Spin Self-Rephasing and Very Long Coherence Times in a Trapped Atomic Ensemble

We perform Ramsey spectroscopy on the ground state of ultracold ${}^{87}\mathrm{Rb}$ atoms magnetically trapped on a chip in the Knudsen regime. Field inhomogeneities over the sample should limit the $1/e$ contrast decay time to about 3 s, while decay times of $58\pm 12\mathrm{s}$ are actually observed. We explain this surprising result by a spin self-rephasing mechanism induced by the identical spin rotation effect originating from particle indistinguishability. We propose a theory of this synchronization mechanism and obtain good agreement with the experimental observations. The effect is general and may appear in other physical systems.

Figure 1

Two classes of atoms (red and blue) precess at different rates. Their Bloch vectors were initially parallel, but have started to dephase (a). The ISRE then makes both vectors rotate around their sum (grey) (b). When this rotation reaches $\pi$, the fast-precessing spin (red) lags behind the slow one (blue), which tends to rephase them as in a spin echo (c).

Figure 2

First experiment: (a) Ramsey fringes in the time domain for ${\Delta }_0/2\pi \approx 0.08\mathrm{Hz}$. The normalized transition probability is plotted as a function of the Ramsey free evolution time. The initial contrast is 89%, probably limited by an inhomogeneous Rabi frequency. At $T_R=5\mathrm{s}$, the contrast is 82% resulting in a $1/e$ time of $58\pm 12\mathrm{s}$. (b) Total number of detected atoms. Although the total time $T_T$ is kept constant, this number depends on $T_R$ because the loss rates of $|0⟩$ and $|1⟩$ differ. The data is fitted by $N_0+N_1=N_T/2(1+e^{-T_R/\tau })$ with $N_T=24.8(2)\times {10}^3$ atoms and $\tau =8.7(2)\mathrm{s}$.

Figure 3

Second experiment: Ramsey fringe contrast as a function of $T_R$ for increased inhomogeneity ${\Delta }_0/2\pi \approx 1.2\mathrm{Hz}$. From bottom to top, colors correspond to densities $\overline{n}=\{0.2,0.8,1.1,1.9,2.6\}\times {10}^{12}{\mathrm{cm}}^{-3}$ (or $N_{\mathrm{at}}=\{7.9,28,41,70,95\}\times {10}^3$). For each $\overline{n}$, the data is normalized to the initial contrast at $T_R=0$. Solid lines are numerical solutions of the kinetic equation for ${\Delta }_0/2\pi \approx 2\mathrm{Hz}$ taking into account the ISRE (${\omega }_{\mathrm{ex}}/2\pi \approx 4.5\mathrm{Hz}\times \overline{n}$) and lateral collisions (${\gamma }_c\approx 2.1{\mathrm{s}}^{-1}\times \overline{n}$). The red dotted line is a calculation for $\overline{n}=2.6$ (same density as the red data points) with lateral collisions (${\gamma }_c=5{\mathrm{s}}^{-1}$), but without ISRE. The red dashed line is a calculation without lateral collisions but with ISRE (${\omega }_{\mathrm{ex}}/2\pi =12\mathrm{Hz}$).