Time-Resolved Measurement of Landau-Zener Tunneling in Periodic Potentials

We report time-resolved measurements of Landau-Zener tunneling of Bose-Einstein condensates in accelerated optical lattices, clearly resolving the steplike time dependence of the band populations. Using different experimental protocols we were able to measure the tunneling probability both in the adiabatic and in the diabatic bases of the system. We also experimentally determine the contribution of the momentum width of the Bose condensates to the temporal width of the tunneling steps and discuss the implications for measuring the jump time in the Landau-Zener problem.

Figure 1

Time-resolved measurement of LZ tunneling. (a) Experimental protocol [shown in the band-structure representation of energy $E(q)$ versus quasimomentum $q$]. Left: The lattice is accelerated, (partial) tunneling occurs. Right: The acceleration is then suddenly reduced and the lattice depth increased so as to freeze the instantaneous populations in the lowest two bands; finally, further acceleration is used to separate, and measure, these populations in momentum space. (b) Experimental results for $V_0=1E_{\mathrm{rec}}$ and $F_0=0.383$ ($a_{\mathrm{LZ}}=13.52{\mathrm{ms}}^{-2}$), giving $T_B=0.826\mathrm{ms}$. The solid and dashed lines are a numerical simulation of our experimental protocol and an exponential decay curve for our system’s parameters, respectively.

Figure 2

(a) LZ survival probability in the adiabatic basis for a fixed force $F_0=1.197$ ($a_{\mathrm{LZ}}=42.25{\mathrm{ms}}^{-2}$) and different lattice depths (filled squares: $V_0=2.3E_{\mathrm{rec}}$; open circles: $V_0=1.8E_{\mathrm{rec}}$; open squares: $V_0=1E_{\mathrm{rec}}$; filled circles: $V_0=0.6E_{\mathrm{rec}}$). The dashed lines are sigmoid fits to the experimental data. Inset: Survival probability in both the adiabatic (open squares) and diabatic (filled triangles) bases for $V_0=1E_{\mathrm{rec}}$ and $F_0=1.197$. (b) Step height $h$ as a function of the inverse adiabaticity parameter $1/\gamma$ for varying lattice depth and $F_0=1.197$ (open symbols), and for varying force with fixed $V_0=1.8E_{\mathrm{rec}}$ (filled symbols). The dashed line is the prediction of Eq. (3) for the LZ tunneling probability.

Figure 3

LZ transition for different momentum widths of the condensate. (a) Survival probability for $\Delta p/p_B=0.2$ (filled squares) and $\Delta p/p_B=0.6$ (open squares). The solid and dashed lines are the results of a numerical simulation and of a sigmoid fit, respectively. (b) Step width $\Delta t_{\mathrm{LZ}}$ as a function of $\Delta p$. The open symbols (connected by a solid line for clarity) are the results of a numerical simulation.