Coherent mode coupling and spin-changing collisions<\/p>\n
We employed a low-noise 6.8-GHz microwave source62<\/a> to drive Rabi oscillations between the F = 1 and F = 2 manifolds. The transition frequencies were defined by an actively stabilized homogeneous magnetic field of 0.955\u2009G. The microwave system was developed and tested for spin-squeezed samples of more than 104 atoms in previous studies, where it contributed no relevant technical noise. Rabi pulse lengths are typically of the order of 100\u2009\u03bcs. We implemented a spin-distillation scheme63<\/a> during the evaporative cooling, such that the BEC occupies the \\(\\left(1,0\\right)\\) level, and the side modes \\(\\left(1,\\pm 1\\right)\\) are initially occupied by only a few atoms at most. Before the spin-changing collisions, we removed these atoms by transferring them to \\(\\left(2,\\pm 1\\right)\\) and exposing the ensemble to resonant cooling light. We repeated this cleaning sequence three times.<\/p>\n We aimed to prepare BECs with a small number of atoms of around 250 to avoid saturating the CCD camera during illumination and to reduce the effect of certain noise contributions, as discussed later. To maintain a high spin-dynamics rate \u03a9 = 2\u03c0 \u00d7 2.2\u2009Hz, we increased the trap frequencies by raising the beam powers of the optical dipole trap from 23\u2009mW and ~320\u2009\u03bcW to 200\u2009mW and 2\u2009mW after BEC creation. The spin-dynamics rate scales as \\({\\it\\varOmega} \\propto {\\bar{\\omega }}^{6\/5}{N}_{\\mathrm{BEC}}^{2\/5}\\) with the geometric mean \\(\\overline{\\omega }\\) of the trap frequencies and the numbers of atoms in the BEC NBEC. We applied a microwave dressing field on the clock transition to shift the Zeeman energy to resonance at q = \u210f\u03a9 (ref. 43<\/a>), where \u210f is the reduced Planck constant. After 120\u2009ms, we found a mean number of 7.5 atoms in the levels \\(\\left(1,\\pm 1\\right)\\). The distribution of the occupation numbers did not follow an exponential decay as predicted by equation (1<\/a>). We ascribe this discrepancy to a non-constant spin-dynamics rate due to fluctuations in the numbers of atoms and trap frequencies.<\/p>\n To couple the levels \\(\\left(1,\\pm 1\\right)\\), we applied a sequence of three microwave Rabi pulses (Extended Data Fig. 1<\/a>). Between two \u03c0-pulses, which transferred the atoms from \\(\\left(1,-1\\right)\\) to \\(\\left(2,0\\right)\\) and back, we used a pulse of variable length on \\(\\left(2,0\\right)\\leftrightarrow \\left(1,+1\\right)\\) to control the coupling ratio of the twin-Fock modes \\(\\left(1,\\pm 1\\right)\\). Effective pulse lengths of 7.66\u2009\u03bcs, 10.8\u2009\u03bcs, 15.4\u2009\u03bcs, 18.8\u2009\u03bcs and 85.0\u2009\u03bcs at a Rabi frequency of \\({\\it\\varOmega }_{{\\rm{R}},{\\sigma }^{-}}=2{\\rm{\\pi }}\\times 2.94\\,\\mathrm{kHz}\\) resulted in small rotation angles between 0.14\u2009rad and 0.35\u2009rad and the HOM coupling corresponding to an angle of \u03c0\/2.<\/p>\n As the transition frequencies for \\(\\left(1,0\\right)\\leftrightarrow \\left(2,\\pm 1\\right)\\) differed from those for \\(\\left(1,\\pm 1\\right)\\leftrightarrow \\left(2,0\\right)\\) by only \u0394 = 2\u03c0 \u00d7 2.66\u2009kHz, the pulse sequence also transferred some BEC atoms to F = 2. These atoms were removed by a cooling light exposure during free fall, 2.5\u2009ms before the detection beams were turned on. We observed that removing atoms this way can cause losses of \\(\\left(1,\\pm 1\\right)\\) atoms. We, thus, kept the fraction of BEC atoms transferred to F = 2 small by using two different techniques. For the coupling pulse, we chose a microwave antenna that couples 4.7 times less to \u03c3+ than to \u03c3\u2212 transitions, that is \\({\\it\\varOmega }_{{\\rm{R}},{\\sigma }^{+}}={\\it\\varOmega }_{{\\rm{R}},{\\sigma }^{-}}\/4.7\\), resulting in a maximally transferred fraction of \\({\\varOmega }_{{\\rm{R}},{\\sigma }^{+}}^{2}\/({\\varOmega }_{{\\rm{R}},{\\sigma }^{+}}^{2}+{\\varDelta }^{2})\\approx 2 \\%\\). For the two \u03c0-pulses, we chose the relative phase such that all BEC atoms that were transferred by the first pulse were transferred back to \\(\\left(1,0\\right)\\) by the second pulse.<\/p>\n Fluorescence detection<\/p>\n For detection, the atomic ensemble was released from the crossed-beam optical dipole trap, exposed to a magnetic field gradient pulse during free fall and then illuminated by six laser beams in an optical-molasses configuration (Extended Data Fig. 2a<\/a>). The magnetic field gradient was generated by a single coil aligned with the homogeneous magnetic quantization field used for the coherent state manipulation. Then, 0.5\u2009ms after release, a capacitor was discharged through the coil over a period of 6\u2009ms, resulting in a peak current of 430\u2009A and a gradient of about 40\u2009G\u2009cm\u22121 at the position of the atoms. A spatial separation of 470\u2009\u03bcm was reached for adjacent modes. At the end of the pulse, 6.5\u2009ms into free fall, both magnetic fields were turned off. After another 3.5\u2009ms, during which time the fields settled down, the laser beams of the optical molasses were turned on.