Dissipative Preparation of Entanglement in Optical Cavities

We propose a novel scheme for the preparation of a maximally entangled state of two atoms in an optical cavity. Starting from an arbitrary initial state, a singlet state is prepared as the unique fixed point of a dissipative quantum dynamical process. In our scheme, cavity decay is no longer undesirable, but plays an integral part in the dynamics. As a result, we get a qualitative improvement in the scaling of the fidelity with the cavity parameters. Our analysis indicates that dissipative state preparation is more than just a new conceptual approach, but can allow for significant improvement as compared to preparation protocols based on coherent unitary dynamics.

Figure 1

(a) Level diagram of a single atom with laser detuning $\Delta$ and cavity detuning $\delta$ from two photon resonance. The optical pumping laser for the two atoms differs by a relative phase of $\pi$. (b) The effective two qubit system. The driving ${\Omega }_{\mathrm{MW}}$ causes rapid transitions between the three triplet states. The atoms decay through the cavity from $|00⟩$ to $|S⟩$ and from $|S⟩$ to $|11⟩$ with effective decay rates ${\kappa }_{\mathrm{eff},1}$ and ${\kappa }_{\mathrm{eff},2}$, where ${\kappa }_{\mathrm{eff},1}\gg {\kappa }_{\mathrm{eff},2}$. The spontaneous emission rates ${\gamma }_{\mathrm{eff},i}$ will tend to reduce the fidelity by redistributing information to the triplet states. (c) Fidelity as a function of the cooperativity $C=g^2/\kappa \gamma$. The inset gives a more accurate account of the scaling ($1-F\le 3.5C^{-1}$) for different values of $\kappa /\gamma$.

Figure 2

The main figure shows the population of the singlet state (thick dashed line) and of the three triplet states (full lines) as a function of time for a random initial state. The curves were plotted for $C=50$, $\kappa =\gamma /2$, $\Omega =5{\Omega }_{\mathrm{MW}}/2$, $g=20\Omega$, and $\Delta$, $\delta$ are such that they maximize the fidelity for small $\Omega$. In this parameter regime, the stationary state has a 92% fidelity with respect to the singlet state. The inset shows the maximal fidelity as a function of the gap size for $C=50$ and $\kappa =\gamma /2$. The main figure corresponds to the cross on the curve in the inset.