Spin squeezing in dipolar spinor condensates

We study the effect of dipolar interactions on the level of squeezing in spin-1 Bose-Einstein condensates by using the single mode approximation. We limit our consideration to the SU(2) Lie subalgebra spanned by spin operators. The biaxial nature of dipolar interactions allows for dynamical generation of spin-squeezed states in the system. We analyze the phase portraits in the reduced mean-field space in order to determine positions of unstable fixed points. We calculate numerically the spin squeezing parameter showing that it is possible to reach the strongest squeezing set by the two-axis countertwisting model. We partially explain scaling with the system size by using the Gaussian approach and the frozen spin approximation.

Figure 1

Dipolar coefficients (a) $\beta /2|c_d/c_2|$ and (b) $\alpha /|c_d/c_2|$ calculated with the Gaussian ansatz for the SMA wave function (8). For ${}^{87}\mathrm{Rb}$ one has $|c_d/c_2|\simeq 0.09$. Both functions are bounded from below and above. A region of parameters where one of the integrals dominates can always be found. The same result was obtained in Ref. [42].

Figure 2

Mean-field phase portraits for (a) $\alpha =1,\beta =0$; (b) $\alpha =1,\beta =1$; (c) $\alpha =1,\beta =3$; (d) $\alpha =1,\beta =10$. Darker regions indicate lower energy. Stable fixed points are marked with green dots, and unstable ones with blue hexagons. The dashed lines mark positions of the nonisolated fixed points.

Figure 3

The inverse of the squeezing parameter ${\xi }^{-2}$ as a function of $\beta$ with $\alpha =1$ calculated numerically for $c_2^{\prime }<0,N=25$ and the effective Hamiltonian (9). The initial state for the evolution is (a) $\left|\theta =0,\phi =0\right.〉$ or (b) $\left|\theta =\pi /2,\phi =0\right.〉$. The qualitative change of the squeezing parameter around $3\alpha /\beta =1$ corresponds to changing stability of the mean-field fixed points between a stable center to an unstable saddle, or vice versa.

Figure 4

The best squeezing ${\xi }_{\mathrm{best}}^2$ (a) and the best squeezing time $t_{\mathrm{best}}$ (c) as a function of $\beta$ with $\alpha =1$ for the initial spin coherent state located on the north pole of the Bloch sphere $|\theta =0,\phi =0\rangle$. The best squeezing (b) and the best squeezing time (d) as a function of $\beta$ with $\alpha =1$ for the initial spin coherent state located at the equator of the Bloch sphere $|\theta =\pi /2,\phi =0\rangle$. Solid lines are numerical results for the effective Hamiltonian (9), dashed lines are numerical results for the simplified model ${\widehat{\mathcal{H}}}_{\mathrm{sim}}=3\alpha {\widehat{J}}_z^2+\beta ({\widehat{J}}_x^2-{\widehat{J}}_y^2)$, while dot-dashed lines are scaling laws obtained within the FSA and BBGKY, as summarized in Tables 1 and 2.