Coherent Delocalization of Atomic Wave Packets in Driven Lattice Potentials

Atomic wave packets loaded into a phase-modulated vertical optical-lattice potential exhibit a coherent delocalization dynamics arising from intraband transitions among Wannier-Stark levels. Wannier-Stark intraband transitions are here observed by monitoring the in situ wave-packet extent. By varying the modulation frequency, we find resonances at integer multiples of the Bloch frequency. The resonances show a Fourier-limited width for interrogation times up to 2 s. This can also be used to determine the gravity acceleration with ppm resolution.

Figure 1

Intraband site-to-site tunneling resonantly sets in when the temporal modulation frequency ${\nu }_M$ of the lattice phase is an integer multiple of the Bloch frequency ${\nu }_B$, which corresponds to the potential energy between adjacent sites due to the gravity acceleration $g$.

Figure 2

Wave-packet expansion as the lattice is modulated in phase at four different frequencies ${\nu }_M=n\times {\nu }_B$, where (▪) $n=1$, (●) $n=2$, (▴) $n=3$, and (◆) $n=4$. The widths are normalized to the unperturbed initial value ${\sigma }_0$. Inset: Atomic density profiles for the first ($n=1$) harmonic modulation at increasing modulation times ($20\to 140\mathrm{ms}$).

Figure 3

Resonance width as function of the modulation time $T$. The resonance is probed in the region ${\nu }_M\simeq {\nu }_B$. The solid line (red online) is the hyperbola $(\pi T{)}^{-1}$ expected from a Fourier-limited resonance width in a two-level system. Inset: resonance spectrum for 50 ms excitation time. The fitting function is of the form of Eq. (2).

Figure 4

Resonance spectra at the 1st, 2nd, and 4th harmonic of the Bloch frequency ${\nu }_B$. The excitation time is set to 2 s. Within the error bars the fitted center line frequencies are in integer multiple ratio.