Generation of Macroscopic Singlet States in a Cold Atomic Ensemble

We report the generation of a macroscopic singlet state in a cold atomic sample via quantum nondemolition measurement-induced spin squeezing. We observe 3 dB of spin squeezing and detect entanglement with $5\sigma$ statistical significance using a generalized spin-squeezing inequality. The degree of squeezing implies at least 50% of the atoms have formed singlets.

Figure 1

(a) Experimental geometry. Near-resonant probe pulses pass through a cold cloud of ${}^{87}\mathrm{Rb}$ atoms and experience a Faraday rotation by an angle proportional to the on-axis collective spin ${\widehat{F}}_z$. The pulses are initially polarized with a maximal Stokes operator ${\widehat{S}}_x$ recorded on a reference detector (${\mathrm{PD}}_3$). Rotation toward ${\widehat{S}}_y$ is detected by a balanced polarimeter consisting of a wave plate (WP), polarizing beam splitter (PBS), and photodiodes (${\mathrm{PD}}_{1,2}$). (b) Pulse sequence: A first QND measurement measures the ${\widehat{F}}_z$ angular momentum component of the input atomic state, and the second and third QND measurements in $1/3$ and $2/3$ of Larmor precession cycles measure ${\widehat{F}}_y$ and ${\widehat{F}}_x$, respectively. (c) $\widehat{\mathbf{F}}$ precesses about a magnetic field ($\mathbf{B}$) along the direction [111], making all components accessible to measurement via stroboscopic probing.

Figure 2

Selection of a macroscopic spin singlet. From the initial spin distribution [blue data in (a)], we select data with $|{\widehat{\mathbf{F}}}^{(1)}-⟨{\widehat{\mathbf{F}}}^{(1)}⟩{|}^2/N_A<C$ [orange data in (a)], where $C$ is a chosen cutoff parameter. We then analyze the second QND measurement ${\widehat{\mathbf{F}}}^{(2)}$ of the same data [orange data in (b)] to detect spin squeezing and entanglement. We illustrate this with data from a sample with $N_A=1.1\times {10}^6$ atoms and $C=1$. Axes in (a) and (b) have units of ${10}^3$ spins. In (a), the solid blue circle has a radius $\sqrt{C{N}_A}$. In (b), the solid blue circles have a radius equal to the $1\sigma$ Gaussian rms of an input ideal TSS with ${\xi }^2=2$, including detection noise, and the dashed blue circles the same for a state with ${\xi }^2=1$. The solid orange circles in (b) indicates the measured $1\sigma$ Gaussian rms of the selected data. In the insets, we plot a histogram of the first and second measurements. The selected data are plotted in orange, and the dashed blue line in (b) indicates the cutoff. (c) Spin-squeezing parameter ${\xi }^2$ (orange diamonds) calculated from the second QND measurement of the selected data as a function of the cutoff parameter $C$. The shaded gray region indicates ${\xi }^2<1$, i.e., spin squeezing according to the criterion given in Eq. (1). For reference, the same parameter calculated from the first QND measurement is also plotted (black circles). Inset: Number of selected data points included as a function of the cutoff parameter.

Figure 3

Individual spin measurements. Histograms of the measurements of each of the three spin components are shown in the first round (blue circles) and second round (green squares) of the stroboscopic probe. We also show the conditional spin distribution $F_k^{(2)}-\chi F_k^{(1)}$ (orange diamonds), where $\chi \equiv \mathrm{cov}(F_k^{(1)},F_k^{(2)})/(\mathrm{\Delta }{F}_k^{(1)}{)}^2$ is the degree of correlation. The shaded gray regions indicate the expected distribution for an ideal input TSS, including detection noise. For presentation purposes, an offset (between 5 and $10\times {10}^3$ spins) has been subtracted from the data (see the Supplemental Material [35]).

Figure 4

Noise scaling of total variance $\tilde{\mathcal{V}}(N_A)$ of the first (blue circles) and second (green squares) QND measurements of the atomic spin distribution and of conditional variance ${\tilde{\mathcal{V}}}_{2|1}$ (orange diamonds). Dashed lines are a quadratic fit, indicating the presence of technical noise in the input atomic state (upper shaded blue region). The solid black line indicates the expected quantum noise $\tilde{\mathcal{V}}=2f{N}_A$ due to an ideal TSS. The lower shaded gray region represents a region with ${\tilde{\mathcal{V}}}_{2|1}<f{N}_A$, indicating spin squeezing and entanglement. The dot-dashed orange curve is a fit to the expected conditional noise reduction with the SNR of the QND measurement as a free parameter. Inset: Semilog plot of the detected spin-squeezing parameter. The dot-dashed curve is the expected spin squeezing calculated from the fitted SNR. Horizontal and vertical error bars represent $1\sigma$ statistical errors, and readout noise has been subtracted from the data (see the Supplemental Material [35]).