The standard quantum formalism introduced at the undergraduate level treats measurement as an instantaneous collapse. In reality however, no physical process can occur over a truly infinitesimal time interval. A more subtle investigation of open quantum systems lead to the theory of continuous measurement and quantum trajectories, in which wave function collapse occurs over a finite time scale associated with an interaction. Within this formalism, it becomes possible to ask many new questions that would be trivial or even ill-defined in the context of the more basic measurement model. In this thesis, we investigate both theoretically and experimentally what fundamentally new capabilities arise when an experimental apparatus can resolve the continuous dynamics of a measurement. Theoretically, we show that when one can perform feedback operations on the timescale of the measurement process, the resulting tools provide significantly more control over entanglement generation, and in some settings can generate it optimally. We derive these results using a novel formalism which encompasses most known quantum feedback protocols. Experimentally, we show that continuous measurement allows one to observe the dynamics of a system undergoing simultaneous non-commuting measurements, which provides a reinterpretation of the Heisenberg uncertainty principle. Finally, we combine the theoretical focus on quantum feedback with the experimental capabilities of superconducting circuits to implement a feedback controlled quantum amplifier. The resulting system is capable of adaptive measurement, which we use to perform the first canonical phase measurement.
The standard quantum formalism introduced at the undergraduate level treats measurement as an instantaneous collapse. In reality however, no physical process can occur over a truly infinitesimal time interval. A more subtle investigation of open quantum systems lead to the theory of continuous measurement and quantum trajectories, in which wave function collapse occurs over a finite time scale associated with an interaction. Within this formalism, it becomes possible to ask many new questions that would be trivial or even ill-defined in the context of the more basic measurement model. In this thesis, we investigate both theoretically and experimentally what fundamentally new capabilities arise when an experimental apparatus can resolve the continuous dynamics of a measurement. Theoretically, we show that when one can perform feedback operations on the timescale of the measurement process, the resulting tools provide significantly more control over entanglement generation, and in some settings can generate it optimally. We derive these results using a novel formalism which encompasses most known quantum feedback protocols. Experimentally, we show that continuous measurement allows one to observe the dynamics of a system undergoing simultaneous non-commuting measurements, which provides a reinterpretation of the Heisenberg uncertainty principle. Finally, we combine the theoretical focus on quantum feedback with the experimental capabilities of superconducting circuits to implement a feedback controlled quantum amplifier. The resulting system is capable of adaptive measurement, which we use to perform the first canonical phase measurement.
Developments over the last two decades have opened the path towards quantum technologies in many quantum systems, such as cold atoms, trapped ions, cavity-quantum electrodynamics (QED), and circuit-QED. However, the fragility of quantum states to the effects of measurement and decoherence still poses one of the greatest challenges in quantum technology. An imperative capability in this path is quantum feedback, as it enhances the control possibilities and allows for prolonging coherence times through quantum error correction. While changing parameters from shot to shot of an experiment or procedure can be considered feedback, quantum mechanics also allows for the intriguing possibility of performing feedback operations during the measurement process itself. This broader approach to measurements leads to the concepts of weak measurement, quantum trajectories, and numerous types of feedback with no classical analogs. These types of processes are the primary focus of this review. We introduce the concept of quantum feedback in the context of circuit-QED, an experimental platform with significant potential in quantum feedback and technology. We then discuss several experiments and see how they elucidate the concepts of continuous measurements and feedback. We conclude with an overview of coherent feedback, with application to fault-tolerant error correction.
Measurement and control of quantum systems play an important role in science and technology, from the foundations of quantum mechanics to practical applications. This thesis presents theoretical and experimental work that exploits quantum measurement and measurement-based feedback control to study some new problems in the contexts of solid state physics and optical quantum systems. In the first part, we consider the dynamics of open quantum systems conditioned on continuous measurements. We revisit the fact that dynamical quantum events are detector dependent by framing the problem as a quantum correlation test of the Einstein-Podolsky-Rosen (EPR) steering type. In this regards, a no-go theorem for quantum diusion is derived. In order to study a realistic experimental situation we scrutinise dierent physical systems, Einstein-Podolsky-Rosen steering inequalities and measurement protocol. We show that the no-go result for quantum diusion is not universal and does not apply to quantum jumps by devising a novel adaptive measurement scheme. Since the objectiveness of pure-state dynamical models is ruled out by quantum entanglement, then it is reasonable to say that quantum jumps are more quantum than quantum diusion. In the rst part we also study a stochastic feedback control of quantum transport. We show that using a feedback control strategy it is possible to restrict the dynamics of a double quantum dot system to a non-orthogonal two-state ensemble. The feedback control approach can be less experimentally challenging than an adaptive measurement scheme, as the latter requires implementing local oscillators, which might be dicult to realise in mesoscopic physics.
