Time-resolved measurement of Landau-Zener tunneling in different bases

A comprehensive study of the tunneling dynamics of a Bose-Einstein condensate in a tilted periodic potential is presented. We report numerical and experimental results on time-resolved measurements of the Landau-Zener tunneling of ultracold atoms introduced by the tilt, which experimentally is realized by accelerating the lattice. The use of different protocols enables us to access the tunneling probability, numerically as well as experimentally, in two different bases, namely, the adiabatic basis and the diabatic basis. The adiabatic basis corresponds to the eigenstates of the lattice, and the diabatic one to the free-particle momentum eigenstates. Our numerical and experimental results are compared with existing two-state Landau-Zener models.

Figure 1

Energy levels as a function of time. The dashed lines show the so-called diabatic levels, i.e., the energies of states in the absence of the interaction. The solid lines demonstrate the so-called adiabatic levels, i.e., the eigenstates of the system corresponding to the instantaneous Hamiltonian.

Figure 2

Time-resolved measurements of the adiabatic survival probability for the Bose-Einstein condensate tunneling in an optical lattice at fixed dimensionless force $F_0=1.197$ and different lattice depths: $V/E_{\mathrm{rec}}=2.3$ (filled squares), $1.8$ (open circles), $1$ (open squares) and $0.6$ (filled circles); all parameters are introduced in detail in Sec. 3 A. For the Bose-Einstein condensate evolution the crossing time is $t=T_{\mathrm{Bloch}}/2$, where the step of the survival probability is one-half of the final value. The dashed and dot-dashed lines are the results of numerical simulations using the cutoff and adiabatic method, respectively (see Sec. 3 C). The lattice depth for the numerical simulations was corrected by up to $\pm 15\%$ with respect to the experimentally measured values to give the best possible agreement.

Figure 3

Band structure in the optical lattice potential and experimental protocols for measuring LZ dynamics in the adiabatic and diabatic bases. After the initial loading into the lattice and acceleration for a time $t_{\mathrm{LZ}}$ (top), measurements of the instantaneous populations in the two states are performed (bottom) as explained in the text. In the top figures, the adiabatic (solid lines) and diabatic (dashed lines) energies for an optical lattice of depth $V_0=1$ are shown.

Figure 4

Comparison between the time evolution of the Bose-Einstein condensate survival probability, in the (a) diabatic and (b) adiabatic basis, for different initial conditions, prepared at a temporal distance $\Delta t=T_B/2$ from the crossing point. The dashed line is the evolution in a possible experimental setup, i.e., the evolution following an initial preparation in the ground state of the adiabatic basis; the solid line is the evolution following an initial preparation in the ground state of the diabatic basis. Parameters are $F_0=1.197$ and $V_0=2.3$, corresponding to $\gamma =2.3$, leading to a jump time in both adiabatic and diabatic bases 1.9 times the Bloch period $T_B$.

Figure 5

Comparison between the time evolution of the Bose-Einstein condensate adiabatic survival probability, starting from initial ground states prepared at different time distances from the transition point. Survival probabilities measured for $F_0=1.197$ and $V_0=0.3$ in (a), and for $F_0=1.197$ and $V_0=3.0$ in (b). The dotted lines show the evolution obtained evolving the survival probability from the ground state in the adiabatic basis at $t=-\infty$; the solid lines illustrate the evolution obtained evolving the survival probability from the ground state which simulates a possible experimental initial state, i.e., the ground state in an adiabatic basis, at a finite time from the transition point.

Figure 6

(a) Momentum distribution at time $t=10T_B$ (dashed lines) starting from the initial momentum distribution (solid lines) under the action of a force directed toward negative $p$ values. The vertical dash-dotted line shows the cutoff value $p_c=3p_{\mathrm{rec}}$ in the definition of Eq. (10). (b) Temporal evolution of the survival probability in the adiabatic basis using the mentioned definition. Simulation parameters: $V_0=2.07$, $F_0=1.197$.

Figure 7

Time-resolved measurements of LZ tunneling in the diabatic basis. (a) Fixed force $F_0=1.197$ with different lattice depths $V_0=1$ (filled circles), $1.8$ (open circles), $2.3$ (filled squares), and $4$ (open squares). (b) Fixed lattice depth $V_0=1.8$ with different forces: $F_0=2.394$ (open circles), $1.197$ (filled squares), and $0.599$ (open squares). The dashed lines are the results of numerical simulations based on Eq. (12), which nicely reproduce the experimental data.