Partitioning dysprosium's electronic spin to reveal entanglement in non-classical states

Quantum spins of mesoscopic size are a well-studied playground for engineering non-classical states. If the spin represents the collective state of an ensemble of qubits, its non-classical behavior is linked to entanglement between the qubits. In this work, we report on an experimental study of entanglement in dysprosium's electronic spin. Its ground state, of angular momentum $J=8$, can formally be viewed as a set of $2J$ qubits symmetric upon exchange. To access entanglement properties, we partition the spin by optically coupling it to an excited state $J'=J-1$, which removes a pair of qubits in a state defined by the light polarization. Starting with the well-known W and squeezed states, we extract the concurrence of qubit pairs, which quantifies their non-classical character. We also directly demonstrate entanglement between the 14- and 2-qubit subsystems via an increase in entropy upon partition. In a complementary set of experiments, we probe decoherence of a state prepared in the excited level $J'=J+1$ and interpret spontaneous emission as a loss of a qubit pair in a random state. This allows us to contrast the robustness of pairwise entanglement of the W state with the fragility of the coherence involved in a Schr\"odinger cat state. Our findings open up the possibility to engineer novel types of entangled atomic ensembles, in which entanglement occurs within each atom's electronic spin as well as between different atoms.


I. INTRODUCTION
Entanglement is a hallmark of non-classical behavior in compound quantum systems. Minimal entangled systems of qubit pairs, as realized with correlated photon pairs, play a central role in testing the foundations of quantum mechanics [1,2]. Entanglement can also be engineered in many-particle systems [3], such as an ensemble of interacting atoms [4]. In this case, the atoms are not individually addressable, and quantum correlations are indirectly revealed by measuring global properties, such as a squeezed spin projection quadrature [5][6][7][8] or via the quantum enhancement of magnetic sensitivity [9][10][11]. State of the art experiments on photonic systems [12], superconducting qubits [13], trapped ions [14] and Rydberg atom arrays [15] can now produce highly entangled states of tens of individually identifiable qubits, in which entanglement is more readily observable.
Besides quantum state tomography, a wide array of methods have been developed for the detection of entanglement [16,17]. In two-qubit systems, the degree of entanglement is quantified by the concurrence [18,19]. Its direct measurement remains challenging since it requires non-linear operations on the prepared state [20][21][22][23], and it was so far only achieved for photon pairs in pure quantum states [21]. In the case of multipartite systems, the study of entanglement is cumbersome due to the existence of distinct classes of entanglement [24]. It is often revealed using entanglement witnesses, by measuring the fidelity with respect to a given entangled state [25] -the method being limited to simple enough target states. In this work, we study the entanglement properties associated with non-classical states of the electronic spin of dysprosium atoms, of angular momentum J = 8 in its ground state. Quantum states with non-classical correlations have been extensively studied in single largespin systems, including two-photon states [26], ground state atomic spins [27,28], molecules [29], and Rydberg atoms [30]. In the formal analogy between a spin J and a set of 2J qubits symmetric upon exchange [31], non-classicality goes hand in hand with entanglement between the virtual qubits. However, as long as the angular momentum J is conserved, the qubit ensemble cannot be partitioned, and the relevance of entanglement is disputable. Here, we use an optical coupling to an excited electronic state of angular momentum J = J − 1 to partition the 16-qubit ensemble associated with the spin J, giving access to entanglement. The virtual absorption of a photon is interpreted as the annihilation of a qubit pair in a state defined by the light polarization, leaving a set of 14 qubits in the excited electronic level (see Fig. 1a). We use this partition to probe entanglement in non-classical spin states, either by characterizing nonclassical behavior of qubit pairs via the measurement of concurrence, or by revealing an increase of entropy upon partition. We extend this protocol to probe decoherence in states prepared in an excited electronic level J = J +1 (see Fig. 1b). There, the spontaneous emission of a photon drives the system to the electronic ground state J, which corresponds to the removal of a qubit pair randomly drawn from the initial state. We reveal the robustness of pairwise entanglement with respect to qubit loss, as well as the fragility of coherence in Schrödinger cat states.
