Entanglement and decoherence of massive particles due to gravity

We analyze the dynamics of a gravity-induced entanglement for $N$ massive particles. Considering the linear configuration of these particles, we investigate the entanglement between a specific pair of particles under the influence of the gravitational interaction between the massive particles. As the particle number increases, the specific particle pair decoheres more easily due to the gravitational interaction with other particles. The timescale of the gravity-induced decoherence is analytically determined. We also discuss the entanglement dynamics of initially entangled particles, which exemplify the monogamy of the gravity-induced entanglement.

Figure 1

Sketch of our model of the $N$-particle system. Each particle is placed at a distance $d$ from its adjacent particles, and initially prepared in a superposition state with the position separated by distance $L$. The direction of the separation between each particle $d$ is orthogonal to the direction of the separation of the particle with its superposed position $L$. The $i$th particle from the left ($1\le i\le N$) has mass $m_i$. The particles interact with one another through gravity.

Figure 2

The negativity $\mathcal{N}$ computed from the partial transposed matrix Eq. (9) at $N=2$ as a function of the dimensionless time ${\mathrm{\Delta }}_{12}t/\hslash$. Because $\mathcal{N}$ is greater than or equal to zero, the state of two particles is entangled except when $\mathcal{N}=0$.

Figure 3

Behavior of $\mathcal{N}$ as a function of the dimensionless time ${\mathrm{\Delta }}_{12}t/\pi \hslash$ for $L/d=1$ when the three particles have the same mass. In the case of $N=3$, $\mathcal{N}$ either takes positive values or is zero; hence, the first and second particles are always entangled except when $\mathcal{N}=0$.

Figure 4

Left: behavior of negativity, $\mathcal{N}$ as a function of the dimensionless time ${\mathrm{\Delta }}_{12}t/\pi \hslash$. Here, we assume ten particles ($N=10$) with the same mass and $L/d=1$. The negativity takes positive values for only a short period of $t\lesssim 2\pi \hslash /{\mathrm{\Delta }}_{12}$. Right: behavior of the corresponding four eigenvalues (red solid line, blue dashed line, green dotted line, and purple dot-dash line). The negativity is positive if at least one of the eigenvalues is negative, and zero if all eigenvalues are positive.

Figure 5

Same as Fig. 4 but with $L/d={10}^{-3}$ (upper left and right panels), and with $L/d={10}^3$ (lower left and right panels).

Figure 6

Alignment of particles in two dimensions. Particles with mass $m$ are placed at lattice points with coordinates $(x,y)=d(i,j)$ and integers $i$ and $j$, where $d$ is the separation distance between the neighboring lattice points. We assume that the total number of particles is $N\times N$ and that each particle is specified by $\mathit{n}=(i,j)$. This figure shows that each particle is in a superposition of localized states along the $z$ axis separated by a distance $L$.

Figure 7

Same as Fig. 4 but for the two-dimensional case with $N=4$.