Spin Squeezing via One-Axis Twisting with Coherent Light

We propose a new method of spin squeezing of atomic spin, based on the interactions between atoms and off-resonant light which are known as paramagnetic Faraday rotation and the fictitious magnetic field of light. Since the projection process, squeezed light, or special interactions among the atoms are not required in this method, it can be widely applied to many systems. The attainable range of the squeezing parameter is $\zeta \gtrsim S^{-2/5}$, where $S$ is the total spin, which is limited by additional fluctuations imposed by coherent light and the spherical nature of the spin distribution.

Figure 1

System of our proposal. A linearly polarized light pulse passes through an atomic ensemble and the polarization plane is rotated. The rotation angle is proportional to $S_z$ and converts to the circular polarization components after passing through the $\lambda /8$ plate twice. When the pulse passes through the atomic ensemble again, the pulse induces a nonlinear rotation to the atomic ensemble around the $z$ axis as a fictitious magnetic field. See text for details.

Figure 2

State evolutions expressed as the quasiprobability distribution for $S=20$. The value of QPD for the $(\theta ,\varphi )$ direction is represented by the gray scale on the unit sphere, which is normalized by the maximum value. (a) The initial spin state. (b) The spin state after the first interaction, where we have set $(\alpha t_1{)}^{2}J/2=0.1$ and $t_2=0$; in other words, $\mu =0$ and ${\mu }^{\prime }=0.1$. (c) The spin state after the second interaction, where we have set $(\alpha t_1{)}^{2}J/2=(\alpha t_2{)}^{2}J/2=0.1$; in other words, $\mu ={\mu }^{\prime }=0.2$. The spin squeezing is realized along the $z^{\prime }$ axis.

Figure 3

(a) Variances of the $z^{\prime }$ (circle below 1) and the $y^{\prime }$ (circle above 1) components for $S=20$ as a function of $\mu$ (${\mu }_{\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f}}=0.117$, ${\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}=0.236$). They are normalized as $2⟨\Delta {\tilde{S}}_{z^{\prime }}^2⟩/S$ and $2⟨\Delta {\tilde{S}}_{y^{\prime }}⟩/S$, respectively. We also plot the approximate value of $2⟨\Delta S_{z^{\prime }}^2⟩/S$ (solid line). (b) Values of ${\mu }_{\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f}}$ (triangle) and ${\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ (square) as a function of $S$, which are the required values of $\mu$ to obtain the half variance of the SQL and the attainable minimum variance, respectively. We also plot the approximate solutions of ${\mu }_{\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f}}$ (dashed line) and ${\mu }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ (solid line). To obtain both (a) and (b), we have assumed $\mu ={\mu }^{\prime }$.