This paper is organized as follows. We present in section II the experimental protocol used to measure the properties of qubit pairs extracted from the electronic spin, based on the polarization dependence of the lightspin interaction. In section III, we investigate the nonclassical character of these qubit pairs via the measurement of the concurrence of the reduced two-qubit density matrix, and apply it to a W state and a squeezed state. In section IV, we investigate the increase of entropy upon the 14|2 partition as a proof of entanglement for W and Schrödinger cat states, by studying the mixed nature of the reduced two-qubit density matrix. Finally, in section V, we study the decoherence upon the loss of a qubit pair triggered by spontaneous emission. We show that nonclassical pairwise correlations are robust with respect to the extraction of qubits. On the contrary, the coherence of a Schrödinger cat state is completely destroyed upon qubit loss, due to the complete which path information carried by the spontaneously emitted photon's polarization. In another superposition state, we show the existence of a quantum jump leaving the path information hidden, such that maximal-order coherence remains visible.

A. Probing pairs via light coupling
The electronic ground state J = 8 can be interpreted as the sum of 2J = 16 virtual spin-1/2s, in a state symmetric upon exchange. We discuss here the partition of this qubit ensemble, prepared in a state ρ, through the coupling to an excited electronic level, of angular mo-mentum J = 7. As sketched in Fig. 1a, the coupling to the excited manifold is induced by light close to the optical transition, via the absorption of a photon. The photon polarization defines an L = 1 quantum state | that can be considered as a symmetric 2-qubit state. We restrict here to the case of a circular polarization σ − , which corresponds to qubits polarized in |↓↓ z . Since the excited state contains only 2J = 14 qubits, two qubits are removed upon photon absorption. The conservation of angular momentum requires these removed qubits to be polarized in |↑↑ z , the time-reversed state of the absorbed photon's polarization. The excited state ρ can be then written as a projected state ρ = ↑↑| z ρ |↑↑ z . The probability for a pair chosen from the 16 qubits to be polarized in |↑↑ z then reads Q pair (e z ) = Tr ρ , defining the pair Husimi function along the direction e z .
To probe this behavior, we measure the light shift V induced by an off-resonant light beam close to the considered optical transition. The light shift, being induced by virtual photon absorption processes, is proportional to the pair Husimi function, as where d = J − 1||d||J is the reduced dipole matrix element, E is the light electric field amplitude, and ∆ is the detuning from resonance.

B. Application to Dicke states
We illustrate our method by measuring the value of the Husimi function Q pair (e z ) for an arbitrary Dicke state |m (with −J ≤ m ≤ J), that we denote Q m hereafter.
All our experiments are performed on a cloud of 1.0(1) × 10 5 dysprosium atoms (of the bosonic isotope 162 Dy), held in an optical dipole trap at a temperature T = 0.54(3) µK. The results described in this work can be understood by considering a single atom, with the ensemble acting as an averaging mechanism only. The experimental scheme for the Q m measurement is shown in Fig. 2a. We prepare the atoms in a coherent state |m = J n polarized along a direction n, parametrized by the spherical angles (θ, φ). The polar angle θ determines the projection probabilities Π m along the Dicke states |m , which are significant for values of m close to J cos θ. We then push the atomic cloud by applying an off-centered laser beam, with circular σ − polarization and blue detuning with respect to an optical transition at 696 nm. The intensity gradient then leads to a force along x proportional to the light shift (Fig. 2a). After this kick, a magnetic field gradient is applied to spatially separate the different m-components along z, which allows us to retrieve the light shift experienced by each Dicke state independently. After a 2.3 ms time-of-flight, we image the atoms and measure the x-displacement for each Dicke state |m that is significantly populated, and hence their values Q m . A typical absorption image is shown in Fig. 2b. Repeating this measurement for various angles θ, we measure the light shifts for all projections m, and infer the Q m values shown in Fig. 2c [32].
