Observation of Photon-Assisted Tunneling in Optical Lattices

We have observed tunneling suppression and photon-assisted tunneling of Bose-Einstein condensates in an optical lattice subjected to a constant force plus a sinusoidal shaking. For a sufficiently large constant force, the ground energy levels of the lattice are shifted out of resonance and tunneling is suppressed; when the shaking is switched on, the levels are coupled by low-frequency photons and tunneling resumes. Our results agree well with theoretical predictions and demonstrate the usefulness of optical lattices for studying solid-state phenomena.

Figure 1

Photon-assisted tunneling. (a) In an optical lattice at rest, the ground-state levels are resonantly coupled, leading to a tunneling energy $J$. (b) When a linear potential is applied, e.g., by accelerating the lattice, the levels are shifted out of resonance and tunneling is suppressed. (c) If the lattice is now periodically shaken at an appropriate frequency, the levels can again be coupled through photons of energy $\hslash \omega$ and tunneling is partially restored.

Figure 2

(a) Suppression of tunneling by a linear potential. Shown here is the normalized effective tunneling parameter $|J_{\mathrm{eff}}/J|$ as a function of the linear potential $F{d}_L$ in units of the tunneling energy. When $F{d}_L/J\approx 1$, the ground-state levels are shifted out of resonance and tunneling is suppressed almost completely. The solid line is a Lorentzian fit with a half-maximum half-width of 0.13. (b) Photon-assisted tunneling resonances in a shaken lattice. For a fixed linear potential $F{d}_L/h=380\mathrm{Hz}$, the condensate width is plotted as a function of the normalized shaking frequency $\hslash \omega /F{d}_L$. The fixed shaking parameter was $K_0=1.8$ for the one-photon resonance (solid circles) and $K_0=3.1$ for the two-photon resonance (open circles), corresponding to the first maximum of the ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ Bessel functions, respectively. For a), $V_0/E_{\mathrm{rec}}=5$ and $J/h=270\mathrm{Hz}$, while for b) $V_0/E_{\mathrm{rec}}=8$ and $J/h=120\mathrm{Hz}$.

Figure 3

Photon-assisted tunneling as a function of the shaking parameter $K_0$. Shown here are the one-photon resonance at $\omega /2\pi =380\mathrm{Hz}$ (a) and the two-photon resonance at $\omega /2\pi =190\mathrm{Hz}$ (b). In both graphs, the full and open squares are the measurements for $N\approx 5\times {10}^4$ and $N\approx 0.5\times {10}^4$, respectively. The solid lines are the moduli of the ${\mathcal{J}}_1(K_0)$ and ${\mathcal{J}}_2(K_0)$ Bessel functions, respectively, whereas the dashed lines are the squares of these functions. The lattice depth $V_0/E_{\mathrm{rec}}=5$ and the constant force $F{d}_L/h=380\mathrm{Hz}$ in this experiment, with a free expansion time $t=100\mathrm{ms}$.