In quantum mechanics, standard or strong measurement approaches generally result in the collapse of an ensemble of wavefunctions into a stochastic mixture of eigenstates. On the other hand, continuous or weak measurements have the propensity to dynamically control the evolution of quantum states over time, guiding the trajectory of the state into non-trivial superpositions and maintaining state purity. This kind of measurement-induced state steering is of great theoretical and experimental interest for the harnessing of quantum bits or "qubits", which are the fundamental unit of the emerging quantum computer. We explore continuous measurement-based quantum state stabilization through linear feedback control for a single quantum bit. By applying a time-varying Rabi drive that includes a linear feedback term, we show that the fixed points of the continuous measurement may be relocated. Analytical derivations of Itô Stochastic Master Equations being ideal models, we employ them to derive the projected ensemble average while utilizing numerical simulations to characterize the stability of the set of possible fixed points, as well as their modified collapse time-scales. We include the effects of realistic experimental non-idealities, such as environmental energy relaxation, dephasing, time-delay, and inefficient measurement. Ultimately, we discuss potential experimental implementations from collaborating universities.
Feedback control in classical systems is an indispensable, ubiquitous tool. The theoretical basis for achieving optimal classical control is well understood, and crucially relies on a very classical assumption: that measurements of the state of a system under control need not perturb that state. In a quantum context this assumption is fundamentally invalid. Although many aspects of the theory of quantum feedback control are relatively well developed, the technological basis for feedback control of a single quantum system has only very recently matured. We demonstrate the experimental realization of a quantum feedback control protocol, perpetually stabilizing the coherent Rabi oscillations of a superconducting qubit. This is the first utilization of quantum feedback control for stabilizing a dynamical process, and the first application of quantum feedback in a solid-state system of any kind. This demonstration comprises the first half of this thesis. The feedback protocol is predicated on the ability to make high-fidelity quantum measurements, which are enabled by quantum-limited Josephson parametric amplifiers (JPAs). The design and realization of the novel Josephson traveling-wave parametric amplifier (JTWPA) comprises the second half of this thesis. The JTWPA achieves order-of-magnitude improvements over state of the art JPAs in bandwidth and signal power handling while providing quantum-limited noise performance, potentially enabling the simultaneous readout of dozens of superconducting qubits and the generation of broadband multi-mode squeezing in the microwave domain.
As a key method in dealing with uncertainties, feedback has been understood fairly well in classical control theory. But for quantum control systems, the capability of measurement-based feedback control (MFC) has not been investigated systematically. In contrast to the control of classical systems where the measurement effect is negligible, the quantum measurement will cause a quantum state to collapse, which will inevitably introduce additional uncertainties besides the system initial uncertainty. Therefore, there is a complicated tradeoff between the uncertainty introduced and the information gained by the measurement, and thus a theoretical investigation of the capability of MFC is of fundamental importance. In this paper, inspired by both the Heisenberg uncertainty principle for quantum systems and the investigation of the feedback capability for classical systems, we try to answer the following three basic questions: (i) How to choose the measurement channel appropriately? (ii) Is the MFC still superior to the open loop control in dealing with the system information uncertainties? and (iii) What is the maximum capability or limitation of the MFC? These questions will be answered theoretically by establishing several theorems concerning the asymptotic reachability of eigenstates of a typical class of Hamiltonian control mode.
The advance of experimental techniques in circuit quantum electrodynamics over last decadehas made it possible to monitor the evolution of qubits under continuous measurements in\nreal time . This kind of measurements, as compared to the discrete ones, provides a stream\nof information for feedback to alter the system's quantum dynamics continuously in a desired\nmanner, thus allowing broader possibilities. We obtain the information at a cost, the backaction on the states. However, the back-action turns out not to be just something disturbing \nand harmful to the control but instead can be harnessed, together with the control, to steer the system to our targets. \nThis thesis discusses one such application, namely to use measurements to create entanglement from remote qubits under locally optimal control. It is illustrated that by carefully designing symmetric measurement and feedback operators, we can construct efficient Markovian or non-Markovian protocols which generate N-qubit W,\ngeneral Dicke and Greenberger-Horne-Zeilinger states with high fidelity above 94% up to\nN = 100. In some cases, the fidelity even approaches unity. Following that, we describe how\nto take advantages of the continuous stream of information to protect the unitary dynamics of\na quantum system subject to noises. We develop a practical protocol for continuous operation\nof a quantum error correcting code for protection of coherent evolution due to an encoded\nHamiltonian against environmental errors, using the three qubit bit-flip code and bit-flip\nerrors as a canonical example. We demonstrate, for both quantum memory and adiabatic\nHamiltonians, that the protocol, once optimized, can significantly reduce logical errors with\na performance which is close to the theoretical limit of the bit-flip code's capability. In the\nadiabatic Hamiltonian case, we also propose some novel correction operators instead of the\nconventional ones to further reduce the logical errors.