Entanglement and extreme spin squeezing for a fluctuating number of indistinguishable particles

We extend the criteria for $k$-particle entanglement from the spin-squeezing parameter presented in Sørensen and Mølmer [Phys. Rev. Lett. 86, 4431 (2001)] to systems with a fluctuating number of particles. We also discuss how other spin-squeezing inequalities can be generalized to this situation. Further, we show how, by employing additional degrees of freedom, it is possible to extend the bounds to the case when the individual particles cannot be addressed. As a by-product, this allows us to show that in spin-squeezing experiments with cold gases the particles are typically distinguishable in practice. Our results justify the application of the Sørensen-Mølmer bounds in recent experiments on spin squeezing in Bose-Einstein condensates.

Figure 1

Illustration of Observation 2 with particles of $d_{\mathrm{in}}=2$ internal states. The light-gray structure (blue online) indicates entanglement between particles. (a) Fully distinguishable particles ($d_{\mathrm{ex}}=N$). Depicted is a three-particle entangled state of $N=8$ particles. Entanglement is present between the particles with energy level $\gamma$ equal to 2, 3, and 5 (counting upward from the lowest level with $\gamma =1$), and between particles 7 and 8. (b) Fully indistinguishable particles ($d_{\mathrm{ex}}=1$). Depicted is a state which does not factorize. (c) Mixed situation ($1<d_{\mathrm{ex}}<N$). Depicted is a four-particle entangled state of $N=11$ particles in $d_{\mathrm{ex}}=5$ external levels. There are two groups of four fully entangled particles: In the external level $\gamma =2$, where the four particles are indistinguishable and in a nonfactorisable state; and in levels 4 and 5. In the latter case, the group of particles in level $\gamma =4$ is distinguishable from the particle in level $\gamma =5$. The two particles in the lowest level $\gamma =1$ are in a symmetric separable state of the form $|\phi \rangle \otimes |\phi \rangle$.

Figure 2

Plot of the spin-squeezing parameters ${\xi }_{\alpha }^2$ [dashed line, Eq. (A6)] and ${\xi }_{\beta }^2$ [solid line, Eq. (A8)]. While ${\xi }_{\alpha }^2$ never goes below the value 1 (dotted line), ${\xi }_{\beta }^2$ is below 1 for a large interval.

Figure 3

Initially, all particles are in the lowest energy level of the trap. The cloud is then released. If it is diluted enough along the horizontal direction, it is likely that at most one particle falls into each of the boxes, which represent the single detectors.