<\/p>\n The molasses consisted of six millimetre-sized, circularly polarized beams, detuned by one natural linewidth \u0393 to the \\(F=2\\to {F}^{{\\prime} }=3\\) cooling transition of the 87Rb D2 line 52S1\/2 \u2192 52P3\/2. The optical set-up was the same as that described in ref. 46<\/a>, but with the beam diameters of the four horizontal beams reduced from 3.1\u2009mm to 1.1\u2009mm. This adaptation reduced the amount of stray light present during the illumination process, which was a main contributor to the counting noise. Beam intensities close to the saturation intensity for isotropically polarized light, Isat = 3.6\u2009mW\u2009cm\u22122, resulted in an expected isotropic photon scattering rate of R = 1.04 \u00d7 107 photons per second. Considering the photon collection efficiency of the detection lens system of 3.9% and the properties of the CCD camera employed (Pixis 1024BR eXcelon WaterCool, 1,024 \u00d7 1,024), which are given by a quantum efficiency of 0.98 primary electrons per incident photon and a set amplification gain of 0.92 digital counts per electron, we expected 1,530 counts during the illumination time of 4.2\u2009ms per atom. From the recorded data, we found 978 counts per atom in \\(\\left(1,-1\\right)\\), 1,580 counts per atom in \\(\\left(1,0\\right)\\) and 830 counts per atom in \\(\\left(1,+1\\right)\\). As the molasses beams were aligned onto the position of the \\(\\left(1,0\\right)\\) atoms after 10\u2009ms of free fall, these atoms experience the most intense and best intensity-balanced light field and, thus, emitted the most fluorescence photons. The twin-Fock modes probably experienced a slight intensity-imbalance of counter-propagating beams, as the horizontal displacement of 470\u2009\u03bcm was not negligible compared with the Gaussian beam waists of about 550\u2009\u03bcm.<\/p>\n Image evaluation<\/p>\n The CCD camera array has 1,024 \u00d7 1,024 physical pixels. Incident photons lead to the accumulation of charge during the illumination time. To reduce the level of electronic noise, arrays of 8 \u00d7 8 were combined into a super-pixel (px in the following) before read-out, resulting in an image of size 128\u2009px \u00d7 128\u2009px. For the three modes \\(\\left(1,-1\\right)\\), \\(\\left(1,0\\right)\\) and \\(\\left(1,+1\\right)\\), we added up the brightness values of the pixels in constant areas (\u2018masks\u2019) around the respective bright spot so that we had a single camera-count value for each image and mode, denoted s0 and s\u00b1 for mF = 0 and mF = \u00b1 1, respectively. We found the best results for round areas with radii of 5\u2009px. Each mask had 69 (super-)pixels, with one (super-)pixel covering a physical area of 34.4\u2009\u03bcm \u00d7 34.4\u2009\u03bcm at the focus plane of the detection objective. We took one image for each run of the experimental apparatus. No background image was taken.<\/p>\n A quantization of the signals, that is an accumulation of the obtained count numbers at evenly spaced values, was visible in the raw data. Importantly, the mean signals of the zero-atom peaks were easily evaluated by a fit with a single Gaussian. To obtain the data shown in Fig. 2b<\/a>, we calculated and subtracted the contributions of two systematic effects (Extended Data Fig. 3<\/a>). First, a very small but still relevant fraction of the mF = 0 atoms moved into the regions of the mF = \u00b11 atoms during illumination, which caused correlations of the zero-atom signals \\({s}_{\\pm }^{({N}_{\\pm }=0)}\\) and the \\(\\left(1,0\\right)\\) signal s0, with correlation coefficients 1.48 \u00d7 10\u22123 (mF = \u22121) and 1.76 \u00d7 10\u22123 (mF = +1). Second, we found that the zero-atom signals \\({s}_{\\pm }^{({N}_{\\pm }=0)}\\), identified from 400 adjacent images, drifted slightly over the duration of the measurement, with the maximal and minimal observed signals differing by 370\u2009counts.<\/p>\n Finally, the occurrences of the count values were fitted with a sum of evenly spaced Gaussian functions:<\/p>\n $$G(s)=\\mathop{\\sum }\\limits_{n=0}^{{n}_{\\max }+1}{a}_{n}\\,\\exp \\left(\\frac{{(s-(ng+b))}^{2}}{2{\\sigma }_{n}^{2}}\\right),$$<\/p>\n \n (6)\n <\/p>\n where an are the peak heights, \u03c3n the peak widths and g the signal per atom. The position of the zero-atom peak b was close to zero due to the applied signal drift correction. For quantization, each camera-count value per image and mode was assigned the integer number of atoms n of the closest peak, resulting in the quantization intervals depicted in Fig. 2b<\/a> and Extended Data Fig. 2b<\/a>, respectively. Note that this quantization technique extends to numbers of atoms much larger than \\({n}_{\\max }\\), the number for the last peak that could be fitted.<\/p>\n For the fitting procedure, we weighted the occurrences within each peak with the inverse of the total number of the detection events for the peak. To ensure convergence, the \\({n}_{\\max }+1\\) peak was assigned a fixed width (which we predicted iteratively from the widths of the previous peaks). The fits with equation (6<\/a>) yielded atomic fluorescence signals g of 832.5(34)\u2009counts per atom for the mF = +1 mode and 975.8(16)\u2009counts per atom for the mF = \u22121 mode. The counting noise was captured in the widths \u03c3n. A noise model of the molasses detection predicted electronic camera noise and background light fluctuations (from shot noise and non-constant beam powers) to be the dominant contributors to the zero-atom signal noise, denoted \u03c30. The most relevant contribution that scales with the number of atoms was from atoms leaving the detection volume during the illumination time due to the slowed, but not spatially restricted, atom movement in the optical molasses. This is captured by c1 (ref. 44<\/a>), and we get<\/p>\n $${\\sigma }_{n}^{2}={\\sigma }_{0}^{2}+{c}_{1}^{2}n.