An ansatz for the nonlinear Demkov-Kunike problem for cold molecule formation

Abstract

Abstract. We study nonlinear mean-field dynamics of ultracold molecule formation in the case when the external field configuration is defined by the level-crossing Demkov-Kunike model, characterized by a bellshaped coupling and finite variation of the detuning. Analyzing the fast sweep rate regime of the strong interaction limit, which models a situation when the peak value of the coupling is large enough and the resonance crossing is sufficiently fast, we construct a highly accurate ansatz to describe the temporal dynamics of the molecule formation in the mentioned interaction regime. The absolute error of the constructed approximation is less than for the final transition probability while at certain time points it might increase up to . Examining the role of the different terms in the constructed approximation, we prove that in the fast sweep rate regime of the strong interaction limit the temporal dynamics of the atom-molecule conversion effectively consists of the process of resonance crossing, which is governed by a nonlinear equation, followed by atom-molecular coherent oscillations which are basically described by a solution of the linear problem, associated with the considered nonlinear one. IntroductionAfter the realization of Bose-Einstein condensation (BEC) [1] in dilute gases of neutral alkalimetal atoms, the next challenge was to achieve a BEC of molecules. Molecules have complex internal structure and, therefore, more degrees of freedom, thus they offer a vast range of properties not available in the case of atoms. However, in contrast to atomic condensates, achieving molecular condensates via standard laser cooling techniques [2] is practically impossible, because of the rovibrational spectrum of the molecules. Consequently, the Raman photoassociation [3] and magnetic Feshbach resonance [4] have become the standard tools to create cold molecules starting from ultracold atomic gases [5][6][7].So far, the theoretical efforts to describe the dynamics of cold molecule formation (e.g., see Refs. [8][9][10][11][12][13][14][15][16][17][18][19][20]) have mostly been focused on the treatment of the Landau-Zener (LZ) model [21]. In the case of photoassociation the LZ model describes a situation when the two quantum states are coupled by an external optical field of constant amplitude and a variable frequency, which is linearly changed in time. But this situation has some drawbacks; it is unrealistic to have a constant coupling that never turns off or infinite energies at . However, there exists a model that has all the virtues of the LZ model and is free from its shortcomings. Such a model is the first Demkov-Kunike (DK) quasilinear level-crossing model of a bell-shaped pulse (vanishing at