Spin squeezing in a spin-orbit-coupled Bose-Einstein condensate

We study the spin squeezing in a spin-1/2 Bose-Einstein condensates (BEC) with Raman-induced spin-orbit coupling (SOC). Under the condition of two-photon resonance and weak Raman coupling strength, the system possesses two degenerate ground states, using which we construct an effective two-mode model. The Hamiltonian of the two-mode model takes the form of the one-axis-twisting Hamiltonian, which is known to generate spin squeezing. More importantly, we show that the SOC provides a convenient control knob to adjust the spin nonlinearity responsible for spin squeezing. Specifically, the spin nonlinearity strength can be tuned to be comparable to the two-body density-density interaction, and hence is much larger than the intrinsic spin-dependent interaction strength in conventional two-component BEC systems such as ${}^{87}\mathrm{Rb}$ and ${}^{23}\mathrm{Na}$ in the absence of the SOC. We confirm the spin squeezing by carrying out a fully beyond-mean-field numerical calculation using the truncated Wigner method. Additionally, the experimental implementation is also discussed.

Figure 1

(a) Dispersion of the single-particle Hamiltonian, where the two degenerate ground states $k_x=\pm k_0$ for an effective spin-1/2 system. (b) Dependence of the two-mode spin nonlinearity strength $\chi$ on $(g-g_{\uparrow \downarrow })$ and $\mathrm{\Omega }$.

Figure 2

Squeezing dynamics for a BEC in boxed potential. (a) Spin evolution at $\mathrm{\Omega }=2E_r$, $N={10}^3$, and $gn\approx 0.63E_r$ with $E_r=5\times {10}^3\left(2\pi \right)\mathrm{Hz}$, where dotted lines, dashed lines, and solid lines denote the evolution of $f_{\text{min}}$, ensemble average of spin evolution ${\overline{f}}_{\text{min}}$, and the ensemble standard deviation $\mathrm{\Delta }{f}_{\text{min}}$ along the direction with minimal ensemble variance. In the figure, $-\mathrm{\Delta }{f}_{\text{min}}$ is also plotted such that the vertical distance between $\pm \mathrm{\Delta }{f}_{\text{min}}$ can clearly demonstrate the spin squeezing. (b) Similar evolution with panel (a) but at $\mathrm{\Omega }=0$, where no squeezing effect is observed. (c) Squeezing parameters as functions of $t$, where the dotted line, the dashed line, and the solid line indicate the numerical result ${\xi }_{\text{TWM}}^2$, the analytical result ${\xi }_{\text{OAT}}^2$, and the fitting curve ${\xi }_{\text{fit}}^2$, respectively. (d) Logarithmic plot of the ensemble averaged momentum distribution $n\left(k_x\right)$.

Figure 3

(a) Analytical ${\chi }_{\text{OAT}}$ and the numerically extracted ${\chi }_{\text{TWM}}$ as functions of $\mathrm{\Omega }$, where $N={10}^3$ and $gn\approx 0.63E_r$ are fixed. (b) Dependence of the optimal squeezing parameter on $N$, where solid line and hollow circles correspond to the analytical result and the numerical result, respectively. In the calculation, we fix $\mathrm{\Omega }=2E_r$.

Figure 4

Squeezing dynamics of a BEC in the presence of an optical lattice. (a) Spin evolution with dotted lines, dashed lines, and solid lines denoting the evolution of $f_{\text{min}}$, ensemble average of spin evolution ${\overline{f}}_{\text{min}}$, and the ensemble standard deviation $\mathrm{\Delta }{f}_{\text{min}}$ along the direction with minimal ensemble variance. (b) Evolution of the squeezing parameters obtained by the TWM ${\xi }_{\text{TWM}}^2$ (solid line) and from the effective model ${\xi }_{\text{J}}^2$ (dashed line) in Eq. (14). In our calculation, we take $V_0=0.04E_r$, $\mathrm{\Omega }=2E_r$, $N={10}^3$, and $gn\approx 0.63E_r$ with $E_r=5\times {10}^3\left(2\pi \right)\mathrm{Hz}$, and the resulting tunneling strength and the nonlinearity satisfy $\left|J\right|/\chi \approx 31.8$.