## Introduction

Future progress in quantum technologies is based on the engineering and manipulation of physical systems with highly non-classical behavior1, such as quantum coherence2, entanglement3, and quantum-enhanced metrological sensitivity4,5. These properties generally come together with an inherent fragility due to decoherence via the coupling to the environment, which makes the generation of highly non-classical states challenging6. An archetype of such systems consists in an object prepared in a coherent superposition of two distinct quasi-classical states, realizing a conceptual instance of Schrödinger cat7. Such states have been realized in systems of moderate size—referred to as ‘mesoscopic’ hereafter—with trapped ions8,9, cavity quantum electrodynamics (QED) systems10,11,12, superconducting quantum interference devices13, optical photons14,15,16,17, and circuit QED systems18,19. Non-classical behavior can also be achieved with other types of quantum systems, including squeezed states20,21,22,23,24,25,26,27,28,29,30,31.

Inspired by the hypothetical cat state |dead〉+|alive〉 introduced by Schrödinger in his famous Gedanken experiment, one usually refers to a cat state in quantum optics as a superposition of quasi-classical states consisting in coherent states of the electromagnetic field, well separated in phase space and playing the role of the |dead〉 and |alive〉 states7. Such cat states can be dynamically generated in photonic systems, e.g. using a Kerr non-linearity18,32. For a spin J, a quasi-classical coherent state is represented as a state $$| \pm J\rangle _{\widehat {\mathbf{u}}}$$ of maximal spin projection m = ±J along an arbitrary direction $$\widehat {\mathbf{u}}$$. It constitutes the best possible realization of a classical state of well-defined polarization, as it features isotropic fluctuations of the perpendicular spin components, of minimal variance $$\Delta J_{\widehat {\mathbf{v}}} = \sqrt {J/2}$$ for $$\widehat {\mathbf{v}} \bot \widehat {\mathbf{u}}$$33. A cat state can then be achieved for large J values, and it consists in the coherent superposition of two coherent spin states of opposite magnetization, which are well separated in phase space. We mention that the Hilbert space dimension of 2J + 1 scales linearly with the separation between the two coherent states of the superposition. Such cat states can be created under the action of non-linear spin couplings34,35,36,37. These techniques have been implemented with individual alkali atoms, using laser fields to provide almost full control over the quantum state of their hyperfine spin38,39,40,41,42. However, the small spin size involved in these systems intrinsically limits the achievable degree of non-classical behavior.

Non-classical spin states have also been created in ensembles of one-electron and two-electron atoms5. When each atom carries a spin-1/2 degree of freedom, a set of N atoms evolving identically can collectively behave as an effective spin J = N/2, that can be driven into non-classical states via the interactions between atoms34,35,36,37,43. In such systems, spin-squeezed states have been realized experimentally20,21,22,25,26,28,29,30,31, as well as non-gaussian entangled states44. Yet, cat states remain out of reach due to their extreme sensitivity to perturbations in such systems. This behavior results from the large size 2N of the Hilbert space (when taking into account non-symmetric quantum states), which scales exponentially with the system size N, resulting in a large number of decoherence channels (e.g. losing a single particle fully destroys their quantum coherence).

In this work, we use samples of dysprosium atoms, each of them carrying an electronic spin of mesoscopic size J = 8. We exploit the AC Stark shift produced by off-resonant light38 to drive non-linear spin dynamics. Each atomic spin independently evolves in a Hilbert space of dimension 2J + 1 = 17, much smaller than the dimension 2N ~ 105 of an equivalent system of N = 16 spins 1/2. We achieve the production of quantum superpositions of effective size 13.9(1.1) (as defined hereafter), close to the maximum allowed value 2J = 16 for a spin J. As this size can be considered large, but not macroscopic according to the original Schrödinger idea, we will hereafter refer to such quantum superpositions as Schrödinger kitten states45. We provide a tomographic reconstruction of the full density matrix of these states and monitor their decoherence due to the dephasing induced by magnetic field noise.

