Nonadiabatic wavepacket dynamics: $k$-space formulation

The time evolution of wavepackets in crystals in the presence of a homogeneous electric field is formulated in $k$ space in a numerically tractable form. The dynamics is governed by separate equations for the motion of the wave form in $k$ space and for the evolution of the underlying Bloch-type states. A one-dimensional tight-binding model is studied numerically and both Bloch oscillations and Zener tunneling are observed. The long-lived Bloch oscillations of the wavepacket center under weak fields are accompanied by oscillations in its spatial spread. These are analyzed in terms of a $k$-space expression for the spread having contributions from both the quantum metric and the Berry connection of the Bloch states. We find that when sizable spread oscillations do occur, they are mostly due to the latter term.

Figure 1

Upper panel: band structure of the one-dimensional tight-binding model of Eq. (29), for the choice of parameters $\gamma =-U=1$. Lower panel: quantum metric [Eq. (21)] for the lowest band.

Figure 2

(Color online) Bloch oscillations of a wavepacket prepared in the lowest band. Upper panel: time evolution of the weights of the wavepacket on the tight-binding basis orbitals, for tight-binding parameters $\gamma =-U=1$. Lower panel: amplitude $A$ of the Bloch oscillations versus the bandwidth $W$.

Figure 3

Time evolution of the total wavepacket spread ${\left(\Delta x\right)}^2$ (solid lines) and of the metric contribution $⟨G_k⟩$ (dashed lines), for two different sets of tight-binding parameters. Upper panel: $\gamma =-U=1$, resulting in a bandwidth $W\sim 0.5$ and a minimum band gap $E_g^{\left(0\right)}\sim 1.1$. Lower panel: $\gamma =0.5$ and $U=-1$, for which $W\sim 0.1$ and $E_g^{\left(0\right)}\sim 1.2$.

Figure 4

Dependence of the time-averaged contributions to the wavepacket spread [Eq. (20)] on the wave form width $\Delta k$, for $\gamma =-U=1$. The two dashed lines are fits to the values from the numerical simulation (symbols), while the solid line equals the right-hand side of Eq. (27).

Figure 5

(Color online) Upper panel: time evolution of a wavepacket prepared in the lowest band, with child packets in the second and third bands appearing as a result of Zener tunneling. The tight-binding parameters are $\gamma =1$ and $U=-0.3$. Lower panel: electric field as a function of time. The field is switched on from $t=0$ to $t=0.86{\tau }_B$, where ${\tau }_B$ is the Bloch period for the saturation field ${\mathcal{E}}_0$.

Figure 6

Band occupancy $P_n\left(t\right)$ for the simulation shown in Fig. 5.