In this paper, a novel continuous non-smooth control strategy is proposed to achieve finite-time stabilization of ladder quantum systems. We first design a universal fractional-order control law for a ladder n-level quantum system using a distance-based Lyapunov function, and then apply the Filippov solution in the sense of differential inclusions and the LaSalle's invariance principle to prove the existence and uniqueness of the solution of the ladder system under the continuous non-smooth control law. Both asymptotic stability and finite-time stability for the ladder system is rigorously established by applying Lyapunov stability theory and finite-time stability criteria. We also derive an upper bound of the time required for convergence to an eigenstate of the intrinsic Hamiltonian. Numerical simulations on a rubidium ladder three-level atomic system validate the effectiveness of the proposed method.
In this paper, a novel continuous non-smooth control strategy is proposed to achieve finite-time stabilization of ladder quantum systems. We first design a universal fractional-order control law for a ladder n-level quantum system using a distance-based Lyapunov function, and then apply the Filippov solution in the sense of differential inclusions and the LaSalle's invariance principle to prove the existence and uniqueness of the solution of the ladder system under the continuous non-smooth control law. Both asymptotic stability and finite-time stability for the ladder system is rigorously established by applying Lyapunov stability theory and finite-time stability criteria. We also derive an upper bound of the time required for convergence to an eigenstate of the intrinsic Hamiltonian. Numerical simulations on a rubidium ladder three-level atomic system validate the effectiveness of the proposed method.
In this paper, we consider the fixed-time stabilization control problem of quantum systems modeled by Schrödinger equations. Firstly, the Lyapunov-based fixed-time stability criterion is extended to finite-dimensional closed quantum systems in the form of coherence vectors. Then for a two-level quantum system with single control input, a non-smooth fractional-order control law is designed using the relative state distance. By integrating the fixed-time Lyapunov control technique and the bi-limit homogeneity theory, the quantum system is proved to be stabilized to an eigenstate of the inherent Hamiltonian in a fixed time. Comparing with existing methods in quantum system control, the proposed approach can guarantee stabilization in a fixed time without depending on the initial states.
We develop a model-free framework for stabilizing quantum states using only empirical finite-difference evaluations of a measurement-derived Lyapunov observable. The controller requires no knowledge of the Hamiltonian, dissipative structure, or generator of the dynamics, and relies solely on discrete measurement data. The approach combines three key elements: sign-based Lyapunov descent, adaptive gain amplification, and a finite-difference analogue of LaSalle's invariance principle. We provide rigorous conditions under which these mechanisms guarantee asymptotic stabilization along the sampling instants in the drift-free case and practical input-to-state stability (ISS) in the presence of unknown drift and noise. The resulting feedback law is simple, derivative-free, and experimentally feasible. A qubit example illustrates the complete closed-loop scheme and the predicted ISS-type behavior. Although demonstrated on a single qubit, the theory applies to arbitrary finite-dimensional quantum systems and offers a foundation for further developments in stochastic, subspace, and multi-qudit model-free quantum control.
The global stabilization of an eigenstate for finite dimensional stochastic quantum systems with non-diagonal free Hamiltonian and non-regular measurement operator is studied in this paper. A two-part switching control strategy is proposed, in which a constant control is used to steer the system state to enter the convergence domain; another control law, which is designed based on Lyapunov stability theorem is used to attract the system state in convergence domain to the target state. The convergence of the target eigenstate from an arbitrary initial state is strictly proved. Moreover, the numerical experiments on a three dimensional stochastic quantum system are implemented to demonstrate the effectiveness of the control proposed.
Rapid stabilization of general stochastic quantum systems is investigated based on the rapid stability of stochastic differential equations. We introduce a Lyapunov–LaSalle-like theorem for a class of nonlinear stochastic systems first, based on which a unified framework of rapidly stabilizing stochastic quantum systems is proposed. According to the proposed unified framework, we design the switching state feedback controls to achieve the rapid stabilization of single-qubit systems, two-qubit systems, and N-qubit systems. From the unified framework, the state space is divided into two state subspaces, and the target state is located in one state subspace, while the other system equilibria are located in the other state subspace. Under the designed state feedback controls, the system state can only transit through the boundary between the two state subspaces no more than two times, and the target state is globally asymptotically stable in probability. In particular, the system state can converge exponentially in (all or part of) the state subspace where the target state is located. Moreover, the effectiveness and rapidity of the designed state feedback controls are shown in numerical simulations by stabilizing GHZ states for a three-qubit system.
For closed quantum systems, if controlled systems are strongly regular and all other eigenstates are directly coupled to a target state, then such control systems can be asymptotically stabilized by means of the Lyapunov-based control. However, when the controlled systems are not strongly regular, or when there exists at least one eigenstate that is directly uncoupled to the target state, the situations will become complicated. This paper proposed a method based on an implicit Lyapunov function to overcome these two degenerate cases. Also, the proposed method is suitable for multi-control Hamiltonian systems. The convergence of closed-loop systems is analyzed by the LaSalle invariance principle. Finally, numerical simulation experiments on a 4-level system are done. The experiment results show the effectiveness of the implicit Lyapunov control method for degenerate cases and multi-control Hamiltonians.
The exponential stabilization of the eigenstates for two-level quantum systems based on continuous noise-assisted feedback control is investigated in this paper. Due to the symmetric structure of state space, quantum systems may converge to an undesired eigenstate under continuous measurement. In order to overcome the convergence obstacle, the continuous noise-assisted feedback strategy, in which the control signal is driven by an exogenous Brownian noise, is proposed to make the quantum systems converge to the target eigenstates exponentially, and the noise gain is designed as a continuously differentiable function of quantum state, which is negatively related to the distance between system state and the target eigenstate. In particular, the exponential convergence is proved based on the stability theory of stochastic differential equations. Moreover, the numerical simulations on a two-level quantum system are performed to verify the effectiveness of noise-assisted feedback control and study the influences of physical parameters and control parameters on convergence rate.