$$<\/p>\n \n (7)\n <\/p>\n The fits with equation (7<\/a>) to the obtained Gaussian widths of the peaks are presented in Fig. 2c<\/a> and Extended Data Fig. 2c<\/a>. For the mF = \u22121 mode, we found \u03c30 = 0.1466(9) atoms, incoherently increasing by \\({c}_{1}=0.107(3)\\,{\\rm{atoms}}\/\\sqrt{\\,{\\rm{atoms}}}\\), whereas for mF = +1, we found \u03c30 = 0.168(4)\u2009atoms and \\({c}_{1}=0.164(15)\\,\\,{\\rm{atoms}}\/\\sqrt{{\\rm{atoms}}}\\). From this, we estimated the detection fidelities, that is the chance of correctly counting the number of atoms, as the integral of the normalized Gaussian functions over the respective quantization intervals. For up to N = 12 atoms, we obtained fidelities of >79% for the mF = \u22121 mode and >60% for the mF = +1 mode. The larger coefficient c1 for mF = +1 can be explained by a less optimal beam intensity balance at the position of these atoms after the spatial separation.<\/p>\n Calculating fidelities \\({\\mathcal{F}}\\)<\/p>\n For two probability distributions p(Jz) and q(Jz), the fidelity is given as \\({\\mathcal{F}}={\\left({\\sum }_{{J}_{z}}\\sqrt{p({J}_{z})q({J}_{z})}\\right)}^{2}\\). For the numbers given in Fig. 3<\/a>, we compared the experimentally observed probabilities \\({p}^{\\exp }({J}_{z};N)\\) with the probability distributions q(Jz) of the ideal quantum states, where Jz = \u2212N\/2, \u2212N\/2 + 1, \u2026, N\/2 represents all possible values for a given even number of atoms N. The experimental probabilities \\({p}^{\\exp }({J}_{z};N)\\) were obtained from the number of occurrences of the value Jz during the full measurement. The probability distributions for the ideal states are given by \\(q({J}_{z})={\\delta }_{0,{J}_{z}}\\) for the twin-Fock states, where \\({\\delta }_{0,{J}_{z}}\\) is the Kronecker delta, and by equation (3<\/a>) for the states after HOM interference. Expressed in terms of Jz and N:<\/p>\n $$q(\\,{J}_{z})=\\left\\{\\begin{array}{cc}\\left(\\begin{array}{l}N\/2+{J}_{z}\\\\ (N\/2+{J}_{z})\/2\\end{array}\\right)\\left(\\begin{array}{l}N\/2-{J}_{z}\\\\ (N\/2-{J}_{z})\/2\\end{array}\\right){\\left(\\frac{1}{2}\\right)}^{N}, & {J}_{z}+N\/2\\,\\mathrm{even},\\\\ 0, & {J}_{z}+N\/2\\,\\mathrm{odd}.\\end{array}\\right.\\,$$<\/p>\n Parity<\/p>\n The parity operator for a single mode assigns a value of +1 to even occupation numbers and \u22121 to odd occupation numbers64<\/a>. As all states can be written as a superposition of Fock states \\(\\left|n\\right\\rangle\\), this property fully defines the operator. It can be written as \\({\\it\\varPi }_{\\mathrm{single}\\,\\mathrm{mode}}={(-1)}^{\\widehat{n}}\\), with \\(\\widehat{n}=\\left|n\\right\\rangle \\left\\langle n\\right|\\) the occupation number operator.<\/p>\n For our two-mode system, in principle, two parity operators exist, \\({\\it\\varPi }_{+}={(-1)}^{{\\widehat{N}}_{+}}\\) and \\({\\it\\varPi }_{-}={(-1)}^{{\\widehat{N}}_{-}}\\). However, for an even total number of atoms N = N+ + N\u2212, we note that (\u22121)N \u2261 1, such that<\/p>\n $${\\it\\varPi }_{z\\,}:={(-1)}^{N\/2-{J}_{z}}={(-1)}^{{N}_{-}}={(-1)}^{{N}_{+}}$$<\/p>\n is well defined and describes the occupation number parity for both of the measured modes. Similarly, we can define parity operators for all spin components Jl, with l = x, y and z, as<\/p>\n $${\\it\\varPi }_{l\\,}:={(-1)}^{N\/2-{J}_{l}}.$$<\/p>\n \u03a0x,y describe parity measurements after rotating the state by 90\u00b0 on the generalized Bloch sphere, that is after HOM interference. Their relation to the single-particle operators is \\({\\it\\varPi }_{l}={\\sigma }_{l}^{\\otimes N}\\), with \u03c3l = 2jl twice the l component of the single-particle spin-1\/2 operator, which can be written as a Pauli matrix and has eigenvalues of \u00b11.<\/p>\n Probabilistic noise model of the measurements<\/p>\n We deliberately avoid using assumptions about the underlying noise sources in the analysis of the recorded population data. In the absence of any noise correction technique, the results presented directly describe the performance of our system.<\/p>\n To compare the results with theoretical expectations and for consistency checks, we developed a numerical model that describes the combined effect of various noise contributions on the population probabilities p\u03b8(Jz; N). These probabilities for the possible outcomes (Jz; N) are modelled as an array, \\({p}_{\\theta }^{{\\rm{model}}}({J}_{z};N)\\), like those depicted in the insets of Fig. 2d,e<\/a>. To ensure that truncation effects were negligible, the model considered up to N = 20 atoms per mode, which is a range much larger than that used for Figs. 3<\/a>\u20135<\/a>. We checked that the results were stable with respect to changes in the array size. The calculation of the probabilities follows these steps:<\/p>\n Start with the probabilities of a superposition of twin-Fock states. The distribution of the total numbers of atoms N follows the recorded average of all measurements.<\/p>\n<\/li>\n Change the probabilities within the subspaces of constant N according to a rotation by the angle \u03b8. For example, \u03b8 = 90\u00b0 results in Holland\u2013Burnett states.<\/p>\n<\/li>\n Undesired transfers of atoms in the BEC reservoir into the detected modes mF = \u00b11 can occur after the mode interference when the magnetic fields change quickly before and during the strong magnetic field gradient pulse for spatial separation.