Manipulation of a quantum system can be viewed in the framework of a control landscape defined as the physical objective as a functional of the control. Control landscape analyses have thus far considered linear quantum dynamics. This paper extends the analysis of control landscape topology to nonlinear quantum dynamics with the objective of steering a finite-level quantum system from an initial state to a final target state. The analysis rests on the assumptions that (i) the final state is reachable from the initial state, (ii) the differential mapping from the control to the state is surjective, and (iii) the control resources are unconstrained. Under these assumptions, landscape critical points (i.e., where the slope vanishes) for nonlinear quantum dynamics only appear as the global maximum and minimum; thus, the landscape is free of traps. Moreover, the landscape Hessian (i.e., the second derivative with respect to the control) at the global maximum has finite rank, indicating the presence of a large level set of optimal controls that preserve the value of the maximum. Extensive numerical simulations on finite-level models of the Gross-Pitaevskii equation confirm the trap-free nature of the landscape as well as the Hessian rank analysis, using either an applied electric field or a tunable condensate two-body interaction strength as the control. In addition, the control mechanisms arising in the numerical simulations are qualitatively assessed. These results are a generalization of previous findings for the linear Schr\"odinger equation, and show promise for successful control in a wide range of nonlinear quantum dynamics applications.
Nonlinear quantum systems behave radically different from their linear counterparts and obstructions, identified as separatrix crossing and bifurcations in the phase space, can prevent their efficient control by standard approaches using adiabatic passage. We provide the exact resonant solution by solving the time-dependent nonlinear Schr\"odinger equation in terms of a Jacobi elliptic integral, interpreting the dynamics with the motion of a planar pendulum. This is referred to as nonlinear Rabi oscillation, which shows an obstruction to complete population transfer above a certain critical value of the nonlinearity strength, corresponding to an oscillating regime of the pendulum. Below this threshold, the so-called
2K pulse induces a complete population transfer, associated with a rotating regime of the pendulum, and representing the nonlinear counterpart of the $\ensuremath{\pi}$ pulse. We next propose a fast, robust, and high-fidelity control technique by a two-stage optimal inverse engineering strategy. It consists in dynamically compensating (diagonal) third-order nonlinearities, referred to as inverse linearization, combined with robust inverse optimization (RIO) developed for linear systems. We show that a moderate nonlinearity strength dramatically improves robustness.
This paper introduces Topological Quantum Control (TQC), a framework integrating Uhlmann fidelity-based robustness metrics with AI-driven pulse optimization for attosecond molecular dynamics. Using the Lindblad master equation to model decoherence, we employ Proximal Policy Optimization (PPO) to navigate non-convex control landscapes. Validated on a three-level Lambda system under extreme noise stress-tests, TQC improves target yield from 62% to 87% compared to standard optimal control. The approach demonstrates enhanced resilience against environmental decoherence, offering a reproducible, open-source proof-of-principle for robust quantum control in complex, open systems.