Quasideterministic generation of maximally entangled states of two mesoscopic atomic ensembles by adiabatic quantum feedback

We introduce an efficient, quasideterministic scheme to generate maximally entangled states of two atomic ensembles. The scheme is based on quantum nondemolition measurements of total atomic populations and on adiabatic quantum feedback conditioned by the measurements outputs. The high efficiency of the scheme is tested and confirmed numerically for ideal photodetection as well as in the presence of losses.

Figure 1

(Color online) Level structure of the atoms. $\mid g⟩$ and $\mid f⟩$ are metastable states, $\mid g⟩$ is coupled off resonantly by the electromagnetic field, whose annihilation operator is ${\widehat{a}}_I$, to the excited state $\mid e⟩$. Here $\gamma$ is the spontaneous transition rate and $\Delta$ is the detuning between the coupling field and the atomic transition frequency.

Figure 2

(Color online) Schematic experimental configuration. Atoms occupying the internal state $\mid g⟩$ in the two samples interact with the light field, incident from the left. The presence of the cavities guarantees a high spectral and directional resolution of the entering photon. The phase shift of the light field due to the interaction with the atoms is registered by the different photocurrents in the two detectors. These signals can be combined with that produced by a signal generator and hence sent back to the two ensembles.

Figure 3

(Color online) Numerical simulations of the evolution in the presence of feedback of the overlap $\left(\mathrm{Tr}\left[{\varrho }_n{\varrho }_{\mathit{me}}\right]\right)$ between the state of the two atomic samples $(N=10)$ and the maximal entangled state in a case in which the feedback scheme is successful ($S$, dotted line) in generating ${\varrho }_{\mathit{me}}$ and in a case in which the scheme is broken ($B$, solid line) by “anomalous” feedback angles (see text); inset: Evolution of the entanglement $E$ in the $S$ case (dotted line) and in the $B$ case (solid line). The values of the parameters used in the simulations are $\eta =1$, $\chi =0.03$ (achievable with highly collimated beams), $\Omega =\pi ∕10$, and the total number of incoming photons is $n_{\mathrm{ph}}=5\times {10}^4$.

Figure 4

(Color online) (a) Numerical simulations of the evolution of $⟨{\widehat{J}}_x^{+}⟩$ (solid line) and of $\delta {\widehat{J}}_z^{+}+\delta {\widehat{J}}_y^{-}$ (dotted line) in the $S$ case; inset (a) evolution of the feedback angle $\lambda$ during the $S$ simulation; (b) Evolution of $⟨{\widehat{J}}_x^{+}⟩$ (solid line) and of $\delta {\widehat{J}}_z^{+}+\delta {\widehat{J}}_y^{-}$ (dotted line) in the $B$ case. Notice that after the point in which $⟨{\widehat{J}}_x^{+}⟩<\delta {\widehat{J}}_z^{+}+\delta {\widehat{J}}_y^{-}$, the variances explodes. This corresponds to exploding values of $\lambda$ around that point [inset (b)]. Parameter values are the same as in Fig. 3.

Figure 5

(Color online) Evolution of $C_{\mathrm{LUR}}$ in a $S$ simulation (dotted line) and in a $B$ simulation (solid line). Positive values of $C_{\mathrm{LUR}}$ provide a quantitative estimate of the amount of entanglement. Parameter values are the same as in Fig. 3 except that now $\eta =0.9$.

Figure 6

(Color online) (a) Evolution of the overlap in the $S$ case (dotted line) and in the $B$ case (solid line); (b) evolution of the purity $\mathrm{Tr}\left[{\varrho }_n^2\right]$ in the $S$ case (dotted line) and in the $B$ case (solid line). Notice that a successful feedback scheme has also a purification action. Obviously when the scheme is broken the purification effect is lost. Parameter values are the same as in Fig. 5.