Thermal Entanglement Between a Jaynes-Cummings Atom and an Isolated Atom

Abstract

We studied the entanglement of a quantum system consisting of a Jaynes-Cummings atom, thermal lossless cavity and an isolated atom. The analytical expressions of the atom-atom negativity for separable and entangled initial atomic states were obtained. The influence of a detuning between the atomic transition frequency and the field frequency and direct dipole-dipole interaction on an atom-atom entanglement is examined. We showed that for a separable initial atomic states a detuning might cause high atom-atom entanglement in the presence of the dipole-dipole interaction. We also obtained that for an entangled initial atomic state a detuning causes a stabilization of an entanglement oscillations both for model with dipole-dipole interaction and model without such interaction.

Appendix

The elements of density matrix (16) for initial entangled state (3) are

$$\begin{array}{@{}rcl@{}} \rho_{11}(t)&=&\cos^{2}\theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{42,n-1}(t)|^{2} + \sin^{2}\theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{43,n-1}(t)|^{2}\\ &&+ \cos\theta \sin\theta \left( \sum\limits_{n = 1}^{\infty} p_{n} C_{42,n-1}(t) C^{*}_{43,n-1}(t) +\sum\limits_{n = 1}^{\infty} p_{n} C_{43,n-1}(t) C^{*}_{42,n-1}(t)\right),\\ \rho_{22}(t)&=&\cos^{2} \theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{22,n-1}(t)|^{2}+\sin^{2} \theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{23,n-1}(t)|^{2} \\ &&+ \cos\theta \sin \theta \left( \sum\limits_{n = 1}^{\infty} p_{n} C_{22,n-1}(t) C^{*}_{23,n-1}(t) + \sum\limits_{n = 1}^{\infty} p_{n} C_{23,n-1}(t) C^{*}_{22,n-1}(t)\right)\\ &&+ p_{0} \left( \cos^{2} \theta |C_{22}(t)|^{2} + \sin^{2} \theta |C_{23}(t)|^{2} \right.\\ &&\left. + \cos \theta \sin\theta C_{22}(t) C^{*}_{23}(t)+ \cos \theta \sin \theta C_{23}(t) C^{*}_{22}(t)\right),\\ \rho_{33}(t)&=& \cos^{2}\theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{32,n-1}(t)|^{2} + \sin^{2} \theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{33,n-1}(t)\\ &&+ \cos\theta \sin \theta \left( \sum\limits_{n = 1}^{\infty} p_{n} C_{32,n-1}(t) C^{*}_{33,n-1}(t) + \sum\limits_{n = 1}^{\infty} p_{n} C_{33,n-1}(t) C^{*}_{32,n-1}(t) \right)\\ &&+ p_{0} (\cos^{2}\theta |C_{32}(t)|^{2} +\sin^{2}\theta |C_{33}(t)|^{2}\\ &&+\cos \theta \sin \theta C_{32}(t) C^{*}_{33}(t)+ \cos\theta \sin \theta C_{33}(t) C^{*}_{32}(t), \end{array} $$

$$\begin{array}{@{}rcl@{}} \rho_{44}(t)&=& \cos^{2}\theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{12,n-1}(t)|^{2} +\sin^{2} \theta \sum\limits_{n = 1}^{\infty} p_{n} |C_{13,n-1}(t)|^{2}\\ &&+ \cos\theta \sin \theta \left( \sum\limits_{n = 1}^{\infty} p_{n} C_{12,n-1}(t) C^{*}_{13,n-1}(t) + \sum\limits_{n = 1}^{\infty} p_{n} C_{13,n-1}(t) C^{*}_{12,n-1}(t)\right)\\ &&+p_{0} (\cos^{2}\theta |C_{12}(t)|^{2} + \sin^{2}\theta |C_{13}(t)|^{2}\\ &&+\cos\theta \sin\theta C_{12}(t) G^{*}_{13}(t)+ \cos\theta \sin \theta C_{13}(t) C^{*}_{12}(t) ),\\ \rho_{23}(t)&=& \cos^{2} \theta \sum\limits_{n = 1}^{\infty} p_{n} C_{22,n-1}(t) C^{*}_{32,n-1}(t) + \sin^{2} \theta \sum\limits_{n = 1}^{\infty} p_{n} C_{23,n-1}(t]) C^{*}_{33,n-1}(t)\\ &&+ \cos\theta \sin \theta \left( \sum\limits_{n = 1}^{\infty} p_{n} C_{22,n-1}(t) C^{*}_{33,n-1}(t) +\sum\limits_{n = 1}^{\infty} p_{n} C_{23,n-1}(t) C^{*}_{32,n-1}(t)\right)\\ &&+ p_{0} (\cos^{2} \theta C_{22}(t) C^{*}_{32}(t)+ \sin^{2}\theta C_{23}(t) C^{*}_{33}(t)\\ &&+\cos\theta \sin\theta C_{22}(t) C^{*}_{33}(t)+ \cos\theta \sin \theta C_{23}(t) C^{*}_{32}(t)). \end{array} $$