Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom

Coherent superposition states of a mesoscopic quantum object play a major role in our understanding of the quantum to classical boundary, as well as in quantum-enhanced metrology and computing. However, their practical realization and manipulation remains challenging, requiring a high degree of control of the system and its coupling to the environment. Here, we use dysprosium atoms—the most magnetic element in its ground state—to realize coherent superpositions between electronic spin states of opposite orientation, with a mesoscopic spin size J = 8. We drive coherent spin states to quantum superpositions using non-linear light-spin interactions, observing a series of collapses and revivals of quantum coherence. These states feature highly non-classical behavior, with a sensitivity to magnetic fields enhanced by a factor 13.9(1.1) compared to coherent spin states—close to the Heisenberg limit 2J = 16—and an intrinsic fragility to environmental noise.


Supplementary Note 1: Ideal spin dynamics
We present here the expected dynamics for a pureĴ 2 x coupling. By decomposing the initial state |−J z on the x basis {|m x }, we find the evolved state as The magnetization and spin projection variance along z were calculated analytically in Ref. [1], as corresponding to the dashed red lines in Fig. 2b and c. The collapse of quantum coherence results from the relative dephasing between the various m 2 ωt phases. For large J values, the evolution of m z and ∆J 2 z are well captured by a gaussian decay on a timescale t c = 1/( √ 2Jω), as We show in the Supplementary Figure 1 the evolution of the magnetization and spin projection variance for J = 8. We find that the gaussian approximations of Eqs. (4) and (5) reproduce very well the exact formulas (2) and (3) during the entire collapse (0 < ωt < 0.3π). This gaussian decay of the magnetization and spin projection variance ceases to be valid close to ωt = π/2, at which all even (odd) m phase factors get rephased together, leading to a quantum superposition between |−J and |J states. More generally, for ωt = nπ/2 (n integer), one expects the formation of the quantum states (6) |ψ = (−1) n/2 J , n even.
Supplementary Note 2: Spin dynamics modeling The non-linear spin dynamics results from the spindependent light shifts caused by laser light whose wavelength is close to the optical transition at λ 0 = 626 nm, that couples the electronic ground state to an excited level of angular momentum J = J + 1. The light frequency is detuned from resonance by ∆ = −2π × 1. The light-induced spin coupling, whose structure depends on the light polarizationû, is obtained using second-order perturbation theory aŝ where c is the speed of light, Γ = 0.85(3) µs −1 is the resonance linewidth [2], ω 0 = 2πc/λ 0 , and I is the light intensity. TheĴ 2 x coupling used to create the superposition state is achieved for a linear polarizationû =x. We discuss below several types of experimental imperfections affecting the spin dynamics.

Finite extent of the atomic sample
We first take into account the variation of intensity and polarization over the atomic sample. We focus on the atomic sample a gaussian beam linearly polarized along x and propagating along y, of waist w 50 µm at the atom position. Due to the beam focusing, we expect a slight polarization ellipticity away from the optical axiŝ u x+iθx/wŷ, where θ = λ/(πw) 4 mrad is the beam divergence. The dynamics shown in Fig. 2 is consistent with a 1/e cloud size σ = 7.3(3) µm, in agreement with the size calculated from the trap geometry and the gas temperature. Given this cloud extent, we expect r.m.s. intensity variations between atoms δI/I = 6% and an ellipticity typically corresponding to a Stokes parameter s 3 = 10 −3 .

Quantization magnetic field
We include the effect of the applied magnetic field of amplitude B = 18.5(3) mG, leading to a linear Zeeman coupling. We fit its orientation from the measured spin dynamics, consistent with an angular mismatch between the quantization field directionb (of components [0.09, −0.11, 0.98] x,y,z ) and the z axis.

Imperfect spin polarization
The atomic gas is prepared in the absolute ground state |−J z under a strong magnetic field B z = 0.5 G. The field is ramped to the final value B z = 18.5(3) mG in 20 ms, during which dipole-dipole interactions lead to a slight promotion of 3% of the atoms into the state |−J + 1 z . We take into account the spin dynamics undergone by these atoms.
Light shift correction to second-order perturbation theory Given the small detuning from resonance, we calculate the first correction to second-order perturbation theory in the light shift. For a light field linearly polarized along x, we obtain the expression (10) The first correcting term leads to a renormalization of the coupling frequency ω, corresponding to a ∼ 20% reduction of ω for our experimental parameters. The expected (renormalized) value ω = 2π × 1.95(10) MHz agrees well with the value ω = 2π × 1.98(1) MHz fitted from the spin dynamics. The second term ∝Ĵ 4 x leads to a slight modification of the spin dynamics, but its effect remains below the experimental noise.