Our measurements are consistent with an absence of light shift for the states |m = −J and |m = −J + 1 , i.e. these states are dark with respect to the J → J = J −1 optical transition for σ − polarized light. In terms of the underlying qubits, the states |m = −J only contains |↓ z -polarized qubits, while the state |m = −J + 1 has a single qubit in |↑ z . In both cases, a qubit pair cannot be found polarized in |↑↑ z , hence Q −J = Q −J+1 = 0.
More generally, a Dicke state |m is composed of J − m qubits in |↓ z and J + m qubits in |↑ z [33]. The probability to pick a pair |↑↑ z simply reads in good agreement with our measurements. We use these measurements to probe the Husimi function of states lacking z rotation symmetry. For this, we measure their projection probabilities Π m (n) along n by combining a spin rotation and a Stern-Gerlach projective measurement along z. We then infer the Husimi function by weighting these probabilities with the Q m values, as In the following, we use the theoretical values of Eq. 1 rather than the measured ones to avoid propagating systematic errors.

C. Coherent and W states
We first apply the above protocol to the quasiclassical coherent spin state |m = −J and the W state |m = −J + 1 . The coherent state can be viewed as a set of 2J qubits polarized in |↓ z , forming a non-entangled product state. The W state, which hosts a single qubit in |↑ z , is a paradigmatic state of a fundamental class of entanglement [24], which has been realized and studied in various settings [34][35][36][37][38][39][40][41].
In our experiment, the atoms are initially spinpolarized in the coherent state |m = −J . To produce the W state, we apply a resonant radiofrequency π-pulse towards |m = −J + 1 , while the other Dicke states are made off-resonant using a quadratic light shift. This light shift is produced using the 696 nm laser beam with a σ − polarization, leading to positive energy shifts for all Dicke states |m , except for m = −J and −J + 1. We reach a maximum W state fidelity of 0.91(1), with residual overlaps on other Dicke states below 4% [42].
We report in Fig. 3a,b the measured projection probabilities Π m (θ) for these two states. For a given projection m, the coherent state probabilities feature a single peak centered on the expected maximum at θ m = acos(m/J), shown as red lines. For the W state probabilities, we observe a double-peaked distribution for all non-stretched states m = ±J. This behavior results from the interference between two processes, depending on whether the spin |↑ z is projected on |↑ θ or |↓ θ . The first (second) process dominates for θ 0 (θ π), and the two processes destructively interfere at θ m , as observed in our data.
We combine these measurements to infer the pair Husimi functions using Eq. 2, finding good agreement with theory for both states (see Fig. 3c). In particular, for the coherent state our data matches well the probability Q pair (θ) = sin 4 (θ/2) that two qubits in |↓ z are projected in |↑ θ . In the following sections we use these measurements to probe entanglement properties.

III. NON-CLASSICALITY OF QUBIT PAIRS
Our first characterization of entanglement of the 2Jqubit state consists in revealing the non-classical character of qubits pairs extracted from it. The collective state ρ pair of a qubit pair symmetric upon exchange can be written as the state of an angular momentum L = 1. Drawing an analogy with quantum optics [43,44], it will be called classical if it can be expressed as a statistical mixture of quasi-classical coherent states [45], as where ||n is a spin-1 coherent state pointing along n, and w n ≥ 0, n w n = 1. Coherent states are the only pure states that satisfy the equality for arbitrary measurement axis n. Then it follows by convexity that Z(n) ≥ 0 for classical states. As shown in [45], the existence of a strictly negative value Z(n) constitutes a necessary and sufficient criterion of nonclassicality.
