Detection of a weak magnetic field via cavity-enhanced Faraday rotation

We study the sensitive detection of a weak static magnetic field via Faraday rotation induced by an ensemble of spins in a bimodal degenerate microwave cavity. We determine the limit of the resolution for the sensitivity of the magnetometry achieved using either single-photon or multiphoton inputs. For the case of a microwave cavity containing an ensemble of nitrogen-vacancy defects in diamond, we obtain a magnetometry sensitivity exceeding $5.2\mathrm{n}\mathrm{T}/\sqrt{\mathrm{Hz}}$ utilizing a single-photon probe field, while for a multiphoton input we achieve a subfemtotesla sensitivity using a coherent-probe microwave field with power of $P_{\text{in}}=1\mathrm{n}\mathrm{W}$.

Figure 1

Schematics of setup for detection of a weak static magnetic field $B$. An ensemble of spins couples to a microwave (mw) cavity. The horizontal-polarized mw field ${\alpha }_{\text{in}}$ inputs into the mw cavity and then is reflected off the cavity $\left({\alpha }_{\mathrm{out}}\right)$ to the detector. The output field reflected off the cavity is detected by the polarization-resolving photon detector D. The polarization of the output field is rotated by an angle ${\phi }_F$ due to the Faraday effect. The cavity has the resonance frequency ${\omega }_r$ and the intrinsic loss ${\kappa }_{\text{i}}$. The input mw field couples to the cavity with an external coupling rate ${\kappa }_{\text{ex}}$.

Figure 2

Configuration describing the interaction between the cavity mode ${\widehat{a}}_{\pm }$ and an ensemble of ${\mathrm{NV}}^{-}$ centers. Two degenerate cavity modes, the right circular-polarized mode ${\widehat{a}}_{+}$ and the left circular-polarized mode ${\widehat{a}}_{-}$, are detuned from the zero-strain splitting by ${\mathrm{\Delta }}_q$. We shift the transition $|m_s=1\rangle \leftrightarrow |m_s=0\rangle \left(\right|m_s=-1\rangle \leftrightarrow |m_s=0\rangle )$ by a bias static magnetic field $B_0$, which creates a static frequency shift $A={\mu }_B{g}_e{B}_0$. The weak static magnetic field $\delta B$ causes another small shift $\delta ={\mu }_B{g}_e\delta B$.

Figure 3

Probabilities $P\left(\xi \right|\delta /{\kappa }_i)$ (a) and derivation of probabilities $\partial P\left(\xi \right|\delta )/\partial \delta$ (b) of three outputs as a function of the magnetic-field-induced level shift $\delta$. ${\mathrm{\Delta }}_b={\mathrm{\Delta }}_a={\mathrm{\Delta }}_q=0,A=0,{\kappa }_{ex}={\kappa }_i,G={\kappa }_i,\gamma ={10}^{-3}$. Blue lines indicates the probability of the vertical polarization $\left(P_V\right)$, red lines for the horizontal-polarized outcome $\left(P_{\text{H}}\right)$, and green lines for the probability of detecting zero photon $\left(P_{⌀}\right)$. Small circles are the average over 500 simulations, with ${\mathrm{\Delta }}_q$ randomly varying within $10\%{\kappa }_i$, around the mean frequency, ${\mathrm{\Delta }}_q=0$.

Figure 4

Fisher information as a function of the magnetic-field-induced level shift $\delta$. ${\mathrm{\Delta }}_r={\mathrm{\Delta }}_q=0,A=0,G={\kappa }_{ex}={\kappa }_{\text{i}},\gamma ={10}^{-3}{\kappa }_{\text{i}}$.

Figure 5

Sensitivity scaled by $\frac{\sqrt{{\kappa }_{\text{i}}}}{{\mu }_B{g}_e}$ as a function of the external coupling ${\kappa }_{\text{ex}}$ and the cavity-spin coupling $G$. $G/{\kappa }_{\text{i}}=(0.02,0.1,0.2)=$ (solid blue line, dashed red line, dot-dashed yellow line). ${\mathrm{\Delta }}_r={\mathrm{\Delta }}_q=0$.

Figure 6

Fisher information scaled by ${\left(\frac{{\mu }_B{g}_e}{{\kappa }_{\text{i}}}\right)}^2$ as a function of the spin-cavity coupling $G$ and the magnetic-field-induced level shift $\delta$. ${\mathrm{\Delta }}_r={\mathrm{\Delta }}_q=0$ and ${\kappa }_{\text{ex}}=10{\kappa }_{\text{i}}$.

Figure 7

Fisher information as a function of the intercavity coupling $J$ and the detuning $\delta$. ${\mathrm{\Delta }}_b={\mathrm{\Delta }}_a={\mathrm{\Delta }}_q=0,A=0,{\kappa }_{\text{ex}}/{\kappa }_{\text{i}}=10,\gamma /{\kappa }_{\text{i}}={10}^{-3},G/{\kappa }_{\text{i}}=0.1$.