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arXiv:2005.12321 · Physical Review A

Robust control of unstable nonlinear quantum systems

J. J. Zhu, Xi Chen, Hans Rudolf Jauslin, S. Guérin

Published Nov 4, 2020• 20 citations
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Abstract

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.

DOI
10.1103/physreva.102.052203

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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 derive a technique of robust and efficient adiabatic passage for a driven nonlinear quantum system, describing the transfer to a molecular Bose-Einstein condensate from an atomic one by external fields. The pulse ingredients are obtained by tracking the dynamics derived from a Hamiltonian formulation, in the adiabatic limit. This leads to a nonsymmetric and nonmonotonic chirp. The efficiency of the method is demonstrated in terms of classical phase space, more specifically with the underlying fixed points and separatrices. We also prove the crucial property that this nonlinear system does not have any solution leading exactly to a complete transfer. It can only be reached asymptotically for an infinite pulse area.
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 2K2K2K 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.
Robust stimulated Raman exact passages are requisite for controlling nonlinear quantum systems, with wide applications, ranging from ultracold molecules to nonlinear optics to superchemistry. Inspired by shortcuts to adiabaticity, we propose the fast-forward scaling of stimulated Raman adiabatic processes with nonlinearity involved, describing the transfer from an atomic Bose-Einstein condensate to a molecular one by controllable external fields. The fidelity and robustness of atom-molecule conversion are shown to surpass those of conventional adiabatic passages, assisted by the fast-forward driving field. Finally, our results are extended to fractional stimulated Raman adiabatic processes for the coherent superposition of atomic and molecular states.
Nontrivial spectral properties of non-Hermitian systems can give rise to intriguing effects that lack counterparts in Hermitian systems. For instance, when dynamically varying system parameters along a path enclosing an exceptional point (EP), chiral mode conversion occurs. A recent study [Phys. Rev. Lett. 133, 113802 (2024)] demonstrates the achievability of pure adiabatic state transfer by specifically selecting a trajectory in the system parameter space where the corresponding evolution operator exhibits a real spectrum while winding around an EP. However, the intended adiabatic state transfer becomes fragile when taking into account the effect of the nonadiabatic transition. In this work, we propose a scheme for achieving robust and rapid adiabatic state transfer in time-modulated two-level non-Hermitian systems by appropriately modulating the system Hamiltonian and time-evolution trajectory. Numerical simulations confirm that a complete adiabatic transfer can always be achieved even under nonadiabatic conditions after one period for different initialized adiabatic states, and the scheme remains insensitive to moderate fluctuations in control parameters. Therefore, this scheme offers alternative approaches for quantum-state engineering in non-Hermitian systems.
Adiabatic quantum operations are ubiquitous in fields such as quantum information processing, quantum control, and magnetic resonance, where their high fidelity and general robustness renders them an essential tool for various applications. This inherent robustness due to adiabaticity can, however, be impaired in cases where the experimental conditions strictly require very fast evolution times. In this thesis, we provide a gradient-based numerical optimization protocol for efficiently engineering adiabatic operations, allowing for the systematic inclusion of robustness criteria against various experimental non-idealities, including inhomogeneities and uncertainties in system parameters, as well as perturbations in the Hamiltonian such as spin-spin interactions. \nThe protocol is implemented in the context of adiabatic passages for magnetic resonance, and \nit is shown that in addition to conventional adiabatic pulses, it can also generate exceptionally fast operations that although slightly deviate from adiabaticity in limited instances, still lead to a high fidelity. \nThe effectiveness of our method for addressing perturbations is also demonstrated by designing a fast adiabatic passage for dipolar-coupled electrons, and numerically comparing its performance with a pulse without such considerations.