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Physical Review A

Nonlinear Landau-Zener tunneling

Biao Wu, Qian Niu

Published Jan 5, 2000• 396 citations
Publisher

Abstract

A self-interacting two-level system depending on an external parameter is investigated. The most striking feature exhibited in this system is the presence of a nonzero tunneling probability in the adiabatic limit for large enough interaction strength. Possible experimental observation of this breakdown of adiabaticity using a Bose-Einstein condensate in an optical potential is suggested.

DOI
10.1103/physreva.61.023402

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A self-interacting two-level system depending on an external parameter is investigated. The most striking feature exhibited in this system is the presence of a nonzero tunneling probability in the adiabatic limit for large enough interaction strength. Possible experimental observation of this breakdown of adiabaticity using a Bose-Einstein condensate in an optical potential is suggested.
The basic properties of the nonlinear Landau-Zener tunnelling,a new quantum tunnelling phenomenon is reviewed,which is discovered in interacting two-level systems,such as Bose-Einstein condensates in optical lattices and molecule magnets.It is found that the nonlinearity induced by the repulsive interaction can enhance the tunnelling probability compared to the usual linear Landdau-Zener tunnelling.When the nonlinearity is strong enough,the adiabaticity can break down:an infinitely small external linear force can cause finite tunnelling probability.
We study tunneling of a Bose-Einstein condensates confined in a effective double-well potential (a single well with a spatially magnetic modulated scattering length, actually), called pseudo double-well trap, in which the interaction of atoms characterized by the s-wave scattering length a s can be widely tuned with a magnetic-field Feshbach resonance. As a result, corresponding to different nonlinear parameters, the energy levels of the nonlinear Schrödinger equation can have complex structures in their dependence on the bias between the wells. We discuss the emergence of looped levels, which lead to a breakdown of adiabaticity that the Landau-Zener transition probability does not vanish even in the adiabatic limit. Moreover, we also find that the Landau-Zener tunneling in the pseudo trap show many striking properties distinguished from that of the real trap model (where the barrier is created by the external potential). Possible experimental observation in an opticallyinduced photonic lattices in a photorefractive material is suggested.
We present a comprehensive analysis of the Landau-Zener tunneling of a nonlinear three-level system in a linearly sweeping external field. We find the presence of nonzero tunneling probability in the adiabatic limit (i.e., very slowly sweeping field) even for the situation that the nonlinear term is very small and the energy levels keep the same topological structure as that of the linear case. In particular, the tunneling is irregular with showing an unresolved sensitivity on the sweeping rate. For the case of fast-sweeping fields, we derive an analytic expression for the tunneling probability with stationary phase approximation and show that the nonlinearity can dramatically influence the tunneling probability when the nonlinear ``internal field'' resonate with the external field. We also discuss the asymmetry of the tunneling probability induced by the nonlinearity. Physics behind the above phenomena is revealed and possible application of our model to triple-well trapped Bose-Einstein condensate is discussed.
We investigate the Landau-Zener tunneling (LZT) of a self-interacting two-level system in which the coupling between the levels is nonreciprocal. In such a non-Hermitian system, when the energy bias between two levels is adjusted very slowly, i.e., in the adiabatic limit, we find that a quantum state can still closely follow the eigenstate solution until it encounters the exceptional points (EPs) at which two eigenvalues and their corresponding eigenvectors coalesce. In the absence of the nonlinear self-interaction, we can obtain explicit expressions for the eigenvectors and eigenvalues and analytically derive the adiabatic LZT probability from invariants at EPs. In the presence of the nonlinear interaction, the dynamics of the adiabatic evolutions are explicitly demonstrated with the help of classical trajectories in the plane of the two canonical variables of the corresponding classical Josephson Hamiltonian. We show that the adiabatic tunneling probabilities can be precisely predicted by the classical action at EPs in the weak nonreciprocal regime. In a certain region of strong nonreciprocity, we find that interestingly, the nonlinear interaction effects can be completely suppressed so that the adiabatic tunneling probabilities are identical to their linear counterparts. We also obtain a phase diagram for large ranges of nonreciprocity and nonlinear interaction parameters to explicitly demonstrate where the adiabaticity can break down, i.e., the emergence of the nonzero tunneling probabilities even in adiabatic limit.
We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well as in the mean-field approximation. Novel nonlinear eigenstates emerge in the mean-field description, which leads to a breakdown of adiabaticity: The Landau-Zener transition probability does not vanish even in the adiabatic limit. It is shown that the emergence of nonlinear eigenstates and thus the breakdown of adiabaticity corresponds to quasi-degenerate avoided crossings of the many-particle levels. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit generalized Landau-Zener formula. A comparison to numerical results for the many-particle system and the mean-field approximation shows an excellent agreement.
We investigate adiabatic Landau-Zener tunnelling in nonlinear three-level system. The levels and tunnelling probability are studied with respect to the nonlinearity. For weak nonlinearity, the structures of the levels are smooth and similar to their linear counterparts. However the eigenstates of the middle-level will be unstable for some system parameters, which leads to the adiabaticity to break down and the tunnelling probability chaotic. For strong nonlinearity, the structures of the levels are distorted, including double-ring-like distortion in lower-level and butterfly-like distortion in upper-level. The adiabaticity will be broken down more badly. Finally we give a possible application of the system in triple-well trapped Bose-Einstein condensates.