Spin squeezing in an ensemble of nitrogen-vacancy centers in diamond

Abstract

Spin squeezed states provide a seminal example of how the structure of quantum mechanical correlations can be controlled to produce metrologically useful entanglement. Such squeezed states have been demonstrated in a wide variety of artificial quantum systems ranging from atoms in optical cavities to trapped ion crystals. By contrast, despite their numerous advantages as practical sensors, spin ensembles in solid-state materials have yet to be controlled with sufficient precision to generate targeted entanglement such as spin squeezing. In this work, we present the first experimental demonstration of spin squeezing in a solid-state spin system. Our experiments are performed on a strongly-interacting ensemble of nitrogen-vacancy (NV) color centers in diamond at room temperature, and squeezing (-0.5 $\pm$ 0.1 dB) is generated by the native magnetic dipole-dipole interaction between NVs. In order to generate and detect squeezing in a solid-state spin system, we overcome a number of key challenges of broad experimental and theoretical interest. First, we develop a novel approach, using interaction-enabled noise spectroscopy, to characterize the quantum projection noise in our system without directly resolving the spin probability distribution. Second, noting that the random positioning of spin defects severely limits the generation of spin squeezing, we implement a pair of strategies aimed at isolating the dynamics of a relatively ordered sub-ensemble of NV centers. Our results open the door to entanglement-enhanced metrology using macroscopic ensembles of optically active spins in solids.

Figures (4)

• Figure 1: Twisting dynamics of a strongly-interacting, two-dimensional NV ensemble.(a) Schematic depiction of a two-dimensional ensemble of NV centers. NVs are confined to a $\sim7$ nm layer (green) formed via nitrogen delta-doping during diamond growth. Each NV exhibits an electronic spin-1 ground state and we encode an effective two-level system in the $\{\ket{m_s = 0}, \ket{m_s = -1}\}$ subspace. NV centers are randomly positioned within the diamond lattice, leading to the presence of strongly coupled dimers. (b) Depicts the orientation of our diamond lattice with one subgroup of NV centers aligned in the [111] lattice direction, perpendicular to the delta-doped layer. Black, blue, and white spheres represent vacancies, nitrogen, and carbon atoms, respectively. (c) Shows the dynamics of a spin polarized initial state under $H_\textrm{XXZ}$. Initial states (represented by the yellow arrow on the Bloch sphere) are offset above (or below) the $x$-axis by an angle $\varphi_o$. $H_\textrm{XXZ}$ induces effective "twisting" dynamics where an initial state precesses about the $z$-axis at a rate proportional to $\expval{S_z}$; the average precession angle in the equatorial plane is given by $\varphi_p$. Colored lines connect data points for different initial states at the same evolution time. (d) Analogous data from (c) showing the precession angle as a function of time for initial states with different offsets above (or below) the $x$-axis.
• Figure 2: Interaction-enabled readout of the quantum projection noise.(a) Our experimental sequence consists of two steps. First, a spin squeezed state is generated via evolution under $H_\textrm{XXZ}$ for a time, $t_\textrm{g}$, starting from an initial spin-polarized state. After a variable global rotation by angle $\theta$ around the $x$-axis, the squeezed state’s anisotropic spin projection noise is read out via quench dynamics (for a time $t_\textrm{r}$) under $H_\textrm{XXZ}$. Throughout our experiments, we utilize an XY-8 dynamical decoupling sequence in order to isolate the NV ensemble from other paramagnetic defects in the diamond lattice (see Methods). (b) Schematic depiction of the intuition underlying interaction-enabled spin projection noise readout. The red patch represents the Wigner quasiprobability distribution (projection noise) of the collective spin in the plane spanned by $S_y$ and $S_z$. For a many-body spin system undergoing effective twisting dynamics, this distribution becomes sheared at a rate $\sim \chi \sqrt{\text{Var}(S_z)}$. This shearing yields a concomitant shrinking of the total spin length, $S_x$, and enables a measurement of the state's quantum variance, $\text{Var}(S_z)$, via the decay timescale of $\langle S_x (t) \rangle$. (c) Interaction-enabled readout in the one-axis twisting model. Initial states with different spin projection noise, $\text{Var}(S_z)$ (top), lead to different decay timescales for $\langle S_x(t) \rangle$ (bottom). A state exhibiting a smaller projection noise (blue) exhibits a slower decay than the initial spin-polarized state (gray), while a state exhibiting a larger projection noise (red) exhibits a faster decay. For the OAT model, the