<\/p>\n These extra particles are modelled by a convolution of the probability array with Poisson distributions with parameters a\u00b1.<\/p>\n<\/li>\n When those BEC atoms that were transferred to \\(\\left(F=2,{m}_{F}=\\pm 1\\right)\\) during the Rabi coupling sequence (Extended Data Fig. 1<\/a>) are removed by a short resonant light pulse after the magnetic field gradient, losses in \\(\\left(F=1,{m}_{F}=\\pm 1\\right)\\) can occur due to collisions with the accelerated atoms. This effect is clearly visible when we purposely remove very many atoms.<\/p>\n These losses are described by convolutions with binomial distributions with probabilities 1 \u2212 l\u00b1.<\/p>\n<\/li>\n The calibration of the detection, that is the definition of the quantization intervals shown in Fig. 2b<\/a> and Extended Data Fig. 2b<\/a>, will have some error. Looking at measurement outcomes for twin-Fock states with very high numbers of atoms N, we notice that the detection predicts slightly asymmetric numbers N\u2212 \u2248 1.052N+.<\/p>\n This is modelled by applying chances of \\(\\sqrt{1.052}-1\\) per atom to overpredict N\u2212 by 1 and underpredict N+ by 1. Note that we cannot use this observation to refine the assignment of the number of atoms when analysing the experimental data. That calibration method would assume, rather than demonstrate, the generation of twin-Fock states.<\/p>\n<\/li>\n Finally, the finite detection resolution, that is counting noise, is considered by miscounting probabilities according to the overlap of the Gaussian peaks from equation (6<\/a>) with adjacent quantization intervals.<\/p>\n<\/li>\n<\/ol>\n The model has only four free parameters, namely a\u00b1 and l\u00b1.<\/p>\n For each value of the rotation angle \u03b8 (0\u2009rad, 0.14\u2009rad, 0.20\u2009rad, 0.28\u2009rad and 0.35\u2009rad), we fitted the model to the full array of measured experimental probabilities \\({p}_{\\theta }^{\\exp }({J}_{z};N)\\) for 0\u2009\u2264\u2009N\u00b1\u2009\u2264\u200920, normalized such that<\/p>\n $$\\mathop{\\sum }\\limits_{N=0}^{20}\\mathop{\\sum }\\limits_{{J}_{z}=-N\/2}^{N\/2}{p}_{\\theta }^{\\exp }(\\,{J}_{z};N\\,)=1.$$<\/p>\n \n (8)\n <\/p>\n Note that this analysis includes odd values of N, which are the primary effect of the noise contributions. For the fitting, we minimized the Hellinger distance between \\({p}_{\\theta }^{{\\rm{model}}}({J}_{z};N)\\) and \\({p}_{\\theta }^{\\exp }({J}_{z};N)\\) using a differential evolution algorithm.<\/p>\n For each \u03b8, we performed 3,816 experimental repetitions, each resulting in a pair (N+, N\u2212) of detected atoms. We obtained fitting parameters a+ = 0.0551(63), a\u2212 = 0.0218(18), l+ = 0.042(22)% and l\u2212 = 1.1(8)% as the mean values for the five angles \u03b8. The uncertainties are the statistical standard deviation. We attribute the increased chance for losses in mF = \u22121 to the asymmetry of the coupling sequence, as a small fraction of atoms might not be transferred back to \\(\\left(F=1,{m}_{F}=-1\\right)\\) by the second \u03c0-pulse due to magnetic field fluctuations.<\/p>\n The obtained parameters indicate that losses have only a minor impact on our system. The primary noise sources were finite detection resolutions, as depicted in Fig. 2c<\/a> and Extended Data Fig. 2c<\/a>, as well as unintended incoherent transfers from \\(\\left(F=1,{m}_{F}=0\\right)\\) to \\(\\left(F=1,{m}_{F}=\\pm 1\\right)\\), as described by a\u00b1.<\/p>\n For clarity, note that the results of the noise model were not incorporated in the data analysis for Figs. 3<\/a>\u20135<\/a>.<\/p>\n Entanglement witness based on parity<\/p>\n We considered the detection of entanglement in systems with many indistinguishable particles, using the definition of particle entanglement as described, for example, in ref. 6<\/a>. The first important result in this field was the spin-squeezing entanglement condition based on the first and second moments of collective observables5<\/a>. Subsequently, many experiments measured collective quantities40<\/a>,50<\/a>,65<\/a>,66<\/a>,67<\/a>.<\/p>\n In this section, we present entanglement relations that are based on N-particle correlations, rather than the first and second moments of collective observables.<\/p>\n We can use the following witness to detect entanglement. For separable states<\/p>\n $$| \\langle {\\it\\varPi }_{x}\\rangle | +| \\langle {\\it\\varPi }_{y}\\rangle | +| \\langle {\\it\\varPi }_{z}\\rangle | \\le 1$$<\/p>\n \n (9)\n <\/p>\n holds, which can be proved following ideas like those in ref. 68<\/a>. For a product state of the type<\/p>\n $$\\left|{\\varPsi }^{(1)}\\right\\rangle \\otimes \\left|{\\varPsi }^{(2)}\\right\\rangle \\otimes \\ldots \\otimes \\left|{\\varPsi }^{(N)}\\right\\rangle ,$$<\/p>\n \n (10)\n <\/p>\n the left-hand side of equation (9<\/a>) can be bounded from above as<\/p>\n $$\\mathop{\\sum }\\limits_{l=x,y,z}\\left|\\mathop{\\prod }\\limits_{n=1}^{N}\\left\\langle {\\sigma }_{l}^{(n)}\\right\\rangle \\right|\\le \\mathop{\\sum }\\limits_{l=x,y,z}\\left|\\left\\langle {\\sigma }_{l}^{(1)}\\right\\rangle \\left\\langle {\\sigma }_{l}^{(2)}\\right\\rangle \\right|\\le 1,$$<\/p>\n \n (11)\n <\/p>\n where in the first inequality we used that \\(| \\langle {\\sigma }_{l}^{(n)}\\rangle | \\le 1\\). In the second inequality, we used the Cauchy\u2013Schwarz inequality and the fact that the length of the Bloch vector is at most 1 for a qubit. Separable states are mixtures of product states. Hence, the inequality in equation (9<\/a>) is also valid for separable states.