To apply this criterion to our system, we use the connection between the mean values of spin projection and the Husimi function, The parameter α being negative, non-classicality is characterized by the existence of a direction n for which C n is strictly positive. This criterion is equivalent to the bipartite entanglement witness established in [46]. We show in Fig. 3d the distribution C n computed from the measured Husimi functions, for the coherent and W states. For these states, symmetric upon rotations around z, we expect C n to only depend on the polar angle θ of the measurement axis [47]. For the coherent state, the measured C n remains close to zero for all angles θ. Indeed, qubits pairs drawn from this state form themselves a spin-1 coherent state, for which C n vanishes according to Eq. (4). For the W state, C n takes significantly positive values for θ close to 0 and π, showing a non-classical character.
We now show that the distribution C n can be used to quantify the degree of non-classicality of a quantum state, defined by its distance to the set of non-classical states [48]. For a system of two qubits, this geometrical measure can be directly expressed in terms of the concurrence C [49], the most common measure of pairwise entanglement [18,19]. The concurrence can be explicitly written in terms of the density matrix, but it does not correspond to a directly accessible physical observable. Remarkably, the distribution C n can be used to retrieve  the concurrence, as This relation was conjectured and numerically checked for randomly generated states in Ref. [50].
For the W state realized in the experiment, the measured C n takes its maximum for θ = 0 leading to a concurrence C = 0.089 (5). This value is about 71% of the maximum possible value C = 1/J = 0.125 in a system of 2J qubits symmetric upon exchange [51], which would be reached for the W state in the absence of experimental imperfections. In our system, the concurrence is limited by the residual population Π −J+2 0.03 in the Dicke state |m = −J + 2 that originates from spin-changing collisions between atoms in |m = −J + 1 .

B. Pairwise correlations in a squeezed state
Non-classical correlations between qubit pairs play a central role in the squeezing of a spin projection quadrature [52]. In this section we extend the measurement of qubit pair properties to a squeezed spin state, that we produce via a non-linear spin dynamics. We apply a χJ 2 x spin coupling, generated by the spin-dependent light shift of the 696 nm laser beam, using a linear polarization e x [53]. This coupling induces a twisting of the spin distribution, leading to the squeezing of a spin projection quadrature [52], as first implemented in atomic Bose-Einstein condensates [8,9]. In our experiment, we apply a non-linear coupling of strength χ = 2π × 32.1(4) kHz for a duration t 700 ns, in the presence of a z magnetic field B = 75(1) mG.
Contrary to the Dicke states discussed above, the spin projection probabilities are no longer invariant around z. We show in Fig. 4a,b the probabilities Π m (θ, φ) for two azimuthal angles φ min = −0.4(2) rad and φ max = φ min + π/2, which feature minimal and maximal spin projection uncertainties, respectively. For θ = π/2, a minimum spin projection uncertainty ∆J min = 0.92 (16) is measured at φ min (see Fig. 4c), in agreement with the value ∆J min = 0.85 expected for an optimally squeezed state (within the one-axis twisting dynamics). We report in Fig. 4d the corresponding distribution C n . The measured C n takes its maximum for θ = π/2 and φ = φ min , i.e. along the squeezed quadrature direction. This maximum gives a value for the concurrence C = 0.058 (6), in agreement with the expected value of 0.055.
Our measurements can be used to check the direct link between quadrature squeezing and pairwise entanglement [54]. Indeed, for the states reached via the oneaxis twisting dynamics, one expects the concurrence to be expressed in terms of the minimum spin projection uncertainty, as From the measured projection quadrature, we calculate a value of 0.053(5) for the right hand side of Eq. 5, in agreement with the direct measurement of the concurrence.

