Generalized acceleration theorem for spatiotemporal Bloch waves

A representation is put forward for wave functions of quantum particles in periodic lattice potentials subjected to homogeneous time-periodic forcing, based on an expansion with respect to Bloch-like states which embody both the spatial and the temporal periodicity. It is shown that there exists a generalization of Bloch's famous acceleration theorem which grows out of this representation and captures the effect of a weak probe force applied in addition to a strong dressing force. Taken together, these elements point at a “dressing and probing” strategy for coherent wave-packet manipulation, which could be implemented in present experiments with optical lattices.

Figure 1

(a) Quasienergy spectrum of an ac-driven 1D optical lattice (67) with depth $V_0/E_{\mathrm{r}}=5.7$, scaled driving frequency $\hslash \omega /E_{\mathrm{r}}=3.71$, and scaled driving amplitude $K=0$. This figure is obtained by projecting the lowest three energy bands of the undriven lattice to the first quasienergy Brillouin zone, ranging from $\varepsilon /\left(\hslash \omega \right)=-1/2$ to $\varepsilon /\left(\hslash \omega \right)=+1/2$. (b) Quasienergy band structure for $K=0.5$. Here the ac Stark shifts still are comparatively weak, but the time-periodic forcing introduces pronounced avoided crossings among the “lowest” three bands. (c) Quasienergy band structure for $K=3.0$ for the “lowest” five bands, revealing substantial ac Stark shifts.

Figure 2

Time evolution of an initial wave packet (69) under the action of a driving pulse (70) with the smooth squared-sine envelope (71), with maximum scaled amplitude $K_{\max }=3.0$, scaled driving frequency $\hslash \omega /E_{\mathrm{r}}=3.71$, and pulse length $T_{\mathrm{p}}=50T$, where $T=2\pi /\omega$ is the duration of a single cycle. The main frame shows the occupation probabilities of the original Bloch energy bands during the pulse. Apart from the initially populated lowest band $n=1$ (jagged line at the top), both bands $n=2$ and $n=3$ become significantly excited during the pulse (jagged lines at the bottom), the band $n=3$ even to a higher extent than $n=2$ at maximum driving strength. In contrast, when monitoring the same dynamics within the bases provided by the instantaneous spatiotemporal Bloch waves, only the corresponding Floquet band $n=1$ is appreciably occupied, as shown by the horizontal line at the top and magnified in the inset. Observe the scale of the inset's ordinate!

Figure 3

(a) Protocol for achieving almost complete interband population transfer in a dressed optical lattice by means of a weak probe force. The dashed line is the scaled envelope of the dressing ac force. The solid line is the negative, scaled probe force, amplified by a factor of 10. (b) Resulting wave-packet dynamics in the Floquet representation, shown as a contour plot of $|g_1{(k,t)|}^2$. Under the first action of the probe force the wave packet is shifted in accordance with the generalized acceleration theorem (32), until it undergoes Zener-type transitions to other quasienergy bands at the avoided crossings visible in Fig. 1b. After the dressing force is switched off, the second, reversed action of the probe force shifts the remaining part of the packet back to the Brillouin zone center. (c) Comparison of the initial wave packet (dashed line) with the part of the wave function that remains in the lowest energy band at the end of the process (solid line).

Figure 4

(a) Protocol for achieving partial interband population transfer in a dressed optical lattice by means of a weak probe force. The dashed line is the scaled envelope of the dressing ac force, which is the same as in Fig. 3. The solid line is the negative, scaled probe force, amplified by a factor of 10; this force is weaker than the one in Fig. 3. (b) Resulting wave-packet dynamics in the Floquet representation, shown as a contour plot of $|g_1{(k,t)|}^2$. Under the first action of the probe force the wave packet is shifted in accordance with the generalized acceleration theorem (32), but not as far as in Fig. 3, such that it undergoes only partial Zener transitions. After the dressing force is switched off, the second, reversed action of the probe force shifts the remaining part of the packet back to the Brillouin zone center. (c) Comparison of the initial wave packet (dashed line) with the part of the wave function that remains in the lowest energy band at the end of the process (solid line).