<\/p>\n For the ideal Dicke state, for even N, the left-hand side is 3. This condition is based on N-body correlations, unlike previous methods that were based on two-body correlations. Here \u03a0l is the parity operator, which equals \\({\\sigma }_{l}^{\\otimes N}\\) for l = x, y and z.<\/p>\n The witness also detects the Greenberger\u2013Horne\u2013Zeilinger states as entangled. The singlet state, given as \\({[(\\left|01\\right\\rangle -\\left|10\\right\\rangle )\/\\sqrt{2}]}^{\\otimes N\/2}\\), has \\({(\\Delta {J}_{z})}^{2}=0\\) and also \\(\\langle {\\sigma }_{x}^{\\otimes N}\\rangle =1\\) and \\(\\langle {\\sigma }_{y}^{\\otimes N}\\rangle =1\\), if N is divisible by 4. Thus, these operators cannot be used to detect genuine multipartite entanglement.<\/p>\n We summarize the results in Extended Data Table 1<\/a>. We assume \\(\\langle {\\sigma }_{y}^{\\otimes N}\\rangle =\\langle {\\sigma }_{x}^{\\otimes N}\\rangle .\\) The witness detects entanglement in all cases.<\/p>\n Next, we will derive an entanglement condition detecting genuine multipartite entanglement based on the parity operator \u03a0z. As a first step, we will derive a criterion that detects entanglement between two groups of the particles.<\/p>\n Entanglement condition using bipartite correlations<\/p>\n In this section, we present a simple relation using the expectation values of collective observables and the expectation values of N-particle correlations. We use these relations to obtain entanglement conditions based on bipartite correlations that detect entanglement between two groups of particles.<\/p>\n Observation 1<\/p>\n For N-qubit quantum states,<\/p>\n $${\\langle \\,{J}_{x}\\rangle }^{2}\/{j}^{2}+{\\langle \\,{J}_{y}\\rangle }^{2}\/{j}^{2}+{\\langle {\\sigma }_{z}^{\\otimes N}\\rangle }^{2}\\le 1$$<\/p>\n \n (12)\n <\/p>\n holds, where j = N\/2 and<\/p>\n $${J}_{l}=\\frac{1}{2}\\mathop{\\sum }\\limits_{n=1}^{N}{\\sigma }_{l}^{(n)}$$<\/p>\n \n (13)\n <\/p>\n for l = x, y, z.<\/p>\n Proof. The ground state of the Hamiltonian \\(H=B{J}_{x}+K{\\sigma }_{z}^{\\otimes N},\\) where B and K are constants, is of the form \\(\\left|\\Psi \\right\\rangle =\\alpha {\\left|0\\right\\rangle }_{x}^{\\otimes N}+\\beta {\\left|1\\right\\rangle }_{x}^{\\otimes N},\\) which is a generalized Greenberger\u2013Horne\u2013Zeilinger state in the x basis. Then, the relevant expectation value of Jx is \\(\\langle {J}_{x}\\rangle =\\frac{N}{2}{\\langle {\\sigma }_{x}\\rangle }_{\\phi }\\) and the expectation value of the products of \u03c3z matrices is \\(\\langle {\\sigma }_{z}^{\\otimes N}\\rangle ={\\langle {\\sigma }_{z}\\rangle }_{\\phi },\\) where we define the single-qubit state \\(\\left|\\phi \\right\\rangle =\\alpha {\\left|0\\right\\rangle }_{x}+\\beta {\\left|1\\right\\rangle }_{x}.\\) As \\({\\langle {\\sigma }_{x}\\rangle }_{\\phi }^{2}+{\\langle {\\sigma }_{z}\\rangle }_{\\phi }^{2}\\le 1,\\) it follows that \\({\\langle {J}_{x}\\rangle }^{2}\/{j}^{2}+{\\langle {\\sigma }_{z}^{\\otimes N}\\rangle }^{2}\\le 1.\\) Then, assuming that the mean spin is not in the x direction but is in the x\u2013y plane, we arrive at equation (12<\/a>).<\/p>\n Observation 2<\/p>\n For bipartite separable states,<\/p>\n $$\\langle \\,{J}_{x}\\otimes {J}_{x}\\rangle \/(\\,{j}_{1}\\,{j}_{2})+\\langle \\,{J}_{y}\\otimes {J}_{y}\\rangle \/(\\,{j}_{1}\\,{j}_{2})+\\left|\\begin{array}{c}\\left\\langle {\\sigma }_{z}^{\\otimes {N}_{1}}\\otimes {\\sigma }_{z}^{\\otimes {N}_{2}}\\right\\rangle \\end{array}\\right|\\le 1$$<\/p>\n \n (14)\n <\/p>\n holds, where j1 = N1\/2 and j2 = N2\/2.<\/p>\n Proof. We start from equation (12<\/a>) and use the Cauchy\u2013Schwarz inequality. See, for example, ref. 68<\/a>.<\/p>\n Entanglement-depth condition based on parity<\/p>\n In this section, we obtain entanglement criteria for detecting entanglement between two groups of particles. We start from the relations based on bipartite correlations. Then, we obtain entanglement conditions that do not need bipartite correlations but rather need the measurement of collective quantities. Such quantities can be measured even in systems in which we cannot address the particles individually. Finally, we present a relation that detects the entanglement depth and can detect genuine multipartite entanglement.<\/p>\n Observation 3<\/p>\n The following expression is true for bipartite separable states:<\/p>\n $$\\mathop{\\sum }\\limits_{l=x,y}\\left\\langle {\\left(\\,{J}_{l}^{\\,(1)}+{J}_{l}^{\\,(2)}\\right)}^{2}\\right\\rangle \/(2{j}_{1}\\,{j}_{2})+\\left|\\left\\langle {\\sigma }_{z}^{\\otimes N}\\right\\rangle \\right|\\le j(j+1)\/(2{j}_{1}\\,{j}_{2}),$$<\/p>\n \n (15)\n <\/p>\n where j1 = N1\/2, j2 = N2\/2 and j = N\/2.<\/p>\n Proof. We start from equation (14<\/a>). We add to both sides \\({\\sum }_{l=x,y}\\langle {(\\,{J}_{l}^{(1)})}^{2}\\rangle \/(2{j}_{1\\,}{j}_{2})+\\langle {(\\,{J}_{l}^{(2)})}^{2}\\rangle \/(2{j}_{1}{j}_{2})\\) to give:<\/p>\n $$\\mathop{\\sum }\\limits_{l=x,y}\\left\\langle {\\left(\\,{J}_{l}^{(1)}+{J}_{l}^{\\,(2)}\\right)}^{2}\\right\\rangle \/(2{j}_{1}\\,{j}_{2})+\\left|\\left\\langle {\\sigma }_{z}^{\\otimes N}\\right\\rangle \\right|$$<\/p>\n \n (16)\n <\/p>\n $$\\le 1+\\mathop{\\sum }\\limits_{l=x,y}\\left\\langle {\\left(\\,{J}_{l}^{(1)}\\right)}^{2}\\right\\rangle \/(2{j}_{1}\\,{j}_{2})+\\left\\langle {\\left(\\,{J}_{l}^{(2)}\\right)}^{2}\\right\\rangle \/(2{j}_{1}\\,{j}_{2}).$$<\/p>\n \n (17)\n <\/p>\n Finally, we use the inequality \\(\\langle {({J}_{x}^{(n)})}^{2}+{({J}_{y}^{(n)})}^{2}\\rangle \\le {j}_{n}({j}_{n}+1).