Quantum metrology with superposition spin coherent states: Insights from Fisher information

We present a closed-form analytical description for the metrological performance of a generic superposition of spin coherent states (SSCS) used as probes for quantum phase estimation. Working in this framework, we derive transparent analytical expressions for the pertinent quantum Fisher information and identify a general class of spin coherent states enabling quantum metrology with Heisenberg-limit precision. Bloch-sphere analysis of antipodal SSCS shows that the phase-estimation precision attainable with such states increases with the distance from the equator of the Bloch sphere, growing from the shot-noise limit level for the equatorial states all the way up to the Heisenberg limit for SSCS that can be represented as superpositions of the poles of the Bloch sphere.

Figure 1

The CRB $\mathrm{\Delta }{\xi }_{\mathrm{CRB}}$ for the phase estimation error of $j=1$ SSCS (17) as a function of $\theta$ for $\varphi =0$ (dotted blue line), $\varphi =\pi /2$ (dashed black line), and $\varphi =\pi$ (red line).

Figure 2

The CRB $\mathrm{\Delta }{\xi }_{\mathrm{CRB}}$ for the phase estimation error of the $j=1$ SSCS $|\phi ,j\rangle =\mathcal{N}\left(\right|{\theta }_1,\phi ,1\rangle +|{\theta }_2,\phi +\varphi ,1\rangle )$ as a function of ${\theta }_1$ and ${\theta }_2$ for $\varphi =0$ (a) and $\varphi =\pi$ (b).

Figure 3

The CRB as a function of $\theta$ for $j$ = 3/2 (a), 2(b), and 7/2 (c) with $\varphi =0$ (dotted blue line), $\varphi =\pi /2$ (dashed black line), and $\varphi =\pi$ (red line).

Figure 4

The CRB $\mathrm{\Delta }{\xi }_{\mathrm{CRB}}$ for $|\phi ,j\rangle =\mathcal{N}\left(\right|{\theta }_1,\phi ,j\rangle +|{\theta }_2,\phi +\varphi ,j\rangle )$ as a function of ${\theta }_1$ and ${\theta }_2$ for $\varphi =\pi$ with $j=3$ (a) and $j=10$ (b).