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arXiv:quant-ph/0602163 · Physical Review A

Towards a generalized Landau-Zener formula for an interacting Bose-Einstein condensate in a two-level system

Dirk Witthaut, Eva-Maria Graefe, H. J. Korsch

Published Jun 7, 2006• 86 citations
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Abstract

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.

DOI
10.1103/physreva.73.063609

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We investigate the dynamics of a Bose-Einstein condensate in a triple-well trap in a three-level approximation. The interatomic interactions are taken into account in a mean-field approximation (Gross-Pitaevskii equation), leading to a nonlinear three-level model. Additional eigenstates emerge due to the nonlinearity, depending on the system parameters. Adiabaticity breaks down if such a nonlinear eigenstate disappears when the parameters are varied. The dynamical implications of this loss of adiabaticity are analyzed for two important special cases: A three-level Landau-Zener model and the stimulated Raman adiabatic passage (STIRAP) scheme. We discuss the emergence of looped levels for an equal-slope Landau-Zener model. The Zener tunneling probability does not tend to zero in the adiabatic limit and shows pronounced oscillations as a function of the velocity of the parameter variation. Furthermore we generalize the STIRAP scheme for adiabatic coherent population transfer between atomic states to the nonlinear case. It is shown that STIRAP breaks down if the nonlinearity exceeds the detuning.
We present a detailed analysis of the Landau–Zener problem for an interacting Bose–Einstein condensate in a time-varying double-well trap, especially focusing on the relation between the full many-particle problem and the mean-field approximation. Due to the nonlinear self-interaction a dynamical instability occurs, which leads to a breakdown of adiabaticity and thus fundamentally alters the dynamics. It is shown that essentially all the features of the Landau–Zener problem including the depletion of the condensate mode can be already understood within a semiclassical phase-space picture. In particular, this treatment resolves the formerly imputed incommutability of the adiabatic and semiclassical limits. The possibility of exploiting Landau–Zener sweeps to generate squeezed states for spectroscopic tasks is analyzed in detail. Moreover, we study the influence of phase noise and propose a Landau–Zener sweep as a sensitive yet readily implementable probe for decoherence, since the noise has significant effect on the transition rate for slow parameter variations.
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.
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In the framework of a basic semiclassical time-dependent nonlinear two-state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo- and magneto-association of an atomic Bose-Einstein condensate. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom-molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Landau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al. [Phys. Rev. A 69 , 043612 (2004); J. Phys. A 38 , 3505 (2005)] by order of magnitude.