Energy-Resolved Information Scrambling in Energy-Space Lattices

Weakly interacting Fermi gases simulate spin lattices in energy space, offering a rich platform for investigating information spreading and spin coherence in a large many-body quantum system. We show that the collective spin vector can be determined as a function of energy from the measured spin density, enabling general energy-space resolved protocols. We measure an out-of-time-order correlation function in this system and observe the energy dependence of the many-body coherence.

Figure 1

Energy-resolved out-of-time-order correlation (OTOC) measurement. The system is initially prepared in a pure state, with the spins for atoms of energy $E_1,E_2,\dots ,E_N$ polarized along the $-z$ axis. (a) OTOC sequence, after which the spatial profiles of the ${\uparrow }_z$ and ${\downarrow }_z$ states are measured for each cloud by resonant absorption imaging. (b) “Single-shot" spin density profile $S_z(x)$ (blue dots). For this measurement, the scattering length in the Hamiltonian $H(a)$ is $a=4.24a_0$, $\varphi =\pi$, and $\sigma =345\mu \mathrm{m}$. (c) An inverse-Abel transform of the spatial profile (blue dots) extracts the single-shot energy-resolved spin density $S_z(E)$ (red dots). An Abel transform of $S_z(E)$ yields the red dashed curve shown in (b), consistent with the data.

Figure 2

Total collective spin projection $S_z$ versus rotation angle $\varphi$ without energy restriction. (a) $F(\varphi )=\frac{1}{2}(N_{{\uparrow }_z}-N_{{\downarrow }_z})/(N_{{\uparrow }_z}+N_{{\downarrow }_z})$ (blue dots) for a measured scattering length $a_{\mathrm{meas}}=4.24a_0$. The red solid curve is the fit of Eq. (8), which determines the magnitudes of the coherence coefficients $|B_m|$ (b) and corresponding phases ${\phi }_m$ (c). (d) Fit of the mean-field model of Ref. [4] to the data (blue dots) for a global detuning $\mathrm{\Delta }=0$ with $a=a_{\mathrm{meas}}$ (black dashed curve) and with $a=2.63a_{\mathrm{meas}}$ (red solid curve).

Figure 3

Energy-resolved collective spin projection $S_z(E)$ versus rotation angle $\varphi$ for spins of selected energies (left to right) $E/E_F=0$, 0.15, 0.25, 0.5, 0.7. Here, $\mathcal{F}(\varphi )=\frac{1}{2}[n_{\uparrow }(E)-n_{\downarrow }(E)]/[n_{\uparrow }(E)+n_{\downarrow }(E)]$. The top row shows the data (blue dots) for a measured scattering length $a=4.24a_0$. The red solid curve is the fit of Eq. (8), which determines the magnitudes of the coherence coefficients $|B_m|$ (second row) and corresponding phases ${\phi }_m$ (third row). The bottom row shows the fits (red solid curves) of the mean-field model of Ref. [4] to the data (blue dots), using a scattering length 2.63 times the measured value and global detunings, ordered in energy, of $\mathrm{\Delta }(\mathrm{Hz})=0$, 0.8, 0.65, $-0.8$, and 0.15.