Bose-Einstein condensate as a quantum memory for a photonic polarization qubit

A scheme based on electromagnetically induced transparency is used to store light in a Bose-Einstein condensate. In this process, a photonic polarization qubit is stored in atomic Zeeman states. The performance of the storage process is characterized and optimized. The average process fidelity is $1.000\pm 0.004$. For long storage times, temporal fluctuations of the magnetic field reduce this value, yielding a lifetime of the fidelity of $(1.1\pm 0.2)$ ms. The write-read efficiency of the pulse energy can reach $0.53\pm 0.05$.

Figure 1

Atomic level scheme of the $D_1$ line in ${}^{87}$Rb. Probe light (dashed arrows) with an arbitrary superposition of the polarizations ${\sigma }^{+}$ and ${\sigma }^{-}$ couples the initial population ($•$) in the hyperfine ground state $|F,m_F\rangle =|1,0\rangle$ to the $D_1$-line excited states $|F^{\prime },m_F^{\prime }\rangle =|1,\pm 1\rangle$. $\pi$-polarized control light (solid arrows) couples these states to the hyperfine ground states $|F,m_F\rangle =|2,\pm 1\rangle$. This makes it possible to store the probe-light polarization qubit in the qubit space spanned by the atomic states $|F,m_F\rangle =|2,\pm 1\rangle$.

Figure 2

Simplified scheme of the optical beam path. A detailed description is given in the text.

Figure 3

Faraday rotation. The normalized Stokes parameter $S_1/S_0$ oscillates as a function of storage time due to an applied magnetic hold field. The line is a fit of Eq. (5). The best-fit value for the $e^{-1/2}$ damping time is ${\sigma }_{\alpha }=(1.1\pm 0.2)$ ms. Note the breaks on the horizontal axis.

Figure 4

The damping factor $\alpha$ of Eq. (7) as a function of storage time. The lowest damping is obtained when synchronizing the start of the EIT write-read cycle with the 50 Hz ac line voltage ($\blacksquare$). Alternatively, we can use an open-loop feed-forward compensation ($•$). Without any compensation, the damping is much stronger ($▴$). The lines show Gaussian fits according to Eq. (8).

Figure 5

Write-read efficiency $\eta$ vs Rabi frequency of the control light ${\Omega }_c$. All data were taken at the two-photon resonance. Data taken on the single-photon resonance ${\Delta }_c=0$ ($\circ$) display a clear maximum as a function of ${\Omega }_c$. Data taken at a single-photon detuning of ${\Delta }_c=2\pi \times 70$ MHz ($•$) display a lower maximum efficiency. The lines are fits of the simple model from the Appendix to the data.

Figure 6

Decay of the write-read efficiency $\eta$ as a function of storage time in a pure BEC. The line shows a Gaussian fit according to Eq. (13), yielding a best-fit value ${\sigma }_{\eta }=0.48$ ms for the $e^{-1/2}$ time of $\eta$. The dominant mechanism that sets this time scale is given by spatial filtering caused by the single-mode fiber combined with photon recoil incurred during the storage process.

Figure 7

Decay of the normalized write-read efficiency $\eta$ in the presence of a noticeable uncondensed fraction of the gas. Thermal motion causes a rapid initial decay of $\eta$. The decay settles to the long-lived level of $\eta$ that is caused by the BEC fraction. The data sets were taken for different BEC fractions and, correspondingly, for different temperatures.

Figure 8

Simple theoretical estimate for the write-read efficiency $\eta$ vs Rabi frequency of the control light ${\Omega }_c$. The dashed, dotted, and solid lines show ${\eta }_{\mathrm{comp}}$, ${\eta }_{\mathrm{trans}}$, and ${\eta }_{\mathrm{comp}}{\eta }_{\mathrm{trans}}$, respectively. Parameters are ${\tau }_p=94$ ns, $t_0=230$ ns, $1/\Gamma =26$ ns, and $d_p\left(L\right)=127$. The pronounced maximum in the solid line arises because for small ${\Omega }_c$, the EIT transmission window in frequency space is too narrow for the incoming probe pulse, whereas for large ${\Omega }_c$, the group velocity is not sufficiently reduced to spatially compress the complete pulse into the BEC. The dash-dotted line shows the result of averaging over the transverse profile of the BEC.

Figure 9

Linear susceptibility $\chi$ as a function of the two-photon detuning ${\delta }_2$. The solid and dotted lines show the predictions of Eq. (A1) for the real and imaginary parts of $\chi$ at ${\Omega }_c=3.3\Gamma =2\pi \times 20$ MHz. (a) At the single-photon resonance, ${\Delta }_c=0$. (b) At a single-photon detuning of ${\Delta }_c=11.4\Gamma =2\pi \times 70$ MHz. Dash-dotted lines (red) show the lowest-order approximations of Eq. (A4). The range where these approximations become poor sets an upper bound for the frequency range which is useful for light storage. This range is much narrower in (b) than in (a). This explains the reduction of the efficiency observed at ${\Delta }_c=2\pi \times 70$ MHz. Note that the scales on the horizontal axes differ by one order of magnitude.