Einstein-Podolsky-Rosen Correlations of Ultracold Atomic Gases

We demonstrate that collective continuous variables of two species of trapped ultracold bosonic gases can be Einstein-Podolsky-Rosen-correlated (entangled) via inherent interactions between the species. We propose two different schemes for creating these correlations—a dynamical scheme and a static scheme analogous to two-mode squeezing in quantum optics. We quantify the correlations by using known measures of entanglement and study the effect of finite temperature on these quantum correlations.

Figure 1

(a) State-preparation scheme (see the text). (b) Dynamics of the entanglement defined by the two-mode squeezing criterion for both sudden and slow intermode coupling barrier ramping up, found through exact simulation of Eq. (1) in Ref. 25 [unless stated otherwise, the simulations are for $N_1=N_2=20$, $J=1\mathrm{kHz}$, and $(E_c{)}_{AA}=(E_c{)}_{BB}=(E_c{)}_{AB}=1\mathrm{Hz}$]. The plot includes the case of a coupled system ($E_{c12}=E_c$, solid blue line), an uncoupled system ($E_{c12}=0$, dashed green line), slow ramp-up (dash-dotted black line), and the classical limit (dotted red line). The trade-off between entanglement and nonlinear phase diffusion is better for the sudden coupling. (c) The same as (b) (except for slow barrier ramp-up) for the negativity entanglement measure (see the text). (d) Maximally achievable negativity (solid blue line) reached through the dynamics as a function of charging energy $E_c$, assuming all coefficients are equal: $(E_c{)}_{AA}=(E_c{)}_{BB}=(E_c{)}_{AB}\equiv E_c$ [32, 33]. The separability limit is indicated by the red dashed line.

Figure 2

(a) Schematic sequence for the creation of “nonlocal” two-mode entanglement in analogy with the BS approach. (b) Single-mode squeezing dynamics as a function of time, for $N=100$. Decoherence effects: the variance of $n_{+}$ (c) and of ${\varphi }_{-}$ (d) as a function of time, in the presence of proper dephasing. We subtract the variance of the Hermitian dynamics to single out the effect of dephasing. The coherence time is estimated to be $\sim 100\mathrm{ms}$ at 20 nK.