Multipath interferometer with ultracold atoms trapped in an optical lattice

We study an ultracold gas of $N$ bosons trapped in a one-dimensional $M$-site optical lattice perturbed by a spatially dependent potential $g{x}^j$, where the unknown coupling strength $g$ is to be estimated. We find that the measurement uncertainty is bounded by $\Delta g\propto \frac{1}{N(M^j-1)}$. For a typical case of a linear potential, the sensitivity improves as $M^{-1}$, which is a result of multiple interferences between the sites, an advantage of multipath interferometers over two-mode setups. Next, we calculate the estimation sensitivity for a specific measurement where, after the action of the potential, the particles are released from the lattice and form an interference pattern. If the parameter is estimated by a least-squares fit of the average density to the interference pattern, the sensitivity still scales like $M^{-1}$ for linear potentials. We finally discuss the role of useful entanglement of the initial state in the lattice to beat the shot-noise limit.

Figure 1

Ultimate scaling of the sensitivity for various nonlinear potentials given by Eq. (20). Red squares, blue circles, black diamonds and green triangles correspond to $j=-2,-1,1,2$, respectively. The black dashed line denotes the sensitivity of a standard $j=1,M=2$ two-mode interferometer.

Figure 2

The sensitivity $mN{\Delta }^2\theta$ of the parameter estimation from the fit to the density of the interference pattern with $M=2$ and $N=200$ calculated using the state with a fixed [open blue circles; Eq. (37)] and an unfixed number of atoms [solid black line; Eq. (38)] as a function of the phase-squeezing parameter. The sensitivities shown are normalized to the SNL. The dashed red line shows the ultimate precision of the phase estimation given by the QFI from Eq. (15). The horizontal black dotted line shows the shot-noise limit.

Figure 3

The sensitivity $mN{\Delta }^2\theta$ of the parameter estimation from the fit to the density of the interference pattern for $M=2,4,6,8,10$, and 12 (different colored solid lines) with a fixed average number of atoms equal to $\langle \widehat{N}\rangle =5\times {10}^4$. The sensitivities shown are normalized to the SNL and multiplied by the factor $\frac{1}{3}(M^2-1)$. The dashed red line shows the ultimate precision of the phase estimation given by the QFI from Eq. (15). The horizontal black dotted line shows the shot-noise limit.