\\)<\/p>\n Next, we will show how to use the criterion given in equation (15<\/a>) to detect genuine multipartite entanglement of the N-qubit system.<\/p>\n Observation 4<\/p>\n States violating the inequality given in equation (15<\/a>) for j1 = k\/2 and j2 = (N \u2212 k)\/2 have at least (k + 1)-particle entanglement, where we assume that k\u2009\u2265\u2009N\/2. A violation for k = N \u22121 means genuine multipartite entanglement (equation (4<\/a>)).<\/p>\n Proof. The violation of equation (15<\/a>) for j1 = k\/2 means that the state cannot be written as a mixture of states of the form<\/p>\n $$\\left|{\\varPsi }_{1}\\right\\rangle \\otimes \\left|{\\varPsi }_{2}\\right\\rangle ,$$<\/p>\n \n (18)\n <\/p>\n where \\(\\left|{\\varPsi }_{1}\\right\\rangle\\) has k qubits and \\(\\left|{\\varPsi }_{2}\\right\\rangle\\) has (N \u2212 k) qubits. Note that this is true for any separation of the qubits into groups of k and (N \u2212 k) qubits. The states given in equation (18<\/a>) are called biseparable states, as they are possibly multipartite entangled states that are separable with respect to a bipartition.<\/p>\n Without loss of generality, let us consider the case \\(\\langle {\\sigma }_{z}^{\\otimes N}\\rangle \\ge 0\\). Then, let us rewrite the inequality given in equation (15<\/a>) as<\/p>\n $$\\mathop{\\sum }\\limits_{l=x,y}\\left\\langle {\\left(\\,{J}_{l}^{\\,(1)}+{J}_{l}^{(2)}\\right)}^{2}\\right\\rangle +2{j}_{1}\\,{j}_{2}\\left\\langle {\\sigma }_{z}^{\\otimes N}\\right\\rangle \\le j(\\,j+1).$$<\/p>\n \n (19)\n <\/p>\n The product j1 j2 = j1(j \u2212 j1) is largest for j1 = j\/2, and it is monotonically decreasing for a decreasing j1. It is also monotonously decreasing if j1 is increasing from j1 = j\/2. It is the smallest for j1 = 1\/2 and j2 = N\/2 \u2212 1\/2 and for j2 = 1\/2 and j1 = N\/2 \u2212 1\/2. In general, the value of j1(j \u2212 j1) is the same for j1 as for j \u2212 j1. Hence, if the criterion in equation (19<\/a>) is violated by a quantum state for j1\u2009\u2264\u2009j\/2, then it is also violated for any \\({j}_{1}^{{\\prime} }\\) fulfilling \\({j}_{1}\\le {j}_{1}^{{\\prime} }\\le j-{j}_{1}.\\)<\/p>\n Let us now consider the criterion in equation (19<\/a>) with j1 = k\/2, where we assumed that k\u2009\u2265\u2009N\/2. Let us consider pure biseparable states of the form equation (18<\/a>), such that \\(\\left|{\\varPsi }_{1}\\right\\rangle\\) has N\u03a81 qubits and \\(\\left|{\\varPsi }_{2}\\right\\rangle\\) has N\u03a82 qubits, and N\u03a81\u2009\u2265\u2009N\u03a82, which also implies N\u03a81\u2009\u2265\u2009N\/2. Such a state can contain at most N\u03a81-particle entanglement. Then, pure biseparable states of the form of equation (18<\/a>) with N\u03a81\u2009\u2264\u2009k cannot violate the criterion. Moreover, as equation (19<\/a>) is linear in the expectation values, such a criterion cannot be violated, even by states that are the mixtures of pure states of the type given in equation (18<\/a>), \\(\\left|{\\varPsi }_{1}\\right\\rangle\\) having k or fewer qubits. Simple arguments then show that the criterion in equation (19<\/a>) cannot be violated by k-producible states, that is, states with at most k-particle entanglement. Thus, a state violating the criterion must have at least (k + 1)-particle entanglement or, equivalently, it must have at least an entanglement depth of (k + 1). It can be proven that states with at most l-particle entanglement with l < N\/2 cannot violate the main condition given in equation (4<\/a>) for any k.<\/p>\n If a state violates the criterion given in equation (19<\/a>) for j1 = 1\/2, then such a state cannot be a mixture of biseparable states of the form of equation (18<\/a>) with \\(\\left|{\\varPsi }_{1}\\right\\rangle\\) and \\(\\left|{\\varPsi }_{2}\\right\\rangle\\) being quantum states of one or more qubits. Thus, the quantum state must be genuine multipartite entangled.<\/p>\n Observation 4 can be used to detect (k + 1)-particle entanglement such that k\u2009\u2265\u2009N\/2. The expectation values used for the entanglement criterion are shown in Extended Data Table 1<\/a>. In the table, we also give the value of the parameter<\/p>\n $${\\mathcal{J}}=\\frac{\\left\\langle \\,{J}_{x}^{2}+{J}_{y}^{2}\\right\\rangle }{N(N+2)\/4},$$<\/p>\n \n (20)\n <\/p>\n which characterizes how symmetric the state is. \\({\\mathcal{J}}=1\\) corresponds to perfect bosonic symmetry.<\/p>\n Entanglement-depth condition based on collective measurements of spin components<\/p>\n Let us use the definitions of the angular momentum components<\/p>\n $${J}_{l}=\\frac{1}{2}\\mathop{\\sum }\\limits_{n=1}^{N}{\\sigma }_{l}^{(n)}$$<\/p>\n \n (21)\n <\/p>\n for l = x, y and z. An entanglement condition is defined in ref. 50<\/a>. We use a somewhat stronger inequality in ref. 52<\/a>, such that for states with at most k-particle entanglement, the following inequality holds<\/p>\n $${(\\Delta {J}_{z})}^{2}\\ge {J}_{\\max }{F}_{k\/2}\\left(\\sqrt{\\frac{\\left\\langle \\,{J}_{x}^{2}+{J}_{y}^{2}\\right\\rangle -{J}_{\\max }(k\/2+1)}{{J}_{\\max }(\\,{J}_{\\max }-k\/2)}}\\right),$$<\/p>\n \n (22)\n <\/p>\n where the maximal spin length is defined as \\({J}_{\\max }=N\/2,\\,{F}_{j}(.)\\) is defined in ref. 69<\/a>. If the above condition is violated, we have at least (k + 1)-particle entanglement.