Squeezing and Entanglement of Density Oscillations in a Bose-Einstein Condensate

The dispersive interaction of atoms and a far-detuned light field allows nondestructive imaging of the density oscillations in Bose-Einstein condensates. Starting from a ground state condensate, we investigate how the measurement backaction leads to squeezing and entanglement of the quantized density oscillations. We show that properly timed, stroboscopic imaging and feedback can be used to selectively address specific eigenmodes and avoid excitation of nontargeted modes of the system.

Figure 1

(a) A BEC imprints a phase shift on a planar coherent light field depending on the atomic density along the BEC axis. The phase shifts are spatially detected by an encapsulating ($l_L>l_x$) array of homodyne detectors (pixels) of widths $l_D=l_x/10$. (b) The atomic modes can be selectively addressed and squeezed, illustrated here, where first the 3rd mode, and subsequently, modes 1 and 5 are squeezed. (c) The absolute value of the covariance matrix elements for the first three odd modes. Continuous probing (i) squeezes and correlates a swathe of atomic modes, while stroboscopic probing (ii) can generate correlations between only selected modes. The same level of entanglement of modes 1 and 3 is achieved in (i) and (ii), while the strong squeezing and the addressing of the 5th mode in (i) is absent in (ii). See text for the pulse sequences. The simulations are for the $D_2$ line (${\sigma }_{+}$ polarized) of 1000 ${}^{87}\mathrm{Rb}$ atoms in the $|F,F_z⟩=|2,2⟩$ state with ${\omega }_x=2\pi \times 150\mathrm{Hz}$, ${\omega }_{\perp }=100{\omega }_x$, and $\mu =2\hslash {\omega }_x$.

Figure 2

The squeezing of the 3rd mode after stroboscopically probing (${\kappa }^2=100{\omega }_x/2\pi$) for 100 trap periods is shown, while the temporal evolution of the quadrature variance $\mathrm{var}[{\widehat{x}}_3^0]$ is illustrated in the inset. Dotted lines indicate the results of the noninteracting atomic system for comparison. For small $\mathrm{\Delta }\varphi$ (pulse duration), $\mathrm{var}[{\widehat{x}}_3^0]$ follows the QND result (red line, see text), while for larger $\mathrm{\Delta }\varphi$, the squeezing is faster but suboptimal. Excellent selectivity, Hellinger distance $D_{\mathcal{H}}\simeq 0$, and little crosstalk, Purity $P\simeq 1$, are observed for small $\mathrm{\Delta }\varphi$.

Figure 3

(a) Modes 1 and 3 are entangled by a pulse sequence of 100 trap periods with ${\varpi }_1={\omega }_1+{\omega }_3$, $\mathrm{\Delta }\phi /2\pi =0.03$, and ${\kappa }^2=30{\omega }_x/2\pi$. The logarithmic negativity $E_{13}$ slightly outperforms the corresponding QND result $E_{\mathrm{QND}}^{13}$. The selectivity within the $j=1,3,5$ subsystem is excellent ($D_{\mathcal{H}}\simeq 0$) and the subsystem’s purity is $P\sim 97\%$. (b) The mode functions (not to scale) of the condensate $f_0(x)$, entangled modes $f_{1,3}^{-}(x)$, and the overlap $\sim f_0^2(x)f_1^{-}(x)f_3^{-}(x)$.

Figure 4

A stochastic trajectory of $⟨{\widehat{x}}_3(t)⟩$ and $⟨{\widehat{p}}_3(t)⟩$, and the standard deviations of 1000 trajectories ${\sigma }_{⟨{\widehat{x}}_3⟩}(t)$ and ${\sigma }_{⟨{\widehat{p}}_3⟩}(t)$, corresponding to Fig. 2 ($\mathrm{\Delta }\varphi /2\pi =0.15$). The gray regions indicate when the measurement occurs. (a) Without feedback, the trajectories diffuse and oscillate, and approximately ${\sigma }_{⟨{\widehat{x}}_3⟩}(t)={\sigma }_{⟨{\widehat{p}}_3⟩}(t)=\sqrt{{\nu }_3{\tau }_T/2}$ (red line). (b) With feedback, the desired damping of displacements is demonstrated.