IV. PROBING ENTANGLEMENT VIA THE SUBSYSTEM ENTROPY
So far, we studied the entanglement of 2J-qubit states via the non-classical character of their qubit pairs. In this section, we access entanglement more directly, by probing whether a given state of the spin J = 8 is separable with respect to the 14|2 partition performed by the photon absorption. For this, we use the fact that for a separable state, the global state is more disordered than its parts [55]. More precisely, we quantify disorder via the Renyi entropy of infinite order (also called the min-entropy), defined as [56] S ∞ (ρ) = − log λ max (ρ), where λ max is the maximum eigenvalue of the density matrix ρ. This eigenvalue corresponds to the maximum possible overlap of ρ with a pure state. To reveal entanglement within a state ρ of the collective spin J, it is thus sufficient to show that the entropy of the reduced pair state ρ pair has a strictly higher entropy than the one of the original state ρ, i.e. if the conditional entropy satisfies [55] The evaluation of the pair state entropy S ∞ (ρ pair ) is based on the tomography of the pair density matrix [57]. Full information on the density matrix is contained in the Husimi function Q pair (n). We fit the measured Husimi function by a spherical harmonic expansion and infer the density matrix as where the L m and Q m matrices correspond to the L = 1 angular momentum components and quadrupole moments, respectively (see Appendix B). We apply this protocol to the W state, taking into account the slight variation of the Husimi function Q pair (n) with respect to the azimuthal angle φ in the prepared state [47]. We infer a density matrix Diagonalization of the reconstructed density matrix gives a maximum eigenvalue λ max (ρ pair ) = 0.882 (5). We now consider the global spin-J state. The projection probability Π −J+1 = 0.91(1) with the Dicke state |m = −J + 1 provides a lower bound on the maximum overlap λ max (ρ) with pure states.
Its negative value shows that the prepared state is not separable with respect to a 14|2 partition, and is thus entangled.
The cat state considered here is the coherent superposition of two quasi-classical spin states |m = ±J [75]. To produce it, we use the one-axis twisting dynamics discussed above, with a stronger non-linear coupling χ = 2π × 1.25 MHz and a reduced magnetic field B = 53.7(1) mG. After showing quadrature squeezing at short times (t ∼ 10 ns), the spin quadratures collapse to a featureless spin distribution, before a revival at a time t cat = π/(2χ) = 200 ns, at which the system forms a coherent superposition of stretched states |m = ±J [53].
In Fig. 5a, we show the measured probabilities Π m (n) for various polar angles θ with a fixed azimuthal angle φ. For θ = 0, we confirm dominant population of the two stretched states, with Π −J = 0.38 (2) and Π J = 0.42 (2). When varying θ, the distribution is a superposition of the contributions of each of the two coherent states forming the cat state. Interestingly, we observe an interference between the two distributions when they overlap, i.e. for θ π/2. As shown in Fig. 5c, the interference pattern depends on the azimuthal angle φ, with an alternation between even-and odd-m projections of period 2π/(2J) [76].
We first test whether a qubit pair extracted from this state features non-classical behavior. We expect the distribution C n to be rotationally invariant around z, and thus study its variation with the polar angle θ in Fig. 5b [77]. Our measurements are consistent with C n < 0 for all angles θ, showing that the reduced two-qubit state is classical. This measurement highlights the well-known property of this state that any of its subsystems is classical.
We now extend the tomography protocol to the cat state, and obtain the reduced two-body density matrix  obtained for a perfect cat state. We compute the maximum eigenvalue λ max (ρ pair ) = 0.53(1) of the reconstructed matrix.
In order to reveal entanglement in the prepared state, we evaluate its overlap with perfect cat states |cat(α) = (|m = −J + e iα |m = J )/ √ 2, which constitute a family of pure quantum states. The simple form of these states in the Dicke basis allows us to express the overlap with a state ρ as where the diagonal elements ρ m,m correspond to the spin projection probabilities Π m . The overlap O α takes its maximum value O for α = − arg ρ −J,J , with We present two protocols giving a lower bound on the extremal coherence |ρ −J,J |, both based on the measurement of an observable A defined on the spin J. We consider its mean value in a state obtained after the cat state preparation, followed by a Larmor rotation around z of angle φ, as The extremal coherence can be singled out by measuring the Fourier coefficient A 2J = |a J,−J ρ −J,J | at frequency 2J [74,76]. We will use observables that can take values in the interval [−1, 1] only, such that |a J,−J | ≤ 1.
