Heralded Generation of an Atomic NOON State

We report the heralded generation of an atomic NOON state by observation of phase super resolution in a motion-sensitive spin-wave (SW) interferometer. The SW interferometer is implemented by generating a superposition of two SWs and observing the interference between them, where the interference fringe is sensitive to the atomic collective motion. By heralded generation of a second order NOON state in the SW interferometer, we observe the interference pattern which provides strong evidence of phase super resolution. The demonstrated SW interferometer can in principle be scaled up to a highly entangled state, and thus is of fundamental importance, and might be used as an inertial sensor.

Figure 1

Experimental setup of the SW interferometer. The inset at the top shows the relevant Zeeman levels for the $|5S_{1/2}⟩\to |5P_{1/2},F^{\prime }=2⟩$ transition of ${}^{87}\mathrm{Rb}$ atoms. Before the experimental cycles, we optically pump the atoms in $|g⟩$ by applying two pumping lights (see 11), one of which is shining in from the lateral side ($\pi$-pump) to introduce radiation pressure force. A weak ${\sigma }^{-}$ polarized write pulse is applied to generate two modes of SW and Stokes fields via spontaneous Raman transition $|g⟩\to |e⟩\to |s⟩$. The Stokes fields are collected at the angle of $\pm 0.6°$ relative to the write beam, combined on a PBS, and directed into a single mode fiber via a fiber coupler ($C_S$). After a controllable delay, a strong ${\sigma }^{+}$ polarized read beam induces the transition $|s⟩\to |e^{\prime }⟩\to |g⟩$, converting the two SWs into two anti-Stokes fields, which are overlapped in another PBS and then directed to a coupler ($C_{\mathrm{AS}}$). Passing through two filter cells (FC), respectively, the anti-Stokes photon from $C_{\mathrm{AS}}^{\prime }$ and Stokes photon from $C_S^{\prime }$ are then sent to the polarization analyzers combined with half wave plate (HWP), polarized-beam splitter (PBS), and single photon detectors ($D$), as illustrated in the inset at bottom. Filter cells are properly pumped in order to absorb the remaining leakage from read or write beams while being transparent for the signals.

Figure 2

The fidelity of anti-Stokes field on $|+⟩$ as a function of $\delta t$. Solid circles (open circles) represents the measured fidelity of anti-Stokes field on $|+⟩$ on condition of a click of Stokes field on state $|+⟩$ ($|-⟩$). The power of the $\pi$-Pump is: (a) 6 mW, (b) 4.5 mW, (c) 3 mW, (d) 1.5 mW, (e) 0.75 mW, (f) 0 mW. The experimental data are jointly fitted by using $f_{+|+}(\delta t)=[0.5+a{sin}^2(\pi \frac{\delta t}{T}+{\varphi }_0)e^{-\delta t^2/{\tau }^2}]/(1+a{e}^{-\delta t^2/{\tau }^2})$ and $f_{+|-}(\delta t)=[0.5+a{cos}^2(\pi \frac{\delta t}{T}+{\varphi }_0)e^{-\delta t^2/{\tau }^2}]/(1+a{e}^{-\delta t^2/{\tau }^2})$ (see 11 for detail). The evolution period $T=\frac{\pi }{\Delta \mathbf{k}·{\mathbf{v}}_{\mathbf{c}}}$ is measured to be (a) $317\pm 18\mu \mathrm{s}$, (b) $330\pm 15\mu \mathrm{s}$, (c) $378\pm 14\mu \mathrm{s}$, (d) $555\pm 47\mu \mathrm{s}$, (e) $591\pm 30\mu \mathrm{s}$, (f) $1177\pm 152\mu \mathrm{s}$. Error bars represent statistical errors, which are $\pm 1$ s.d.

Figure 3

The atomic velocity as a function of the $\pi$-pump power. The initial velocity induced by the unbalance of other lasers is about $0.03\mathrm{m}/\mathrm{s}$. The velocity acquired in the pumping stage increases with the pumping power until reaching a plateau of about $0.12\mathrm{m}/\mathrm{s}$. Error bars represent statistical errors, which are $\pm 1$ s.d.

Figure 4

Comparison of the performance of the first and second order NOON state. (a) The normalized coincidence count for the second order NOON state as a function of $\delta t$. The experimental data are fitted by using $c(\delta t)=b{sin}^2(\pi \frac{\delta t}{T^{\prime }}+{\varphi }_0^{\prime })e^{-2\delta t^2/{\tau }^2}+d{e}^{-\delta t^2/{\tau }^2}$. The optimization fitting gives an evolution period of $T^{\prime }=(73\pm 7)\mu \mathrm{s}$ (See 11 for details). (b) The interference pattern for $N=1$ under the same condition. The lines are the fitted with the same method as in Fig. 2. The fitting evolution period is $T=(220\pm 17)\mu \mathrm{s}$. Error bars represent statistical errors, which are $\pm 1$ s.d.