Method for preparing two-atom entangled states in circuit QED and probing it via quantum nondemolition measurements

We propose a probabilistic scheme to prepare a maximally entangled state between a pair of two-level atoms coupled to a leaking cavity mode in circuit QED, without requiring precise time control of the system evolution and initial atomic state. We show that the steady state of this dissipative system is a mixture of two parts, where the atoms are either in their ground state or in a maximally entangled one. Then, by applying a weak probe field to the cavity mode, we are able to discriminate those states without disturbing the atomic system, i.e., to perform a quantum nondemolition measurement via the cavity transmission. In this scheme, one has maximum cavity transmission only when the atomic system is in an entangled state, so that a single click in the detector is enough to be sure that the atoms are in a maximally entangled state. Our scheme relies on an interference effect as it happens in an electromagnetically induced transparency phenomenon so that it works even in the limit of a cavity decay rate much stronger than the atom-field coupling.

Figure 1

(a) Pictorial experimental setup. A pair of two-level atoms coupled to a leaking cavity mode. Once the system reaches the steady state, the weak probe field is switched on and the cavity transmission is monitored. (b) Energy-level diagram of the whole system with the decay rates and the probe field.

Figure 2

(a) Cavity transmission vs detuning between the probe and the cavity field, setting $\lambda =0.2\kappa$, $\varepsilon =0.05\lambda$, and ${\Gamma }_c=0.1\lambda$. We observe that even for $\lambda \sqrt{2}\lesssim \kappa /2$ and ${\Gamma }_c\ne 0$, the cavity transmission is close to zero for ${\Delta }_p=0$ when ${\rho }_{ss}\to |G,0\rangle$ (dashed line), the circles are for ${\Gamma }_c=0$, and the solid line represents the cavity transmission when ${\rho }_{ss}\to |D,0\rangle$ (empty cavitylike). Time evolution of the concurrence $C$ (dashed line) and normalized transmission $T$ (solid line) for a single trajectory when (b) ${\rho }_{ss}\to |G,0\rangle$ and (c) ${\rho }_{ss}\to |D,0\rangle$, assuming $|\varphi \left(0\right)\rangle ={|g\rangle }_1\otimes {|e\rangle }_2\otimes {|0\rangle }_c$.