Improvement of the entanglement generation in atomic states using a single-mode field in the Tavis–Cummings model

Abstract

Assuming the interaction of two separable two-level atoms with a single-mode field in the Tavis–Cummings model, we investigate the entanglement generation in the atomic state using the concurrence measure. Specifically, even and odd coherent states are selected as the initial states of the field and the entanglement of the final atomic state is computed and compared with the corresponding one when the ordinary coherent state is the initial field state. It is observed that even and odd coherent states induce more entanglement in the atomic state, so that the Bell states can be obtained by optimizing the parameters. In addition, we observe a direct connection between the non-Gaussianity of field states and the entanglement of atomic states. Also, a mixture of separable and entangled states is generated with the passage of time and the range of entanglement variation decreases.

Graphical abstract

Appendix A: Calculation of the coefficients of the atom–field state in the Tavis–Cummings model

Using (2)—(5), the atom–field state at time t is given by:

$$ \left| {\Psi_{a - f} (t)} \right\rangle = e^{{ - i\,\,\left( {\hat{H}_{1} + \hat{H}_{2} } \right)\,\,t}} \left| {\Psi_{a} (0)} \right\rangle \left| {\Psi_{f} (0)} \right\rangle $$

(A.1)

where $\hat{H}_{1}$ and $\hat{H}_{2}$ are as follows:

$$ \hat{H}_{1} = \hbar \left( {\omega \hat{a}^{\dag } \hat{a} + \frac{{\omega_{a} }}{2}\sum\limits_{i = A,B} {\hat{\sigma }_{i}^{z} } } \right) $$

(A.2)

$$ \hat{H}_{2} = \hbar \left( {g\sum\limits_{i = A,B} {\left( {e^{i\theta } \hat{\sigma }_{i}^{ + } \hat{a} + e^{ - i\theta } \hat{\sigma }_{i}^{ - } \hat{a}^{\dag } } \right)} } \right) $$

(A.3)

With the use of Backer-Campbell-Hausdorff lemma, the commutator of $\hat{H}_{1}$ and $\hat{H}_{2}$ is [43]:

$$ \left[ {\hat{H}_{1} ,\hat{H}_{2} } \right] = \left( {\omega - \omega_{a} } \right)g\left( { - e^{i\theta } \sum\limits_{i = A,B} {\hat{\sigma }_{i}^{ + } \hat{a} + e^{ - i\theta } \,\sum\limits_{i = A,B} {\hat{\sigma }_{i}^{ - } } \hat{a}^{\dag } } } \right) $$

(A.4)

Assuming the resonance of the atomic and field frequencies $\left( {\omega = \omega_{a} } \right),$ the commutator in (A.4) vanishes. Thus, (A.1) can be written as:

$$ \left| {\Psi_{a - f} (t)} \right\rangle \, = \hat{U}_{1} \left( t \right)\hat{U}_{2} \left( t \right)\left| {\Psi_{a} (0)} \right\rangle \left| {\Psi_{f} (0)} \right\rangle $$

(A.5)

where $\hat{U}_{1} \left( t \right)$ and $\hat{U}_{2} \left( t \right)$ are given by:

$$ \begin{gathered} \hat{U}_{1} \left( t \right) = \exp \left[ { - i\frac{{\hat{H}_{1} t}}{\hbar }} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\begin{array}{*{20}c} {e^{{ - i\omega \,t\left( {\,\hat{a}^{\dag } \hat{a}\, + 1} \right)}} } & 0 & 0 & 0 \\ 0 & {e^{{ - it\omega \,\hat{a}^{\dag } \hat{a}}} } & 0 & 0 \\ 0 & 0 & {e^{{ - it\omega \,\hat{a}^{\dag } \hat{a}}} } & 0 \\ 0 & 0 & 0 & {e^{{ - i\omega \,t\left( {\,\hat{a}^{\dag } \hat{a} - 1} \right)}} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$

(A.6)

$$ \hat{U}_{2} \left( t \right) = \exp \left[ { - i\frac{{\hat{H}_{2} t}}{\hbar }} \right] = \hat{U}_{2A} \otimes \hat{U}_{2B} $$

(A.7)

In the writing of the matrix (A.6), the orthonormal set $\left\{ {\left| {00} \right\rangle ,\;\left| {01} \right\rangle ,\;\left| {10} \right\rangle ,\;{\text{and}}\;\left| {11} \right\rangle } \right\}$, has been used. With the assumption of the similarities of the two atoms A and B, $\hat{U}_{2A}$ and $\hat{U}_{2B}$ are given by:

$$ \begin{gathered} \hat{U}_{2A} = \hat{U}_{2B} = e^{{ - ig\,t\left( {e^{i\theta } \hat{\sigma }_{i}^{ + } \hat{a} + e^{ - i\theta } \hat{\sigma }_{i}^{ - } \hat{a}^{\dag } } \right)}} \hfill \\ \,\,\,\,\,\,\,\,\,\, = \left( {\begin{array}{*{20}c} {\cos \left( {gt\sqrt {\hat{a}^{\dag } \hat{a}} } \right)} & {\frac{{ - ie^{ - i\theta } \hat{a}^{\dag } }}{{\sqrt {\hat{a}\hat{a}^{\dag } } }}\sin \left( {gt\sqrt {\hat{a}\hat{a}^{\dag } } } \right)} \\ {\frac{{ - ie^{i\theta } }}{{\sqrt {\hat{a}\hat{a}^{\dag } } }}\sin \left( {gt\sqrt {\hat{a}\hat{a}^{\dag } } } \right)\hat{a}} & {\cos \left( {gt\sqrt {\hat{a}\hat{a}^{\dag } } } \right)} \\ \end{array} } \right) \hfill \\ \end{gathered} $$