## Results

### Experimental protocol

Our experimental scheme is sketched in Fig. 1a. We use an ultracold sample of about 105 164Dy atoms, initially spin-polarized in the absolute ground state |−Jz, under a quantization magnetic field $${\mathbf{B}} = B\widehat {\mathbf{z}}$$, with B = 18.5(3) mG (see Methods). The non-linear spin dynamics results from spin-dependent energy shifts induced by a laser beam focused on the atomic sample. The laser wavelength is chosen close to the 626-nm resonance line, such that the light shifts are proportional to the polarizability tensor of a J = 8 to J′ = 9 optical transition. For a linear light polarization along x, the light shift operator reduces to a coupling $$\propto J_x^2$$ (up to a constant), and we expect the spin dynamics to be described by the Hamiltonian38

$$\hat H = \hbar \omega _{\mathrm{L}}\hat J_z + \hbar \omega \hat J_x^2,$$
(1)

where the first term corresponds to the Larmor precession induced by the magnetic field, and the second term is the light-induced spin coupling. The light beam intensity and detuning from resonance are set such that the light-induced coupling frequency ω = 2π × 1.98(1) MHz largely exceeds the Larmor precession frequency ωL = 2π × 31.7(5) kHz. In such a regime the Hamiltonian of Eq. (1) takes the form of the so-called one-axis twisting Hamiltonian, originally introduced for generating spin squeezing21,22,43. We drive the spin dynamics using light pulses of duration t ~ 10 ns to 1. Once all laser fields are switched off, we perform a projective measurement of the spin along the z-axis in a Stern–Gerlach experiment (see Fig. 1c). Measuring the atom number corresponding to each projection value m allows to infer the projection probabilities Πm, −J ≤ m ≤ J.

### Quantum state collapses and revivals

We first investigated the evolution of the spin projection probabilities Πm as a function of the light pulse duration t. As shown in Fig. 2, we find the spin dynamics to involve mostly the even |mz states. This behavior is expected from the structure of the $$\hat J_x^2$$ coupling, which does not mix the even-|mz and odd-|mz sectors.

Starting in |−Jz, we observe for short times that all even-|mz states get gradually populated. The magnetization $$m_z \equiv \langle \hat J_z\rangle$$ and spin projection variance $$\Delta J_z^2$$ relax to almost constant values mz = −0.3(2) and $${\mathrm{\Delta }}J_z^2 = 33(1)$$ in the whole range 0.2π < ωt < 0.36π. This behavior agrees with the expected collapse of coherence induced by a non-linear coupling. To understand its mechanism in our system, we write the initial state in the x basis, as

$$| - J\rangle _z = \mathop {\sum}\limits_m ( - 1)^mc_m|m\rangle _x,\quad c_m = 2^{ - J}\sqrt {\left( {\begin{array}{*{20}{c}} {2J} \\ {J + m} \end{array}} \right)} .$$
(2)

In this basis, the non-linear coupling $$\hat J_x^2$$ induces m-dependent phase factors, leading to the state

$$|\psi (t)\rangle = \mathop {\sum}\limits_m ( - 1)^m{\mathrm{e}}^{ - {\mathrm{i}}m^2\omega t}c_m|m\rangle _x.$$
(3)

The variations between the accumulated phase factors lead to an apparent collapse of the state coherence46. The collapse timescale tc can be estimated by calculating the typical relaxation time of the magnetization, yielding $$t_{\mathrm{c}} = 1/(\sqrt {2J} \omega )$$, i.e. ωtc = 0.08π37,43 (see the Supplementary Note 1 and Supplementary Fig. 1).

For longer evolution times, we observe the occurence of peaks in mz(t) or $${\mathrm{\Delta }}J_z^2$$, that we interpret as the formation of states with significant quantum coherence18,47,48. After a quarter of the period, i.e. ωt = π/2, all odd-m (and all even-m) phase factors in Eq. (3) get in phase again, leading to the superposition

$$|\psi _{{\mathrm{kitten}}}\rangle = {\mathrm{e}}^{{\mathrm{i}}\pi /4}(| - J\rangle _z - {\mathrm{i}}|J\rangle _z)/\sqrt 2 ,$$
(4)

between maximally polarized states of opposite orientation35,37, that we refer to as a ‘kitten’ state14. We observe that, for durations 0.45π < ωt < 0.49π, the magnetization remains close to zero while the variance in the spin projection features a peak of maximal value $${\mathrm{\Delta }}J_z^2 = 57.1(2)$$ (see Fig. 2).

For pure quantum states, such a large variance is characteristic of coherent superpositions between states of very different magnetization. However, from this sole measurement we cannot exclude the creation of an incoherent mixture of |±Jz states. We observe at later times revivals of magnetization that provide a first evidence that the state discussed above indeed corresponds to a coherent quantum superposition. The first revival occurs around ωt = π, and corresponds to a re-polarization of the spin up to mz = 6.0(1), with most of the atoms occupying the state |Jz. We detect another revival of magnetization around ωt = 2π, corresponding to a magnetized state close to the initial state (mz = −6.0(2)). Between these two revivals, we observe another superposition state (large spin projection variance $${\mathrm{\Delta }}J_z^2 = 47.0(6)$$) around ωt = 3π/2.

