Detection and engineering of spatial mode entanglement with ultracold bosons

We outline an interferometric scheme for the detection of bimode and multimode spatial entanglement of finite-temperature interacting Bose gases. Whether entanglement is present in the gas depends on the existence of the single-particle reduced density matrix between different regions of space. We apply the scheme to the problem of a harmonically trapped repulsive boson pair and show that while entanglement is rapidly decreasing with temperature, a significant amount remains for all interaction strengths at zero temperature. Thus, by tuning the interaction parameter, the distribution of entanglement between many spatial modes can be modified.

Figure 1

(Color online) Schematic showing a trapped wave function split into two spatial modes $a$ and $b$. The modes are combined at a 50:50 beam splitter and the particles in the output modes $c$ and $d$ are counted.

Figure 2

Amount of entanglement between two spatial regions occupied by a boson pair as specified in the text as a function of temperature. The values of the dimensionless interaction parameter increase from the top curve to the bottom curve as $g_{1D}=0,2,5,10,\infty$. Temperature is scaled in units of $\hslash \omega /k_B$.

Figure 3

The system is split into three spatial modes $a$, $b$, and $c$. Mode $a$ is defined as the region $-\infty <x<-L_b/2$, mode $b$ as $-L_b/2<x<L_b/2$, and mode $c$ as $L_b/2<x<\infty$. Entanglement is investigated between neighboring modes $a$ and $b$ ($b$ and $c$) (left-hand side) and also between the outer two modes $a$ and $c$ (right-hand side) at $T=0$ as the interaction is varied. Three different central mode lengths $L_b=4,2.8,1.4$ are taken.