Dynamics of a many-particle Landau-Zener model: Inverse sweep

We consider dynamics of a slowly time-dependent Dicke model, which represents a many-body generalization of the Landau-Zener model. In particular, the model describes narrow Feshbach resonance passage in an ultracold gas of Fermi atoms. Adiabaticity is destroyed when a parameter crosses a critical value, even at very slow sweeping rates of a parameter. The dynamics crucially depends on direction of the sweep. We apply our recent analysis (A. P. Itin and P. Törmä, e-print arXiv:0901.4778) to the “inverse” sweep through the resonance, corresponding (in a context of Feshbach resonance passage) to dissociation of molecules. On a level of the mean-field approximation, the dynamics is equivalent to a molecular condensate formation from Bose atoms within a two-mode model. Mapping the system to a Painlevé equation allows us to calculate deviation from adiabaticity at very slow sweeps analytically.

Figure 1

Phase portraits of Hamiltonian (2). From (a) to (i): $\gamma =-20,-4,-1.8,-0.9,0.0,0.9,1.8,20,100$, correspondingly. Saddle points are denoted by asterisks; the bold trajectory is the separatrix. Shaded areas illustrate definitions of the classical actions (see main text).

Figure 2

(Color online) Change in the classical action for the backward sweep with zero initial action (i.e., $\Delta I=I_f-I_{-}=I_f$). Inset shows change in the action for small values of $ϵ$ and a linear fit which gives $I_f/ϵ=0.220631$. Theoretical prediction (7) is $\frac{\text{ln}2}{\pi }\approx 0.220636$.