form of this decay can be analytically derived, $\langle S_x(t) \rangle \sim e^{-2\chi^2 \text{Var}(S_z) t^2} \equiv e^{-(t/T_2)^2}$. (inset) Depicts the mapping between the decay timescale, $T_2$, and the quantum variance (normalized by the variance of the initial spin-polarized state, see Methods). (d) Interaction-enabled readout in our disordered dipolar XXZ model. Unlike the OAT model, there does not exist a simple analytic mapping between $\text{Var}(S_z)$ and the decay timescale of $\langle S_x(t) \rangle$. Thus, we numerically simulate the quench dynamics under $H_\mathrm{XXZ}$ for different initial states (top) using the discrete-cluster truncated Wigner approximation. The dynamics of $\langle S_x(t) \rangle$ exhibit a similar qualitative dependence on the state's spin projection noise, $\mathrm{Var}(S_z)$, as in panel (c). (inset) Depicts the one-to-one mapping between the fitted $T_2$ decay timescale and the spin projection noise, $\text{Var}(S_z)$, of the initial state. (e) A spin-polarized initial state, $\ket{\mathbf{x}}$, is evolved under $H_\text{XXZ}$ for a time $t_\mathrm{g} = 3.2\ \mu\mathrm{s}$. We measure the subsequent quench dynamics for different rotation angles $\theta$ (inset), and the $S_x(t_r)$ decay is normalized by $S_x(t_\mathrm{g} = 0)$. For each angle, the state's projection noise is squeezed along a different axis, and we redefine the time, $t_\mathrm{r} \rightarrow t_\mathrm{r}^\mathrm{eff}$, in order to account for this (see Methods). Consistent with our numerical simulations (d), certain rotation angles yield a longer effective decay timescale (blue), while other rotation angles yield a shorter effective decay timescale (red). For each angle, a timescale $T_2$ is extracted from the decay in the striped fitting time window by fitting the data to $e^{-(t/T_2)^{2/3}}$. These timescales (purple data) are shown as a function of $\theta$ in (f). The shaded purple region surrounding the data correspond to changes in the extracted $T_2$ as a function of changing the fitting window (from $3.2-25$$\mu$s to $3.2-35$$\mu$s), and demonstrates the robustness of our approach. By using the numerically determined mapping (d), one can immediately convert the decay timescale to the quantum variance of the state (green). Note that the mapping we utilize accounts for experimental imperfections such as the polarization fidelity of the initial state (see Methods). (g) In order to probe the dynamics of spin squeezing, we evolve the spin-polarized initial state for different times, $t_\textrm{g}$. At each time, we perform our interaction-enabled readout protocol, and extract the normalized quantum variance, $\mathrm{Var}(S_\theta) / \mathrm{Var}(S_{\theta=0})$ as a function of $\theta$. This allows us to determine the minimum variance as a function of $t_\textrm{g}$ (red data). We also directly measure the decay of the collective spin length as a function of $t_\textrm{g}$ (navy data). Taken together, this enables us to compute the dynamics of the squeezing parameter, $\xi^2$ as a function of the preparation time (h). We find that $\xi^2$ exhibits a monotonic increase as a function of time; thus, despite the presence of an anisotropic spin projection noise distribution, the prepared state does not exhibit metrologically useful spin squeezing.
• Figure 3: Reduction of positional disorder via lattice engineering.(a) Highlights the difference in the distribution of dipolar interaction strengths, $P(J)$, for an ordered two-dimensional array (left) and a positionally disordered ensemble (right). For the disordered ensemble, $P(J)$ exhibits heavy tails corresponding to strongly-interacting clusters of spins, such as "dimers" (outlined in gray). $J = 0$ is indicated by the vertical dashed line. (b) Numerical (cluster DTWA) simulations of the squeezing dynamics of an ordered versus disordered NV ensemble with $N=100$. (top) Shows the minimum quantum variance and the collective spin length as a function of the squeezing generation time, $t_\textrm{g}$. (bottom) Shows the squeezing parameter as a function of $t_\textrm{g}$. Consistent with our experimental observations [Fig. \ref{['fig:fig2']}h], the disordered case (dotted lines) exhibits a monotonically increasing squeezing parameter, $\xi^2$. By contrast, for the lattice case (solid lines), the squeezing parameter is below unity. (c,d) Schematic illustrating two lattice-engineering approaches for reducing the amount of positional disorder in our NV ensemble. Both strategies attempt to isolate NVs in the central region of the interaction spectrum, $P(J)$, thereby eliminating strongly-coupled dimers in the tails of $P(J)$. The first approach (c), leverages shelving to the $|m_s = +1\rangle$ state. In particular, after optical pumping to $|m_s = 0 \rangle$, a weak microwave pulse with high-frequency selectivity is used to drive NVs in the central region of