Determination of Chern numbers with a phase-retrieval algorithm

Ultracold atoms in optical lattices form a clean quantum simulator platform which can be utilized to examine topological phenomena and test exotic topological materials. Here we propose an experimental scheme to measure the Chern numbers of two-dimensional multiband topological insulators with bosonic atoms. We show how to extract the topological invariants out of a sequence of time-of-flight images by applying a phase-retrieval algorithm to matter waves. We illustrate advantages of using bosonic atoms as well as efficiency and robustness of the method with two prominent examples: the Harper-Hofstadter model with an arbitrary commensurate magnetic flux and the Haldane model on a brick-wall lattice.

Figure 1

Preparation of an initial eigenstate in the regime of nontrivial topology, if initially ${\mathit{k}}_{in}\approx {\mathit{k}}_D$. To avoid excitations to another band, before we change parameters of the system across the topological quantum phase transition, we have to apply a weak constant force ${\mathit{F}}_1$ to shift the quasimomentum ${\mathit{k}}_{\text{aux}}={\mathit{k}}_{in}+\frac{\mathrm{\Delta }{t}_1}{\hslash }{\mathit{F}}_1$ far away form the Dirac point ${\mathit{k}}_D$. At ${\mathit{k}}_{\text{aux}}$ the upper band is not populated if the change of parameters across the topological phase transition is performed sufficiently slowly. Subsequently, i.e., after the change of system parameters to a topological regime, once again we apply a constant force to transfer a system into any final quasimomentum $\mathit{k}={\mathit{k}}_{\text{aux}}+\frac{\mathrm{\Delta }{t}_2}{\hslash }{\mathit{F}}_2$.

Figure 2

Harper-Hofstadter model is a 2D square lattice with tunneling amplitudes $J,J^{\prime }$, in $x,y$, pierced by uniform artificial magnetic field. A particle traveling along $y$ acquires the Peierls phase. We denote (magnetic) elementary cells by green rectangles for two magnetic fluxes through plaquette, $\varphi =1/3$ (left panel) and $\varphi =1/5$ (right panel). The corresponding energy spectra are calculated for $J/J^{\prime }=1/2$. Chern numbers associated to energy bands are indicated.

Figure 3

Reconstruction of the Chern number for the lowest band of the Harper-Hofstadter model with magnetic flux $\varphi =1/3$ and $\varphi =1/5$. Upper panel: Mean (over distinct quasimomenta) of logarithm of sorted retrieval errors $\varepsilon$ [see Eq. (8)]. On average, about 90% of independent phase-retrieval runs converge successfully $(\varepsilon \approx {10}^{-6}{10}^{-7})$. Lower panel: The reconstructed Chern numbers as a function of a percentage of rejected retrievals. The calculations give the proper value $c_1=-1$ within error bars even if all unsuccessful retrievals are selected. Although the better results are obtained for a higher mesh size ($8\times 8$), after rejecting about 10% of the worst retrievals, a very coarse mesh ($4\times 4$) already gives a perfect agreement with the model.

Figure 4

Haldane model on a brick-wall lattice: two interpenetrating square lattices, with real and complex tunnelings to the nearest and next-nearest-neighboring sites, respectively. Arrows denote directions of the tunnelings.

Figure 5

Topological phase diagram of the lowest band in the Haldane model, obtained from simulated TOF images using a phase-retrieval algorithm. Black lines indicate phase transitions at $\pm 3\sqrt{3}sin\theta$, predicted by Haldane [48]. The quality of the phase diagram depends on the percentage of rejected phase-retrieval outputs. Upper panel: When all the phase-retrieval runs are taken into account, only general features of the phase diagram are reproduced. Lower panel: Rejection of 50% worst results (according to the retrieval error $\varepsilon$), already leads to an exact reconstruction of the phase diagram and more rejections do not change the picture.

Figure 6

Analysis of the influence of experimental imperfections on the retrieved Chern numbers. All presented results are related to the lowest energy band of the Harper-Hofstadter model with the flux $\varphi =1/3$. We consider a finite lattice consisting of $7\times 7$ elementary magnetic cells ($7\times 21$ lattice sites); cf. Fig. 2. Panel (a): average values of the obtained Chern number $c_1$ of the lowest band as a function of the Brillouin zone meshing. It turns out that it is sufficient to discretize the first BZ with a $4\times 4$ mesh only in order to obtain the correct value of $c_1$. Panel (b): impact of the excitation of the system to the second energy band. In the Harper-Hofstadter model with $q=3$ bands, the lowest energy band corresponds to $c_1=-1$ while the Chern number of the second band is $c_2=2$ (see Fig. 2). For $8\times 8$ meshing, when the occupation of the second band exceeds ${\left|\beta \right|}^2\approx 0.12$, the obtained Chern number of the lowest band becomes incorrect, i.e., it switches from $c_1=-1$ to 0. When ${\left|\beta \right|}^2\gtrsim 0.86$, the system is actually in the second band and the value of the obtained Chern number equals 2 as expected. For a $4\times 4$ mesh, this limit is much smaller: ${\left|\beta \right|}^2\le 0.02$, and the Chern number of the higher band is not correctly reproduced. Panel (c): dependence of average values of the Chern number on the signal-to-noise ratio. In order to successfully reproduce the Chern number, the signal-to-noise has to be greater than about 5.5. It is evident that discarding more retrieval results reduces the limitation. Panel (d): average values of the Chern number for different resolutions of an experimental imaging system. Finite resolution is simulated by convolution of the atomic density after time of flight with the Gaussian function of width ${\sigma }_r$. Horizontal axis shows ${\sigma }_r$ in units of the width ${\sigma }_{\text{peak}}$ of the highest Bragg peak observed in the atomic density after TOF. If ${\sigma }_r/{\sigma }_{\text{peak}}\lesssim 0.4$, the retrieved Chern number is correct.

Figure 7

Illustration of the result of the projection ${\mathit{P}}_w={\mathit{P}}_{w_2}\circ {\mathit{P}}_{w_1}$ that is used in the phase-retrieval algorithm. The Harper-Hofstadter model with the flux $\varphi =1/3$ is considered. The plot shows a cut of the probability density along the $x$ direction before (dashed line) and after (solid line) the projection. The projection reestablishes the translational symmetry of the system. Horizontal lines help to see that due to the projection, the probability densities in all sublattice sites become equal.

Figure 8

Comparison of different versions of the phase-retrieval algorithm. The combination of the HIO and ER methods leads to a smaller error $\varepsilon$ [see Eq. (B1)] than the ER method alone but to a much larger error than in the case when the projection ${\mathit{P}}_w={\mathit{P}}_{w_2}\circ {\mathit{P}}_{w_1}$ is applied every third iteration (see the discussion in the text). Fringes correspond to one cycle of the $\text{HIO}\left(20\right)+\text{ER}\left(1\right)=21$ iterations. The last 30 iterations correspond to the pure ER method which allows one to reduce the error at the end of the retrieval process.