Interferometry with independently prepared Bose-Einstein condensates

Whenever the value of an unknown parameter $\theta$ is extracted from a series of experiments, the result is inevitably burdened by the uncertainty $\Delta \theta$. If the system that is the subject of measurement consists of unentangled particles, this uncertainty is bounded by the shot-noise limit. To overcome this limitation, it is necessary to use a properly entangled state, which is usually prepared in a dedicated procedure. We show that quantum correlations arising from the indistinguishability of bosons are a sufficient resource for the sub-shot-noise interferometry. To this end, we consider an interferometer, which operates on two independently prepared Bose-Einstein condensates with fluctuating numbers of particles. We calculate the sensitivity obtained from the measurement of the number of atoms and compare it with the ultimate achievable bound. Our main conclusion is that even in the presence of major atom number fluctuations, an interferometer operating on two independently prepared condensates can give very high precision. These observations indicate a new possibility for an interferometer operating below the shot-noise limit.

Figure 1

Left panel: The QFI (solid lines) and the Fisher information (dashed lines) for the measurement of populations as a function of the width $\sigma$ for $\overline{N}=750$ particles normalized to the SNL and averaged over a statistical ensemble of disorders $\xi$, with statistical properties described in the text. In all cases, the shades show the related statistical widths. From the bottom to the top: $\varepsilon =0,0.3,1$. For $\varepsilon =0.3$ and $\varepsilon =1$, short horizontal black lines at large $\sigma$ indicate the limiting values of the QFI calculated using Eq. (12). Right panel: The exemplary probability distributions for $\sigma =50$. Note that in the absence of disorder, the QFI crosses the SNL at $\sigma =\sqrt{750/2}=19.4$, as expected from the result below Eq. (10).

Figure 2

The probability distribution $p_N\left(n\right|\theta )$ from Eq. (13) calculated with Gaussian $P_{a/b}$ with mean $\overline{N}=100$ atoms and width equal to $\sigma =0.1$ (a), $\sigma =10$ (b), and $\sigma =20$ (c). The panels show $p_{N=100}\left(n\right|\theta )$ as a function of $n$ (radial variable) and $\theta \in [0,2\pi ]$ (polar variable). The insets are enlargements of the regions marked with orange squares. When $\sigma$ grows [from (a) to (c)], fine structures of the probability vanish, giving a smaller value of the Fisher information (14). However, when the disorder is introduced to the probability, according to Eq. (11), the fine structures are restored, as shown in panel (d) for $\sigma =20$ and $\varepsilon =1$.