The polarization and the fundamental sensitivity of $^{39}$K ($^{133}$Cs)-$^{85}$Rb-$^{21}$Ne co-magnetometer

We have studied the polarization of the 85Rb and the fundamental sensitivity of two types of hybrid optical pumping spin exchange relaxation free (SERF) co-magnetometers based on 39K-85Rb-21Ne and 133Cs-85Rb-21Ne with taking the wall relaxation rate into account,which has a big effect on the polarization and fundamental sensitivity,but it is often neglected in the experiments,respectively.The polarization of the 85Rb varies with the thirteen systemic parameters: the number density of buffer gas 21Ne and quench gas N2,frequency rate of pump beam,cell temperature,the photon number flux,mole fraction of 85Rb,cell effective radius,pump beam spot radius,power density of pump beam,input rotation velocity in the x and y direction,external magnetic field in the y and z direction,respectively.The number density of the 21Ne,photon number flux,pump beam spot radius and frequency rate will affect the polarization of the 85Rb greatly by affecting the the pumping rate of pump beam,a magnetic field produced by 85Rb atom spins and experienced by the 21Ne spins.We obtain a general formula on the fundamental sensitivity of the hybrid optical pumping SERF co-magnetometers due to shot-noise,where the fundamental sensitivity varies with the twelve systemic parameters: the number density of buffer gas and quench gas,frequency rate of pump beam,photon number flux,mole fraction of 85Rb,cell effective radius,pump beam spot radius,power density of pump beam,external magnetic field,measurement volume,cell temperature and measurement time.The highest fundamental sensitivity $7.5878\times10^{-11}$ $rad/s/Hz^{1/2}$ of 133Cs-85Rb-21Ne co-magnetometer is higher than the ultimate theoretical sensitivity $2\times10^{-10}$ $rad/s/Hz^{1/2}$ of K-21Ne co-magnetometer.The fundamental sensitivity limit of the hybrid optical pumping SERF co-magnetometers is superior to $3.7334\times10^{-12}$ $rad/s/Hz^{1/2}$.