Spin Squeezing in a Quadrupolar Nuclei NMR System

We have produced and characterized spin-squeezed states at a temperature of $26°\mathrm{C}$ in a nuclear magnetic resonance quadrupolar system. The experiment was carried out on ${}^{133}\mathrm{Cs}$ nuclei of spin $I=7/2$ in a sample of lyotropic liquid crystal. The source of spin squeezing was identified as the interaction between the quadrupole moment of the nuclei and the electric field gradients present within the molecules. We use the spin angular momentum representation to describe formally the nonlinear operators that produce the spin squeezing on a Hilbert space of dimension $2I+1=8$. The quantitative and qualitative characterization of this spin-squeezing phenomenon is expressed by a squeezing parameter and squeezing angle developed for the two-mode Bose-Einstein condensate system, as well as by the Wigner quasiprobability distribution function. The generality of the present experimental scheme points to potential applications in solid-state physics.

Figure 1

Experimental results of spin squeezing under the one-axis twisting model. The NSCS $|\zeta (\pi /2,\pi )⟩$ was evolved under the Hamiltonian ${\mathcal{H}}_{\mathrm{NMR}}^s$. The dynamics of spins was monitored in time steps of $\delta \tau =3\mu \mathrm{s}$ over the time interval [$0,{\nu }_Q^{-1}$]. (a) Theoretical Wigner quasiprobability distribution function computed from a density matrix at time steps ${\tau }_k$, where $k=4,17,40$, and 44. (b) Experimental Wigner quasiprobability distribution function calculated from the tomographed density matrix. (c) Dynamics of the squeezing parameter ($\xi$) from theoretical prediction (black solid line) and experimental results (dark green dots). The error bars are discussed in [51]. Blue open circles correspond to analysis of the Wigner quasiprobability distribution function of (a) and (b). (d) Dynamics of the squeezing angle (${\alpha }_{\xi }$).