Analysis of dephasing mechanisms in a standing-wave dipole trap

We study in detail the mechanisms causing dephasing of hyperfine coherences of cesium atoms confined by a far-off-resonant standing-wave optical dipole trap [S. Kuhr et al., Phys. Rev. Lett. 91, 213002 (2003)]. Using Ramsey-spectroscopy and spin-echo techniques, we measure the reversible and irreversible dephasing times of the ground-state coherences. We present an analytical model to interpret the experimental data and identify the homogeneous and inhomogeneous dephasing mechanisms. Our scheme to prepare and detect the atomic hyperfine state is applied at the level of a single atom as well as for ensembles of up to 50 atoms.

Figure 1

(Color online) Experimental setup.

Figure 2

(Color online) State-selective detection of a single atom. The graphs show the fluorescence signal of the atom during state preparation and detection, binned in time intervals of $5\mathrm{ms}$. The bars above the graphs show the timing of the lasers. Graphs (a)(i) and (b)(i) show the signals of a single atom, prepared in $F=3$ and $F=4$, respectively. Graphs (a)(ii) and (b)(ii) show the added signal of about 150 events.

Figure 3

(Color online) Atom counting. Initial and final numbers of atoms are inferred from their fluorescence in the MOT. Shown are the integrated APD counts binned in time intervals of $1\mathrm{ms}$ and accumulated over 10 repetitions with 20 atoms each.

Figure 4

(Color online) Rabi oscillations on the $\mid F=4,m_F=0⟩\to \mid F=3,m_F=0⟩$ clock transition recorded at a trap depth $U_0=1.0\mathrm{mK}$. Each data point results from 100 shots with about 60 initial atoms. The line is a fit according to Eq. (6).

Figure 5

(Color online) Ramsey fringes recorded for two different trap depths (a) $U_0=0.1\mathrm{mK}$ and (b) $0.04\mathrm{mK}$. Their decay with time constants $T_2^{*}=4.4\pm 0.1\mathrm{ms}$ and $20.4\pm 1.1\mathrm{ms}$, respectively, is governed by inhomogeneous dephasing caused by the energy distribution in the trap. Each data point results from 30 shots with about 50 initial atoms. The damped oscillation is a fit with $P_{3,\text{Ramsey}}\left(t\right)$ and the envelopes are the functions $B\pm A\alpha (t,T_2^{*})$ [see Eqs. (26, 29)].

Figure 6

Spin echoes. Shown are spin echoes recorded for three different trap depths: (a) $U_0=1.0\mathrm{mK}$, (b) $0.1\mathrm{mK}$, and (c) $0.04\mathrm{mK}$. We observe a decrease of the maximum spin-echo amplitude with increasing time of the $\pi$ pulse, with longer decay times in lower trap depths. All spin echoes are fitted using Eq. (37). In (a) and (b), the first curve is a Ramsey signal, recorded with otherwise identical parameters.

Figure 7

(Color online) Decay of the spin echo visibility, extracted from the signals of Fig. 6. The fits (red lines) are the Gaussians of Eq. (45). The dashed and dotted lines are the best and worst case predictions inferred from the measured pointing instability of the trapping laser shown in Fig. 9.

Figure 8

Allan deviation of the intensity fluctuations according to Eq. (47).

Figure 9

(Color online) Measuring the pointing instability. (a) The dipole trap beams having a frequency difference of $\Delta \nu =10\mathrm{MHz}$ are overlapped on a fast photodiode. (b) Allan deviation of the amplitude of the resulting beat signal.