Quantum control theory and applications: A survey <s> Stochastic master equations <s> The goal of this article is to provide a largely self-contained introduction to the modelling of controlled quantum systems under continuous observation, and to the design of feedback controls that prepare particular quantum states. We describe a bottom-up approach, where a field-theoretic model is subjected to statistical inference and is ultimately controlled. As an example, the formalism is applied to a highly idealized interaction of an atomic ensemble with an optical field. Our aim is to provide a unified outline for the modelling, from first principles, of realistic experiments in quantum control. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Stochastic master equations <s> Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Stochastic master equations <s> We present a pedagogical treatment of the formalism of continuous quantum measurement. Our aim is to show the reader how the equations describing such measurements are derived and manipulated in a direct manner. We also give elementary background material for those new to measurement theory, and describe further various aspects of continuous measurements that should be helpful to those wanting to use such measurements in applications. Specifically, we use the simple and direct approach of generalized measurements to derive the stochastic master equation describing the continuous measurements of observables, give a tutorial on stochastic calculus, treat multiple observers and inefficient detection, examine a general form of the measurement master equation, and show how the master equation leads to information gain and disturbance. To conclude, we give a detailed treatment of imaging the resonance fluorescence from a single atom as a concrete example of how a continuous position measurement arises in a physical system. <s> BIB003
In feedback control, we generally need to continuously monitor a quantum system to obtain feedback information. The evolution of a quantum system under continuous measurements of an observable X can be described by the following master equation BIB003 : where κ is a parameter related to the measurement strength, X = tr(Xρ), dW is a Wiener increment with zero mean and variance equal to dt and satisfies the following relationship to the measurement output y Equation (14) is usually called a stochastic master equation (SME). Such a stochastic master equation can be obtained as a filtering equation from quantum filtering theory (for details, see BIB001 , BIB002 ). It is worth noting that the state ρ in the SME (14) is a conditional state since it uses information from the continuous measurements. It should be pointed out that (14) is only a typical form of SME and there exist many different types of SMEs which depend on different measurement processes  . Stochastic master equations have been derived for some quantum optical systems under continuous measurements. For example, we consider an atomic ensemble interacting with an electromagnetic field considered in BIB001 . Consider the atomic Hamiltonian H(t) = ∆F z + u(t)F y , where ∆ is the atomic detuning and u(t) is the strength of a magnetic field in the y-direction, and F z and F y are the collective dipole moments of the ensemble (for details, see BIB001 ). If we neglect spontaneous emission, the corresponding SME is BIB001  where s is related to experimental parameters such as ∆, and dW satisfies