Wave-function-renormalization effects in resonantly enhanced tunneling

We study the time evolution of ultracold atoms in an accelerated optical lattice. For a Bose-Einstein condensate with a narrow quasimomentum distribution in a shallow optical lattice the decay of the survival probability in the ground band has a steplike structure. In this regime we establish a connection between the wave-function-renormalization parameter $Z$ introduced by P. Facchi, H. Nakazato, and S. Pascazio [Phys. Rev. Lett. 86, 2699 (2001)] to characterize nonexponential decay and the phenomenon of resonantly enhanced tunneling, where the decay rate is peaked for particular values of the lattice depth and the accelerating force.

Figure 1

Energy diagram for a particle in a periodic potential vs either the time, in units of Bloch time $T_{\mathrm{B}}$ defined in Eq. (14), or the atomic momentum, $p_{\mathrm{at}}/2p_{\mathrm{rec}}$, where $p_{\mathrm{rec}}=\hslash \pi /d_L$ units. Under the application of an external force, the quasimomentum increases with time, and at the avoided crossing between two bands, at the edge of the Brillouin zone, where $p_{\mathrm{at}}=p_{\mathrm{rec}}$, the condensate tunnels with a probability amplitude $p_{ij}$ and survives with an amplitude $s_{ij}$, defined in Eq. (22). Between two avoided crossings of the lowest two bands a relative phase $\phi$, defined in Eq. (17), is acquired, which is graphically displayed as a shaded area. The final survival probability at a given time is the sum over all possible routes, just like a path integral in momentum space.

Figure 2

Adiabatic survival probability in the lowest band $P\left(t\right)$ vs time, obtained by numerically solving the Schrödinger equation for the atomic evolution under the Hamiltonian of Eq. (11). The initial state has initial quasimomentum $0.2p_{\mathrm{rec}}$ in the Brillouin zone and a negligible quasimomentum width. (a) Potential depth $V_0=1.5$ and phase $\phi \simeq 4\pi$ from Eq. (15); (b) $V_0=4.5,\phi \simeq 2\pi$. The solid blue line represents the unperturbed time evolution, and the dashed green line represents the time evolution with the phase change discussed in Sec. 5b after each Bloch period. The red crosses are an extrapolation of the first time step to the following periods. Plateaus manifest for $V_0=1.5$ (a). The validity of this picture is discussed in the text and in the Appendix.

Figure 3

Average band gap $\langle \Delta E\rangle$ vs $V_0$ (both in units of $E_{\mathrm{rec}}$). Comparison between numerical results, analytical results from Eq. (19), and the harmonic oscillator approximation (20). In the small-$V$ regime the band integration yields a good approximation, while for larger $V$, where the coupling becomes continuous, the harmonic oscillator approximation is more effective.

Figure 4

Decay rate $\gamma$ and wave-function renormalization $Z$ vs $\langle \Delta E\rangle /F_0=\phi /2\pi$. Comparison among analytical results, obtained by exact diagonalization of the reduced evolution operator $U$ in Eq. (22) (red dashed lines), numerical simulations based on Eq. (11) (solid lines), and Landau-Zener prediction (dotted lines for $\gamma$). $V_0=2$ in the first two panels, and $V_0=4$ in the bottom two panels.

Figure 5

$P\left(t\right)$: experimental results (squares) and numerical solution of the Schrödinger equation describing the atomic cloud within the accelerated optical lattice (blue dashed line). The red solid lines are exponential fits to the experimental data based on $P_{\mathrm{Z}}\left(t\right)$ from Eq. (30), whose crossing with the $y$ axis yields the value of $Z$. In (a) $V_0=5.8E_{\mathrm{rec}}$, $F_0=5$, and in (b) $V_0=1E_{\mathrm{rec}}$, $F_0=0.383$. Both cases yield $Z<1$. The slope of the exponential decay gives the decay rate $\gamma$.

Figure 6

(a) Scaling plot of $Z-1$ vs $\phi$ of Eq. (17), derived from RET experimental results (squares) using a narrow atomic quasimomentum distribution. The experimental point at $\phi =4.8$ is obtained from the data of Fig. 5a, and the point at $\phi =34$ is obtained from the data of Fig. 5b. Lines are the theoretical predictions for $V_0=1,2,3,4$ (dotted, dashed, dot-dashed, and solid lines, respectively). The RET coupling yields the oscillating behavior of $Z$ vs $\phi$, with the oscillation amplitudes increasing at lower $V_0$ for a fixed $\phi$. (b) Theoretical prediction for $Z-1$ in an Austin-type experiment, with a broad atomic quasimomentum distribution, at $V_0=3,3.5$, and 4 (solid, dashed, and dotted lines, respectively).

Figure 7

Adiabatic coupling strength $c\left(t\right)$, defined in Eq. (A4) and normalized to maximal coupling, plotted vs time and optical lattice depth. Comparison between (a) the Lorentzian ansatz and (b) numerical results based on Eq. (11). The assumption of short tunneling events at the avoided crossings is valid for $V_0\lesssim 4.5$ in units of $E_{\mathrm{rec}}$ (shallow lattice).