory of quantum feedback networks has recently been developed with the aim of showing how quantum input-output components may be connected together so as to control, stabilize, or enhance the performance of one of the subcomponents. In this paper, we show how the degree to which an idealized component (a degenerate parametric amplifier in the strong-coupling regime) can squeeze input fields may be enhanced by placing the component in loop in a simple feedback mechanism involving a beam splitter. We study the spectral properties of output fields, placing particular emphasis on the elastic and inelastic components of the power density.
We show that, for arbitrary quantum feedback networks consisting of several quantum mechanical components connected by quantum fields, the limit of adiabatic elimination of fast oscillator modes in the components and the limit of instantaneous transmission along internal quantum field connections commute. The underlying technique is to show that both limits involve a Schur complement procedure. The result shows that the frequently used approximations, for instance, to eliminate strongly coupled optical cavities, are mathematically consistent.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.
This research focuses on advanced noise suppression technologies for high-precision measurement systems, particularly addressing the limitations of classical noise reducing approaches. The noise level of laser sources is a crucial factor that directly affects the measurement sensitivity in applications such as gravitational wave detection and biomedical imaging. Classical feedback control technologies are effective but often encounter a bottleneck resulting from the classical noise suppression limits. To cope with these challenges, a novel method integrating quantum squeezed light with classical feedback control systems to reduce intensity noise is proposed in this work. By employing an amplitude-squeezed light field, a quantum-enhanced feedback control model is developed, thereby theoretically examining the influence of both the feedback loop gain and the degree of squeezing on the noise suppression performance. The results show that the injection of squeezed light significantly reduces the intensity noise, approaching the shot noise limit (SNL), thereby improving the system sensitivity beyond the classical noise reduction boundaries. Specifically, –10 dB squeezed state injection into the feedback system yields an additional noise suppression of approximately 8.97 dB, exceeding what is achievable using classical feedback alone. This demonstrates that the potential of the proposed method can enhance measurement precision close to the quantum noise limits without increasing the laser power. The analysis highlights the asymmetric noise suppression effects between the inner feedback loop and outer feedback loop. Although the outer loop benefits significantly from the squeezed light injection and achieves noise levels that are unattainable by classical feedback methods, the inner loop shows relatively minor improvements. This asymmetry is attributed to the inherent characteristics of quantum squeezing and the limitations of the feedback loop design. Further investigation into the individual noise components reveals that the primary contributors to the intensity noise include input noise, photodetector noise, and beam splitter-induced vacuum fluctuations. The injection of squeezed light effectively mitigates these vacuum fluctuations, which are typically a major noise source in high-precision laser systems. Theoretical research results show that the use of squeezed light in feedback control systems can effectively enhance noise suppression, equivalent to a nine fold increase in detected optical power, without the physical drawbacks of increasing laser power such as thermal noise. In conclusion, this study provides a theoretical validation of combining quantum squeezed states with classical feedback control to exceed classical noise suppression limits. The integration of a –10 dB squeezed state demonstrates a significant noise reduction, showing that this hybrid approach could revolutionize noise management in precision measurement applications. The results pave the way for further exploring quantum-enhanced control technologies in fields such as gravitational wave detection, quantum communication, and advanced optical sensing, providing a pathway for improving sensitivity and noise suppression without increasing additional power requirements.
In the control of classical mechanical systems, feedback has been applied to the generation of desired nonlinear dynamics, e.g., in chaos control. However, how much this can be done is still an open problem in quantum mechanical systems. This paper presents a scheme of enhancing nonlinear quantum effects via the recently developed coherent feedback techniques, which can be shown to outperform the measurement-based quantum feedback scheme that can only generate pseudo-nonlinear quantum effects. Apart from the advantages of our method, an unsolved problem is that the decoherence rate is also increased by the quantum amplifier, which may be solved by introducing, e.g., an integral device or an nonlinear quantum amplifier. Such a proposal is demonstrated via two application examples in quantum optics on chip. In the first example, we show that nonlinear Kerr effect can be generated and amplified to be comparable with the linear effect in a transmission line resonator (TLR). In the second example, we show that by tuning the gains of the quantum amplifiers in a TLR coherent feedback network, the resulting nonlinear effects can generate and manipulate non-Gaussian “light” (microwave field) which exhibits fully quantum sub-Poisson photoncount statistics and photon antibunching phenomenon. The scheme opens up broad applications in engineering nonlinear quantum optics on chip. Particularly, in this study, the concept of feedback nonlinearization which is very useful for quantum feedback control systems is introduced. This is in contrast to the feedback linearization concept used in classical nonlinear feedback control systems.