Figure 1

(Color online) Schematic representation of a beam splitter for a BEC in a double-well potential. The blue region is the condensate density distribution obtained by solving a one- dimensional Gross-Pitaevskii equation; the red line is the potential barrier. The beam splitter is realized in three steps: (a) two independent condensates, Eq. (1), are created in the double well; (b) the height of the potential barrier separating the two condensates is decreased, allowing a tunneling between the two wells for a certain time $t$; and (c) the potential barrier is raised to suppress the tunneling.Reuse & Permissions

Figure 2

(Color online) Relative number and phase distributions in the linear limit, plotted at different times: (a) and (b) $t=0$; (c) and (d) $t=\pi ∕\left(8K\right)$; (e) and (f) $t=\pi ∕\left(4K\right)$, corresponding to a $\pi ∕2$ pulse or $50∕50$ beam splitter. Here $N=40$, and $K\left(t\right)$ is kept constant during the dynamics. As illustrated by Eq. (8), the linear dynamics depends on the product $Kt$.Reuse & Permissions

Figure 3

(Color online) (a) Dynamical evolution of $K\left(t\right)∕K\left(0\right)=[1-{\cos }^n(t\pi ∕\tau )]$, with $\tau =0.4$ and different values of $n$. (b) Width of the phase distribution as a function of time for the different $K\left(t\right)$ plotted in (a). The width of the phase distribution has a minimum at $t\approx 0.4$ for a sudden displacement of the double well [case $n\to \infty$, with $K\left(t\right)∕K\left(0\right)=\Theta (\tau -t)$]. For finite values of $n$ (slow displacement), the minimum value of the width is reached at longer times and with larger values (lower sensitivities). In the inset, we plot the phase distribution width as a function of time in logarithmic scale. The best scenario is obtained for the fastest splitting. Here we considered $K=0.5$, $N=40$, and a constant $E_c=0.4$. Time is in units $\pi ∕\left(4K\right)$.Reuse & Permissions

Figure 4

(Color online) Width of the phase distribution Eq. (7) as a function of time [in units $\pi ∕\left(4K\right)$] and for different value of the interaction energy $E_c$. The linear case $(E_c=0)$ is represented by the solid red line. By increasing $E_c$ the minimum is attained at smaller times and at corresponding larger widths. We notice that, independently of $E_c$, at $t=0$ we have a flat probability distribution [see Fig. 2b] corresponding to a phase width $2\pi$ and complete phase uncertainty. Here $K=0.5$ and $N=40$.Reuse & Permissions

Figure 5

(Color online) Relative number and phase distributions at the optimal working point (minimum phase width), for different values of $E_c$: (a) and (b) $E_c=0.04$; (c) and (d) $E_c=4$; (e) and (f) $E_c=400$. These distributions can be compared with the ones in Figs. 2e, 2f calculated in the linear regime, $E_c=0$, and at the optimal time $\pi ∕\left(4K\right)$. Here the parameters are the same as in Fig. 4, $N=40$ and $K=0.5$.Reuse & Permissions

Figure 6

(Color online) Optimal splitting time [in units $\pi ∕\left(4K\right)$] as a function of $E_c$ and for different values of $N$. Points are numerical results, lines are guides to the eye. Here $K=0.5$.Reuse & Permissions

Figure 7

(Color online) Scaling parameter $\beta$ [defined in Eq. (10)] as a function of $K∕E_c$. Points represent results of the numerical simulations, the blue line is a guide to the eye. The red vertical dashed lines delineate three regimes: Rabi, Josephson, and Fock. Horizontal green dotted lines indicate the two relevant limits of quantum interferometry: the Heisenberg limit $\beta =1$ and the standard quantum limit $\beta =0.5$. Note that the Heisenberg limit is reached asymptotically in the number of particles. Here we considered (a) $N=40$ and (b) $N=80$. The Josephson regime becomes larger when increasing the number of particles. In the Fock regime it is not possible to define a scaling of the phase uncertainty since the phase distribution is almost flat and the phase sensitivity is $\Delta \theta \sim 2\pi$.Reuse & Permissions

Figure 8

(Color online) Time oscillation of ${\mid c_{N∕2}\left(t\right)\mid }^2$, for different values of $E_c$ and in the self-trapping (a) and non-self-trapping (b) regimes. Time is in units $\pi ∕\left(4K\right)$.Reuse & Permissions

Figure 9

(Color online) Maximum oscillation amplitude of ${\mid c_{N∕2}\left(t\right)\mid }^2$ as a function of $E_c∕K$ and for different numbers of particles. We can clearly distinguish between the self-trapped and non-self-trapped regimes. Points are numerical results, lines are guides to the eye. Solid black line corresponds to $N=10$, dashed yellow to $N=20$, dotted green to $N=40$, dot-dashed blue to $N=80$, and dot-dot-dashed red to $N=160$. In the inset, we plot the value of ${(E_c∕K)}_{\mathrm{cr}}^{\mathrm{JF}}$ corresponding to a maximum oscillation amplitude equal to 0.99, as a function of $N$. We define this as the critical point between the self-trapping and non-self-trapping regimes. Points represent numerical results. The dotted line is the result of a three-mode approximation of the $c_n\left(t\right)$ dynamics, as given by Eq. (13).Reuse & Permissions

Figure 10

(Color online) $C_{\mathrm{NOON}}$, defined in Eq. (15), as a function of $E_c∕K$. Points correspond to results of numerical simulations using different numbers of particles, and lines are guides to the eye: solid red line refers to $N=160$, dashed blue to $N=80$, dotted green to $N=40$, dot-dashed yellow to $N=20$, and dot-dot-dashed black to $N=10$. The sharp decrease of $C_{\mathrm{NOON}}$ defines a critical value of $E_c∕K$, which is reported, as a function of the number of particles, in the inset. The dashed line in the inset is given by Eq. (16) and is obtained by numerical fitting.Reuse & Permissions