Heisenberg-scaled magnetometer with dipolar spin-1 condensates

We propose a scheme to realize a Heisenberg-scaled magnetometer using dipolar spin-1 condensates. The input state of magnetometer is prepared by slowly sweeping a transverse magnetic field to zero, which yields a highly entangled spin state of $N$ atoms. We show that this process is protected by a parity symmetry such that the state preparation time is within the reach of the current experiment. We also propose a parity measurement with a Stern-Gerlach apparatus which is shown to approach the optimal measurement in the large atom number limit. Finally, we show that the phase estimation sensitivity of the proposed scheme roughly follows the Heisenberg scaling.

Figure 1

Schematic plot of the scheme.

Figure 2

Transverse field dependence of the lowest 10 energy levels of the Hamiltonian (2) for $N=20$ and $c=-0.145$. The magnetic field strength $B_x$ is a dimensionless quantity.

Figure 3

(a) $\mathcal{F}$ and ${\mathcal{L}}^2$ as functions of $N$ with the Hamiltonian $H_0$. (b) Comparison of the reduced Fisher information obtained with Hamiltonian $H_0$ and $H$. The sweeping rate is $|v_B|=10N$.

Figure 4

(a) $\mathcal{F}$ and ${\mathcal{L}}^2$ as functions of $v_B$ for the Hamiltonian $H_0$ with $N=1000$. (b) Comparison of the reduced Fisher information obtained via Hamiltonian $H_0$ and $H$ with $N=100$. The sweeping rate $v_B$ is a dimensionless quantity.

Figure 5

$\mathcal{F}$ vs $\delta B_z$ for various pulse numbers. Other parameters are $N=2000$ and $|v_B|=10N$. Here $\delta B_z$ is a dimensionless quantity and, for simplicity, the system is evolved with the Hamiltonian $H_0$.