The observed spin dynamics qualitatively agrees with the one expected for a pure $$\hat J_x^2$$ coupling43 (dashed red line in Fig. 2), while a more precise modeling of the data—taking into account the linear Zeeman coupling produced by the applied magnetic field, as well as a fit of experimental imperfections (see Methods)—matches well our data (blue line in Fig. 2).

### Probing the coherence of the superposition

In order to directly probe the coherences we follow another experimental protocol allowing us to retrieve the spin projection along directions lying in the xy equatorial plane, corresponding to observables $$\hat J_\phi \equiv {\mathrm{cos}}\phi \hat J_x + {\mathrm{sin}}\phi \hat J_y$$ (see Methods). The coherence of the state |ψkitten〉, involving the opposite coherent states |±Jz, cannot be probed using a linear spin observable, such as the magnetization, but requires interpreting the detailed structure of the probability distributions Πm(ϕ)49. By expanding the coherent states |±Jz on the eigen-basis |mϕ of the spin component $$\hat J_\phi$$, we rewrite the state as

$$|\psi _{{\mathrm{kitten}}}\rangle = \frac{{{\mathrm{e}}^{{\mathrm{i}}\pi /4}}}{{\sqrt 2 }}\mathop {\sum}\limits_m \left[ {{\mathrm{e}}^{ - {\mathrm{i}}(J\phi + m\pi )} - {\mathrm{e}}^{{\mathrm{i}}\left( {J\phi + \frac{\pi }{2}} \right)}} \right]c_m|m\rangle _\phi$$
(5)

where the cm coefficients were introduced in Eq. (2). For the particular angles ϕ = (p + 1/4)π/J (p integer), the two terms in brackets cancel each other for odd m values. Alternatively, for angles ϕ = (p − 1/4)π/J we expect destructive interferences for even m8,49. This behavior can be revealed in the parity of the spin projection

$$P(\phi ) \equiv \mathop {\sum}\limits_m ( - 1)^m{\Pi}_m(\phi ) = {\mathrm{sin}}(2J\phi ),$$
(6)

which oscillates with a period 2π/(2J).

As shown in Fig. 3a, the experimental probability distributions Πm(ϕ) feature strong variations with respect to the angle ϕ. The center of mass of these distributions remains close to zero, consistent with the zero magnetization of the state |ψkitten〉. We furthermore observe high-contrast parity oscillations agreeing with the above discussion and supporting quantum coherence between the |±Jz components (see Fig. 3c).

Information on maximal-order coherences can be unveiled using another measurement protocol, which consists in applying an additional light pulse identical to the one used for the kitten state generation50. When performed right after the first pulse, the second pulse brings the state |ψkitten〉 to the polarized state |Jz, which corresponds to the second revival occuring around ωt = π in Fig. 2. An additional wait time between the two pulses allows for a Larmor precession of angle ϕ around z, leading to the expected evolution

$$|\psi (\phi )\rangle = {\mathrm{cos}}(J\phi )|J\rangle _z + {\mathrm{sin}}(J\phi )| - J\rangle _z,$$
(7)
$$m_z(\phi ) = J\,{\mathrm{cos}}(2J\phi ).$$
(8)

We vary the wait time and measure corresponding probability distributions Πm(ϕ) (Fig. 3b) and magnetization mz(ϕ) (Fig. 3c) consistent with Eqs. (7) and (8), respectively. This non-linear detection scheme reduces the sensitivity to external perturbations, as it transfers information from high-order quantum coherences onto the magnetization, much less prone to decoherence. It also decreases the requirements on the detection noise51,52,53,54,55.

### A highly sensitive one-atom magnetic probe

The Larmor precession of the atomic spins in small samples of atoms can be used for magnetometry combining high spatial resolution and high sensitivity56. While previous developments of atomic magnetometers were based on alkali atoms, multi-electron lanthanides, such as erbium or dysprosium intrinsically provide an increased sensitivity due to their larger magnetic moment, and potentially a substantial quantum enhancement when probing with non-classical spin states57.

