We present a stochastic projection formalism for the description of quantum dynamics in bosonic or spin environments. The Schrödinger equation in the coherent state representation with respect to the environmental degrees of freedom can be reformulated by employing the Feshbach partitioning technique for open quantum systems based on the introduction of suitable non-Hermitian projection operators. In this picture the reduced state of the system can be obtained as a stochastic average over pure state trajectories, for any temperature of the bath. The corresponding non-Markovian stochastic Schrödinger equations include a memory integral over the past states. In the case of harmonic environments and linear coupling the approach gives a new form of the established non-Markovian quantum state diffusion stochastic Schrödinger equation without functional derivatives. Utilizing spin coherent states, the evolution equation for spin environments resembles the bosonic case with, however, a non-Gaussian average for the reduced density operator.
Principles of monitoring non-Markovian open quantum systems are analyzed. We use a field representation of the environment [C. W. Gardiner and M. J. Collett, Phys. Rev. A 31, 3761 (1985)] for the separation of its memory and detector part, respectively. We claim that the system-plus-memory compound becomes Markovian; the detector part is tractable by standard Markovian monitoring. Because of non-Markovianity, only the mixed state of the system can be predicted; the pure state of the system can be retrodicted. We present the corresponding non-Markovian stochastic Schr\"odinger equation.
For the
n-qubit stochastic open quantum systems, based on the Lyapunov stability theorem and LaSalle's invariant set principle, a pure state switching control based on on-line estimated state feedback (short for OQST-SFC) is proposed to realize the state transition the pure state of the target state including eigenstate and superposition state. The proposed switching control consists of a constant control and a control law designed based on the Lyapunov method, in which the Lyapunov function is the state distance of the system. The constant control is used to drive the system state from an initial state to the convergence domain only containing the target state, and a Lyapunov-based control is used to make the state enter the convergence domain and then continue to converge to the target state. At the same time, the continuous weak measurement of quantum system and the quantum state tomography method based on the on-line alternating direction multiplier (QST-OADM) are used to obtain the system information and estimate the quantum state which is used as the input of the quantum system controller. Then, the pure state feedback switching control method based on the on-line estimated state feedback is realized in an
n-qubit stochastic open quantum system. The complete derivation process of
n-qubit QST-OADM algorithm is given; Through strict theoretical proof and analysis, the convergence conditions to ensure any initial state of the quantum system to converge the target pure state are given. The proposed control method is applied to a 2-qubit stochastic open quantum system for numerical simulation experiments. Four possible different position cases between the initial estimated state and that of the controlled system are studied and discussed, and the performances of the state transition under the corresponding cases are analyzed.