## All Together Now

In quantum entanglement, correlations between particles mean that the measurement of one determines the outcome of the other(s). Generally, when trying to exploit quantum entanglement, the larger the number of entangled particles, the better. However, the size of entangled systems has been limited. **Haas et al.** (p. 180, published online 27 March; see the Perspective by

**Widera**) prepared a small ensemble of ultracold atoms into a collective entangled state. Starting from one internal quantum state, the system of cold atoms was excited with a weak microwave pulse leading to a small excitation probability. Because it is not known which atom is promoted into the excited state, the detection of one quantum of excitation projects the system into an entangled quantum state, called a W-state. A fast repeat-until-success scheme produced such W-states quasi-deterministically. Using such a technique was able to yield entangled states of more than 40 particles. The relatively large ensemble-entangled states could potentially in the future find use in quantum sensing or enhanced quantum metrology applications.

## Abstract

Multiparticle entanglement enables quantum simulations, quantum computing, and quantum-enhanced metrology. Yet, there are few methods to produce and measure such entanglement while maintaining single-qubit resolution as the number of qubits is scaled up. Using atom chips and fiber-optical cavities, we have developed a method based on nondestructive collective measurement and conditional evolution to create symmetric entangled states and perform their tomography. We demonstrate creation and analysis of entangled states with mean atom numbers up to 41 and experimentally prove multiparticle entanglement. Our method is independent of atom number and should allow generalization to other entangled states and other physical implementations, including circuit quantum electrodynamics.

For entanglement-enabled technologies as well as from a fundamental point of view, an important goal is to scale up the number of entangled particles in many-qubit systems. In a bottom-up approach, individual addressing and universal quantum gates allow full control on the quantum state, and, in principle, any entangled state can be produced. However, because the number of gate operations scales up with particle number, experiments based on this method (such as ion traps) are currently limited to less than 20 entangled qubits (*1*).

Many important entangled states can be produced in a “top-down” approach with collective interactions or quantum nondemolition (QND) measurements, the complexity of which does not increase with particle number. In atomic ensembles, spin-squeezed (*2*, *3*) and twin Fock states (*4*) have been produced by collisional interactions; collective QND measurement (*5*–*8*) and cavity-mediated interactions (*9*) have been used to produce spin squeezing and have been proposed for Schrödinger cats (*10*) and Dicke states, including twin Fock states (*11*). However, state-of-the-art QND measurements in ensembles are still far from the single-particle resolution that would be required, for example, to reach the Heisenberg limit of quantum metrology (*12*). Moreover, the full quantum state of a system cannot be experimentally determined nor efficiently analyzed beyond 10 to 20 qubits, in general (*13*), so that new methods that specifically identify and characterize relevant forms of entanglement in large systems are required.

Here, we use a cavity-based measurement that distinguishes one particular many-particle quantum state from the orthogonal subspace of all other states. This allows us to prepare many-body entangled states projectively and to directly measure their quasiprobability distribution with high resolution. Consider an ensemble of *N* atoms, all strongly coupled to a single mode of a high-finesse cavity (*14*–*18*) (Fig. 1, A and B). The cavity and probe beam are tuned for detecting the hyperfine state, or (*19*). Transmission through the cavity is observed only when all *N* atoms are in . A single atom in makes the cavity fully reflecting (*20*), and no further substantial changes occur when more than one atom is in . Due to the very strong coupling of the atom-resonator system (*21*), this is true for all atomic positions, so that the atoms are indistinguishable when probed by the cavity mode. Furthermore, we set the probe power such that the total probability for a spontaneous emission event is much smaller than one (*19*), limiting the amount of atom-distinguishing information that leaks into the environment. Thus, to good approximation, measuring the cavity transmission is a projective measurement with two eigenvalues: high transmission corresponding to and low transmission to the orthogonal subspace containing all other states, where at least one atom is in . This measurement enables the generation of multiparticle entanglement as follows: Atoms are prepared in a premeasurement state , then are measured as described. Low transmission signals preparation of (meaning that the cycle must be repeated), whereas high transmission prepares the system in , where *c* is a normalization factor. For a suitable choice of , can have interesting nonclassical properties. Here, we prepare a that is a good approximation of(1)known as the W state or the first Dicke state. It represents a fundamental class of entangled states (*22*), which are robust against particle loss and enable some meteorological gain over nonentangled states (*23*).

