Multiple-time-scale Landau-Zener transitions in many-body systems

Motivated by recent cold-atom experiments in optical lattices, we consider a lattice version of the Landau-Zener problem. Every single site is described by a Landau-Zener problem, but due to particle tunneling between neighboring lattice sites this on-site single-particle Landau-Zener dynamics couples to the particle motion within the lattice. The lattice, apart from having a dephasing effect on single-site Landau-Zener transitions, also implies, in the presence of a confining trap, an intersite particle flow induced by the Landau-Zener sweeping. This gives rise to an interplay between intra- and intersite dynamics. The adiabaticity constraint is therefore not simply given by the standard one, the Hamiltonian rate of change relative to the gap of the on-site problem. In experimentally realistic situations, the full system evolution is well described by Franck-Condon physics; e.g., nonadiabatic excitations are predominantly external ones characterized by large phononic vibrations in the atomic cloud, while internal excitations are very weak as close-to-perfect on-site transitions take place.

Figure 1

Ground-state distributions $|{\psi }_{\alpha \mathbf{j}}{|}^2$ (a)–(f) of the two $p_x$ (left) and $p_y$ orbitals (right) for different “detunings”; $\lambda t=-0.001$ (a) and (b), $\lambda t=0$ (c) and (d), and $\lambda t=0.001$ (e) and (f). The lower plot (g) shows the total atomic imbalance $Z_{\mathrm{tot}}\left(\lambda t\right)$. Apart from the regime $\lambda t\in \pm 5\times {10}^{-4}$, the system ground state populates approximately only a single on-site orbital state (full polarization). The dimensionless parameters used are $\omega =0.003,t_1=-0.09,t_2=0.0045$, and $U=0.38$. The latter three numerical values correspond to an optical lattice with an amplitude of 17 recoil energies which is chosen to be an experimentally relevant situation [15]. The trap frequency is such that approximately a few hundred lattice sites are populated.

Figure 2

Intrinsic excitations $P_{\mathrm{iex}}\left(\lambda \right)$ (a) and variance of the squeezing parameter $\delta F_y\left(\lambda \right)$ (b). The open circles mark calculated values, while the solid line is a fifth-order polynomial least-squares fit. Combining the results of the two plots, near full adiabatic population transfer is only encountered for $\lambda <{10}^{-10}$. The parameters are the same as for Fig. 1; i.e., they correspond to a lattice amplitude of 17 recoil energies.

Figure 3

The final (full) distribution $Q_{\mathbf{j}}\left(t_f\right)=|{\psi }_{x\mathbf{j}}\left(t_f\right){|}^2+{\left|{\psi }_{y\mathbf{j}}\left(t_f\right)\right|}^2$ of the two orbitals in a situations where nonadiabatic excitations occurs mainly in the external degrees of freedom ($P_{\mathrm{iex}}=0.026$), i.e., in phononic vibrations of the particle distribution. The distribution is still squeezed in the $y$ direction despite the fact that almost only the $p_y$-orbital states are populated which clearly indicates that the transition takes place in the Franck-Condon regime. The sweep velocity $\lambda =1.68\times {10}^{-9}$ and the rest of the parameters are as in Fig. 1.

Figure 4

The “spectral function” $S_{\alpha }\left(\nu \right)$ for the $x$ and $y$ vibrations of the full atomic distribution $Q_{\mathbf{j}}$. The Fourier transform is taken in the interval preceding the LZ transition. The parameters are as in Fig. 3.

Figure 5

The Franck-Condon mechanism. We imagine that initially the particle's state is ${\psi }_{xi}\left(x\right)$ (schematically represented by a $p_x$ orbital in the plot) and a coupling $\Omega$ between the $p_x$ and $p_y$ orbitals is turned on. If the coupling realizes a $\pi$-pulse transition, the particle will reside solely on the upper $p_y$-orbital branch after the turn-off of the coupling. In the Franck-Condon regime the state will be approximately ${\psi }_{yi}\left(x\right)$ (a $p_y$-orbital state at site $i$); i.e., the effective coupling strengths $|{\Omega }_{i,i}|\gg |{\Omega }_{i,i\pm 1}|$ and the pulse is short such that tunnelings are negligible during the transfer. It is clear that for a pulse with a long duration the external evolution will affect the final state on the $p_y$-orbital branch.