Generating Controllable Atom-Light Entanglement with a Raman Atom Laser System

We introduce a scheme for creating continuous variable entanglement between an atomic beam and an optical field, by using squeezed light to outcouple atoms from a Bose-Einstein condensate via a Raman transition. We model the full multimode dynamics of the atom laser beam and the squeezed optical field and show that, with appropriate two-photon detuning and two-photon Rabi frequency, the transmitted light is entangled in amplitude and phase with the outcoupled atom laser beam. The degree of entanglement is controllable via changes in the two-photon Rabi frequency of the outcoupling process.

Figure 1

Internal energy levels of our three-level atom. A condensate of state $|1⟩$ atoms confined in a trapping potential is coupled to free space via a Raman transition. The two fields of the Raman transition are a probe beam [annihilation operator $\widehat{E}(x,t)$] and a control field, which we have assumed is strong compared to the probe beam and can be approximated by a classical field [$\Omega (x,t)$] which is detuned from the excited state by an amount $\Delta$.

Figure 2

Density of the condensate atoms $|{\varphi }_0(x){|}^2$ (dotted-dashed line), atom laser beam $⟨{\widehat{\psi }}_2^{†}(x){\widehat{\psi }}_2(x)⟩$ (solid line), and the density of the probe optical field $⟨{\widehat{E}}^{†}(x)\widehat{E}(x)⟩$ multiplied by the ratio of the speed of light to the mean atomic speed $mc/2\hslash k_0$ (dashed line), at $t=40\mathrm{ms}$ for (a) ${\Omega }_{23}=1.5\times {10}^8\mathrm{rad}{\mathrm{s}}^{-1}$ and (b) ${\Omega }_{23}=0.75\times {10}^8\mathrm{rad}{\mathrm{s}}^{-1}$ after 40 ms. The probe beam and the atom laser beam have been multiplied by 100 in order to fit them on the same scale as the condensate.

Figure 3

Variances of the amplitude quadratures for the atom laser beam (solid line) and probe beam (dotted-dashed line) for (a) ${\Omega }_{23}=0.75\times {10}^8\mathrm{rad}{\mathrm{s}}^{-1}$ and (b) ${\Omega }_{23}=1.5\times {10}^8\mathrm{rad}{\mathrm{s}}^{-1}$. The initial state of the probe beam was an amplitude squeezed state with a squeezing parameter $r=2.0$. In case (b), the squeezing is almost completely transferred to the atom laser beam, which destroys the squeezing in the optical beam. In case (a), the squeezing is only partially transferred, and some squeezing remains in the optical beam.

Figure 4

$V_{\inf }({\widehat{X}}^{-})$ (dotted-dashed line), $V_{\inf }({\widehat{X}}^{+})$ (dashed line), and $V_{\inf }({\widehat{X}}^{+})V_{\inf }({\widehat{X}}^{-})$ (solid line) for ${\Omega }_{23}=0.75\times {10}^8\mathrm{rad}{\mathrm{s}}^{-1}$, and an initial amplitude squeezed optical state with the squeezing parameter $r=2.0$.