Partitioning dysprosium's electronic spin to reveal entanglement in nonclassical states

Quantum spins of mesoscopic size are a well-studied playground for engineering nonclassical states. If the spin represents the collective state of an ensemble of qubits, its nonclassical behavior is linked to entanglement between the qubits. In this paper, we report on an experimental study of entanglement between two subsystems of 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 $J^{\prime }=J-1$, 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 nonclassical 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 $J^{\prime }=J+1$ and interpret spontaneous emission as a loss of a qubit pair in a random state. This allows us to contrast the robustness of nonclassical pairwise correlations of the W state with the fragility of the coherence involved in a Schrödinger 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. Qubit ensembles with large entanglement depth could then be realized with a few atoms only, facilitating the scaling up of quantum-enhanced sensors.

Figure 1

Scheme of the experiments manipulating qubit pairs in the electronic spin of dysprosium. An electronic spin of angular momentum $J$ can be viewed as a set of $2J$ virtual qubits symmetric upon exchange. (a) The coherent coupling to an excited state $J^{\prime }=J-1$ with ${\sigma }_{-}$ polarized light probes the probability to find a qubit pair polarized in ${|\uparrow \uparrow \rangle }_z$. (b) The spontaneous emission from an excited state $J^{\prime }=J+1$ removes a random pair of qubits.

Figure 2

Husimi function measurement for Dicke states. (a) Scheme of the light shift measurement. We measure the force induced on the atoms by an off-centered laser beam, blue detuned with respect to the optical resonance. (b) Image of an atomic gas prepared in a coherent state of polar angle $\theta \simeq {100}^{\circ }$. The atoms are kicked along $x$ by the laser beam. Subsequently, we apply a magnetic field gradient separating the magnetic sublevels $|m\rangle$ along $z$ during time of flight. The dashed line indicates the mean $x$ position in the absence of the repulsive laser beam. (c) Probability $Q_m$ for a qubit pair taken in the Dicke state $|m\rangle$ to be in ${|\uparrow \uparrow \rangle }_z$, deduced from the kick amplitudes. In all figures, error bars represent the $1\sigma$ statistical uncertainty (here smaller than the blue circles). The black lines are the theoretical values of Eq. (1).

Figure 3

Qubit pair properties of coherent and W states. (a) and (b) Measured spin projection probabilities ${\mathrm{\Pi }}_m$ as a function of the polar angle $\theta$, for a coherent spin state (a) and for the W state (b). The red vertical lines indicate the expected maxima for the coherent state, also corresponding to minima for the W state. The top panels represent the considered spin-$J$ states on the Bloch sphere, where the red circles indicate the spanned measurement projection axis. (c) Pair Husimi function $Q_{\text{pair}}$ computed from the (a) and (b) data (blue circles and red squares, respectively). The lines correspond to the expected functions $Q_{\text{pair}}\left(\theta \right)$ for the coherent and W states (red and blue lines). (d) Distribution ${\mathcal{C}}_{\mathbf{n}}$ of nonclassical correlations as a function of the polar angle $\theta$. The points ${\mathcal{C}}_{\mathbf{n}}>0$ measured for the W state evidence nonclassicality.

Figure 4

Qubit pair properties for a squeezed state. (a) and (b) Measured spin projection probabilities ${\mathrm{\Pi }}_m$ for a squeezed spin state, as a function of the polar angle $\theta$ with azimuthal angles ${\varphi }_{\text{min}}$ (a) and ${\varphi }_{\text{max}}$ (b). (c) Spin projection uncertainty $\mathrm{\Delta }{J}_{\mathbf{n}}$ computed from the (a) and (b) data (blue circles and red squares, respectively). The lines correspond to the projection uncertainties expected for the targeted spin state. (d) Distribution ${\mathcal{C}}_{\mathbf{n}}$ of nonclassical correlations as a function of $\theta$.

Figure 5

Characterization of entanglement in a Schrödinger cat state. (a) Measured spin projection probabilities ${\mathrm{\Pi }}_m$ for a cat state, as a function of the polar angle $\theta$. The azimuthal angle $\varphi =0.86\left(5\right)$ rad is chosen such that the two coherent-state Husimi functions destructively interfere for odd $m$ values around $\theta =\pi /2$. (b) Distribution ${\mathcal{C}}_{\mathbf{n}}$ inferred from the probabilities shown in (a) (blue circles). The solid line is the expected variation for a perfect cat state. (c) Projection probabilities ${\mathrm{\Pi }}_m$ measured along equatorial directions ($\theta =\pi /2$) parametrized by the azimuthal angle $\varphi$. (d) Evolution of the mean parity $\langle \mathcal{P}\rangle$ deduced from (c). (e) Projection probabilities ${\mathrm{\Pi }}_m$ measured after a Larmor rotation of angle $\varphi$ followed by a second nonlinear evolution. (f) Evolution of the mean sign of even projections $\langle \mathrm{\Sigma }\rangle$ deduced from (e). The solid lines in (d) and (f) are fits with a Fourier series.