<\/p>\n The criterion in ref. 50<\/a> can be improved in a different way in certain cases. We can detect (k + 1)-particle entanglement with a new condition:<\/p>\n $${(\\Delta {J}_{z})}^{2}\\ge {J}_{\\max }{F}_{k\/2}\\left(\\frac{\\sqrt{\\left\\langle \\,{J}_{x}^{2}+{J}_{y}^{2}\\right\\rangle -X}}{{J}_{\\max }}\\right).$$<\/p>\n \n (23)\n <\/p>\n Here we used that the maximum of \\({(\\Delta {J}_{x})}^{2}+{(\\Delta {J}_{y})}^{2}\\) for pure states that are at most k-particle entangled is70<\/a>,71<\/a><\/p>\n $$X=\\lfloor N\/k\\rfloor R(k)+R(r),$$<\/p>\n \n (24)\n <\/p>\n where we define<\/p>\n $$R(n)=\\left\\{\\begin{array}{ll}n\/2(n\/2+1), & \\,{\\rm{even}}\\,n,\\\\ n\/2(n\/2+1)-1\/4, & \\,{\\rm{odd}}\\,n.\\end{array}\\right.$$<\/p>\n \n (25)\n <\/p>\n For an n-qubit state, \\({(\\Delta {J}_{x})}^{2}+{(\\Delta {J}_{y})}^{2}\\le R(n)\\) holds. Moreover,<\/p>\n $$r=N-\\lfloor N\/k\\rfloor k.$$<\/p>\n \n (26)\n <\/p>\n We used both conditions on the experimental data. For k = 1, we used the condition for entanglement in ref. 72<\/a>. We chose the results from the method that gave a larger entanglement depth. The results are shown in Extended Data Fig. 4<\/a>, and the expectation values used for the criterion are given in Extended Data Table 1<\/a>.<\/p>\n Note that for states in the Jz = 0 subspace, \u3008Jx\u3009 = \u3008Jy\u3009 = 0, and for states with at most k-particle entanglement:<\/p>\n $$\\left\\langle \\,{J}_{x}^{2}+{J}_{y}^{2}\\right\\rangle \\le \\lfloor N\/k\\rfloor R(k)+R(r)$$<\/p>\n \n (27)\n <\/p>\n holds. Any state that violates equation (27<\/a>) is at least (k + 1)-particle entangled.<\/p>\n Entanglement witness for a state with an indefinite number of particles<\/p>\n For separable states with a given number of particles, we have72<\/a><\/p>\n $$(N-1){(\\Delta {J}_{z})}^{2}-\\left\\langle \\,{J}_{x}^{2}+{J}_{y}^{2}\\right\\rangle +\\frac{N}{2}\\ge 0.$$<\/p>\n \n (28)\n <\/p>\n Note that the left-hand side is identical to zero for N = 0 and N = 1.<\/p>\n Then, if the number of particles is fluctuating, we use73<\/a><\/p>\n $$\\begin{array}{rcl}{(\\Delta {J}_{z})}^{2} & – & \\left\\langle {(N-1)}^{-1}{J}_{x}^{2}\\right\\rangle -\\left\\langle {(N-1)}^{-1}{J}_{y}^{2}\\right\\rangle \\\\ & + & \\frac{1}{2}\\left\\langle {(N-1)}^{-1}N\\right\\rangle \\ge 0,\\end{array}$$<\/p>\n \n (29)\n <\/p>\n for which the left-hand side is \u22121.205(44), more than 27 standard deviations below zero. To compute the expectation values, we used the data for N = 2, 3, 4, \u2026, 12 particles, which includes odd numbers of particles.<\/p>\n Extracting the Fisher information from the Hellinger distance<\/p>\n From the recorded occurrences for (Jz; N) for the five small rotation angles \u03b8, we estimated the Fisher information FN when using the prepared N-atom twin-Fock states from the experimental probabilities \\({p}_{\\theta }^{\\exp }({J}_{z};N)\\), here normalized such that \\({\\sum }_{{J}_{z}}{p}_{\\theta }^{\\exp }({J}_{z};N)=1\\). Measured occurrences for N = 2 and N = 10 are displayed in Extended Data Fig. 5<\/a>. The data clearly show that the rate at which the measured experimental probabilities change with the rotation angle \u03b8 is much larger for the N = 10 atom state.<\/p>\n To estimate the uncertainties in the analysis, we performed a Monte Carlo resampling of the measurement occurrences, which we used to calculate the experimental probabilities \\({p}_{{\\theta }_{1,2}}^{\\exp }({J}_{z};N)\\). We assumed multinomial distributions with event probabilities given by \\({p}_{{\\theta }_{1,2}}^{\\exp }({J}_{z};N)\\) for the N + 1 possible outcomes of Jz. Subsequently, we computed the Hellinger distance of the recomputed distributions according to equation (5<\/a>) and obtained the final value \\({d}_{{\\rm{H}},\\mathrm{fit}}^{2}({\\theta }_{1},{\\theta }_{2};N)\\) as the mean and its uncertainty as the standard deviation.<\/p>\n For each possible choice of reference angle \u03b81, we fitted the obtained values using the quadratic function<\/p>\n $${d}_{{\\rm{H}},\\mathrm{fit}}^{\\,2}({\\theta }_{1},{\\theta }_{2};N\\,)=\\frac{{F}_{N}({\\theta }_{1})}{8}{({\\theta }_{1}-{\\theta }_{2})}^{2}+b,$$<\/p>\n \n (30)\n <\/p>\n where FN(\u03b81) and b are free fitting parameters.<\/p>\n Finally, we computed the weighted average of the Fisher information obtained for different \u03b81 as<\/p>\n $${\\bar{F}}_{N}=\\mathop{\\sum }\\limits_{{\\theta }_{1}}\\frac{{w}_{{\\theta }_{1}}}{{\\sum }_{\\theta }{w}_{\\theta }}{F}_{N}({\\theta }_{1})\\,{\\rm{,\\; with}}$$<\/p>\n \n (31)\n <\/p>\n $${w}_{{\\theta }_{1}}={\\left(\\frac{1}{\\Delta {F}_{N}({\\theta }_{1})\/{F}_{N}({\\theta }_{1})}\\right)}^{2}.$$<\/p>\n \n (32)\n <\/p>\n Here the weights \\({w}_{{\\theta }_{1}}\\) were computed from the relative uncertainty of FN(\u03b81). For the uncertainty of \\({\\bar{F}}_{N}\\), we used the averaged uncertainties of the fitting parameters FN(\u03b81) rather than the standard error of the mean, as the five values FN(\u03b81) are not statistically independent.<\/p>\n We note that the resampling method introduced a statistical bias. As \\({d}_{{\\rm{H}},\\mathrm{fit}}^{2}({\\theta }_{1},{\\theta }_{2};N)\\) is a convex function, this bias was positive. Comparing the obtained mean values with those directly calculated from the measured experimental probabilities, we see that the bias depends only on the available sample sizes and not directly on \u03b8. As these sample sizes were almost independent of \u03b8, the free fitting parameter b can account for the introduced bias.