Distributed Entangled State Production by Using Quantum Repeater Protocol

Abstract

We consider entangled state production utilizing a full optomechanical arrangement, based on which we create entanglement between two far three-level V-type atoms using a quantum repeater protocol. At first, we consider eight identical atoms (1,2,⋯ ,8), while adjacent pairs (i,i + 1) with i = 1,3,5,7 have been prepared in entangled states and the atoms 1, 8 are the two target atoms. The three-level atoms (1,2,3,4) and (5,6,7,8) distinctly become entangled with the system including optical and mechanical modes by performing the interaction in optomechanical cavities between atoms (2,3) and (6,7), respectively. Then, by operating appropriate measurements, instead of Bell state measurement which is a hard task in practical works, the entangled states of atoms (1,4) and (5,8) are achieved. Next, via interacting atoms (4,5) of the pairs (1,4) and (5,8) and operating proper measurement, the entangled state of target atoms (1,8) is obtained. In the continuation, entropy and success probability of the produced entangled state are then evaluated. It is observed that the time period of entropy is increased by increasing the mechanical frequency (ω_M) and by decreasing optomechanical coupling strength to the field modes (G). Also, in most cases, the maximum of success probability is increased by decreasing G and via decreasing ω_M.

Appendices

Appendix A: The used approach to obtain the effective Hamiltonian (4)

As mentioned prior to (4), following the approach explained in Refs. [65, 66], the effective Hamiltonian (4) can be obtained. In this approach, at first using the Hamiltonian (2) and the following Baker-Hausdorff lemma:

$$ \begin{array}{@{}rcl@{}} \hat{H}^{\text{int}}(t)&=&e^{i \hat{H}_{0} t} \hat{H}_{1}e^{-i \hat{H}_{0} t}\\ &=&\hat{H}_{1}+it\left[\hat{H}_{0},\hat{H}_{1} \right]+\frac{(it)^{2}}{2!} \left[\hat{H}_{0},\left[\hat{H}_{0},\hat{H}_{1} \right]\right]+\cdots \end{array} $$

(23)

the Hamiltonian in the interaction picture is obtained. As stated in Refs. [65, 66], after calculating the Hamiltonian in the interaction picture with the following form:

$$ \begin{array}{@{}rcl@{}} \hat{H}^{\text{int}}(t)=\sum\limits_{n=1}^{N}\hat{h}_{n} \exp({-i\omega_{n}t})+\hat{h}_{n}^{\dagger} \exp({i\omega_{n}t}), \end{array} $$

(24)

the effective Hamiltonian can be obtained as below:

$$ \begin{array}{@{}rcl@{}} \hat{H}^{\text{eff}}(t)=\sum\limits_{m,n=1}^{N}\frac{1}{\hbar{\omega}_{{mn}}}[\hat{h}_{m}^{\dagger},\hat{h}_{n}] \exp\left( {i\left[\omega_{m}-\omega_{n}\right]}t\right). \end{array} $$

(25)

In (24), (25), ω_n(m) > 0 is frequency and N is the total number of different harmonic terms making up the interaction Hamiltonian. Also, ω_mn is defined by the following equation:

$$ \begin{array}{@{}rcl@{}} \frac{1}{{\omega}_{{mn}}}=\frac{1}{2}\left( \frac{1}{\omega_{m}}+\frac{1}{\omega_{n}}\right), \end{array} $$

(26)

which may be considered as the harmonic average of ω_m and ω_n. In our case, the effective Hamiltonian (4) has been obtained using (3), (24), (25) and (26).

Appendix B: Calculating the coefficients of entangled state (5)

In this section, the differential equations related to the state (5) with the help of effective Hamiltonian (4) and the time-dependent Schrödinger equation are obtained as $\dot {X}=SX$, where $\dot {X}$ and S are respectively defined as follow,

$$ \begin{array}{@{}rcl@{}} \dot{X}&=& \frac{d}{dt} \left( \begin{array}{ccccccccccc} A_{1}(t) & A_{2}(t) & A_{3}(t) & A_{4}(t) & A_{5}(t) & A_{6}(t) & A_{7}(t) & A_{8}(t) & A_{9}(t) & A_{10}(t) & A_{11}(t) \end{array} \right)^{T},\\ \end{array} $$

(27)

$$ S= \left( \begin{array}{ccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & i\frac{G \lambda_{1}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}} & i\frac{ (2{\lambda^{2}_{1}}+ G^{2})}{\omega_{M}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -2 i\frac{ {\lambda^{2}_{1}}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & i\frac{G \lambda_{1}}{\omega_{M}} & -i\frac{ ({\lambda^{2}_{1}}-G^{2})}{\omega_{M}} & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & 2 i\frac{ G \lambda_{1}}{\omega_{M}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & i\frac{G \lambda_{1}}{\omega_{M}} & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & -i\frac{ ({\lambda^{2}_{1}}-G^{2})}{\omega_{M}} & 2 i\frac{G \lambda_{1}}{\omega_{M}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 i\frac{G \lambda_{1}}{\omega_{M}} & 2 i\frac{G \lambda_{1}}{\omega_{M}} & 4 i\frac{ ({\lambda^{2}_{1}}+ G^{2})}{\omega_{M}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & -i\frac{{\lambda^{2}_{1}}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i\frac{G \lambda_{1}}{\omega_{M}} & i\frac{G \lambda_{1}}{\omega_{M}} & i\frac{(2 {\lambda^{2}_{1}}+ G^{2})}{\omega_{M}} \end{array} \right). $$

(28)

The equation $\dot {A}_{1}(t)=0$ easily results in $A_{1}(t)=\frac {1}{2}$. Also, paying attention to the considered initial conditions, we find that A₂(t) = A₉(t), A₃(t) = A₁₀(t), A₄(t) = A₁₁(t) and A₆(t) = A₇(t). Now, using Laplace transform and with the help of Mathematica software, the coefficients A₄(t), A₅(t), A₈(t) and then the coefficients A₂(t), A₃(t) and A₆(t) can be calculated, however, due to their complex and lengthy forms we ignore presenting their explicit forms.

Appendix C: Calculating the coefficients of entangled state (11)

The differential equations related to state (11) using the effective Hamiltonian (10) and the time-dependent Schrödinger equation are obtained as $\dot {Y}=SY$, where S has been defined in (28) and $\dot {Y}$ is defined as follows,

$$ \begin{array}{@{}rcl@{}} \dot{Y}&=& \frac{d}{d\tau} \left( \begin{array}{ccccccccccc} {B^{i}_{1}}(\tau) & {B^{i}_{2}}(\tau) & {B^{i}_{3}}(\tau) & {B^{i}_{4}}(\tau) & {B^{i}_{5}}(\tau) & {B^{i}_{6}}(\tau) & {B^{i}_{7}}(\tau) & {B^{i}_{8}}(\tau) & {B^{i}_{9}}(\tau) & B^{i}_{10}(\tau) & B^{i}_{11}(\tau) \end{array} \right)^{T}. \end{array} $$

(29)

Paying attention to (17), (20), it may be seen that, knowing the coefficients ${B^{i}_{2}}(\tau )$, ${B^{i}_{3}}(\tau )$, ${B^{i}_{9}}(\tau )$ and $B^{i}_{10}(\tau )$ is enough for our purpose, and these coefficients can be obtained after calculating ${B^{i}_{4}}(\tau )$ and $B^{i}_{11}(\tau )$ using Laplace transform and with the help of Mathematica software.