The coefficient A 2J then provides a lower bound on the extremal coherence |ρ −J,J |. The first observable we consider is the parity P of the spin projection along an equatorial direction n ⊥ e zan observable commonly used to characterize cat states [61,62,66,67,[72][73][74]. We fit its oscillation, shown in Fig. 5d, with a Fourier series, from which we get the Fourier coefficient P 2J = 0.26(1). The second observable uses a non-linear evolution, obtained by repeating the one-axis twisting evolution used to produce the cat state [53,[78][79][80][81] (see the scheme of Fig. 5e). In the absence of imperfections, the system is brought to a superposition sin(Jφ) |m = −J + cos(Jφ) |m = J , which allows us to extract the maximal coherence from the projection probabilities in stretched states only. The projection probabilities measured with this protocol are shown in Fig. 5e. In practice, we observe residual probabilities in other projection values m, with m even only, as expected from parity symmetry. We thus use an observable Σ defined as the sign of the spin projection on even states, with Σ = m even sgn(m)Π m . Fig. 5f, gives a Fourier coefficient Σ 2J = 0.247 (5). The advantage of the second method will become clear when we consider a more complex quantum state in the next section.

Its oscillation, shown in
The two protocols lead to comparable estimates of the extremal coherence. Using the measured probabilities Π ±J quoted above, we infer a lower bound on the overlap O ≥ 0.66 (2) and thus on the eigenvalue λ max (ρ). Together, these measurements provide a conditional entropy which proves entanglement more evidently than for the W state.
We note that the requirement O > λ max (ρ pair ) = 0.53 (1), that we used to demonstrate the non-separability of the 14|2 partition, is consistent with the entanglement witness O > 0.5 extensively used for cat states [25].

V. DECOHERENCE UPON QUBIT LOSS
We now consider the removal of a pair of qubits randomly drawn from the electronic spin, irrespective of its quantum state. For this purpose, we prepare a quantum state of interest ρ in an excited level of angular momentum J = 9, corresponding to a symmetric state of 2J = 18 qubits (see Fig. 1 b). The spontaneous emission of a photon drives the system to the ground state J = 8, which has two missing qubits. Since the emitted photon can carry an arbitrary polarization, the process allows for three independent quantum jumps associated with the polarizations e − , e z , e + , with e ± = (e x ± ie y )/ √ 2. The ground-state density matrix then reads ρ = eu=e−,ez,e+ e u | ρ |e u , which can be simply written as ρ = Tr 2 ρ , corresponding to the loss of an arbitrary qubit pair.

A. Robustness of pairwise quantum correlations
We first investigate the effect of particle loss on a W state prepared in an excited electronic level of angular momentum J = J + 1, coupled to the ground state with an optical transition of wavelength 626 nm. To produce the state |m = −J + 1 in the excited level, we start in the coherent state |m = −J of the lowest energy manifold, and use π polarized resonant light to couple the system to the desired state (see Fig. 6a). As shown in Fig. 6b,  of 0.98 for a pulse duration t pulse 62 ns -the excited state lifetime being τ exc 1.2 µs [82].
Following the light pulse, we wait for spontaneous emission to occur before measuring the spin state in the ground level. We observe significant populations only in the states |m = −J and |m = −J + 1 , as expected from the selection rule |m − m| ≤ 1. The state |m = −J + 1 is dominantly populated, showing that, in most cases, the |↑ excitation of the W state is not removed upon the loss of a qubit pair. The projection probabilities, shown in Fig. 6d, are close to the expected values Π −J+1 = 1/(J + 1) and Π −J = 1 − Π −J+1 , with a residual difference mostly explained by the imperfect state preparation.
