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Physical Review A

Control of nonlinear quantum systems: From nonlinear Rabi oscillations to robust two-stage optimal inverse engineering

J. J. Zhu, S. Guérin

Published Dec 2, 2024• 3 citations

Abstract

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.

DOI: 10.1103/physreva.110.063503

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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

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pulse induces a complete population transfer, associated with a rotating regime of the pendulum, and representing the nonlinear counterpart of the

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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.

Adiabatic passage is a standard tool for achieving robust transfer in quantum systems. We show that in driven nonlinear quantum systems adiabatic passage becomes highly nonrobust when the target is unstable. This occurs for a generic (1:2) resonance, for which the complete transfer corresponds to a hyperbolic fixed point featuring an adiabatic connectivity strongly sensitive to small perturbations of the model. By inverse engineering, we devise high-fidelity and robust partially nonadiabatic trajectories. They localize at the approach of the target near the stable manifold of the separatrix, which drives the dynamics towards the target in a robust way. These results can be applicable to atom-molecule Bose-Einstein condensate conversion and to nonlinear optics.

We develop an inverse geometric optimization technique that allows the derivation of optimal and robust exact solutions of low-dimension quantum control problems driven by external fields. We determine in the dynamical variable space optimal trajectories constrained to robust solutions by Euler-Lagrange optimization; the control fields are then derived from the obtained robust geodesics and the inverted dynamical equations. We apply this method, referred to as robust inverse optimization (RIO), to design optimal control fields producing a complete or half population transfer and a not quantum gate robust with respect to the pulse inhomogeneities. The method is versatile and can be applied to numerous quantum control problems, e.g., other gates, other types of imperfections, Raman processes, or dynamical decoupling of undesirable effects.

Using optimal control, we establish and link the ultimate bounds in time (referred to as the quantum speed limit) and energy of two- and three-level quantum nonlinear systems which feature 1:2 resonance. Despite the unreachable complete inversion, by using the Pontryagin maximum principle, we determine the optimal time, pulse area, or energy for a given arbitrary accuracy. We show that the third-order Kerr terms can be absorbed in the detuning in order to lock the dynamics to the resonance. In the two-level problem, we determine the nonlinear counterpart of the optimal $\ensuremath{\pi}$-pulse inversion for a given accuracy. In the three-level problem, we obtain an intuitive pulse sequence similar to the linear counterpart but with different shapes. We prove the (slow) logarithmic increasing of the optimal time as a function of the accuracy.

Coherent quantum control shows promise in preparing and manipulating internal states within a quantum system coherently. Specifically, it aims to smoothly transition the system from an initial state to a desired target state using external electromagnetic fields. This transfer of population between ground states is crucial in various quantum processes, including applications in photochemistry for controlling reactions with fast laser fields, enhancing Nuclear Magnetic Resonance images, quantum optics and quantum information science. Quantum information processing manipulates data with unitary operators called quantum logic gates, which control state transitions in qubits. These gates are vital for future quantum computers, requiring high efficiency and robustness despite experimental limitations and errors, like inconsistent control fields. The primary goal in quantum computing is to transfer population from the ground state to the target state with a high fidelity , good robustness and minimum interaction time between the electromagnetic fields and the system. Currently, creating a control method that meets all three criteria is a significant challenge in quantum information. However, by merging reverse engineering using the Lewis-Riesenfeld dynamic invariant with optimal control theory, we've shown that designing such a control field can be simplified into a Lagrange optimization problem with integral constraints. This hybrid method, called RIO (Robust Inverse Optimization), is adaptable and applicable to diverse quantum control challenges.