Suppressing excitations in the nonlinear Landau-Zener model

TL;DR

This work investigates how effectively nonlinear quantum dynamics can suppress excitations and coherences in a generalized Landau–Zener framework. By formulating a nonlinear Schrödinger evolution with a state-dependent term, it defines a generalized energy spectrum from stationary-state expectations and shows that the nonlinear term acts as an approximate shortcut to adiabaticity for the linear problem. Through Bloch-vector analysis and time-dependent driving, the authors demonstrate that stronger nonlinearities reduce transition probabilities and coherence buildup, with the final state converging to the ground-state energy more rapidly as drive time decreases. They support these findings with measures of coherence, including the relative entropy of coherence and the Wigner–Yanase skew information, and discuss the broader implications for nonlinear control in quantum dynamics.

Abstract

Many complex quantum systems can be described by effectively nonlinear dynamics. While such dynamics have many appealing characteristics, they also make the analysis significantly more involved. This is due to the fact that only a few analytical treatments exist, and that the language of quantum mechanics is built for linear operators. For instance, the very formulation of the quantum adiabatic theorem requires the underlying dynamics to be linear. In this work we show that in a generalized Landau-Zener model, nonlinear dynamics can be leveraged to suppress excitations and coherences of the corresponding linear scenario. To this end, we introduce a generalized "energy spectrum", which is defined by the expectation values of the energy under the stationary states. As a main result, we show that the nonlinear term in the evolution equation acts like an effective shortcut to adiabaticity for the linear Landau-Zener problem.

Figures (7)

• Figure 1: (left panel) Generalized energy "spectrum" for the nonlinear Landau-Zener model with Gross-Pitaevskii nonlinearity, $\tilde{\kappa}(z)=\kappa\, z$. Gray lines correspond to the spectrum of the linear Landau-Zener model. Stationary Bloch vector of the "excited" state (middle panel) and "ground" state (right panel). Parameters are $J=1$ and $\kappa=4$.
• Figure 2: (left panel) Generalized energy "spectrum" for the nonlinear Landau-Zener model for Bose liquids with a logarithmic term, $\tilde{\kappa}(z)=\kappa\, \mathrm{arctan}(z)$. Gray lines correspond to the spectrum of the linear Landau-Zener model. Stationary Bloch vector of the "excited" state (middle panel) and "ground" state (right panel). Parameters are $J=1$ and $\kappa=4$.
• Figure 3: Probability to find the nonlinear Landau-Zener model in the excited state, $\mathcal{P}_\tau=(1+z_{\tau/2})/2$, for a Gross-Pitaeveskii nonlinearity, $\tilde{\kappa}(z)=\kappa\, z$, and $\alpha=100$. Red, dashed line depicts the Landau-Zener formula Eq. \ref{['eq:LZ']}.
• Figure 4: Probability to find the nonlinear Landau-Zener model in the excited state, $\mathcal{P}_\tau=(1+z_{\tau/2})/2$, for a logarithmic nonlinearity, $\tilde{\kappa}(z)=\kappa\, \mathrm{arctan}(z)$, and $\alpha=100$. Red, dashed line depicts the Landau-Zener formula Eq. \ref{['eq:LZ']}.
• Figure 5: Expected energy \ref{['eq:energy']} normalized by the minimal ground state energy (obtained for $\tau\gg1$) at $t=\tau/2$ for a Gross-Pitaeveskii nonlinearity, $\tilde{\kappa}(z)=\kappa\, z$, and $\alpha=100$.