Light intensity response time
The light pulse shape is controlled using an acousto-optic modulator, leading to a finite response time in the 10-100 ns range. By solving numerically the spin dynamics with the actual pulse shape, we estimate that the finite response time leads to minor differences compared to square pulses of same area.

Incoherent light scattering
We model the effect of incoherent light scattering, taking into account Rayleigh and Raman scatterings using a Monte Carlo wavefunction method [3] . For the detuning chosen in our experiments, we estimate a light scattering probability of 0.7% for the light pulse duration required to produce the superposition state. We compare the effects of the different imperfections in Tab. I. For each imperfection, we calculate the resulting decrease of the metrological gain G with respect to the maximum value of 2J = 16. The dominant imperfection stems from the finite extent of the atomic gas, leading to inhomogeneous light intensity and to polarization ellipticity. Combining all effects together, we estimate a maximum metrological gain of G = 14.5, consistent with the measured value G = 13.9(1.1).

Supplementary Note 3: Metrological gain versus measurement scheme
We discuss in the main text two methods to probe magnetic fields using the kitten state, based on the evolution of the projection probability distributions Π m (φ) along equatorial directions shown in Fig. 3a. The first method uses the oscillatory behavior of the parity P (φ) of these distributions, and the second is based on the variation of the Hellinger distance d H (φ, φ ) between probability distributions for different phases φ and φ .
A similar analysis can be performed using the nonlinear detection scheme, which leads to the probability distributions Π m (φ) shown in Fig. 3b. We first exploit the measured magnetization oscillations shown in Fig. 3c. We evaluate the measurement precision using the general formula (9) with the observableÔ =Ĵ z , leading to the expression of the metrological gain G = 2J(A/∆J z ) 2 , where A = 6.0(2) is the oscillation amplitude and ∆J 2 z = 49(1) is the spin projection variance (measured for the m z 0 data). A second measurement scheme consists in quantifying the variations of the probability distributions Π m (φ) using the Hellinger distance, following the same procedure than for the data of Fig. 3a.
We show in Tab. II the metrological values obtained from the four measurement schemes discussed above. For both data sets we find that exploiting the variations of the Hellinger distance d H (φ, φ ) leads to the highest G values, both of them being compatible with the upper bound 2∆J 2 z /J = 14.3(1). Supplementary Note 4: Dephasing due to magnetic field fluctuations As discussed in the main text, we mainly attribute the Supplementary Tab. II. Metrological gain values corresponding to the measurement schemes discussed in the main text. The first scheme consists in measuring the spin projection on equatorial directions (see Fig. 3a), and using either the parity oscillations or the variations of the full probability distributions. The second scheme uses a non-linear evolution, and using the magnetization oscillations or the variations of the full probability distributions (see Fig. 3b).
observed decoherence to magnetic field fluctuations along z. The dephasing due to this classical noise can be modeled using standard techniques from nuclear magnetic resonance [4][5][6] . We write the magnetic field as B z (t) = B 0 z + b(t) with b(t) = 0. Such a field induces a noise in the Larmor rotation angle For a coherent state prepared on the equator, this noise leads to the decay of transverse spin J ⊥ (t) = J e iδφ(t) . For a superposition state |ψ kitten it reduces the extremal coherence as |ρ −J,J | = e i2Jδφ(t) . A purely Markovian evolution would be expected for white magnetic field noise, leading to phase diffusion behavior [7]. The Markovian approximation corresponds to phase diffusion without memory, corresponding to b(t)b(t ) = 2D δ(t − t ) [8] . For gaussian noise statistics, we use e inδφ(t) = e −n 2 δφ(t) 2 /2 , with a diffusive phase noise δφ(t) 2 = 2D(µ B g J / ) 2 t, leading to exponential damping of coherences, of 1/e times τ 0 = ( /µ B g J ) 2 D for the transverse spin J ⊥ and τ = τ 0 /(2J) 2 for the extremal coherence |ρ −J,J |.
In our experiment, we rather expect typical magnetic field variations to occur on timescales much larger than the ∼ 100 µs decoherence timescale. In this regime, the magnetic field can be considered as static during a single realization of the experiment, and decoherence arises from shot-to-shot fluctuations. We then expect δφ(t) = µ B g J bt/ , which obviously leads to the relationship e i2Jδφ(t) = e iδφ(2Jt) .
This relationship implies that the damping of extremal quantum coherences after a duration t is equal to the damping of the coherence of coherent states after a duration 2Jt, consistently with our observations. Since each atom carries a magnetic moment ∼ 10 µ B , we also expect an additional magnetic field created by the atomic sample itself, but we estimate its contribution to the coherence damping rate to be one order of magnitude smaller than external magnetic field effects.