Orbital angular momentum spectrum and entanglement in a rotating accelerated reference frame

The definition of a particle varies across different theories. The quantum field theory in curved spacetime shows that from the perspective of a linearly accelerated observer, an inertial empty space may be full of thermal particles. This effect is known as the Unruh effect. When the degrees of freedom of orbital angular momentum (OAM) are considered, all OAM modes share the same expected particle number. Here, we examine the OAM spectrum in a rotating accelerated reference frame to see how the spectrum differs from the linear accelerated case. When the observer starts to rotate, not all OAM modes are allowed and some negative energy modes show up. To understand how a rotating accelerated observer actually perceives these particles, the Unruh-DeWitt detector and its detailed balance are studied. This relation is studied both in the comoving inertial frame and in the rest frame. Based on these results, the OAM entanglement degradation is explored in two-dimensional and high-dimensional cases, respectively. The results indicate that the entanglement dimension and the highest order of OAM modes are mainly related to the acceleration and the rotation, respectively. It is then demonstrated that these results can be generalized to all stationary trajectories.

Figure 1

The curves for expected particle number. The energy $\mathcal{E}$, angular velocity $\omega$, and the acceleration $a$ are in the units of $\mathrm{J}$, $\mathrm{J}/\hslash$, and $c\mathrm{J}/\hslash$, respectively. The acceleration $a$ is set to 10. The energy $\mathcal{E}$ and the OAM $l$ are, respectively, set to (a) 1 and 3, and (b) $-1$ and $-3$.

Figure 2

The expected particle numbers for (a) $l=0$, (b) $l=3$, and (c) $l=10$ modes. For better illustration, we truncate the particle number at 10 for $l=3$ and $l=10$ modes. The energy $\mathcal{E}$ is set to 1.

Figure 3

Distributions for the expected particle number with $\mathcal{E}=1$ and $a=10$. The angular velocities are set to (a) $0$, (b) $0.1$, (c) 1, (d) $10$, (e) $-0.1$, and (f) $-1$.

Figure 4

Distributions for the expected particle number with $\mathcal{E}=-1$ and $a=10$. The angular velocities are set to (a) 0, (b) $0.5$, (c) 2, (d) 10, (e) $-0.5$, and (f) $-2$.

Figure 5

The integral ranges for three different cases. The red lines indicate the integral ranges for the two integrals in the transition rates. Note that the points of $\pm l\omega$ are excluded from the integrals.

Figure 6

Probability that the detector is in the states $|e,l⟩$. $E$ and $a$ are set to 1 and 10, respectively.

Figure 7

Curves for (a) purities and (b) negativities with OAM $l=1$, 2, 5, 9, and 20. We set $\mathcal{E}=1$ and $a=1$.

Figure 8

Curves for purities and negativities with maximal OAM $M=1$, 2, 5, 9, and 20. We set $\mathcal{E}=1$ and $a=1$.