Dipole–dipole interaction between trapped two-level ions interacting with a quantized field in the Lamb–Dicke regime

Abstract

To achieve more accuracy in a system containing N trapped two-level ions interacting with a single-mode quantized field, in this paper we will consider the dipole–dipole interaction (DDI) between the trapped ions. In more detail, we first suggest an appropriate form of a quantized DDI Hamiltonian between two trapped ions that has arisen from the spontaneous emission of the excited one. Accordingly, we have considered the spontaneous emission as the origin of dissipation in our proposed model. Then, the case in which the trapped ions interact with a single-mode quantized field in the Lamb–Dicke regime as well as the first vibrational sideband is considered, in which the DDI between the trapped ions is also taken into account. Using the state vector, which is analytically obtained for the whole system under our consideration, we evaluate the amount of entanglement between the field and the two trapped ions by utilizing the linear entropy. We observe that the entanglement depends on the distance between the two trapped ions; i.e., decreasing the distance led to decrement in the amount of entanglement. In the continuation, we extend the proposed model at first to three (N=3) and then to more trapped ions (N>3), and the most general entangled state vectors related to the outlined systems are obtained, for which the same scenario (for two trapped ions) is repeated. Comparing the latter cases (N≥3) with the previous one (N=2), we conclude that the entanglement is sensitive to the number of interacting trapped ions in addition to the distances between them in such a way that by increasing the number of trapped ions, the degree of entanglement between them and the field is increased. These results help one to control the entanglement between the field and the trapped ions in a more accurate manner by tuning the distances between the trapped ions and their numbers.