Echo Spectroscopy and Quantum Stability of Trapped Atoms

We investigate the dephasing of ultra cold ${}^{85}\mathrm{R}\mathrm{b}$ atoms trapped in an optical dipole trap and prepared in a coherent superposition of their two hyperfine ground states by interaction with a microwave pulse. We demonstrate that the dephasing, measured as the Ramsey fringe contrast, can be reversed by stimulating a coherence echo with a $\pi$ pulse between the two $\frac{\pi }{2}$ pulses, in analogy to the photon echo. We also demonstrate that “echo spectroscopy” can be used to study the quantum dynamics in the trap even when more than ${10}^6$ states are thermally populated and to study the crossover from quantum to classical dynamics.

Figure 1

Ramsey fringe contrast (○) and echo signal ($\times$) measured as a function of time between $\pi /2$ pulses. For the echo signal the value 0 represents complete coherence, and the value $1/2$ represents complete dephasing. Trap laser wavelength is 800 nm. A coherent echo ($P_2\ll 1/2$) persists long after the fringe contrast has decayed. The statistical uncertainty of each data set is indicated by error bars on the last point.

Figure 2

Echo signal $P_2$ measured for three different trap laser wavelengths $\lambda$. Lower curve: $\lambda =805\mathrm{n}\mathrm{m}$. A good echo signal ($P_2\ll 1/2$) is seen almost independent of time between pulses. Middle curve: $\lambda =798.25\mathrm{n}\mathrm{m}$. The echo signal oscillates around a value smaller than $1/2$ with partial revivals at $\tau =1/2{\tau }_{\mathrm{o}\mathrm{s}\mathrm{c}}$ and $\tau ={\tau }_{\mathrm{o}\mathrm{s}\mathrm{c}}$, where ${\tau }_{\mathrm{o}\mathrm{s}\mathrm{c}}=3.6\mathrm{m}\mathrm{s}$ is the measured trap oscillation frequency in the transverse direction. Upper curve: $\lambda =796.25\mathrm{n}\mathrm{m}$. After a short time the echo signal completely disappears ($P_2=1/2$), but partly revives again at $\tau =3.3\mathrm{m}\mathrm{s}$, close to ${\tau }_{\mathrm{o}\mathrm{s}\mathrm{c}}$. Error bars on the last point indicate the statistical uncertainty.

Figure 3

$\times$: long time level of measured echo signal as a function of difference between potentials ($\delta V/V$). Solid line: calculation of the ensemble average of $|⟨n^{\prime }=n\mid n⟩{|}^4$ (a measure of the quantum stability of the system) for a 2D harmonic potential with oscillation time of 3.6 ms [16].

Figure 4

MW spectroscopy of atoms in a trap at a wavelength of 805 nm, with a 20 ms MW pulse. MW-power equivalent of (a) a free-space $\pi$ pulse and (b) a free-space $4\pi$ pulse.