The non-classicality of qubit pairs in the final state is probed via the distribution C n introduced in section III A. We remind that C n is obtained from the spin projection probabilities along n. Since its maximum value is expected to be reached along z, we only consider projections along this direction, and obtain C z = 0.104 (3). This value provides a lower bound on the qubit pair concurrence, expected to be C = 1/(J + 1) 0.111 in the initial state. The proximity of the initial state concur-rence and the measured one after decay illustrates that losing qubits does not alter non-classicality of the remaining qubits pairs [24].

B. Fragility of macroscopic coherence
We contrast this behavior with the fragility of entanglement in coherent superpositions of states distant in phase space [83].
We consider two examples, namely a cat state |ψ 1 = (|m = −J + |m = J )/ √ 2, and the superposition |ψ 2 = (|m = −J + 1 + |m = J − 1 )/ √ 2. Their preparation consists in producing a cat state in the ground manifold (|m = −J + |m = J )/ √ 2 (see section IV B), and then applying resonant light to couple it to the excited manifold. The state |ψ 1 is produced using an x-linear polarization e x = (e + + e − )/ √ 2, which dominantly couples the stretched states |m = ±J to states |m = ±J (see Fig. 7a). Couplings to states |m = ±(J − 2) also occur, albeit with very small Clebsch-Gordan coefficients, such that these processes can be neglected [84]. The state |ψ 2 is obtained using a z-linear polarization (see Fig. 7d). In both cases, a coherent Rabi oscillation is observed when varying the pulse duration, and the fidelity of the preparation is limited by the one of the cat state in the ground level. We show in Appendix C that the coherence of the superposition is maintained during Rabi flopping, by studying the states reached after 2π pulses.
We study the effect of qubit loss, triggered by spontaneous emission, on the superposition states |ψ 1 and |ψ 2 . For the cat state |ψ 1 , we only expect population of the stretched states |m = ±J (see Fig. 7b). To check the coherence between them, we measure the sign observable Σ as a function of the azimuthal angle φ, as in section IV B. As shown in Fig. 7d, its oscillation is completely washed out, with a measured Fourier component Σ 2J = 0.006 (10), indicating an absence of coherence. For the superposition state |ψ 2 , we observe dominant projection probabilities in the states |m = ±(J − 1) , corresponding to the spontaneous emission of a σ ∓ polarized photon, respectively (see Fig. 7f). We do not measure any significant variation of these probabilities with the azimuthal angle φ, excluding coherence between them. We also measure residual projection probabilities in the stretched states |m = ±J , which occur via the spontaneous emission of a π polarized photon. The advantage of the sign observable Σ becomes clear here: it allows one to test the coherence between the states |m = ±J , without being perturbed by the atoms populating oddm states. The measured probabilities in stretched states coherently oscillate as a function of the angle φ (see Fig. 7h). More quantitatively, the sign observable, which involves even-m only, evolves with a Fourier component Σ 2J = 0.024 (1).
The complete loss of coherence when starting in the cat state |ψ 1 can be interpreted as follows. The sponta-neous decay involves two orthogonal polarizations, with a σ + polarized photon emitted when starting in the component |m = −J , while a σ − polarized photon is associated to the decay of the state |m = J (see Fig. 7b). The photon polarization thus holds complete which path information on the spin state polarization -a term referring to Einstein's version of the double-slit interference experiment [85,86]. In this case, the coherence between the different paths is erased after spontaneous emission.
For the state |ψ 2 , the most probable quantum jumps correspond to the emission of σ + and σ − polarized photons, which carry information about the state polarization (see Fig. 7e). On the contrary, the quantum jump associated with the emission of a π polarized photon does not give this information, which explains the residual coherence. The measured Fourier coefficient Σ 2J corresponds to 9.7(5)% of the value measured in the absence of the excitation. This reduction is consistent with the probability 1/(J + 1) 11.1% to scatter a π polarized photon for the considered state, showing that this channel fully preserves coherence.

