[Paper Review] On exponential stabilization of N-level quantum angular momentum systems

Weichao Liang, Nina H. Amini+1 more|arXiv (Cornell University)|Feb 15, 2019

Quantum Information and Cryptography33 references4 citations

TL;DR

This paper establishes general sufficient conditions for almost sure exponential stabilization of N-level quantum angular momentum systems to a target eigenstate under continuous-time measurements, using stochastic and geometric control tools. By analyzing quantum trajectory asymptotics and applying stochastic Lyapunov techniques, it proves exponential convergence and designs parametrized feedback laws validated via numerical simulations for three-level systems with exponential convergence rates matching theoretical predictions.

ABSTRACT

In this paper, we consider the feedback stabilization problem for N-level quantum angular momentum systems undergoing continuous-time measurements. By using stochastic and geometric control tools, we provide sufficient conditions on the feedback control law ensuring almost sure exponential convergence to a predetermined eigenstate of the measurement operator. In order to achieve these results, we establish general features of quantum trajectories which are of interest by themselves. We illustrate the results by designing a class of feedback control laws satisfying the above-mentioned conditions and finally we demonstrate the effectiveness of our methodology through numerical simulations for three-level quantum angular momentum systems.

Motivation & Objective

• To address the challenge of stabilizing N-level quantum angular momentum systems to a desired eigenstate under continuous measurement.
• To derive general feedback control laws ensuring almost sure exponential convergence to a target eigenstate.
• To analyze the asymptotic behavior of quantum trajectories as a foundation for stability analysis.
• To demonstrate the effectiveness of the proposed control methodology through numerical simulations for three-level systems.
• To extend prior results on quantum state reduction and feedback stabilization to general N-level systems with explicit parametrized controllers.

Proposed method

• Uses stochastic differential equations to model the quantum filtering dynamics of N-level systems under continuous measurement.
• Applies stochastic Lyapunov techniques to prove exponential convergence of the system to the target eigenstate under specific feedback law conditions.
• Employs the support theorem and asymptotic analysis of quantum trajectories to establish reachability of any neighborhood of the target state from any initial state.
• Derives general conditions on the feedback law that ensure exponential convergence, based on the system's drift and diffusion components.
• Proposes explicit parametrized feedback laws of the form (14) and (16) that satisfy the derived conditions and ensure exponential stabilization.
• Validates the theoretical results through numerical simulations of three-level systems, tracking the evolution of the Lyapunov function and Bures distance to the target state.

Experimental results

Research questions

• RQ1 Under what general conditions on the feedback law does an N-level quantum angular momentum system converge exponentially to a target eigenstate almost surely?
• RQ2 How do the asymptotic properties of quantum trajectories influence the stability of the system under feedback control?
• RQ3 Can the exponential convergence rate be quantified and matched to theoretical predictions in numerical simulations?
• RQ4 What is the role of the measurement back-action and control law structure in achieving stabilization toward a specific eigenstate?
• RQ5 How do the parameters of the feedback law affect the convergence speed and robustness to initial conditions?

Key findings

• The system exhibits almost sure exponential convergence to the target eigenstate when the feedback law satisfies the derived general conditions, with convergence rate proportional to ηM/2.
• For zero control, the system undergoes quantum state reduction with exponential rate ηM/2, as confirmed by the decay of the Lyapunov function and Bures distance in simulations.
• Numerical simulations for three-level systems show that the expected value of the Lyapunov function decays as V(ρ₀)e^(-ηM/2 t), matching theoretical predictions.
• The Bures distance to the target state decays exponentially with rate ηM/2, bounded by (C₂/C₁)d_B(ρ₀, E)e^(-ηM/2 t), with C₁ = 1/2 and C₂ = 3.
• Larger values of α and γ in the feedback law (14) accelerate convergence away from antipodal states and toward the target, respectively.
• The feedback law (16) ensures exponential stabilization even from initial states in the interior of the state space, with trajectories entering the target neighborhood rapidly.