Efficient two-mode interferometers with spinor Bose-Einstein condensates

We consider general three-mode interferometers using a spin-1 atomic Bose-Einstein condensate with macroscopic magnetization. We show that these interferometers, combined with the measurement of the number of particles in each output port, provide an ultrahigh phase sensitivity. We construct effective two-mode interferometers which involve two Zeeman modes showing that they also provide an ultrahigh phase sensitivity but of a bit reduced factor in the corresponding Fisher information. A special case of zero magnetization is shown to persist the efficiency of the two-mode interferometry.

Figure 1

(a) An example of the FI (20) divided by $4N^2$ versus $\theta$ from exact numerical calculations for $M=0.8N$, $q=0$, and $c_2>0$. The FI at $\theta =0$ versus $q$ for the three-mode (black circles) and two-mode (gray crosses) interferometry compared with the QFI for the three-mode (black solid lines) and two-mode (gray dashed lines) interferometry in the ground state of the system for (b) fixed magnetization $M=0.1N$ and $c_2<0$, $M=0.5N$ and (c) $c_2<0$, and (d) $M=0.8N$ and $c_2>0$. The total number of atoms is $N={10}^2$.

Figure 2

(a) An example of $\theta$ dependence of the FI (20) divided by $4N^2$ for ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}={\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}^{\left(A\right)}$ (orange dashed line), the two-mode ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}={\widehat{K}}_y$ (black dash-dotted line) and the three-mode ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}={\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}^{\left(B\right)}$ (blue solid line) interferometric transformations with $M=0.1N$, $\beta ={10}^{-3}$, $q=-0.5$, $c_2<0$. In panels (b)–(d) the two-mode (crossed points) and the three-mode (filled points) versions of the FI divided by $4N^2$ versus ${log}_{10}\beta$ are shown and compared with the two-mode (dashed lines) and three-mode (solid lines) versions of the QFI divided by $4N^2$ for different values of $q$ as indicated by colors: $q=-5$ (purple), $q=-0.5$ if $c_2<0$ or $q=0.5$ if $c_2>0$ (red), and $q=5$ (green). The parameters are (b) $M=0.1N$, $c_2<0$, $M=0.5N$, (c) $c_2<0$, (d) $M=0.8N$, $c_2>0$ with $N=50$.

Figure 3

(a), (b) The FI divided by $4N^2$ from Eq. (20) versus $\theta$ for the two-mode with ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}={\widehat{K}}_y$ in and (c), (d) the three-mode with ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}={\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}^{\left(B\right)}$ interferometric transformations including fluctuations of magnetization of amount $\sigma /\sqrt{N}=0$, $\sqrt{2}/20$, $\sqrt{2}/10$, $\sqrt{2}/5$, $3\sqrt{2}/10$ marked by colored solid lines from top to bottom in panels (a) and (c), and for $\sigma /\sqrt{N}=\sqrt{2}/2$, $7\sqrt{2}/10$, $\sqrt{2}$ marked by colored solid lines from top to bottom in panels (b) and (d), with $N=50$, $m=0.1$, $c_2>0$, $\beta ={10}^{-2}$, and $q=0.5$.

Figure 4

(a) The maximal value of the relative Fisher information $\mathcal{I}\left(\theta \right)/F_Q$ versus fluctuations of magnetization $\sigma /\sqrt{N}$ for the two-mode with ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}={\widehat{K}}_y$ (crossed points) and the three-mode with ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}={\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}^{\left(B\right)}$ (filled points) interferometry with $q=-5$ (marked by purple stars for the two-mode and by diamonds for the three-mode cases), $q=0.5$ (marked by red crosses for the two-mode and by squares for the three-mode cases) and $q=5$ (marked by green “Y” for the two-mode and by circles for the three-mode cases), and for $\beta =10$, $N=50$, $m=0.8$, $c_2>0$. (b) ${\theta }_{\mathrm{opt}}$ versus $\sigma /\sqrt{N}$ for the same parameters as in panel (a). In panels (c) and (d) the same quantities are plotted as in panels (a) and (b), respectively, but for $\beta ={10}^{-2}$. Vertical lines in panels (a) and (c) mark SQL. Colored lines linking the points are added to guide the eye.

Figure 5

The FI at $\theta =0$ (green points) versus ${log}_{10}\beta$ for (a) $q=-5$ and for (b) $q=5$ compared with the QFI (black solid line), with $N=100$, $c_2<0$, $\sigma =0$, $M=0$. The variances of ${\widehat{D}}_{xy}$ (red dashed line), ${\widehat{J}}_x$ (orange dotted line), and $\widehat{Y}$ (purple dot dashed line) are also shown. All quantities are divided by $4N^2$.

Figure 6

The precision from the error-propagation formula (51) (red solid) for a single Dicke state $\widehat{\rho }=|j,m\rangle \langle j,m|$ versus the fractional magnetization $m$ compared with the lower bound set by the QFI (blue dashed). In the figure, $j=10$.