<\/p>\n We further note that the Hellinger method itself is also affected by a statistical bias53<\/a>, especially when the sample sizes are small. To quantitatively determine the effect, we employed the probabilistic noise estimation model to predict probability distributions for all rotation angles \u03b8. From these probabilities, we calculated the Hellinger distance directly, without any form of random sampling or rounding to integer occupation numbers. Thus, no statistical bias was expected. Employing the same fitting techniques as for the measured data, the model gives a scaling \\(\\frac{{N}^{1.81}}{2}+N\\). Adding the fourth-order Taylor expansion term \\(-(\\frac{1}{256}{F}_{N}^{2}-\\frac{1}{192}{F}_{N}){\\theta }^{4}\\) to the fitting function \\({d}_{{\\rm{H}},\\mathrm{fit}}^{2}\\), as expected for twin-Fock states, resulted in \\(\\frac{{N}^{1.87}}{2}+N\\). We further note that the measurement point for \u03b8 = 0.35\u2009rad and N = 14 atoms lies outside the range in which \\({d}_{{\\rm{H}}}^{2}({\\theta }_{1},{\\theta }_{2};N)\\) can be approximated by a Taylor expansion, as there is a non-differentiable point at \u03b8 = 0.321\u2009rad. Neglecting this data point in the fit of FN yielded a scaling of \\(\\frac{{N}^{2.01}}{2}+N\\). Thus, our measurements are compatible with the N2 scaling of the ideal twin-Fock state.<\/p>\n Calculating uncertainties<\/p>\n Unless stated otherwise, the error bars throughout this article represent the standard errors. For further details of our calculations of uncertainties, see ref. 50<\/a>.<\/p>\n For Fig. 4<\/a> and Extended Data Fig. 4<\/a>, the uncertainties were calculated using a Monte Carlo resampling approach. This method is analogous to that used in our Fisher information analysis. For each number of atoms N, we resampled the occurrences of the Jz values (without HOM coupling) and Jx values (after HOM coupling). This resampling used multinomial distributions, where the sample sizes and probabilities were derived from the measured data, specifically \\({p}_{\\theta }^{\\exp }({J}_{z};N)\\), which was normalized such that \\({\\sum }_{{J}_{z}=-N\/2}^{N\/2}{p}_{\\theta }^{\\exp }({J}_{z};N)=1\\). From each obtained sample (indexed by i = 0, 1, \u2026, 10,000), we calculated value pairs \\({\\{\\langle {J}_{x}^{2}+{J}_{y}^{2}\\rangle ,\\langle {\\varPi }_{z}\\rangle \\}}_{i}\\) for equation (4<\/a>) and \\({\\{\\langle {J}_{x}^{2}+{J}_{y}^{2}\\rangle ,{(\\Delta {J}_{z})}^{2}\\}}_{i}\\) for equations (22<\/a>) and (23<\/a>). We then computed two samples {ki} of entanglement-depth values: one based on parity (for Fig. 4<\/a>) and one based on the variance of Jz (for Extended Data Fig. 4<\/a>). The displayed k values are the average of the {ki}, and their uncertainties are displayed using upper and lower standard deviations, as described below in equation (33<\/a>). Extended Data Table 2<\/a> shows the minimally verified entanglement depths for confidence regions of 68% and 95%, that is the largest possible integer kmin such that \\({k}_{i}\\ge {k}_{\\min }\\) still holds for at least 68% or 95% of the samples.<\/p>\n The new entanglement-depth criterion, equation (4<\/a>), is applicable only for k\u2009\u2265\u2009N\/2. For 1.31% (N = 10) and 6.81% (N = 12) of the resampled value pairs, the criterion could not detect entanglement. In these cases, we used the criterion given by equations (22<\/a>) and (23<\/a>) to detect entanglement.<\/p>\n For the asymmetric error bars in Fig. 4<\/a> and Extended Data Fig. 4<\/a>, we used the definitions for the upper and lower variances:<\/p>\n $$\\begin{array}{rcl}{({\\Delta }_{+}x)}^{2} & = & \\frac{2}{M}\\mathop{\\sum }\\limits_{n:{x}_{n}\\ge \\langle x\\rangle }{({x}_{n}-\\langle x\\rangle )}^{2},\\\\ {({\\Delta }_{-}x)}^{2} & = & \\frac{2}{M}\\mathop{\\sum }\\limits_{n:{x}_{n} < \\langle x\\rangle }{({x}_{n}-\\langle x\\rangle )}^{2},\\end{array}$$<\/p>\n \n (33)\n <\/p>\n where xn for n = 1, 2, \u2026, M are a set of values and \u3008x\u3009 is the average. With these definitions, for the variance<\/p>\n $${(\\Delta x)}^{2}=\\frac{{({\\Delta }_{+}x)}^{2}+{({\\Delta }_{-}x)}^{2}}{2}$$<\/p>\n \n (34)\n <\/p>\n holds.<\/p>\n","protected":false},"excerpt":{"rendered":"Coherent mode coupling and spin-changing collisions We employed a low-noise 6.8-GHz microwave source62 to drive Rabi oscillations between…\n","protected":false},"author":3,"featured_media":840153,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_share_on_mastodon":"0"},"categories":[25],"tags":[2267,2266,2271,2270,834,2265,2268,2269,492,28377,159,27105,2264,67,132,68],"class_list":["post-840152","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics","tag-atomic","tag-classical-and-continuum-physics","tag-complex-systems","tag-condensed-matter-physics","tag-general","tag-mathematical-and-computational-physics","tag-molecular","tag-optical-and-plasma-physics","tag-physics","tag-quantum-metrology","tag-science","tag-single-photons-and-quantum-effects","tag-theoretical","tag-united-states","tag-unitedstates","tag-us"],"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/116683073720500434","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/840152","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=840152"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/840152\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/840153"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=840152"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=840152"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=840152"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n