Coherence of a qubit stored in Zeeman levels of a single optically trapped atom

We experimentally investigate the coherence properties of a qubit stored in the Zeeman substates of the $5^2{S}_{1/2},F=1$ hyperfine ground level of a single optically trapped ${}^{87}$Rb atom. Larmor precession of a single atomic spin-1 system is observed by preparing the atom in a defined initial spin state and then measuring the resulting state after a programmable period of free evolution. Additionally, by performing quantum-state tomography, maximum knowledge about the spin coherence is gathered. By using an active magnetic field stabilization and without application of a magnetic guiding field, we achieve transverse and longitudinal dephasing times of $T_2^{*}=75-150\mu \mathrm{s}$ and $T_1>0.5\mathrm{ms},$ respectively. We derive the light-shift distribution of a single atom in the approximately harmonic potential of a dipole trap and show that the measured atomic spin coherence is limited mainly by residual position- and state-dependent effects in the optical trapping potential. The improved understanding enables longer coherence times, an important prerequisite for future applications in long-distance quantum communication and computation with atoms in optical lattices, or for a loophole-free test of Bell’s inequality.

Figure 1

Probability density $p\left(\Delta B_{\sigma }\right)$ of the optically induced magnetic field in a 3D harmonic trap resulting from thermal motion of the atom. The distribution is plotted as a function of the deviation $\Delta B_{\sigma }=B_{\sigma }^0-B_{\sigma }$ of the field $B_{\sigma }$ from its maximal value $B_{\sigma }^0$ at the bottom of the trap. The curve was calculated according to Eq. (16), assuming a trap depth $U_0=k_B\times 650\mu \mathrm{K}$, average atomic temperature $T=150\mu \mathrm{K},$ and $1\%$ fraction of circularly polarized trap light.

Figure 2

Schematic of the experimental setup (not to scale). The dipole trap beam is focused into a vacuum cell by means of a high NA objective. The same objective collects the light emitted by the atom into a single-mode optical fiber. The magnetic field sensor positioned on the cell surface is used to give a feedback signal to compensation coils for field stabilization. For simplicity, only one pair of compensation coils is shown.

Figure 3

Temporal evolution of different atomic spin states. (a) Evolution of the superposition states $\frac{1}{\sqrt{2}}\left(\right|1,-1\rangle \pm |1,+1\rangle )$ in an effective magnetic field of $5.5\mathrm{mG}$ along the quantization axis. (b) Evolution of states $|1,\pm 1\rangle$ in a field compensated to $B\le 2\mathrm{mG}$. Measured were the populations of the spin states $\frac{1}{\sqrt{2}}\left(\right|1,-1\rangle -|1,+1\rangle )$ in (a) and $|1,+1\rangle$ in (b), respectively. The solid lines represent numerical fits of the measured data to the dephasing model in Eq. (7), which mainly incorporates fluctuations of the effective magnetic field due to the vector light shift.

Figure 4

Partial tomographic reconstruction of the quantum state evolution. Shown are the density matrices (real part) for the prepared states $\frac{1}{\sqrt{2}}\left(\right|1,-1\rangle \pm |1,+1\rangle )$ in (a) and (b), and $|1,\mp 1\rangle$ in (c) and (d).

Figure 5

Purity parameter resulting from the partial state tomography, giving a lower bound for the state purity [6].