We interpret below the oscillation of the parity P(ϕ) discussed in the previous section as the footing of a magnetometer with quantum-enhanced precision, based on the non-classical character of the kitten state. According to generic parameter estimation theory, the Larmor phase ϕ can be estimated by measuring a generic observable $$\hat {\cal O}$$ with an uncertainty

$${\mathrm{\Delta }}\phi = \frac{{{\mathrm{\Delta }}\hat {\cal O}}}{{{\mathrm{d}}\langle \hat {\cal O}\rangle /{\mathrm{d}}\phi }}$$
(9)

for a single measurement58. Measuring the angle ϕ using coherent spin states (e.g. in a Ramsey experiment) leads to a minimum phase uncertainty $${\mathrm{\Delta }}\phi _{{\mathrm{SQL}}} = 1/\sqrt {2J} ,$$ corresponding to the standard quantum limit (SQL). For an uncertainty limit on phase measurement Δϕ we define the metrological gain compared to the SQL as the ratio G≡(ΔϕSQLϕ)2, also commonly referred to as the quantum enhancement of measurement precision5. In this framework, the parity oscillation P(ϕ) expected from Eq. (6) for the state kitten〉 yields a metrological gain G = 2J, corresponding to the best precision limit Δϕ = 1/(2J) achievable for a spin J—the Heisenberg limit. From the finite contrast C = 0.74(2) of a sine fit of the measured parity oscillation, we deduce a metrological gain G = 2JC2 = 8.8(4).

A further increase of sensitivity can be achieved using the full information given by the measured probability distributions Πm(ϕ) (see Fig. 3a), i.e. without assuming the parity to be the most sensitive observable to measure phase variations44. In this more general approach, the phase sensitivity is obtained from the rate of change of the probability distribution Πm(ϕ) upon a variation of ϕ, that we quantify using the Hellinger distance $$d_{\mathrm{H}}^2(\phi ,\phi \prime ) \equiv \frac{1}{2}\mathop {\sum}\nolimits_m \left[ {\sqrt {{\Pi}_m(\phi )} - \sqrt {{\Pi}_m(\phi \prime )} } \right]^2$$ between the distributions Πm(ϕ) and Πm(ϕ′). For small angle differences, one expects the scaling behavior $$d_{\mathrm{H}}(\phi ,\phi \prime ) \simeq \sqrt {F/8} |\phi - \phi \prime |$$, where F is the classical Fisher information, which quantifies the measurement sensitivity as $${\mathrm{\Delta }}\phi = 1/\sqrt F$$44,58. For coherent spin states, the Fisher information F = 2J corresponds to a measurement precision at the SQL. More generally, an increase in the slope of the Hellinger distance variation signals a gain in precision compared to the SQL, quantified by the metrological gain G = F/(2J). For the kitten state given by Eq. (5), we expect a metrological gain G = 2J at the Heisenberg limit. We show in Fig. 3d the Hellinger distance computed from the distributions Πm(ϕ) shown in Fig. 3a. Its variation for small angle differences yields a metrological gain G = 13.9(1.1). We thus find that using the full information from the probability distributions—rather than using its parity P(ϕ) only—increases the phase sensitivity.

For a given quantum state used to measure the Larmor phase, we expect the metrological gain to remain bounded by the value of its spin projection variance, as $$G \le 2{\mathrm{\Delta }}J_z^2/J = 14.3(1)$$58. As the measured gain coincides with this bound within error bars, we conclude that the phase measurement based on the Hellinger distance is optimum. We also performed a similar Hellinger distance analysis based on the distributions Πm(ϕ) shown in Fig. 3b leading to a comparable metrological gain G = 14.0(9) (see the Supplementary Note 3). Further increase of sensitivity would require improving the state preparation.

### Tomography of the superposition state

In order to completely characterize the superposition state, we perform a tomographic reconstruction of its density matrix59. The latter involves (2J + 1)2 − 1 = 288 independent real coefficients, that we determine from a fit of the spin projection probabilities Πm measured on the z-axis and on a set of directions uniformly sampling the xy equatorial plane60. The inferred density matrix is plotted in Fig. 4a. Its strongest elements correspond to populations and coherences involving the coherent states |±Jz, as expected for the state |ψkitten〉. We measure a coherence to population ratio 2|ρJ,J|/(ρJ,−J + ρJ,J) = 0.92(8).

