Effects of photon field on entanglement generation in charged particles

The Bose-Marletto-Vedral experiment is a proposal for testing the quantum nature of gravity with entanglement due to Newtonian gravity. This proposal has stimulated controversy on how the entanglement due to Newtonian gravity is related to the essence of quantum gravity and the existence of gravitons. Motivated by this, we analyze the entanglement generation between two charged particles coupled to a photon field. We assume that each particle is in a superposition of two trajectories and that the photon field is initially in a coherent state. Based on covariant quantum electrodynamics, the formula for the entanglement negativity of the charged particles is derived for the first time. Adopting simple analytic trajectories of the particles, we demonstrate the entanglement between them. It is observed that the entanglement is suppressed by the decoherence due to the vacuum fluctuations of the photon field. We also find that the effect of quantum superposition of bremsstrahlung appears in the entanglement negativity formula. The similar structures between the gravity theory and electromagnetic theory suggests that a similar feature may be observed in the entanglement generation by quantum gravitational radiation.

Figure 1

Configuration of trajectories of two charged particles. The length scale of each superposition is $L$, the coordinate time during which each particle is superposed is $T$, and the particles are initially separated by the distance $D$.

Figure 2

Negativity $𝒩$ induced by the Coulomb potential between the charged particles. We adopted $L/cT=0.1$.

Figure 3

Configuration of a single charged particle trajectory.

Figure 4

Linear configuration in the regimes $cT\gg D\sim L$ and $cT\gg D\gg L$. The left panel shows the entire view of the linear configuration.

Figure 5

Negativity $𝒩$ for the linear configuration. (a) is the case $cT\gg D\sim L$ while (b) is the case $cT\gg D\gg L$. We adopted $L/cT=0.1$.

Figure 6

Linear configuration in the $D\gg cT\gg L$ regime.

Figure 7

Minimum eigenvalue ${\lambda }_{\min }[{\rho }_{12}^{{\mathrm{T}}_1}]$ for the linear configuration in the regime $D\gg cT\gg L$. We adopted $L/cT=0.1$.

Figure 8

Parallel configuration in $cT\gg D\gg L$ regime.

Figure 9

Negativity $𝒩$ for the parallel configuration. (a) is the case $cT\gg L\gg D$, whereas (b) is the case $cT\gg D\gg L$. We adopted $L/cT=0.1$.

Figure 10

Parallel configuration in $D\gg cT\gg L$ regime.

Figure 11

Minimum eigenvalue ${\lambda }_{\min }[{\rho }_{12}^{{\mathrm{T}}_1}]$ for the parallel configuration in the regime $D\gg cT\gg L$. We adopted $L/cT=0.1$.

Figure 12

Angular distribution of the photon field induced by each trajectory of the accelerating charged particle 1 for linear configuration (a) and parallel configuration (b) on the $x\text{−}y$ plane at a constant time.