Observation of Density-Induced Tunneling

We study the dynamics of bosonic atoms in a tilted one-dimensional optical lattice and report on the first direct observation of density-induced tunneling. We show that the interaction affects the time evolution of the doublon oscillation via density-induced tunneling and pinpoint its density and interaction dependence. The experimental data for different lattice depths are in good agreement with our theoretical model. Furthermore, resonances caused by second-order tunneling processes are studied, where the density-induced tunneling breaks the symmetric behavior for attractive and repulsive interactions predicted by the Hubbard model.

Figure 1

Oscillation frequency $f_0$ of the doublon number in a tilted lattice as a function of $a_s$ for $V_z=8E_R$ (diamonds), $10E_R$ (squares), and $12E_R$ (circles) with initial on-site occupancy (a) $n=1$ and (b) $n=2$ (see insets). The dashed lines show the prediction of the standard Hubbard model. The solid lines depict the interaction dependence due to density-induced tunneling [see Eq. (1)]. Theoretical predictions correspond to a 4% lower lattice depth [17]. The experimental data of (a) are taken from Ref. [18].

Figure 2

(a) Fourier spectrum of the simulated doublon number dynamics for $E=U$ without (upper panel) and with (lower panel) inclusion of density-induced tunneling. Here, $n=1$, $V_z=10E_R$, and $a_s=400a_0$. The dashed (solid) line shows the predominant frequency component predicted without (with) density-induced tunneling. The shaded areas incorporate a Gaussian broadening with a width $w=4\mathrm{Hz}$ for visualization purposes. The insets show the time trace for the first 30 ms. (b) Mode spectrum as a function of the single-particle tunneling rate $J/h$ tuned via the lattice depth $V_z$ for $a_s=400a_0$ ($w=4\mathrm{Hz}$). (c) Mode spectrum as a function of $a_s$ at fixed $V_z=10E_R$ ($w=1\mathrm{Hz}$). In (b) and (c) the dashed line shows $4J/h$, while the solid line shows $4J_{\mathrm{tot}}/h$, including density-induced tunneling.

Figure 3

Doublon dynamics for on-site occupancy $n=2$. (a) Number of doubly (upper lines) and singly (lower lines) occupied sites as a function of $E$ after $t_h=50\mathrm{ms}$ for $a_s=245(5)a_0$ (circles) and $a_s=354(5)a_0$ (diamonds) at $V_z=12E_R$, giving resonances for the $|2,2⟩\leftrightarrow |3,1⟩$ oscillations at $E_{22}=h\times 940(20)\mathrm{Hz}$ and $E_{22}=h\times 1283(20)\mathrm{Hz}$, respectively (see [17] for details). The dashed lines indicate the calculated $U$ and $U/2$. (b) Measured Fourier spectra extracted from time traces of the doublon number at $V_z=10E_R$ and $E=E_{22}$ for different $a_s$. The solid blue lines result from multiple Gaussians fits to the data [17]. The shaded area projected into the axis plane depicts the simulated Fourier spectra shown in (c). The lines show the expected frequencies for the standard (dashed) and extended (solid) Hubbard model [Eq. (1)]. (c) Simulated Fourier spectra as a function of $a_s$ incorporating a Gaussian broadening with a width $w=1\mathrm{Hz}$. The markers denote peak positions deduced from the experimental data shown in (b). The marker size scales linearly with the area under the respective fits. Open circles indicate broad Gaussian fits with a large ratio of width and amplitude $w/A>0.2$.

Figure 4

(a) Time-averaged number of the sum of doublons and triplons (upper lines) and triplons only (lower lines) as a function of $E$ for $V_z=8E_R$ at $a_s=250a_0$ (blue) and $a_s=-250a_0$ (red). The solid lines show numerical simulations including density-induced tunneling, while for the dashed lines $\mathrm{\Delta }J=0$. The gray area indicates sampling over $E$ in the experiment with an estimated width of $\approx 50\mathrm{Hz}$. (b) Numerical Fourier spectrum of the doublon dynamics as a function of $a_s$ revealing the breaking of symmetry between attractive and repulsive scattering for $E=U/2$. The dashed line indicates $f_0=\nu (J+a_s\mathrm{\Delta }J)(J+2a_s\mathrm{\Delta }J)/U$ with a prefactor $\nu =19$ in accordance with [24]. Experimental time traces at the $U/2$ resonance for (c) $a_s=+250a_0$ and (d) $a_s=-250a_0$ at $V_z=8E_R$ showing an asymmetric behavior between repulsive and attractive interaction caused by density-induced tunneling. The dashed lines are guides to the eye that indicate the steady-state doublon number.