In order to further illustrate the non-classical character of the superposition state, we compute from the density matrix its associated Wigner function W(θ, ϕ)22, defined for a spin over the spherical angles θ, ϕ as

$$W(\theta ,\phi ) = \mathop {\sum}\limits_{\ell = 0}^{2J} \mathop {\sum}\limits_{m = - \ell }^\ell \rho _\ell ^mY_\ell ^m(\theta ,\phi ),$$
(10)

where $$\rho _\ell ^m$$ is the density matrix component on the spherical harmonics $$Y_\ell ^m(\theta ,\phi )$$61. The reconstructed Wigner function, plotted in Fig. 4b, exhibits two lobes of positive value around the south and north poles, associated with the population of the states |±Jz. It also features interferences around the equatorial plane originating from coherences between these two states, with strongly negative values in a large phase space area. This behavior directly illustrates the highly non-classical character of the kitten state.

### Dephasing due to classical noise

We furthermore investigated the environment-induced decay of quantum coherence by following the evolution of density matrices ρ(t) reconstructed after variable wait times t in the 10–100 µs range.

While we do not detect significant evolution of the populations Πm, we observe a decrease of the extremal coherence |ρJ,J|, of 1/e decay time τ = 58 ± 4 µs, which we attribute to fluctuations of the ambient magnetic field. To calibrate such a dephasing process, we study the damping of the amplitude J(t) of a coherent state, initially prepared in the state |Jx and evolving under the applied magnetic field along z and the ambient magnetic field fluctuations (see Methods). As shown in Fig. 5b, the transverse spin amplitude J decays on a 1/e timescale τ0 = 740 ± 80 µs, consistent with residual magnetic field fluctuations in the mG range. The decoherence rate of the kitten state is thus enhanced by a factor τ0/τ = 13(2) compared to a coherent state, which illustrates the intrinsic fragility of mesoscopic coherent superpositions.

Spin decoherence due to magnetic field fluctuations can be modeled similarly to the $$T_2^ \ast$$ decay in nuclear magnetic resonance62 (see the Supplementary Note 4). Using a magnetic probe located close to the atom position, we measure shot-to-shot magnetic field fluctuations on a 0.5-mG range, but their variation on the ~100-μs dephasing timescale remains negligible. In this regime, we expect the dephasing of the state |ψkitten〉 to occur 2J = 16 times faster than for a coherent state, a value close to our measurement.

Finally, we plot in Fig. 5c, d the reconstructed density matrix and its associated Wigner function for the wait time t = 70 ± 3 µs. The weak amplitude of coherences and the shrinking of the negative regions in the Wigner function illustrate the dynamics towards an incoherent statistical mixture6.

## Discussion

In this work, we use spin-dependent light shifts to drive the electronic spin J = 8 of dysprosium atoms under a non-linear one-axis twisting Hamiltonian. The observation of several collapses and revivals of quantum coherence shows that the spin dynamics remains coherent over a full period of the evolution. In particular, the state produced after one quarter of the period consists of a coherent superposition between quasi-classical spin states of opposite orientation, which can be viewed as a mesoscopic instance of Schrödinger cat. While such coherent dynamics could be achieved with individual alkali atoms of smaller spin size39,40, the realization of large-size coherent superpositions with ensembles of spin-1/2 particles is extremely challenging9,17. The high fidelity of our protocol stems from the reduced size 2J + 1 of the available Hilbert space, that scales linearly with the effective distance 2J between the states involved in the superposition. Such scaling contrasts with the exponential scaling in the number of accessible states for ensembles of qubits, which dramatically increases the number of decoherence channels. Similarly, the full tomographic reconstruction of the produced quantum state also crucially relies on this limited size of the Hilbert space. Quantum state tomography of an equivalent 16-qubit ensemble remains inaccessible, unless restricting the Hilbert space to the permutationally invariant subspace63 or using compressed sensing for almost pure states64.

We show that our kitten state provides a quantum enhancement of precision of 13.9(1.1), up to 87(2)% of the Heisenberg limit. So far, such a high value could only be reached in ensembles of thousands of qubits based on multiparticle entanglement25,27,28,29,30. In such systems, while entanglement occurs between a large number of qubits, the quantum enhancement of precision remains small compared to the system size, far from the Heisenberg limit. Our protocol could be extended to prepare kitten states $$(| - K\rangle _z - {\mathrm{i}}|K\rangle _z)/\sqrt 2$$ with |K| ≤ J, by initiating the atoms in |−Kz before applying a non-linear spin coupling identical to the one used in this work. This would allow us to demonstrate the Heisenberg scaling of measurement sensitivity δϕ1/K. We could also implement, using similar techniques, protocols to prepare non-classical states based on adiabatic evolutions65,66,67.

