Tomography of Band Insulators from Quench Dynamics

We propose a simple scheme for tomography of band-insulating states in one- and two-dimensional optical lattices with two sublattice states. In particular, the scheme maps out the Berry curvature in the entire Brillouin zone and extracts topological invariants such as the Chern number. The measurement relies on observing—via time-of-flight imaging—the time evolution of the momentum distribution following a sudden quench in the band structure. We consider two examples of experimental relevance: the Harper model with $\pi$ flux and the Haldane model on a honeycomb lattice. Moreover, we illustrate the performance of the scheme in the presence of a parabolic trap, noise, and finite measurement resolution.

Figure 1

General procedure. (a) A lattice with two sublattice states is quenched abruptly such that both sublattices are energetically separated by $\hslash \omega =v_A-v_B$ and tunneling is suppressed (drawn example: a square lattice with alternating tunneling matrix elements $J$ and $J^{\prime }$). (b) Initially, the system is a band insulator with the lower band occupied completely. At every quasimomentum $\mathit{k}$, the two-dimensional state space is represented by a Bloch sphere, with the state of the lower band lying at $-\widehat{\mathit{h}}(\mathit{k})$. With the quench, two flat bands are created, onto which the state is projected. The resulting dynamics corresponds to a rotation around the $z$ axis of the Bloch sphere with the same frequency $\omega$ for every $\mathit{k}$. Monitoring this dynamics in momentum space allows for reconstruction of the initial position on the Bloch sphere, giving a complete tomography of the initial band-insulating state.

Figure 2

(a)–(d) Time-resolved TOF images for the $\pi$-flux Harper model ($\mathrm{\Delta }=0$ and $k_y^0=\pi /2$). To mimic a realistic experiment, we added normal-distributed noise with a standard deviation of 0.1 of the average signal and assumed a limited resolution of 21 points along $k_x$ and 41 time points. (e)–(f) If edge states are supported, the temporal maximum as a function of quasimomentum winds around the time period once [panels (a),(b)], and ${\widehat{\mathit{h}}}_{k_y^0}(k_x)=({\widehat{h}}_x,{\widehat{h}}_y,0)$ describes a unit circle around the origin (e),(f), contrary to when no edge state is supported (c),(d) and (g),(h), respectively). Red line: exact case. Bullets: values extracted from the noisy, resolution-limited data of the upper row, using Eq. (1) [$k_x$ from 0 (light) to $\pi$ (dark)].

Figure 3

(a) Haldane-like lattice model with real tunneling parameter $J$ and complex tunneling parameter $J^{\prime }{e}^{i\theta }$. (b) $n(\mathit{k},t)$ changes strongly with time ($J^{\prime }=0.3J$, $\theta =0.4\pi$). (c) The winding of the phase ${\phi }_{\mathit{k}}$ is opposite around the two Dirac points [white circles in (b)], independently of $\theta$. (d),(e) In the topologically trivial phase (d, $J^{\prime }=0.3J$ and $\theta =0$), the plotted quantity ${\widehat{h}}_z$ has the same sign at the two Dirac cones, while in the topological phase [(d), $J^{\prime }=0.3J$ and $\theta =0.4\pi$] it has opposite sign; this sign change is clearly visible as a kink in the measured quantity $|{\widehat{h}}_z|$ (lower plots showing $|{\widehat{h}}_z|$ along the dashed lines; lines: ideal case, bullets: data for trapped system with normal-distributed noise of variance 0.05, using realistic parameters [13, 15], $J/\hslash =2\pi \times 0.26\mathrm{kHz}$, $\omega =2\pi \times 10\mathrm{kHz}$, trapping frequency $2\pi \times 50\mathrm{Hz}$, lattice spacing 380 nm). (f) The coarse-grained Chern number computed following Ref. [48] reproduces the exact result already for a limited resolution of $4\times 4$ reciprocal lattice points.