Dephasing due to electromagnetic interactions in spatial qubits

Matter-wave interferometers with micro-particles will enable the next generation of quantum sensors to probe minute quantum phase information. Therefore, estimating the loss of coherence and the degree of entanglement degradation for such interferometers is essential. In this paper, we will provide a noise analysis in frequency-space focusing on electromagnetic sources of dephasing. We will assume that our matter-wave interferometer has a residual charge or dipole which can interact with a neighbouring particle in the ambience. We will investigate the dephasing due to the Coulomb, charge-induced dipole, charge-permanent dipole, and dipole-dipole interactions. All these interactions constitute electromagnetically driven dephasing channels that can affect single or multiple interferometers. As an example, we will apply the obtained formulae to situations with two adjacent micro-particles, which can provide insight for the noise analysis in the quantum gravity-induced entanglement of masses (QGEM) protocol and the C-NOT gate: we will compute the dephasing due to a gas of environmental particles interacting via dipole-dipole and charge-charge couplings, respectively. To obtain simple analytical dephasing formulae, we will employ uniform probability distributions for the impact parameter and for the angles characterizing the relative orientation with respect to the interferometer and a Gaussian distribution for the velocities of the environmental particles. In both cases, we will show that the dephasing rate grows with the number density of particles present in the vacuum chamber, as expected.