Proposal for monitoring and heralding position states of atoms in a one-dimensional waveguide

We study single-photon transport in a single-mode waveguide, in which two-level atoms in spatial superpositions are embedded. We find that the transmission of the photon can be used as a nondestructive probe for the coherence of the superposition. Under certain trapping configurations, the protocol proposed is shown to be independent of which wells the atoms are initially trapped in. Furthermore, we show that by looking at the transmission in a suitable regime, a maximally entangled state can be postselected from an unknown superposition.

Figure 1

Trapping and probing schemes in the case of two atoms in adjacent wells. (a) Single-well configuration ($A=0$) with localized atoms (dark colors). (b) Double-well configuration ($A\ne 0$) with atoms in a spatial superposition (light colors). (c) Probing of the atomic spatial state through the photon transmission and reflection spectra.

Figure 2

Absolute difference between the two extreme probability distributions that can be obtained for $k{l}_t=\pi /2$. Note that it is upper bounded by 1/2. Inset: transmission coefficient for the different configurations. The solid, dashed, and dotted lines represent, respectively, the $d^0$, $d^{+}$, and $d^{-}$ configurations. Note that the last two coincide.

Figure 3

Transmission coefficient in a regime where the extraction of a Bell state $|{\Psi }^{+}\rangle$ can be efficiently performed by probing at $\Delta /\gamma =0.25$, where $\left|t\right|=1$ and $\left|t\right|\approx 0$. The two atoms are in adjacent wells ($n=1$) and $2k{l}_t=-arctan\left(0.25\right)+\pi$. The solid, dashed, and dotted lines represent, respectively, the $d^0$, $d^{+}$, and $d^{-}$ configurations.

Figure 4

Singlet fidelity of the postselected state (thick line) and the initial state $|\varphi \rangle$ (dashed line) as a function of the ratio between the input-pulse bandwidth $\Omega$ and the interaction strength $\gamma$. Plotted for $|c_L|=1/\sqrt{2}$. Notice how if $\Omega \ll \gamma$, $\mathcal{F}=1$ can be achieved.

Figure 5

Contour plot of $|c_L{c}_R{|}^2$ derived from (5) assuming the condition $k{l}_t=\pi /2$ is matched when computing the spectra $|t_{d^0}|$ and $|t_{d^{\pm }}|$. The probability of transmission $P_{|\varphi \rangle }\left(\text{trans.}\right)$, coming from an imperfect implementation, includes a correction term $\varepsilon \in \mathbb{R}$ such that $k{l}_t=\pi /2+\varepsilon$. The $y$ axis is the correct value of the superposition-weights product, and the dashed line represents a perfect implementation for which the value derived from (5) is correct. (a) The two atoms are in adjacent wells ($n=1$). (b) $n=2$. (c) $n=5$. For this last case, in the lower left corner, one would get $|c_L{c}_R{|}^2\approx 1/4$ whereas there is no superposition in reality.

Figure 6

Transmission coefficient at $\Delta /\gamma =0.25$ as a function of the normalized bandwidth $\Omega /\gamma$ for the same parameters as in Fig. 3. The solid, dashed, and dotted lines represent, respectively, the $d^0$, $d^{+}$, and $d^{-}$ configurations.