the interaction spectrum from $|m_s = 0 \rangle \rightarrow |m_s = -1 \rangle$ (light blue). Next, a strong microwave pulse is used to shelve the remaining NVs from $|m_s = 0 \rangle \rightarrow |m_s = +1 \rangle$ (dark blue), where they are decoupled from the subsequent squeezing dynamics. Finally, another microwave pulse is used to bring those NVs in $|m_s = -1 \rangle$ back to the $|m_s = 0 \rangle$ state. The second approach (d), uses a form of adiabatic depolarization. After optical pumping, we initialize the NV spins along the $x$-axis via a $\pi/2$-pulse. Next, we turn on a strong transverse field, $h_x S_x$, and slowly ramp this field down to a final value $h_x^f$. NVs in the central region of the interaction spectrum (with $|J| \lesssim h_x^f$) will maintain their initial $x$-polarization. By contrast, NVs in the tails of the interaction spectrum (with $|J| \gtrsim h_x^f$) will exhibit rapid, dipolar-induced depolarization, which effectively removes them from the subsequent squeezing dynamics (see Methods). (e,f) The functional form of the decay of $\langle S_x(t) \rangle$ can act as a proxy for the amount of positional disorder present in our NV ensemble. As previously shown in Fig. \ref{['fig:fig2']}(d), in the fully disordered case, $\langle S_x(t) \rangle \sim e^{-(t/T_2)^{2/3}}$. In the lattice case [as shown in panel (b)], one finds that $\langle S_x(t) \rangle$ instead scales as $\sim e^{-(t/T_2)^{2}}$. Thus, the stretch exponent of the decay of $\langle S_x(t) \rangle$ provides a metric for characterizing our lattice engineering. For both the shelving approach (e) and the adiabatic depolarization approach (f), we find that the stretch exponent can be improved from $\sim 0.8$ up to $\sim 1.4$, representing a significant reduction in the positional disorder. (g,h) A spin-polarized initial state, $\ket{\mathbf{x}}$, is evolved under $H_\text{XXZ}$ in the presence of lattice engineering (via the shelving approach) for a time $t_\mathrm{g} = 2.4µs$. Here, we demonstrate the read out of the quantum variance of the resulting state via the same interaction-based approach as before. (g) Displays the readout quench dynamics for different rotation angles $\theta$ in direct analogy to Fig. \ref{['fig:fig2']}(e). For each angle, a timescale $T_2$ is extracted from the decay in the striped time window by fitting the data to $e^{-t/T_2}$. We note that although the early-time stretch exponent is larger than one (panels e,f), this stretch exponent crosses over to $2/3$ at late times; thus we pick a stretch exponent of one as a simple interpolation that accurately represents the decay timescale during the striped time window. These timescales (purple data) are shown as a function of $\theta$ in (h). The shaded purple region surrounding the data correspond to changes in the extracted $T_2$ as a function of changing the fitting window (from $2.4-12$$\mu$s to $2.4-16$$\mu$s). By using the numerically determined mapping (explicitly including the lattice engineering, see Methods), one can immediately convert the decay timescale to the quantum variance of the state (green).
• Figure 4: Spin squeezing in a disordered NV ensemble. (a,d) Depict the decay of the collective spin length as a function of the squeezing preparation time, $t_\textrm{g}$, in the presence of lattice engineering. For both the shelving approach (a) and the adiabatic depolarization approach (d), we observe a significantly slower decay, compared to the fully disordered case (black data). For the shelving approach, colored data indicate different amounts of lattice engineering corresponding to the frequency selectivity (i.e. Rabi frequency, $\Omega$) of the microwave pulse, with $\Omega = 310$ kHz (purple data) and $\Omega = 125$ kHz (pink data). For the adiabatic depolarization approach, the transverse field starts at $h_x = 380$ kHz and is ramped down to $h_x = 27$ kHz over $24$$\mu$s. Dashed lines are a guide to the eye. (b,e) Depict the minimum variance as a function of $t_\textrm{g}$, in the presence of lattice engineering (see Methods). Experimental conditions for the two approaches are analogous to panels (a,d). Dashed lines are a guide to the eye. (c,f) Show the squeezing parameter as a function of $t_\textrm{g}$ for both lattice engineering strategies. In both cases, the squeezing parameter drops below unity for an extended period of evolution time. For the shelving method, we observe spin squeezing with an optimal squeezing parameter, $\xi^2 = 0.90(3)$, at a time $t_\mathrm{g} = 1.6\ \mu$s, while for the depolarization method, we observe spin squeezing with an optimal squeezing parameter, $\xi^2 = 0.89(2)$, at a time $t_\mathrm{g} = 1.6\ \mu$s. Shaded regions correspond to numerical predictions from cluster DTWA accounting for experimental imperfections, lattice engineering, and the uncertainty due to the average over positional disorder.