Figure 6

Loss of a qubit pair in a W state. (a) Scheme for the preparation of the W state in the excited electronic level. (b) Evolution of the mean atom velocity acquired due to the photon absorption recoil, as a function of the light pulse duration. The dashed line is a model taking into account spontaneous emission during the pulse. (c)–(e) Top panels: expected states, with a scheme of spontaneous emission in (d) showing the Clebsch-Gordan coefficients for the two possible quantum jumps. Bottom panels: spin projection probabilities in the absence of the resonant light pulse (c), for a $\pi$ pulse (d), and for a $2\pi$ pulse (e). The solid lines are the probabilities expected for a perfect W state, while the dashed lines use the same model as in (b).

Figure 7

Loss of a qubit pair from superposition states. (a) Preparation method for the Schrödinger cat state $|{\psi }_1\rangle$ in the excited electronic level. Given the small values of their Clebsch-Gordan coefficients, we neglect the couplings between $|m=\pm 8\rangle$ and $|m^{\prime }=\pm 7\rangle$. (b) Scheme of the subsequent spontaneous emission. (c) Top panel: spin projection probabilities measured in the $xy$ plane, as a function of the azimuthal angle $\varphi$. Bottom panel: The corresponding sign observable $\langle \mathrm{\Sigma }\rangle$, together with a fit with a Fourier series. The $y$-axis range has been reduced compared with Fig. 5 to highlight the absence of oscillation. (d)–(f) show the same information for the superposition state $|{\psi }_2\rangle =\left(\right|m^{\prime }=-8\rangle +|m^{\prime }=8\rangle )/\sqrt{2}$.

Figure 8

Proposed scheme for entangling several Dy atoms in an optical resonator. An off-resonant optical cavity in the strong-coupling regime couples an ensemble of $N$ atoms together. For ${\sigma }_{+}$ polarized cavity light, the total spin projection along $z$ is conserved, and the cavity mediates the coherent exchange of $\uparrow$ qubit excitations between atoms. Such couplings can be used to stabilize a W state, with a single $\uparrow$ excitation symmetrically shared between the $N\times \left(2J\right)$ qubits.

Figure 9

Deviation from $z$ rotation symmetry in the prepared W state. (a) and (b) Projection probabilities ${\mathrm{\Pi }}_m$ as a function of the polar angle $\theta$, for ${\varphi }_1=0.36\left(5\right)$ rad and ${\varphi }_2={\varphi }_1-\pi /2$. (c) Pair Husimi functions $Q_{\text{pair}}$ inferred from the (a) and (b) data (blue circles and red squares, respectively). The error bars represent the statistical uncertainty from a bootstrap random sampling analysis. The line corresponds to the expected variation for the W state. (d) Distribution ${\mathcal{C}}_{\mathbf{n}}$ as a function of $\theta$. The two azimuthal angles ${\varphi }_1$ and ${\varphi }_2$ are chosen to minimize and maximize the measured ${\mathcal{C}}_{\mathbf{n}}$, respectively.

Figure 10

Deviation from $z$ rotation symmetry in the prepared Schrödinger cat state. (a) Pair Husimi functions $Q_{\text{pair}}$ as a function of the polar angle $\theta$, for ${\varphi }_1=3.3\left(1\right)$ rad and ${\varphi }_2={\varphi }_1-\pi /2$ (blue circles and red squares, respectively). The line corresponds to the expected variation for a perfect cat state. (b) Distribution ${\mathcal{C}}_{\mathbf{n}}$ as a function of $\theta$ deduced from the data in (a).

Figure 11

(a) Scheme of the $2\pi$ Rabi oscillation starting in a Schrödinger cat state of the electronic ground level, for an $x$-polarized laser excitation. (b) Top panel: spin projection probabilities measured in the $xy$ plane, as a function of the azimuthal angle $\varphi$. Bottom panel: the corresponding sign observable $\langle \mathrm{\Sigma }\rangle$, together with a fit with a Fourier series. (c) and (d) show the same information for a $z$-polarized laser excitation.