Our method could also be applied to systems of larger electronic spin J. Dysprosium being the optimum choice among all atomic elements in the electronic ground state, further improvement would require using highly excited electronic levels, such as Rydberg atomic states12, or using ultracold molecules68. By increasing the atom density, one could also use interactions between N atoms of spin J to act on a collective spin of very large size $${\cal J} = NJ$$, allowing to explore non-classical states of much larger size.

## Methods

### Sample preparation and detection

We use samples of about 9(1) × 104 atoms of 164Dy, cooled to a temperature $$T \simeq 2$$ μK using laser cooling and subsequent evaporative cooling in an optical dipole trap69. The dipole trap has a wavelength λ = 1064 nm, resulting in negligible interaction with the atomic spin70. The samples are initially spin-polarized in the absolute ground state |−Jz, with a bias field $$B_z \simeq 0.5\,{\mathrm{G}}$$ along z, such that the induced Zeeman splitting largely exceeds the thermal energy. Before starting the light-induced spin dynamics, we ramp the bias field down to the final value Bz = 18.5(3) mG in 20 ms. We checked that the promotion to higher spin states (with m > −J) due to dipole–dipole interactions remains negligible on this timescale. The optical trapping light is switched off right before the spin dynamics experiments.

After the light-induced spin dynamics, we perform a Stern–Gerlach separation of the various spin components using a transient magnetic field gradient (typically 50 G/cm during 2 ms) with a large bias magnetic field along z. After a 3.5 ms time of flight, the atomic density is structured as 17 separated profiles (see Fig. 1c), allowing to measure the individual spin projection probabilites Πm using resonant absorption imaging, where m is the spin projection along z. The relative scattering cross-sections between |mz sub-levels are calibrated using samples of controlled spin composition.

Spin projection measurements along equatorial directions are based on spin rotations followed by a projective measurement along z. We apply a magnetic field pulse along y, of temporal shape $$B_y(t) = B_y^{{\mathrm{max}}}{\mathrm {sin}}^2(\pi t/\tau )$$, with τ = 3 and $$B_y^{{\mathrm{max}}}$$ adjusted to map the z-axis on the equator. Taking into account the static field along z, we expect the pulse to map the equatorial direction of azimutal angle $$\phi _{\mathrm{i}} \simeq 0.35{\mathrm{rad}}$$ on the z-axis. An arbitrary angle ϕ = ϕi + ϕL can be reached using an additional wait time before the By pulse, allowing for a Larmor precession of angle ϕL. The calculation of the angle ϕLuses the magnetic field component Bz measured using an external probe, allowing to reduce the effect of shot-to-shot magnetic field fluctuations.

### Spin dynamics modeling

Quantitative understanding of the observed spin dynamics requires taking into account experimental imperfections. We include the linear Zeeman coupling induced by the magnetic field applied along z (see Eq.(1)), leading to a small Larmor rotation on the typical timescales used for the light-induced spin dynamics. We also take into account the slight polarization ellipticity expected from the focusing of the laser beam on the atomic sample (beam divergence $$\theta = \lambda /(\pi w) \simeq 4{\mathrm{mrad}}$$). Finally, we improve the spin dynamics modeling by fitting a small angle mismatch $$\simeq 8^\circ$$ between the quantization field and the z-axis. More details on this modeling can be found in the Supplementary Note 2.

### Quantum state tomography

The density matrix of the kitten state is determined from a least-square fit of the measured spin projection probabilities Πm along z and Πm(ϕ) on equatorial directions60. We uniformly sample the equatorial plane using a set of azimutal angles ϕ[ϕ0, ϕ0 + π]. The procedure thus requires variable spin rotation durations (on average 10 μs), which limits the quality of the tomography due to dephasing. To reduce its effect, we use the magnetic field values measured for each experiment with an external probe to compensate for part of the dephasing, which increases the quality of the tomography and extents the coherence times by a factor $$\simeq 3$$. The robustness of the method with respect to measurement noise and finite sampling is tested using a random-weight bootstrap method, from which we define the statistical error bars in Fig. 5.

### Calibration of dephasing

To calibrate the dephasing of coherences due to magnetic field fluctuations, we perform a Ramsey experiment using coherent spin states. We start in the ground state |−Jz, that we bring on the equator using a π/2 magnetic field pulse applied along y. We then let the spin precess around z for a duration t, and subsequently perform a second π/2 pulse before performing a spin projection measurement along z. We observe Ramsey oscillations of the magnetization mz(t) = J(t)cos(ωLt + ϕ), where the local oscillation contrast J(t) corresponds to the transverse spin amplitude shown in Fig. 5b.