We develop an approach for efficient and scalable simulations of measurement and control of quantum systems built upon existing phase-space methods, namely, the truncated Wigner approximation.. We benchmark against existing particle-filter methods by simulating measurement-based feedback cooling in a two-mode system, whose low-dimensional nature permits a computation of an exact solution. The advantage of our method is multimode scalability, which we demonstrate through the successful simulation of measurement-based feedback cooling of an incoherent quasi-one-dimensional thermal ensemble to quantum degeneracy. As the underlying principle of our approach exploits a general correspondence between measurement and coherent feedback, we anticipate it is also applicable across a broad range of other quantum control scenarios.
We develop an approach for efficient and scalable simulations of measurement and control of quantum systems built upon existing phase-space methods, namely, the truncated Wigner approximation.. We benchmark against existing particle-filter methods by simulating measurement-based feedback cooling in a two-mode system, whose low-dimensional nature permits a computation of an exact solution. The advantage of our method is multimode scalability, which we demonstrate through the successful simulation of measurement-based feedback cooling of an incoherent quasi-one-dimensional thermal ensemble to quantum degeneracy. As the underlying principle of our approach exploits a general correspondence between measurement and coherent feedback, we anticipate it is also applicable across a broad range of other quantum control scenarios.
We demonstrate a technique for performing stochastic simulations of conditional master equations. The method is scalable for many quantum-field problems and therefore allows first-principles simulations of multimode bosonic fields undergoing continuous measurement, such as those controlled by measurement-based feedback. As examples, we demonstrate a 53-fold speed increase for the simulation of the feedback cooling of a single trapped particle, and the feedback cooling of a quantum field with 32 modes, which would be impractical using previous brute force methods.
Engineers are looking to produce technologies that exploit the full quantum nature of a system, including quantum computing, quantum communication and quantum metrol- ogy. Quantum control allows engineers to modify the behaviour of the full quantum state of a system, which will be an important part of making these technologies commercially viable. If these technologies are to be competitive, they must be scaled up to large sizes. We produce a novel toolset which can be used to simulate large con- trolled quantum systems efficiently. We target this toolset at a large quantum system that can immediately benefit: Bose-Einstein Condensates (BECs). We first develop the Controlled Fokker-Planck Equation (CFPE), which allows scalable simulation of a controlled BEC. We find using coherent-state-based methods to generate a CFPE produces solutions with severely limited integration times, as monitoring of a BEC is intrinsically involves continuous 'number-like' measurements. To solve this problem, we develop a novel representation based on number state which acts in harmony with the measurement: the Number-Phase Wigner (NPW) representation. The NPW method efficiently and accurately simulates controlled BECs. It converges over 4 orders of mag- nitude longer than its closest coherent-state-based competitor. Using the NPW we are able to perform the first full-field analysis of a BEC under a cooling feedback control. We find that quantum noise produces heating in BECs that was not previously pre- dicted using approximate methods. We remove the effects of quantum noise with the creation of a novel control.
Simulation of conditional master equations is important to describe systems under continuous measurement and for the design of control strategies in quantum systems. For large bosonic systems, such as Bose-Einstein condensates and atom lasers, full quantum-field simulations must rely on scalable stochastic methods. Currently, these methods have a convergence time that is restricted by the use of representations based on coherent states. Here, we show that typical measurements on atom-optical systems have a common form that allows for an efficient simulation using the number-phase Wigner (NPW) phase-space representation. We demonstrate that a stochastic method based on the NPW can converge orders of magnitude longer and more precisely than its coherent equivalent. We then examine how these methods can be used in multimode simulations, demonstrated by a simulation of a two-mode Bose-Hubbard model. Finally, we combine these techniques to demonstrate a full-field simulation of a realistic multimode quantum system controlled by active feedback.
We introduce a general framework, based on collision models and discrete completely positive maps, to describe on an equal footing coherent and measurement-based feedback control of quantum mechanical systems. We apply our framework to prominent tasks in quantum control, ranging from cooling to Hamiltonian control. Unlike other proposed comparisons, where coherent feedback always proves superior, we find that either measurements or coherent manipulations of the controller can be advantageous depending on the task at hand. Measurement-based feedback is typically superior in cooling, while coherent feedback is better at assisting quantum operations. Furthermore, we show that both coherent and measurement-based feedback loops allow one to simulate arbitrary Hamiltonian evolutions, and discuss their respective effectiveness in this regard.
Ultracold atomic Fermi gases are the leading platform for analogue quantum simulation, and provide a promising avenue to study the origin of high-temperature superconductivity in cuprates. However, current experimental approaches to cooling Fermi gases use evaporative cooling, which is limited by poor thermalisation properties of fermions and is non-number-conserving. This prevents the creation of useful analogue simulators of many collective phenomena. This thesis is the first theoretical investigation into the use of continuous-measurement feedback control as an alternative means of cooling an atomic Fermi gas. Since tractable simulation of Fermi gas dynamics requires simplifications to the full quantum field theory, we derive and simulate a fermionic equivalent to the Gross-Pitaevskii equation, generalising a model of feedback-controlled BECs by Haine et al. to multimode ultracold atomic Fermi gases. We demonstrate that in the absence of measurement effects, a suitable control can drive an interacting Fermi gas arbitrarily close to its ground state. However, although control schemes based upon damping spatial density fluctuations work well for single-spatial-mode BECs, we show that they perform poorly for Fermi gases with a large number of atoms due to counter-oscillation of multiple spatial modes, which must exist due to Pauli exclusion. We generalise a feedback-measurement model of BECs by Szigeti et al. to a multimode atomic Fermi gas, and perform stochastic simulations of measured, feedback-controlled fermions in the single-atom and many-atom mean-field limits. The effects of measurement backaction are an important consideration, since in a realistic experiment knowledge of the system state used for feedback must be obtained from measurement, leading to competition between measurement-induced heating and feedback cooling. We show that weaker and less precise measurements cool the system to a lower equilibrium excitation energy, but are unable to place practical lower bounds on measurement strength due to the lack of a system-filter separation. When measurement-induced heating is accounted for, we find that the equilibrium energy per particle scales superlinearly, suggesting that existing control schemes which work well for bosons would not be effective for fermions. In light of this, we propose several avenues of future investigation to overcome this limitation, leaving open the possibility of feedback control of atomic Fermi gases as a pathway to analogue quantum simulation.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.