Spacetime diamonds

We show that the particle-number distribution of diamond modes, modes that are localized in a finite spacetime region, are thermal for the Minkowski vacuum state of a massless scalar field, an analogue to the Unruh effect. The temperature of the diamond is inversely proportional to its size. An inertial observer can detect this thermal radiation by coupling to the diamond modes using an appropriate energy-scaled detector. We further investigate the correlations between various diamonds and find that entanglement between adjacent diamonds dominates.

Figure 1

Diamonds in ($1+1$)-dimensional Minkowski spacetime. Only a chain of diamonds along the $t$ axis is plotted, and they are labeled by integers $n=0,\pm 1,\dots$. The sizes of these diamonds are the same, $2/a$.

Figure 2

Entanglement between Gaussian modes in the first and zeroth diamonds. Left: Quadrature phase $\varphi =0$. Right: Quadrature phase $\varphi =0.2\pi$. The central frequency of the Gaussian mode in the zeroth diamond is set to be ${\omega }_0/a=1.0$, and the bandwidth and central position are the same, ${\sigma }_1/a={\sigma }_0/a=0.02$, $a{v}_1=a{v}_0=0$.