Atom-optics simulator of lattice transport phenomena

We experimentally investigate a scheme for studying lattice transport phenomena, based on the controlled momentum-space dynamics of ultracold atomic matter waves. In the effective tight-binding models that can be simulated, we demonstrate that this technique allows for a local and time-dependent control over all system parameters, and additionally allows for single-site resolved detection of atomic populations. We demonstrate full control over site-to-site off-diagonal tunneling elements (amplitude and phase) and diagonal site energies, through the observation of continuous-time quantum walks, Bloch oscillations, and negative tunneling. These capabilities open up new prospects in the experimental study of disordered and topological systems.

Figure 1

Continuous-time quantum walks on unbounded and bounded lattices. (a) Experimental and simulated time-of-flight absorption images of atomic populations in momentum orders ${\psi }_n$ (with momenta $p_n=2\hslash k)$, before laser addressing and after $500\mu \mathrm{s}$ of evolution with uniform nearest-neighbor couplings $t_n=t\approx 0.3E_R$ and on site energies ${\varepsilon }_{n+1}-{\varepsilon }_n=0$ (for $n\in \{-10,9\})$. The typical uncertainty in $t_n$ is on the few percent level. (b) Integrated (along $z)$ momentum spectra, normalized with respect to the total atom number (color scale at right), are plotted as a function of evolution time $T$ (with units $\hslash /t)$. (c) Normalized populations of the zeroth, first, and second (black circles, open green circles, and red squares) momentum orders vs evolution time. The dotted curves are solutions to the effective dynamics described by Eq. (1), while the dashed curves include off-resonant Bragg coupling terms, as described in the text. (d) Experimental integrated momentum spectra vs evolution time, with uniform $t_n$ truncated such that only five sites are coupled $(t_n=t\approx 0.3E_R$ for $n\in \{-2,1\})$. (e) Standard deviation of the population distributions for an unbounded lattice (black squares, solid theory line) and a bounded lattice (red circles, dashed theory line).

Figure 2

Bloch oscillations in a linear potential gradient. (a) Integrated momentum spectra vs evolution time, with uniform $t_n=t\approx 0.33E_R$ with $n\in \{-10,9\}$ for $\hslash \xi /E_R\approx 1$. (b) By detuning all two-photon transition frequencies from resonance by an equal amount $\hslash \xi$, we simulate dynamics of particle transport in a linear potential of the form ${\varepsilon }_{n+1}={\varepsilon }_n+\hslash \xi$. (c) Standard deviation of the population distributions vs evolution time for detunings of $\hslash \xi /E_R\approx 0.5,1,1.75$ (black circles, open green circles, red squares), along with theoretical predictions (dashed lines). The inset shows the effective Bloch frequency vs detuning, as determined from an oscillatory fit, which is well controlled by the two-photon detunings $\xi$.

Figure 3

Reversal of dynamics through temporal switching of the tunneling phase. (a) Integrated momentum spectra vs evolution time, with the uniform $|t_n|=t\approx 0.3E_R$ where $n\in \{-10,9\}$, with a phase reversal, $t\to -t$, at $\sim 325\mu \mathrm{s}$ (indicated by the red marker). (b)–(d) Normalized populations of the zeroth, first, and second (black circles, open green circles, red squares) momentum orders vs evolution time. As indicated by the dashed vertical lines and shaded regions, the phase is switched one time, three times, and eight times for panels (b), (c), and (d), respectively.