(A.8)

Now, assuming initial atomic state as $\left| {00} \right\rangle$ and using (A.5) – (A.8), we have

$$ \begin{aligned} & \left| {\Psi_{a - f} (t)} \right\rangle\\ & = \,\left( {\begin{array}{*{20}c} {e^{{ - i\left( {\hat{a}^{\dag } \hat{a} + 1} \right)\omega \,t}} \cos^{2} \left( {gt\sqrt {\hat{a}^{\dag } \hat{a}} } \right)} \\ { - ie^{i\theta } e^{{ - i\omega t\hat{a}^{\dag } \hat{a}}} \cos \left( {gt\sqrt {\hat{a}^{\dag } \hat{a}} } \right)\frac{1}{{\sqrt {\hat{a}\hat{a}^{\dag } } }}\sin \left( {gt\sqrt {\hat{a}\hat{a}^{\dag } } } \right)\hat{a}} \\ { - ie^{i\theta } e^{{ - i\omega t\hat{a}^{\dag } \hat{a}}} \frac{1}{{\sqrt {\hat{a}\hat{a}^{\dag } } }}\sin \left( {gt\sqrt {\hat{a}\hat{a}^{\dag } } } \right)\hat{a}\,\cos \left( {gt\sqrt {\hat{a}^{\dag } \hat{a}} } \right)} \\ { - e^{2i\theta } e^{{ - i\left( {\hat{a}^{\dag } \hat{a} - 1} \right)\omega \,t}} \left( {\frac{1}{{\sqrt {\hat{a}\hat{a}^{\dag } } }}\sin \left( {gt\sqrt {\hat{a}\hat{a}^{\dag } } } \right)\hat{a}} \right)^{2} } \\ \end{array} } \right)\left| {\Psi_{f} \left( 0 \right)} \right\rangle \end{aligned}$$

(A.9)

By substituting $\left| {\Psi_{f} \left( 0 \right)} \right\rangle = \sum\limits_{n}^{{}} {g_{n} \left| n \right\rangle }$ in the above equation, the atom–field state is:

$$ \left| {\Psi_{a - f} (t)} \right\rangle = \,\sum\limits_{n = 0}^{\infty } {\left( {a_{n} \left| {00} \right\rangle + b_{n} \left| {01} \right\rangle + c_{n} \left| {10} \right\rangle + {\text{d}}_{n} \left| {11} \right\rangle } \right)\left| n \right\rangle } $$

(A.10)

where $a_{n} ,\;b_{n} ,\;c_{n} \;{\text{and}}\;d_{n}$ are given by:

$$ \begin{gathered} a_{n} = g_{n} \,e^{{ - i\left( {n + 1} \right)\omega \,t}} \cos^{2} \left( {gt\sqrt n } \right) \hfill \\ b_{n} = - ie^{i\theta } g_{n + 1} \,e^{ - in\omega t} \cos \left( {gt\sqrt n } \right)\sin \left( {gt\sqrt {n + 1} } \right) \hfill \\ c_{n} = - ie^{i\theta } g_{n + 1} e^{ - in\omega t} \sin \left( {gt\sqrt {n + 1} } \right)\cos \left( {gt\sqrt {n + 1} } \right) \hfill \\ d_{n} = - e^{2i\theta } g_{n + 2} e^{{ - i\left( {n - 1} \right)\omega \,t}} \sin \left( {gt\sqrt {n + 1} } \right)\sin \left( {gt\sqrt {n + 2} } \right) \hfill \\ \end{gathered} $$

(A.11)

Assuming $\left| {\Psi_{f} \left( 0 \right)} \right\rangle$ is in the coherent, even coherent and odd coherent states, the coefficients, $g_{n}$, are, respectively, given by:

$$ g_{n} = e^{{ - \,\,\frac{{\left| \alpha \right|^{2} }}{2}}} \frac{{\alpha^{n} }}{{\sqrt {n!} }} $$

(A.12)

$$ g_{n} = N_{e} \left( {1 + \left( { - 1} \right)^{n} } \right)e^{{ - \,\,\frac{{\left| \alpha \right|^{2} }}{2}}} \frac{{\alpha^{n} }}{{\sqrt {n!} }} $$

(A.13)

$$ g_{n} = N_{o} \left( {1 - \left( { - 1} \right)^{n} } \right)e^{{ - \,\,\frac{{\left| \alpha \right|^{2} }}{2}}} \frac{{\alpha^{n} }}{{\sqrt {n!} }} $$

(A.14)

where $N_{e}$ and $N_{o}$ have been given in (10). By substituting the above equations in (A.11), (7) and (9) are obtained.