ac Stark shift and multiphotonlike resonances in low-frequency-driven optical lattices

We suggest that Bose-Einstein condensates in optical lattices subjected to ac forcing with a smooth envelope may provide detailed experimental access to multiphotonlike transitions between ac-Stark-shifted Bloch bands. Such transitions correspond to resonances described theoretically by avoided quasienergy crossings. We show that the width of such anticrossings can be inferred from measurements involving asymmetric pulses. We also introduce a pulse tracking strategy for locating the particular driving amplitudes for which resonances occur. Our numerical calculations refer to a currently existing experimental setup [E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H.-C. Nägerl, Phys. Rev. Lett. 104, 200403 (2010)].

Figure 1

Energy band structure of an unperturbed optical cosine lattice with depth $V_0=2.3E_{\mathrm{r}}$. Measured in terms of the “photon” energy $\hslash \omega =0.23E_{\mathrm{r}}$, as corresponding to the driving frequency $\omega /\left(2\pi \right)=300$ Hz, the gap between the lowest two bands is ${\Delta }_1=5.05\hslash \omega$ at the Brillouin zone edge, and ${\Delta }_0=18.24\hslash \omega$ at its center.

Figure 2

Interband transition probabilities after pulses with the squared-sine envelope (3) and length $T_{\mathrm{p}}=60\times 2\pi /\omega$, obtained from numerical solutions of the Schrödinger equation. All atoms which are excited to bands $n>1$ after a pulse are assumed to escape from the lattice.

Figure 3

Quasienergies ${\varepsilon }_n\left(0\right)$ at the center of the Brillouin zone for an optical lattice with depth $V_0/E_{\mathrm{r}}=2.3$, driven with scaled frequency $\hslash \omega /E_{\mathrm{r}}=0.23$. The quasienergy originating from the lowest band $n=1$ is marked by arrows. It exhibits a substantial ac Stark shift and undergoes pronounced avoided crossings for $K>3$.

Figure 4

Quasienergies ${\varepsilon }_n\left(k\right)$ for $k/k_{\mathrm{L}}=0.8$ and $n=1,2,3$. The arrow indicates the quasienergy originating from the lowest energy band $n=1$.

Figure 5

Survival probability of atoms in the lowest band after asymmetric pulses with fast switch-on time $T_{\mathrm{p}}^{\left(1\right)}/2=5T$, and with varying switch-off durations $T_{\mathrm{p}}^{\left(2\right)}/2$. The initial packets were centered around $\langle k\rangle \left(0\right)/k_{\mathrm{L}}=0.8$, so that the dynamics is determined by the spectrum shown in Fig. 4; $K_{\max }=1.2$ for all pulses.

Figure 6

Pulse tracking realized by switching off the squared-sine envelope (3) with $T_{\mathrm{p}}=60T$ abruptly at $t=T_{\mathrm{out}}$ and recording the escape probability at that moment. The onset of interband transitions then reveals the presence of a multiphoton resonance at the instantaneous amplitude reached at $t=T_{\mathrm{out}}$.