Multiphoton interband excitations of quantum gases in driven optical lattices

We report on the observation of multiphoton interband absorption processes for quantum gases in shaken light crystals. Periodic inertial forcing, induced by a spatial motion of the lattice potential, drives multiphoton interband excitations of up to the ninth order. The occurrence of such excitation features is systematically investigated with respect to the potential depth and the driving amplitude. Ab initio calculations of resonance positions as well as numerical evaluation of their strengths exhibit good agreement with experimental data. In addition our findings could make it possible to reach novel phases of quantum matter by tailoring appropriate driving schemes.

Figure 1

(a) Setup of the driven 1D lattice with both beams being linearly polarized along the $z$ axis. (b) Illustration of two-photon transitions into the first excited band at the minima at $q=0$ for $J_0^{\mathrm{eff}}>0$ (black) and $q=\pi /a$ for $J_0^{\mathrm{eff}}<0$ (gray). (c) Multiphoton transition energies (white lines) to the first excited band according to Eq. (6) for $K=3.82$ (see inset). Gray shaded areas depict the maximum possible range of MPA. (d) Typical time-of-flight images of the driven 1D lattice for positive tunneling (top panel), negative tunneling (middle panel), and a heated, incoherent sample (bottom panel).

Figure 2

Systematic investigation of multiphoton spectra in the driven 1D lattice. (a) The effective tunneling matrix element $J^{\mathrm{eff}}$ acquires different values at the four measured driving amplitudes $K$, scaling with the Bessel function ${\mathcal{J}}_0\left(K\right)$. (b) Excitation spectra for the four driving amplitudes depicted in (a) with the maximum optical density encoded in brightness. Solid black lines, numbered on the right-hand side, indicate the calculated positions of MPA to the first excited energy band according to Eq. (6).

Figure 3

Emergence of multiphoton interband transitions for increasing driving amplitude. (a) Multiphoton excitation spectrum obtained at a fixed 1D lattice depth of $9.5E_{\mathrm{rec}}$ for increasing driving amplitudes. The positions and maximum possible widths of the depicted $n\mathrm{th}$-order transitions are indicated above the spectrum. Data at $K\approx 4.0$ (second row from top) have not been measured and are interpolated. (b) Numerical simulation of the observed multiphoton interband excitation spectra. Plotted is the minimum occupation $N_0$ of the lowest band observed during 20 ms of driving at the final value of $K$. Avoided crossings and excitations to higher-energy bands are clearly visible. In (c) the resolution of the simulation data shown in (b) has been reduced to match the resolution of the experimental data. To take into account the linear ramping to the final driving amplitude $K$, excitations present for smaller values of $K$ are kept in the spectra for larger $K$. The resulting spectrum clearly matches the experimental data depicted in (a).

Figure 4

Driving in the triangular lattice. (a) Illustration of the experimental setup, lattice structure, and elliptical forcing. (b) Tunneling renormalization of the diagonal lattice bonds, which depends on the horizontal and vertical frequency modulation amplitudes ${\nu }_{x/y}$. The isotropic renormalization condition of ${\nu }_y=\sqrt{3}{\nu }_x$ is plotted as a dashed line [36]. (c) Excitation spectrum for a fixed isotropic driving amplitude of $K=3.82$, indicated by the solid dot in (b). Expected positions and widths for multiphoton transitions are plotted as solid black lines, similar to Fig. 2. MPA resonances of up to the fifth order can be clearly identified.

Figure 5

Resonance condition for an $n$-photon transition between the bands $\alpha =0$ and $\alpha =1$ in the extended Floquet Hilbert space. The energy levels correspond to the unperturbed quasienergies ${\varepsilon }_{\alpha }^{\text{eff}}\left(q\right)+m\hslash \mathrm{\Omega }$. The states $|\alpha qm\rangle \rangle$ are labeled by the band index $\alpha$, the quasimomentum wave number $q$, and the relative “photon” number $m$.

Figure 6

Single-photon transition.

Figure 7

Two-photon transition.

Figure 8

Three-photon transition.

Figure 9

Minimum occupation of the lowest band during 20 ms of time evolution vs driving frequency $\mathrm{\Omega }$ and driving strength $K$. At $t=0$ the system is prepared in the Bloch state corresponding to the minimum of the effective dispersion relation of the lowest band, that is, with quasimomentum $q=0$ $\left(q=\pi /a\right)$ for $K$ lower (larger) than 2.4.

Figure 10

Same as Fig. 9, but with the initial state shifted away from the minimum by (a) $\mathrm{\Delta }q=0.1\pi$ and (b) $\mathrm{\Delta }q=0.2\pi$.

Figure 11

Occupation of the lowest bands during the time evolution. Plotted is the overlap squared of the time evolved state $\psi \left(t\right)$ with eigenstates ${\varphi }_j$ for (a) $K=3.0$ and $\mathrm{\Omega }=2\pi \times 4.65\mathrm{kHz}$, (b) $K=3.0$ and $\mathrm{\Omega }=2\pi \times 5.00\mathrm{kHz}$, and (c) $K=2.5$ and $\mathrm{\Omega }=2\pi \times 2.90\mathrm{kHz}$. The bands for $\alpha =0$ to 3 are depicted with increasing brightness (see legend). Bands above $\alpha =3$ exhibit no significant occupation and are omitted from the plots.