Squeezing of spin-1 quantum states via a one-axis twisting Hamiltonian

Squeezing of quantum states is of great interest due to its application in high-precision measurement that can exceed standard quantum noise limit described by Heisenberg uncertainty relations. Here, we study the squeezing of quantum systems with $\text{SU}\left(3\right)$ symmetry via a one-axis twisting Hamiltonian and examine the intriguing connections between the squeezing of spin-1/2 and spin-1 systems. There are seven subalgebras of $\text{su}\left(3\right)$ Lie algebra. All these subalgebras are identical with $\text{su}\left(2\right)$ Lie algebra but not all of them have the same anticommutation relations as $\text{su}\left(2\right)$. Interestingly, squeezing parameters corresponding to spin-1 subalgebras depend not only on structure constants but also on anticommutation relations of the subalgebras. Our results are reported for the subalgebras with vanishing anticommutators and nematic squeezing, while in other cases our first-principle calculation recovers known results.

Figure 1

For large $N$, there is a linear dependence between the minimum attainable squeezing parameters and $N^{-2/3}$. This dependence is depicted by (i) a solid yellow line for spin 1/2, (ii) a solid black line for type-1 squeezing of coherent state $|0,0,N\rangle$, (iii) a red dashed line for type-2 squeezing of coherent state $|0,0,N\rangle$, and (iv) a blue dash-dotted line for type-2 squeezing of nematic state $|0,N,0\rangle$. Numerically calculated exact values are plotted by rectangles, circles, triangles, and dots.

Figure 2

The squeezing parameters of two spin-1 particles as a function of $\theta$. The minimum attainable values are depicted by rectangles.