To obtain the W state, we start from and apply a weak coherent microwave pulse on the qubit transition. If the excitation probability *p* is small, this prepares . Measuring as described either projects back onto or prepares . Low transmission heralds successful preparation, and the state is then available for further experiments. Figure 1C shows the expected evolution during the sequence. Note that the system is always in a known, pure state, as in a quantum feedback scheme, and in contrast to single-photon heralded schemes (*24*, *25*).

To fully characterize the produced state, we have developed a tomography technique that measures the Husimi *Q* distribution of the total spin (*26*). In the ideal sequence (Fig. 1C), the state evolves inside the symmetric subspace, which contains the states with maximum pseudo-spin *J* = *N*/2 (Fig. 1D). In this subspace, which is spanned by the Dicke states and can be represented on a generalized Bloch sphere, the *Q* function is defined as (*26*), where is the probability that all atoms are in after a rotation of the state with density matrix ρ. This expression shows that a direct measurement of *Q* is obtained by combining a rotation (performed with a microwave pulse here) with the measurement described above. Indeed, the probability to obtain high transmission for a given state measures its overlap with (*21*, *27*). This method strongly differs from marginal distribution measurements (*3*, *28*) and is similar to one developed in quantum optics (*29*). Note that our binary individual measurement is sufficient to distinguish all symmetric states in the tomography when performed with a sufficient number of repetitions. For states outside the symmetric subspace (*J* < *N*/2), spin conservation under entails that they never transform into and, thus, have a zero contribution to the measured *P*_{0} for all (Fig. 1D) (*21*). Therefore, the norm of *Q* of a given state yields its probability of lying in the symmetric subspace, whereas the shape of fully characterizes the symmetric part of the state.

In our experiment, a small atom number is prepared from an ultracold ensemble close to quantum degeneracy and trapped in a single antinode of a one-dimensional intracavity optical lattice, where each atom is strongly coupled to the resonant cavity mode (*15*, *21*). Figure 2A shows the tomography curve for , the state before the preparation sequence. The measured curve is well approximated by the expected cos^{2}* ^{N}*(θ/2), from which we deduce the mean number of atoms

*N*that contribute to the state. Because of our atom number preparation (

*21*), we expect the prepared atom number to follow a binomial distribution with SD σ = 4.2, 3.6, and 2.8 for

*N*= 41, 23, and 12.

Applying the entanglement preparation method [see (*21*) for experimental details], the tomography results *P*_{0}(θ) of the prepared W states for *N* = 41, 23, 12 are shown in Fig. 2, B to D. The curves feature the characteristic central dip expected for the nonclassical W state. The nonzero value for θ = 0 indicates some remnant population in , which is below 10% for all atom numbers. In addition, the curves have a high contrast, indicating a population in the symmetric subspace above 40% for all atom numbers. This clearly sets them apart from their classical counterparts, the statistical mixtures of all states with one localized excitation, (orange curves in Fig. 2, B to D). For those states, the symmetric subspace population is only 1/*N*, so that the maximum value of *P*_{0}(θ) is small.

Because of the rotational symmetry of the state, a fair estimate of the W state fidelity is obtained from the single polar cut of the *Q* distribution. In fact, we expect that other cuts do not contain additional information because of a slow drift of the magnetic field during the long acquisition time of our data (several days), which randomizes the phases between the different Dicke states. This can be seen in the symmetry of the tomography data shown in Fig. 2 [*P*_{0}(–θ) ≈ *P*_{0}(θ)]. We also measured and verified that it is very similar to *P*_{0}(θ,0). To partially reconstruct the density matrix ρ in the Dicke basis, we make the simplifying assumption that ρ is purely diagonal and deduce the populations ρ* _{ii}* using a maximum likelihood algorithm. We have checked that this assumption does not overestimate the fidelity ρ

_{11}, nor does it underestimate ρ

_{00}for our data (

*21*). The red solid lines in Fig. 2 show the tomography curves corresponding to the reconstructed density matrices displayed as insets, where we truncated the basis to the first three Dicke states (

*21*). A fidelity ρ

_{11}of 0.42, 0.37, and 0.31 was obtained for

*N*= 41, 23, and 12. The 1σ error is below 0.1 in all cases (

*21*).

