Quantum iSWAP gate in optical cavities with a cyclic three-level system

Abstract

In this paper we present a scheme to directly implement the iSWAP gate by passing a cyclic three-level system across a two-mode cavity quantum electrodynamics. In the scheme, a three-level $\Delta$-type atom ensemble prepared in its ground state mediates the interaction between the two-cavity modes. For this theoretical model, we also analyze its performance under practical noise, including spontaneous emission and the decay of the cavity modes. It is shown that our scheme may have a high fidelity under the practical noise.

Appendix: using the time-averaging method to obtain effective hamiltonians

We using the uncoupled time-dependent Schrödinger equation in the interaction picture, as below:

$$\begin{aligned} i\left[ \frac{\partial \vert {\Psi _{1}(t)}\rangle }{\partial t}\right] =H_\mathrm{eff}\vert {\Psi _{1}(t)}\rangle . \end{aligned}$$

(15)

where

$$\begin{aligned} H_\mathrm{eff}=\frac{1}{{i\hbar }}(H^{(1)}_\mathrm{eff}(t)+H^{(2)}_\mathrm{eff}(t))\int (H^{(1)}_\mathrm{eff}(t')+H^{(2)}_\mathrm{eff}(t'))dt'. \end{aligned}$$

(16)

where the indefinite integral is evaluated at time t without a constant of integration. These arguments can be placed on more rigorous footing by considering the evolution of a time-averaged wavefunction.

From the second part of the article, we can know

$$\begin{aligned} H^{(2)}_\mathrm{eff}= & {} \Omega (\vert {+}\rangle \langle {+}\vert -\vert {-}\rangle \langle {-}\vert ), \end{aligned}$$

(17)

$$\begin{aligned} H^{(1)}_\mathrm{eff}= & {} \frac{g_{a}a}{{\sqrt{2}}}(\vert {+}\rangle +\vert {-}\rangle )\langle {g}\vert +\frac{g_{b}b}{{\sqrt{2}}}(\vert {+}\rangle -\vert {-}\rangle )\langle {g}\vert +h.c. \end{aligned}$$

(18)

Therefore, we can obtain the effective Hamiltonian

$$\begin{aligned} \begin{aligned} H_\mathrm{eff}&=\frac{1}{{2\Omega }}[(\vert {+}\rangle \langle {+}\vert -\vert {-}\rangle \langle {-}\vert )(g^{2}_{a}a^{\dag }a+g^{2}_{b}b^{\dag }b)\\&\quad +g_{a}g_{b}(\vert {+}\rangle \langle {+}\vert +\vert {-}\rangle \langle {-}\vert )(a^{\dag }b+ab^{\dag })\\&\quad -2g_{a}g_{b}\vert {g}\rangle \langle {g}\vert (a^{\dag }b+ab^{\dag })]+\hbox {oscillating-terms}. \end{aligned} \end{aligned}$$

(19)

Here, we prepare the initial state of the atom in level $\vert {g}\rangle$, the dynamics generated by $H_\mathrm{eff}$ acting on this state factors out and leaves the atomic state unchanged. This allows us to reduce the dynamics to that of the cavity fields only and if we confine our interest to dynamics which are time-averaged over a period much longer than the period of any of the oscillations present in the effect Hamiltonian then the oscillating terms may be neglected, and we can get the effective Hamiltonian

$$\begin{aligned} H_\mathrm{eff}=-(\zeta a^{\dag }b+\zeta ^{*}b^{\dag }a)\vert {g}\rangle \langle {g}\vert . \end{aligned}$$

(20)

Here $\zeta =\frac{g_{a}g_{b}}{\Omega }$ is the effective coupling constant.