Twisting tensor and spin squeezing

A unified tensor description of quadratic spin squeezing interactions is proposed, covering the single- and two-axis twisting as special cases of a general scheme. A closed set of equations of motion of the first moments and variances is derived in Gaussian approximation, and their solutions are discussed from the prospect of fastest squeezing generation. It turns out that the optimum rate of squeezing generation is governed by the difference between the largest and the smallest eigenvalues of the twisting tensor. A cascaded optical interferometer with Kerr nonlinear media is proposed as one of possible realizations of the general scheme.

Figure 1

A possible scheme realizing the interaction. (a) Modes of two crossed resonators mix at a balanced beam splitter according to $c=(a+b)/\sqrt{2}$, and $d=(a-b)/\sqrt{2}$, and each of the four beams propagates through a nonlinear medium. This setup leads to the Hamiltonian of Eq. (9). (b) Chaining of the mode mixing plus nonlinearities leads to the general form of Hamiltonian (4).

Figure 2

Schematic pictures of the Hamiltonian $H$ (a, c, e) and squeezing rate $Q$ (b, d, f). The geometry of angles $\vartheta$, $\phi$, and $\alpha$ is shown in (b). The shades correspond to the mean value of the Hamiltonian and to the squeezing rate of a spin coherent state with the direction of $\vec{J}$, the lighter shade corresponds to higher values. The Hamiltonian of Eq. (4) is with $\vec{\omega }=0$ and diagonal twisting tensor with $\{{\chi }_{xx},{\chi }_{yy},{\chi }_{zz}\}=\{1,0,0\}$ (OAT: a, b), $\{1,0,0.5\}$ (TACT: c, d), and $\{1,0,0.8\}$ (general twisting: e, f).

Figure 3

Time evolution of the squeezing parameter, diagonal form of the twisting tensor with ${\chi }_x=1$ and ${\chi }_y=0$. (a) One axis squeezing with ${\chi }_z=0$ (green broken line), TACT with ${\chi }_z=0.5$ (full blue line), and general twisting with ${\chi }_z=0.8$ (red dash-dotted line), the number at each line being the number of particles $N$, and the symbol $\infty$ denotes the asymptotics of Eq. (33). (b) OAT with no rotation (green broken line) and with optimized rotation according to Eq. (37) (full blue line). Inset: Husimi function of the state with $N=60$, optimized rotation, and ${\chi }_x\tau =5$.