Nonperturbative approach for the electronic Casimir-Polder effect in a one-dimensional semiconductor

We present the electronic Casimir-Polder effect for a system consisting of two impurities on a one-dimensional semiconductor quantum wire. Due to the charge transfer from the impurity to a one-dimensional conduction band, the impurity states are dressed by a virtual cloud of the electron field. The attractive electronic Casimir force arises due to the overlap of the virtual clouds. The Van Hove singularity causes the persistent bound state (PBS) to appear below the band edge even when the bare impurity state energy is above the band edge. Since the decay rate of the virtual cloud of the PBS in space is small, the Casimir force can be of a very long range. While the overlap of the electronic virtual cloud is consistent with the idea of the radiation reaction, it is shown that also vacuum fluctuations play a role in the electronic Casimir force as a result of the fermionic anticommutation relations. We introduce an effective mass, different from the effective band mass of the conduction band, which is associated with the distance of the energy of the PBS from the band edge where the Van Hove singularity is located and determines the decay rate of the electronic Casimir-Polder force.