Spin squeezing and precision probing with light and samples of atoms in the Gaussian description

We consider an ensemble of trapped atoms interacting with a continuous-wave laser field. For sufficiently polarized atoms and for a polarized light field, we may approximate the nonclassical components of the collective spin angular momentum operator for the atoms and the Stokes vectors of the field by effective position and momentum variables for which we assume a Gaussian state. Within this approximation, we present a theory for the squeezing of the atomic spin by polarization rotation measurements on the probe light. We derive analytical expressions for the squeezing with and without inclusion of the noise effects introduced by atomic decay and by photon absorption. The theory is readily adapted to the case of inhomogeneous light-atom coupling [A. Kuzmich and T.A.B. Kennedy, Phys. Rev. Lett. 92, 030407 (2004)]. As a special case, we show how to formulate the theory for an optically thick sample by slicing the gas into pieces, each having only small photon absorption probability. Our analysis of a realistic probing and measurement scheme shows that it is the maximally squeezed component of the atomic gas that determines the accuracy of the measurement.

Figure 1

Uncertainty of $p_{\mathrm{at}}$ as function of time. The effective coupling is ${\kappa }^2=1.83\times {10}^6{\mathrm{s}}^{-1}$. The lower curve is without inclusion of atomic decay, and the upper curve includes atomic decay with a rate $\eta =1.7577{\mathrm{s}}^{-1}$ and photon absorption with $ϵ=0.028$. These values correspond, for example, to a $2\text{−}{\mathrm{mm}}^2$ interaction area, $2\times {10}^{12}\text{atoms},5\times {10}^{14}\text{photons}{\mathrm{s}}^{-1}$, 10 GHz detuning, and 852 nm light, appropriate for the ${}^{133}\mathrm{Cs}\mathbf{(}6S_{1∕2}(F=4)-6P_{1∕2}(F=5)\mathbf{)}$ transition with decay rate $3.1\times {10}^7{\mathrm{s}}^{-1}$ and corresponding atomic dipole moment $d=2.61\times {10}^{-29}\mathrm{C}\mathrm{m}$. Factors of order unity related to the coupling matrix elements among different states of the actual Zeeman substructure are omitted.

Figure 2

Uncertainty of the maximally squeezed component of the atomic gas $\left(P_{\mathrm{at}}\right)$ as a function of time. The higher-lying curves show the uncertainty in the symmetric collective parameter [Eq. (36)] for $n=10$ uniformly distributed values of ${\kappa }^2$ in $[0.9{\kappa }_0^2;1.1{\kappa }_0^2]$ (middle) and $[0.5{\kappa }_0^2;1.5{\kappa }_0^2]$ (upper). The central effective coupling is ${\kappa }_0^2=1.83\times {10}^6{\mathrm{s}}^{-1}$, and all other parameters are as in Fig. 1. The lower curve is the smallest eigenvalue of the covariance matrix, which is the same for the two ranges of ${\kappa }^2$ to the precision visible in the figure.

Figure 3

Uncertainty of the maximally squeezed component of the gas $\left(P_{\mathrm{at}}\right)$ as function of probing time for varying degrees of photon absorption. The percentage of photons absorbed is indicated as the solid curves. The gas is decomposed in $n$ slices, each absorbing ${ϵ}_i=0.028$ of the light intensity. From the lower to the upper curve the number of such slices attains the values $n=1$, 4, 8, 13, 25, and 50. Other physical parameters are as specified in Fig. 1.

Figure 4

As Fig. 3 but only for $n=4$ and $n=50$. Curves (1) and (3) represent the maximally squeezed component of the gas. Curves (2) and (4) display the uncertainty in the inhomogeneous collective variable $P_{\mathrm{eff}}$ of Eq. (35) as a function of time.

Figure 5

Uncertainty of the parameter $\theta$ as a function of time and for different variances of the coupling strength as specified in the text. The gas is sliced in $n=10$ pieces. The number of atoms and photons are as in the preceding figures. The value of ${\alpha }_i$ is 0.2236. The dashed curves show the limiting uncertainty in the $\theta$ parameter as estimated from Eq. (65) for the standard collective variable $P$ of Eq. (36) with the smallest variance in the coupling strength for the lowest dashed curve and the highest variance for the upper dashed curve. The constant horizontal solid line gives the limiting value of Eq. (64), as obtained by the maximally squeezed component $P_{\mathrm{eff}}$ of Eq. (35), and it is independent of the variance of the coupling strength. The decreasing solid curve is a collection of indistinguishable curves showing the numerical results for all the different variances of the coupling strength (see text).