Entanglement Dynamics of Linear and Nonlinear Interaction of Two Two-Level Atoms with a Quantized Phase-Damped Field in the Dispersive Regime

Abstract

In this paper we study the linear and nonlinear (intensity-dependent) interactions of two two-level atoms with a single-mode quantized field far from resonance, while the phase-damping effect is also taken into account. To find the analytical solution of the atom-field state vector corresponding to the considered model, after deducing the effective Hamiltonian we evaluate the time-dependent elements of the density operator using the master equation approach and superoperator method. Consequently, we are able to study the influences of the special nonlinearity function $f (n) = \sqrt {n}$, the intensity of the initial coherent state field and the phase-damping parameter on the degree of entanglement of the whole system as well as the field and atom. It is shown that in the presence of damping, by passing time, the amount of entanglement of each subsystem with the rest of system, asymptotically reaches to its stationary and maximum value. Also, the nonlinear interaction does not have any effect on the entanglement of one of the atoms with the rest of system, but it changes the amplitude and time period of entanglement oscillations of the field and the other atom. Moreover, this may cause that, the degree of entanglement which may be low (high) at some moments of time becomes high (low) by entering the intensity-dependent function in the atom-field coupling.

Appendix

As previously mentioned, the Liouville operators may be obtained by expression $L_{ij}=\langle i|\dot {{\rho }}(t)|j\rangle$, where subscripts i, j ∈{e, g} belong to degrees of freedom corresponding to atom 1. Here, $\dot {{\rho }}(t)$ is governed by (8). The term L _{e e} is defined as $\langle e_{1}|\dot {\rho }|e_{1}\rangle$. With this definition, the second term of the first line in (8) is vanished due to the terms 〈e₁|g₁〉,〈g₁|e₁〉. Also, the second line in (8) is totally withdrawn due to the terms 〈e₂|e₁〉,〈e₁|e₂〉,〈g₂|e₁〉 and 〈e₁|g₂〉. Thus, we have

$$\begin{array}{@{}rcl@{}} L_{ee}&=&\dot{\rho}_{e_{1}e_{1}}(t)=\langle e_{1}|\dot{\rho}(t)|e_{1}\rangle\\ &=&-i{\Omega}_{1}\{(a^{\dagger} a + 1)f^{2}(a^{\dagger} a + 1)\langle e_{1}|\rho(t)|e_{1}\rangle-\langle e_{1}|\rho(t)|e_{1}\rangle(a^{\dagger} a + 1)f^{2}(a^{\dagger} a + 1) \\ &&+\gamma[2a^{\dagger} a\langle e_{1}|\rho(t)|e_{1}\rangle a^{\dagger} a-\langle e_{1}|\rho(t)|e_{1}\rangle a^{\dagger} aa^{\dagger} a-a^{\dagger} a\langle e_{1}|\rho(t)|e_{1}\rangle]\}. \end{array} $$

(33)

Now, considering superoperators MO = a^†aO and PO = Oa^†a and replacing the arbitrary operator O with ρ(t) which is corresponding to atom 1 degrees of freedom, the above operator can be rewritten as

$$L_{ee}\,=\,-i{\Omega}_{1}\! \left[(M\,+\,1)f^{2}(M\,+\,1)\rho_{ee}\,-\,(P\,+\,1)f^{2}(P\,+\,1)\rho_{ee}\!\right]+\gamma (2MP\rho_{ee}-P^{2}\rho_{ee}-M^{2}\rho_{ee}), $$

where, ρ _{e e} = 〈e₁|ρ(t)|e₁〉. Then, the resulted expression can be rewritten as follows

$$L_{ee}=\left\{-i{\Omega}_{1} \left[(M + 1)f^{2}(M + 1)-(P + 1)f^{2}(P + 1)\right]+\gamma (2MP-P^{2}-M^{2})\right\}\rho_{ee}, $$

The other relations (17)–(19) may be obtained, similarly.