One important limit to the fidelity is false-positive detection during preparation (atoms detected in although they are in ). Discarding runs with ambiguous measurement results reduces this error and increases fidelity, at the cost of reducing success probability. To decide whether the cavity is in the high- or low-transmission state, we analyze both the reflected and transmitted photon numbers, *N*_{R} and *N*_{T}, and compare to predefined thresholds. Figure 3A shows the count distribution of *N*_{R} (*21*). All data was acquired using *N*_{T} = 0, *N*_{R} ≥ 4 as the criterion signaling successful preparation, leaving room for postselection using a higher threshold *N*_{R,}_{min}. Figure 3B shows the fidelity as a function of *N*_{R,}_{min}, confirming that this is an important contribution to the preparation error. The data shown in Fig. 2 correspond to values of *N*_{R,}_{min} [slightly different for each atom number (*21*)] such that the fraction of successful runs is ~10%.

Knowing how much entanglement is present in a many-particle state is difficult, even when the full density matrix is known. We now establish criteria for entanglement in the vicinity of the W state by solely comparing the two populations ρ_{00} and ρ_{11}. Bipartite entanglement being extremely difficult to prove with a W state (*30*), we rather look for multipartite entanglement and for the minimal number of entangled atoms. A state with density matrix ρ contains, at most, *k* entangled particles if ρ can be decomposed as a convex sum of density matrices , where each ρ* _{i}* corresponds to a density matrix containing, at most,

*k*atoms, and

*M*≦

*N*(

*30*). Starting from this expression, we first calculate the upper bound for ρ

_{11}as a function of ρ

_{00}for a fully separable state with

*k*= 1 (

*21*). The bound, which is tight, is shown as the solid thick blue line in Fig. 4. Any state corresponding to a point outside the blue shaded region necessarily contains at least two-particle entanglement. The calculation can be extended to larger

*k*, and the green lines show the bounds for

*k*= 8, 12, and 16. The experimental state with the highest fidelity of 0.42 lies ~1 SD above the

*k*= 12 curve, indicating that it contains at least 13 entangled atoms.

In our present setup, fidelity is limited by decoherence due to differential light shifts in the dipole trap (*21*) and a probability of ~0.2 for spontaneous emission from during the QND detection. For large atom numbers, spontaneous emission from will eventually become dominant. Simulations show that, with state-of-the-art optical cavities, the entanglement process can be scaled up to ensembles with *N* > 10^{4}. Because our method relies only on coherent evolution and collective QND measurement, it can be adapted to many systems and, in particular, to other forms of cavity quantum electrodynamics, including addressable qubits such as ions in optical cavities (*31*) or superconducting qubits in microwave cavities (*32*), as long as they are indistinguishable by cavity measurement. Furthermore, by including multiple rotations and QND detection intervals, or by combining it with other entanglement schemes such as cavity-induced spin squeezing (*6*), our scheme can be extended to a large range of entangled states. In combination with the inherent single-particle resolution, this makes it possible to investigate the fundamental limits of metrologically relevant forms of entanglement and could considerably enhance the precision of interferometric devices based on quantum metrology. In addition, our method is well suited to investigate quantum Zeno dynamics (*33*), where permanent QND observation of a degenerate eigenvalue limits the coherent evolution of the system to a given subspace, enabling preparation of a large variety of entangled states (*34*).

## Supplementary Materials

## References and Notes

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**Acknowledgments:**We thank G. Semerjian for help with the calculations on the entanglement criterion and B. Huard for discussions. This work was supported by the European Union Information and Communication Technologies project QIBEC (Quantum Interferometry with Bose-Einstein Condensates) (GA 284584) and the Integrating Project AQUTE (Atomic Quantum Technologies) (GA 247687). F.H. acknowledges a scholarship by Institut Francilien de Recherche sur les Atomes Froids. Author contributions: F.H. and J.V. performed the experiment; R.G. made contributions in its early stages; and F.H., J.V., J.R., and J.E. contributed to data analysis and interpretation, as well as to the manuscript.