Spin-orbit-coupling-induced spin squeezing in three-component Bose gases

We observe spin squeezing in three-component Bose gases where all three hyperfine states are coupled by synthetic spin-orbit coupling. This phenomenon is a direct consequence of spin-orbit coupling, as can be seen clearly from an effective spin Hamiltonian. By solving this effective model analytically with the aid of a Holstein-Primakoff transformation for a spin-1 system in the low excitation limit, we conclude that the spin-nematic squeezing, a category of spin squeezing existing exclusively in large spin systems, is enhanced with increasing spin-orbit coupling intensity and effective Zeeman field, which correspond to Rabi frequency ${\mathrm{\Omega }}_R$ and two-photon detuning $\delta$ within the Raman scheme for synthetic spin-orbit coupling, respectively. These trends of dependence are in clear contrast to spin-orbit-coupling-induced spin squeezing in spin-1/2 systems. We also analyze the effects of harmonic trap and interparticle interaction with realistic experimental parameters numerically, and find that a strong harmonic trap favors spin-nematic squeezing. We further show spin-nematic squeezing can be interpreted as two-mode entanglement or two-spin squeezing at low excitation. Our findings can be observed in ${}^{87}\mathrm{Rb}$ gases with existing techniques of synthetic spin-orbit coupling and spin-selective imaging.

Figure 1

(a,b) Single-particle phase diagrams of a three-component Bose gas with one-dimensional SOC in the (a) ${\mathrm{\Omega }}_R$–$\epsilon$ plane with $\delta /E_r=1$ and (b) ${\mathrm{\Omega }}_R$–$\delta$ plane with $\epsilon /E_r=6$. The lowest branch of the single-particle dispersion spectrum acquires either one, two, or three local minima in different parameter regimes separated by solid lines. On the dashed lines within regions of multiple minima, two of the local minima are degenerate. Typical examples are shown for the lowest branch of dispersion curves by changing (c) Rabi frequency ${\mathrm{\Omega }}_R$ with $\delta /E_r=1$ and $\epsilon =0$ and (d) quadratic Zeeman energy $\epsilon$ with $\delta /E_r=1$ and ${\mathrm{\Omega }}_R/E_r=2$.

Figure 2

Spin-nematic squeezing parameter ${\xi }_x$ as a function of (a) Rabi frequency ${\mathrm{\Omega }}_R$ with $\delta =0$ and $\epsilon /E_r=6$ and (b) quadratic Zeeman splitting $\epsilon$ with $\delta =0$ and ${\mathrm{\Omega }}_R/E_r=2$. In both figures, results obtained from the effective spin model Eq. (5) are illustrated by blue solid lines, in comparison to the numerical solutions of the GP equation for a pancake-shaped trap with ${\omega }_x={\omega }_y=50$ Hz, ${\omega }_z=1500$ Hz (black dashed) and for a cigar-shaped trap with ${\omega }_x={\omega }_y=5000$ Hz, ${\omega }_z=1500$ Hz (red dotted). Here, we consider a gas of ${}^{87}\mathrm{Rb}$ atoms in the $F=1$ manifold with background interaction and total particle number $N={10}^5$.

Figure 3

(a) Variations of spin-nematic squeezing parameter $\xi$ as a function of two-photon detuning $\delta$ with ${\mathrm{\Omega }}_R/E_r=2$ and $\epsilon /E_r=6$. The analytic result obtained from the effective spin model Eq. (5) within low-density excitation approximation (blue solid) is compared with numerical solutions of the GP equation for a pancake-shaped trap with ${\omega }_x={\omega }_y=50$ Hz, ${\omega }_z=1500$ Hz (black dashed) and for a cigar-shaped trap with ${\omega }_x={\omega }_y=5000$ Hz, ${\omega }_z=1500$ Hz (red dotted). (b) Atom number fractions of the $|-1\rangle$ (black dotted), $|0\rangle$ (blue dashed), and $|+1\rangle$ (red solid) states.

Figure 4

(a) Two-mode entanglement parameter ${\xi }_{\mathrm{DCZ}}^0$ and (b) two-spin squeezing parameter ${\xi }_{\mathrm{UV}}^0$ versus Rabi frequency ${\mathrm{\Omega }}_R$ with other parameters being $\delta =0$ and $\epsilon /E_r=6$. The analytic result obtained from the effective spin model Eq. (5) within low-density excitation approximation (blue solid) is compared with numerical solutions of the GP equation for a pancake-shaped trap with ${\omega }_x={\omega }_y=50$ Hz, ${\omega }_z=1500$ Hz (black dashed) and for a cigar-shaped trap with ${\omega }_x={\omega }_y=5000$ Hz, ${\omega }_z=1500$ Hz (red dotted). The insets show the squeezing parameters as functions of $\theta$. Notice that the optimal squeezings in both criteria are obtained when $\theta =n\pi$ with $n$ an integer.