Energy-resolved spin correlation measurements: Decoding transverse spin dynamics in weakly interacting Fermi gases

We study transverse spin dynamics on a microscopic level by measuring energy-resolved spin correlations in weakly interacting Fermi gases (WIFGs). The trapped cloud behaves as a many-body spin lattice in energy space with effective long-range interactions, simulating a collective Heisenberg model. We observe the flow of correlations in energy space in this quasi-continuous system, revealing the connection between the evolution of the magnetization and the localization or spread of correlations. This work highlights energy-space correlation as an observable in quantum phase transition studies of WIFGs, decoding system features that are hidden in macroscopic measurements.

Figure 1

Correlation function c${}_{ij}^{\perp }$, ensemble-averaged over 30 shots with a selected $\phi$ distribution, at different evolution times with interaction strength $\zeta =1.2$ (a)–(c) and $\zeta =1.8$ (d)–(f). $E_i$ and $E_j$ are in units of effective Fermi energy $E_F$. Only the lowest $70\%$ of energy bins are adopted in data analysis as higher energy groups contain very few particles. The c${}_{ij}^{\perp }$ values shown here and in Figs. 2–2 and 2–2 are amplified by dividing by an energy-dependent attenuation coefficient $\mathrm{\Gamma }\left(E_i\right)$ arising from the finite energy resolution ($\lesssim 0.08\sqrt{E_i}$) to restore the amplitudes to their correct values [20].

Figure 2

Time-dependent transverse magnetization ($\frac{1}{2}{\mathcal{M}}_{\perp }^2$) and correlation gradient (${\mathcal{D}}_m$) at different interaction strengths $\zeta$, along with corresponding c${}_{ij}^{\perp }$ correlation plots. Solid circles in (a) and (f) are $\frac{1}{2}{\mathcal{M}}_{\perp }^2$ obtained by ensemble averaging over multiple shots with the desired $\phi$ distribution. Darker blue or green corresponds to the cases with stronger interaction. A detailed description of data selection and error bar calculation is in Supplemental Material [20]. Dashed lines are predictions from the quasi-classical spin model [12, 19]. Hollow circles in (b) and (g) are correlation gradient ${\mathcal{D}}_m$ extracted from normalized correlation $c$${}_{ij}^{\perp }$ data at corresponding interaction strength and evolution time. Note that the vertical scale in (g) is expanded to show the details. Correlation plots (c)–(e) and (h)–(j) show c${}_{ij}^{\perp }$ at $\tau =80$ ms (left of each pair) and 200 ms (right of each pair). (c) $\zeta =0(a=0a_0)$, (d) $\zeta =0.6(a=2.62a_0)$, (e) $\zeta =1.2(a=5.19a_0)$, (h) $\zeta =1.8(a=8.05a_0)$, (i) $\zeta =2.3(a=10.54a_0)$, (j) $\zeta =5.3(a=23.86a_0)$.

Figure 3

Observing the emergence of spin locking by measuring $\frac{1}{2}{\mathcal{M}}_{\perp }^2$ for various interaction strengths $\zeta$ (top axis) and corresponding scattering lengths $a$ (bottom axis) at (a) 80 ms, (b) 120 ms, (c) 160 ms, and (d) 200 ms. Blue circles are averaged data over multiple shots with the same averaging and error bar calculation for Fig. 2. Bright red curves are predictions with the quasi-classical spin evolution model, and the pink bands correspond to a $2\%$ standard deviation in cloud size ${\sigma }_{Fx}$.