Long-lived periodic revivals of coherence in an interacting Bose-Einstein condensate

We observe the coherence of an interacting two-component Bose-Einstein condensate (BEC) surviving for seconds in a trapped Ramsey interferometer. Mean-field-driven collective oscillations of two components lead to periodic dephasing and rephasing of condensate wave functions with a slow decay of the interference fringe visibility. We apply spin echo synchronous with the self-rephasing of the condensate to reduce the influence of state-dependent atom losses, significantly enhancing the visibility up to $0.75$ at the evolution time of $1.5$ s. Mean-field theory consistently predicts higher visibility than experimentally observed values. We quantify the effects of classical and quantum noise and infer a coherence time of $2.8$ s for a trapped condensate of $5.5\times {10}^4$ interacting atoms.

Figure 1

(Color) Simulations of collective oscillations of a two-component Bose-Einstein condensate: (a) axial density profiles of the components at different evolution times (solid lines) and the two-component ground state (dashed lines); (b) normalized local spin projection after Ramsey interferometry ($N=5.5\times {10}^4$, chemical potential 582 Hz, $\Delta /\left(2\pi \right)=-37$ Hz, no losses). Lines of constant $p_z$ represent wavefronts of relative phase. The mean-field dynamics acts to periodically separate the components and curve the phase wavefronts modulating the contrast of interference fringes.

Figure 2

Ramsey interference fringes in the atom number [(a) and (c)] and phase [(b) and (d)] domains for evolution times of 20 ms [(a) and (b)] and 450 ms [(b) and (d)]. Run-to-run variations in the atom number $N$ cause correlated changes in data points for $P_z$. This allows us to correct the phase of the data for the variations of $N$ in phase domain (b) and (d). Solid lines represent expected fringes according to Eq. (2).

Figure 3

Schematic of interferometric sequence for (a) Ramsey and (c) spin echo. Corresponding plots of the visibility $\mathcal{V}$ [(b), (d)] where data (filled circles) are measured in the phase domain as a function of $T$. Curves correspond to simulations using coupled GPE theory (red dotted line), truncated Wigner model with classical noise (blue dashed), and with additional 10 s exponential decay (solid black).

Figure 4

Phase uncertainty growth in Ramsey (a) and spin echo (b) sequences with the two-component BEC (black circles) and noncondensed cold atoms (red triangles). The blue dashed lines show the effect of quantum noise simulated with the truncated Wigner method. The red dash-dotted lines indicate the contribution from MW frequency fluctuations. The black solid lines show the simulated combined effect of quantum and classical noise.