Quantized Topological Response in Trapped Quantum Gases

In this letter, we propose a quantized topological response in trapped 1D quantum gases. The experimental protocol for the response requires the application of an instant optical pulse to a half-infinite region in an asymptotically harmonic trap and measuring the density distribution. We show that the corresponding linear response is described by a universal quantized formula in the thermal dynamical limit, which is invariant under local continuous deformations of the trapping potential $V$, atom distribution $f_{\mathrm{\Lambda }}$, the spatial envelope of the optical pulse ${\mathrm{\Theta }}_p$, and the measurement region ${\mathrm{\Theta }}_m$. We test the statement by various numerical analysis, the result of which is consistent with the analytical prediction to high accuracy. We further show that a short but finite optical pulse duration only results in a violation of the quantization near the transition time, which suggests that quantized response could be observed in realistic experiments. We also generalize our results to non-linear quantized topological responses for atoms in higher dimensional harmonic traps.