VI. SUMMARY AND OUTLOOK
In this paper, we show that the 2J-qubit ensemble associated with an atomic electronic spin J can be partitioned via the optical coupling to an excited level J = J − 1. Among these qubits, 2J − 2 of them constitute the excited level, and the remaining two are annihilated by the absorbed photon, in a state defined by the light polarization. We investigate this process using atomic dysprosium, and use it to probe entanglement in nonclassical states of spin J = 8. We fully characterize the non-classical character of its reduced two-qubit state, and study the increase of entropy upon partition as a smoking gun for entanglement.
In a second set of experiments, we consider the partition of an angular momentum J = J + 1 of an excited electronic state. There, a random qubit pair is extracted by spontaneous emission towards the ground state J. We show that non-classical pairwise correlations are robust to particle loss. On the contrary, we observe that coherent superpositions of states distant in phase space are very fragile.
In this work, the study of light-spin interaction is limited to measurements of the electronic spin. A first extension would be to collect the spontaneously emitted photon, whose polarization is entangled with the electronic spin, as for experiments performed with trapped ions, atoms in optical cavities or solid-state qubits [87][88][89][90]. One would thus explicitly access the which path information carried by the photon upon spontaneous emission of a Schrödinger cat state. More generally, the photon would allow one to couple qubit pairs from the electronic spin J = 8 to 'flying qubits', which could then be manipulated to entangle distant atoms [91], and generalize quantum communication schemes to a mesoscopic degree . . . . . .  of freedom [92].
Another interesting perspective would be to place the atomic gas in an optical cavity. The electronic spin J of a single atom would be coherently coupled to the cavity light mode, leading to a compound light-spin object [93]. For a large set of atoms, the cavity light would couple the electronic spins [94,95] -a process that can be interpreted as an exchange of qubits between collective spins. This mechanism could be used to create entangled states of many mesoscopic spins [96], which could feature a strong quantum enhancement of magnetic sensitivity [97], or serve as a playground for studies of decoherence.

ACKNOWLEDGMENTS
This work is supported by European Union (ERC TOPODY 756722). We thank Jean Dalibard for stimulating discussions.
Appendix A: Deviation to z rotation symmetry in the W and cat states The W state |m = −J + 1 is invariant upon rotations around z, such that all observables should depend on the polar angle θ only. In practice, the state prepared close to the W state is not perfectly rotationally symmetric, because of a residual coherent admixture with other Dicke states. We measure a small φ variation of the measured probability distributions Π m (n), as well as the pair Husimi function Q pair and distribution C n deduced from them. We show in Fig. 8 the measured data for two azimuthal angles φ 1 = 0.36(5) rad and φ 2 = φ 1 − π/2, for which C n is minimized and maximized, respectively. The data shown in Fig. 3 of the main text corresponds to an average over φ, the error bars taking into account this dispersion.
The cat state |m = −J + |m = J is not rotationally invariant. Yet, its reduced two-body density matrix, given by Eq. 8, is invariant such that the pair Husimi function Q pair and distribution C n should depend on θ only. Like for the W state, we measure a slight variation of these quantities with φ, as shown in Fig. 9. Since we focus on extracting the concurrence from the maximum of C n , we show in the main text the data measured for an azimuthal angle φ 1 = 3.3(1) rad that maximizes C n .
Appendix B: Spin-1 tomography using the pair Husimi function The Husimi function of a spin-1 quantum state ρ expands on the spherical harmonics Y m with = 1, 2 and |m| ≤ , as written in Eq. 6. This linear decomposition allows us to retrieve the density matrix ρ, as given by Eq. 7, where we introduce the operators The preparation of superposition states in the excited electronic state, as studied in section V B, uses coherent Rabi oscillations, starting in a Schrödinger cat state of the ground electronic level (|m = −J +|m = J )/ √ 2. To check that coherence is maintained during the Rabi oscillation, we study it after a 2π excitation, by measuring the oscillation of the sign observable Σ .
As shown in Fig. 10, we find that the coherence |ρ −J,J |, estimated by the Fourier component Σ 2J , is reduced to 0.202(2) (0.211 (6)   citation, i.e. above 80% of the value obtained with no