New spin squeezing and other entanglement tests for two mode systems of identical bosons

Abstract

For any quantum state representing a physical system of identical particles, the density operator must satisfy the symmetrization principle (SP) and conform to super-selection rules (SSR) that prohibit coherences between differing total particle numbers. Here we consider bi-partitite states for massive bosons, where both the system and sub-systems are modes (or sets of modes) and particle numbers for quantum states are determined from the mode occupancies. Defining non-entangled or separable states as those prepared via local operations (on the sub-systems) and classical communication processes, the sub-system density operators are also required to satisfy the SP and conform to the SSR, in contrast to some other approaches. Whilst in the presence of this additional constraint the previously obtained sufficiency criteria for entanglement, such as the sum of the $\skew3\hat{S}_{x}$ and $\skew3\hat{S}_{y}$ variances for the Schwinger spin components being less than half the mean boson number, and the strong correlation test of $|\langle \skew3\hat{a}^{m}\,(\skew3\hat{b}^{\dagger })^{n}\rangle |^{2}$ being greater than $\langle (\skew3\hat{a}^{\dagger })^{m}\skew3\hat{a}^{m}\,(\skew3\hat{b}^{\dagger })^{n}\skew3\hat{b}^{n}\rangle (m,n=1,2,\ldots )$ are still valid, new tests are obtained in our work. We show that the presence of spin squeezing in at least one of the spin components $\skew3\hat{S}_{x}$ , $\skew3\hat{S}_{y}$ and $\skew3\hat{S}_{z}$ is a sufficient criterion for the presence of entanglement and a simple correlation test can be constructed of $|\langle \skew3\hat{a}^{m}\,(\skew3\hat{b}^{\dagger })^{n}\rangle |^{2}$ merely being greater than zero. We show that for the case of relative phase eigenstates, the new spin squeezing test for entanglement is satisfied (for the principle spin operators), whilst the test involving the sum of the $\skew3\hat{S}_{x}$ and $\skew3\hat{S}_{y}$ variances is not. However, another spin squeezing entanglement test for Bose–Einstein condensates involving the variance in $\skew3\hat{S}_{z}$ being less than the sum of the squared mean values for $\skew3\hat{S}_{x}$ and $\skew3\hat{S}_{y}$ divided by the boson number was based on a concept of entanglement inconsistent with the SP, and here we present a revised treatment which again leads to spin squeezing as an entanglement test.