Figure 1

Far-field diffraction image of Stokes photons from permutation-symmetric dilute ensembles. The pair-correlation sum $P$ in the many-atom state of interest manifests as a sharp diffraction peak (for $P>0$) or dip (for $P<0$) along the forward direction, with strength proportional to $\left|P\right|$ and width inversely proportional to the ensemble size.Reuse & Permissions

Figure 2

(a) Phase diagram in the parameter space $(\langle {\widehat{N}}_s\rangle ,\Delta N_s,P)$. States in the surrounded region are all entangled ones. (b) A slice of (a) taken for $\langle {\widehat{N}}_s\rangle =N/2$. The red, blue, and green regions are entangled states violating inequalities (4a), (4b), and (4c), respectively. The gray surfaces in (a) and the black curves in (b) are boundaries between physical and unphysical regions. Positive and negative sections of the $P$ axis use different linear scales. (c) Strength of the diffraction peak (red) or dip (blue) for eigenstates of total spin ${\widehat{J}}^2$ and ${\widehat{J}}_z$. Inequalities (4a) and (4b) are violated in the peak and dip regions, respectively. States violating inequality (4c) form a subset of the dip region, to the left of the dashed curve. (d) Upper (lower): Peak- (dip-) to-background ratio as a function of the collection interval ${\tau }_c$ for a half-spin-excitation state with $P=2.5N$ ($P=-0.35N$), shown as the black curve. The calculation is for $N=4000$ atoms of a two-dimensional (2D) Gaussian distribution with FWHM $A=100\mu$m. Peak or dip (background) strength is evaluated at $\theta =0$ ($\theta =2\pi /k_{0}A$), shown by the red (blue) solid curve. Dashed curves are calculations with the multiple light scattering and dipole-dipole interaction neglected.Reuse & Permissions

Figure 3

Diffraction images (bottom panels) from atomic ensembles in different spin configurations (top panels). (a) Spin-coherent state with in-plane polarization. (b) Evolution in a Zeeman field gradient; a phase gradient of spins is imprinted, resulting in a displacement of the diffraction peak, which can form the principle of a gradiometer. (c) Atomic motions diminish the spin polarization, resulting in decay of the displaced peak. This can be the principle for nondemolition measurement of the atomic temperature and collisional interactions.Reuse & Permissions

Figure 4

Zeeman field gradiometer using a 1D atomic ensemble. (a),(b) Schematic of the setup. The ensemble and a 1D CCD array are placed respectively on the two common focus lines of a group of elliptical cylinder mirrors. The mirrors ensure that the majority of photons is collected by the detector. The smallest ensemble can be an atom pair, giving a direct analog of double-slit interferometry. (c) Signal of $N/2$ probes using atom pairs (upper panel) and single probe using $N$ atoms collectively (lower panel), with each atom placed randomly on the focus line according to a Gaussian distribution with FWHM $A$. The peak strength in the lower panel is enhanced by a factor of $N$. (d) Gradiometer sensitivity at a spatial resolution $A=1$ mm with a resource of $N$ unentangled atoms. The diffraction-based gradiometer using $N$ atoms collectively ($N/2$ atom pairs independently) has a sensitivity of $1/N$ ($1/\sqrt{N}$) scaling, shown by the solid (dashed) black line. The sensitivity of flying-atom Mach-Zehnder interferometry (MZI) gradiometer is shown for Ref. [38]. The probe time ${\tau }_0=A/\Delta v=0.1$ s for the blue line, limited by a finite velocity uncertainty $\Delta v=1$ cm/s, while ${\tau }_0=1$ s for all other lines, limited only by the single-spin dephasing time.Reuse & Permissions

Figure 5

Diffraction pattern of Stokes photons emitted by a chain of ten atoms initially in the spin-coherent state with in-plane polarization ${\otimes }_{j=1}^{10}\frac{{|\uparrow \rangle }_j+{|\downarrow \rangle }_j}{\sqrt{2}}$. With the collection interval $T=5/\Gamma$, $99.3\%$ of all spin excitations are converted into Stokes photons, and with $T=0.1/\Gamma$, $10\%$ of all spin excitations are converted into Stokes photons. Red dashed lines: exact numerical solution of the master equation Eq. (7). Blue solid lines in (a)–(c): perturbative solution keeping the first-order effect of ${\mathcal{L}}_1$ [i.e., Eq. (A9)]. Blue solid lines in (d)–(f): the zeroth-order solution without ${\mathcal{L}}_1$ [i.e., Eq. (A1)].Reuse & Permissions

Figure 6

Diffraction pattern of Stokes photons emitted by a chain of 12 atoms initially in a many-body singlet state (i.e., $J=0$, $M=0$). Red dashed lines: exact numerical solution of the master equation (7). Blue solid lines in (a)–(c): perturbative solution keeping the first-order effect of ${\mathcal{L}}_1$ [i.e., Eq. (A9)]. Blue solid lines in (d)–(f): the zeroth-order solution without ${\mathcal{L}}_1$ [i.e., Eq. (A1)].Reuse & Permissions

Figure 7

Diffraction pattern of Stokes photons emitted by a chain of 12 atoms initially in the many-body singlet state (i.e., $J=0$, $M=0$) with various lengths of the collection time $T$. Red dashed lines: exact numerical solution of the master equation (7). Blue solid lines: perturbative solution keeping the first-order effect of ${\mathcal{L}}_1$ [i.e., Eq. (A9)].Reuse & Permissions

Figure 8

The results including not only the nearest-neighbor ${\Gamma }_1$ term, but also (a) the next-nearest-neighbor ${\Gamma }_2$ and (b) the ${\Gamma }_2$ and ${\Gamma }_3$ terms for a 12-qubit $J=0$, $M=0$ permutation-invariant state. Values of $d$ and $T$ are given in the figure. Red dashed lines: the numerical simulated results. Blue solid lines: the fitting using Eq. (A9). The dipole-dipole interaction $G_j$ is not considered here.Reuse & Permissions

Figure 9

Gradiometer based on flying-atom Mach-Zehnder interferometry.Reuse & Permissions