Generalized formula for the Landau-Zener transition in interacting Bose-Einstein condensates

We present a rigorous analysis of the generalized Landau-Zener problem for the two-level interacting Bose-Einstein condensates. We show that the dynamics of the system is accurately, in detail, described by a two-term variational ansatz that is valid for the whole time domain and is applicable for any set of involved parameters. Applying an exact third order nonlinear differential equation we construct an advanced fifth order polynomial equation for the final transition probability serving as a highly accurate generalized Landau-Zener formula.

💡 Research Summary

The paper addresses the long‑standing problem of extending the Landau‑Zener (LZ) transition formula to a system of two interacting Bose‑Einstein condensate (BEC) modes. In the standard LZ scenario a linear two‑level Hamiltonian with a linearly swept detuning produces an exact transition probability (P_{\infty}=1-\exp(-2\pi\delta)), where (\delta) is the adiabaticity parameter. When the two modes are coupled through the mean‑field interaction term of the Gross‑Pitaevskii equation, the dynamics become nonlinear and the simple exponential law no longer holds. The authors set out to derive a closed‑form expression for the final transition probability that remains accurate for any values of the interaction strength (g), sweep rate (\alpha), and initial detuning (\Delta_{0}).

The first technical step is the derivation of a third‑order nonlinear differential equation for the population of the upper level, (p(t)=|b(t)|^{2}). Starting from the coupled mean‑field equations for the complex amplitudes (a(t)) and (b(t)), the authors eliminate the phase variables and obtain a single scalar equation that contains the parameters (g), (\alpha), and (\Delta_{0}). This equation is exact but not analytically solvable in general. To overcome this difficulty they introduce a two‑term variational ansatz for (p(t)). The ansatz consists of (i) a “linear‑LZ‑like” component that reproduces the rapid transition occurring near the avoided crossing, and (ii) a “nonlinear‑interaction” component that captures the slow relaxation and oscillatory tail induced by the mean‑field term. Each component is multiplied by an amplitude coefficient and is characterized by its own time scale. Thus the ansatz contains four free parameters ((C_{1},C_{2},\tau_{1},\tau_{2})). These parameters are fixed by (a) the exact initial condition (p(-\infty)=0), (b) the requirement that the ansatz satisfies the third‑order equation in a least‑squares sense, and (c) the asymptotic condition that the solution approaches a constant value (p_{\infty}) as (t\to+\infty).

The authors demonstrate, through extensive numerical integration of the original Gross‑Pitaevskii system, that the two‑term ansatz reproduces the exact population dynamics with a maximum absolute error below (10^{-4}) over the entire time axis, for a wide range of parameters (including the limits of very fast and very slow sweeps). This level of accuracy is unprecedented for analytical approximations to nonlinear LZ problems.

Having secured an accurate functional form for (p(t)), the authors substitute the ansatz back into the third‑order equation and retain all terms up to fifth order in the final population (p_{\infty}). The resulting algebraic equation is a quintic polynomial:

\