Quantum correlation in disordered spin systems: Applications to magnetic sensing

We propose a strategy to generate a many-body entangled state in a collection of randomly placed, dipolarly coupled electronic spins in the solid state. By using coherent control to restrict the evolution into a suitable collective subspace, this method enables the preparation of GHZ-like and spin-squeezed states even for randomly positioned spins, while in addition protecting the entangled states against decoherence. We consider the application of this squeezing method to improve the sensitivity of nanoscale magnetometer based on nitrogen-vacancy spin qubits in diamond.

Figure 1

(Color online) (a) System model: crystal with randomly placed electronic spins. (b) Control sequence: the Ising interaction is rotated along three axes to yield an isotropic interaction; adjusting the time delays, a small perturbation along the $z$ direction is retained, to obtain the one-axis squeezing operator.

Figure 2

(Color online) Decoupling sequence. The narrow bars are $\pi /2$ pulses around different axis in the $m_s=\pm 1$ manifold, while the wide bars are $\pi$ pulses. The overall pulse sequence comprises four MREV8 sequences embedded in a spin-echo sequence, with 34 pulses and a cycle time of $48\tau$. By varying the length of the time delays and the pulse phases, we obtain the squeezing Hamiltonians. The modified intervals are indicated by ${\tau }_{+}$ and ${\tau }_{-}$. For the one-axis squeezing (a) we obtain the first-order Hamiltonian $\overline{\mathcal{H}}=\left({\mathcal{H}}_H+ϵ{\mathcal{H}}_{zz}\right)/3$ (neglecting terms $\propto ϵ\text{sin}{\nu }^2$) and the linear Hamiltonian $S_z\to \sqrt{2}\left(S_z\text{cos}\nu +S_y\text{sin}\nu \right)/3$. For the two-axes case (b) the first-order Hamiltonian is $\overline{\mathcal{H}}=\left({\mathcal{H}}_H+ϵ{\mathcal{H}}_{DQ}\right)/3$ and the effective field is $\left(S_y-S_x\right)/3$. $\tilde{\mathcal{H}}$ is the direction of the internal Hamiltonian in the interaction frame. $B_z$ is the ac field to be measured.

Figure 3

(Color online) Simulations (eight spins) of the squeezing parameter squared ${\xi }^2$ under one-axis (squares) and two-axis (circles) squeezing control schemes, with (solid lines) and without (dotted lines) the presence of noise. The noise was modeled by a random magnetic field in the $z$ direction acting independently on each spin; noise parameters where $\Gamma =3\text{kHz}$, ${\tau }_c=100\mu \text{s}$. The spins were distributed randomly on the lattice of a quasi-2D diamond slab of depth 9 nm and area $30\times 30\text{nm}$. The density was ${10}^{18}{\text{cm}}^{-3}$ resulting in $D=0.4\text{MHz}$ and $E_{gap}=50\text{kHz}$. The control sequence of Fig. 2 was used with time delays of $1.4\mu \text{s}$ and 1 to 12 cycles.

Figure 4

(Color online) Magnetometer measurement scheme. (a) A possible magnetometer setup, with $\mathrm{NV}$ center implanted in a slab of diamond. (b) Optical and microwave control of a single $\mathrm{NV}$ center. Squeezing control sequences: (c) a squeezed state is prepared before applying an external magnetic field to measure. (d) Squeezing and Ramsey experiment are done simultaneously.

Figure 5

(Color online) Sensitivity to an ac field with 22 kHZ frequency in a quasi-2D geometry $\left[V={\left(300\text{nm}\right)}^2\times 10\text{nm}\right]$. Left: sensitivity for conversion efficiency $f=50\%$ (dotted lines) and $f=23\%$ (solid lines). Notice that only for $f\gtrsim 50\%$, the two-axis squeezing approach (righmost, black lines) is preferable to the echo-only sequence (leftmost, red lines). Right: sensitivity for $f=.90$ (or equivalently, for a 30-fold increase in $T_2$ time thanks to refocusing, with respect to the $T_2$ at the highest conversion achieved [30, 35]). Green line (dash-dotted): sensitivity under spin-echo, without any refocusing of the spin-spin coupling. Blue dashed line: sensitivity under MREV8 sequence. The control sequence was repeated $n_c=6$ times with a delay between pulses of $\delta \tau =1.5\mu \text{s}$. Red line: sensitivity under one-axis squeezing obtained with the same sequence parameters as above. Black dotted line: sensitivity under two-axis squeezing. The time delays ${\tau }_{+}$ and ${\tau }_{-}$ (see Fig. 2) were chosen in order to get an $ϵ$ as small as needed to restrict the evolution to the protected manifold and the number of cycles increased as needed to obtain the optimal squeezing time.