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Nonlinear two-level Landau-Zener type equations for systems with relevance for Bose-Einstein condensates and nonlinear optics are considered and the minimal time ${T}_{\mathrm{min}}$ to drive an initial state to a given target state is investigated. Surprisingly, the nonlinearity may be canceled by a time-optimal unconstrained driving and ${T}_{\mathrm{min}}$ becomes independent of the nonlinearity. For constrained and unconstrained driving explicit expressions are derived for ${T}_{\mathrm{min}}$, the optimal driving, and the protocol.

Achieving full control of a Bose-Einstein condensate can have valuable applications in metrology, quantum information processing, and quantum condensed matter physics. We propose protocols to simultaneously control the internal (related to its pseudospin-1/2) and motional (position-related) states of a spin-orbit-coupled Bose-Einstein condensate confined in a Morse potential. In the presence of synthetic spin-orbit coupling, the state transition of a noninteracting condensate can be implemented by Raman coupling and detuning terms designed by invariant-based inverse engineering. The state transfer may also be driven by tuning the direction of the spin-orbit-coupling field and modulating the magnitude of the effective synthetic magnetic field. The results can be generalized for interacting condensates by changing the time-dependent detuning to compensate for the interaction. We find that a two-level algorithm for the inverse engineering remains numerically accurate even if the entire set of possible states is considered. The proposed approach is robust against the laser-field noise and systematic device-dependent errors.

Motivated by recent time domain experiments on ultrafast atom ionization, we analyze the transients and timescales that characterize, besides the relatively long lifetime, the decay by tunneling of a localized state. While the tunneling starts immediately, some time is required for the outgoing flux to develop. This short-term behavior depends strongly on the initial state. For the initial state tightly localized so that the initial transients are dominated by over-the-barrier motion, the timescale for the flux propagation through the barrier is close to the B\"uttiker-Landauer traversal time. Then a quasistationary, slowly decay process follows, which sets ideal conditions for observing diffraction in time at longer times and distances. To define operationally a tunnelling time at the barrier edge, we extrapolate backwards the propagation of the wave packet escaped from the potential. This extrapolated time is considerably longer than the timescale of the flux and density buildup at the barrier edge., Comment: 9 pages, 10 figures

In the era of digital quantum computing, optimal digitized pulses are requisite for efficient quantum control. This goal is translated into dynamic programming, in which a deep reinforcement learning (DRL) agent is gifted. As a reference, shortcuts to adiabaticity (STA) provide analytical approaches to adiabatic speed up by pulse control. Here, we select single-component control of qubits, resembling the ubiquitous two-level Landau-Zener problem for gate operation. We aim at obtaining fast and robust digital pulses by combining STA and DRL algorithm. In particular, we find that DRL leads to robust digital quantum control with operation time bounded by quantum speed limits dictated by STA. In addition, we demonstrate that robustness against systematic errors can be achieved by DRL without any input from STA. Our results introduce a general framework of digital quantum control, leading to a promising enhancement in quantum information processing., 6 pages, 3 figures plus Supplemental Materials