Manipulation of collective quantum states in Bose-Einstein condensates by continuous imaging

We develop a Gaussian state treatment that allows a transparent quantum description of the continuous, nondestructive imaging of and feedback on a Bose-Einstein condensate. We have previously demonstrated [A. C. J. Wade et al., Phys. Rev. Lett. 115, 060401 (2015)] that the measurement backaction of stroboscopic imaging leads to selective squeezing and entanglement of quantized density oscillations. Here, we investigate how the squeezing and entanglement are affected by the finite spatial resolution and geometry of the probe laser beam and of the detector and how they can be optimized.

Figure 1

A far-detuned light field propagating along the $z$ axis interacts dispersively with an atomic ensemble, which imprints density-dependent phase shifts across the coherent wave front. The detection of the optical phase shifts by an array of homodyne detectors (pixels) of widths $l_{\mathrm{D}}$ reveals information about the atomic density along the BEC axis. The simulated examples are performed for probing of the $D_2$ line (${\sigma }_{+}$ polarized light) of $1000{}^{87}\mathrm{Rb}$ atoms, prepared in the $|F,m_F\rangle =|2,2\rangle$ state with axial ${\omega }_x=2\pi \times 150\mathrm{Hz}$ and radial ${\omega }_{\perp }=100{\omega }_x$ trapping frequencies, and chemical potential $\mu =2\hslash {\omega }_x$.

Figure 2

The planar coherent light field is discretized into cuboidal mode functions for the specific example of $N_{\mathrm{L}}=2$. During a sequence of time steps of duration $\tau$, sets of light modes along the $x$ direction are detected after interacting with the atomic system.

Figure 3

The diagonal ${\overline{\kappa }}_{jj}^2$ (solid) and off-diagonal ${\overline{\kappa }}_{j\overline{j}}^2$ (dashed) couplings with $\overline{j}=j+2$ are shown for the ideal detector (blue, upper curve) and for $l_{\mathrm{R}}=(0.2977,0.6306,1.0658) [\lambda =(780.24\mathrm{nm},3.5\mu \mathrm{m},10\mu \mathrm{m}\left)\right]$ shown from the upper to the lower solid and dashed curves (green, red, and yellow). The lines are to guide the eye between discrete $j$ values.

Figure 4

The temporal dynamics of $\mathrm{var}\left[{\widehat{x}}_j\right]$ (solid line), $\mathrm{var}\left[{\widehat{p}}_j\right]$ (dashed line), and $\mathrm{covar}[{\widehat{x}}_j,{\widehat{p}}_j]$ (dash-dotted line) is shown for the zeroth (blue), first (green), and second (red) modes, with ${\overline{\kappa }}_{11}^2={\omega }_x$. In the lower panels, the uncertainty ellipse of the Gaussian Wigner function of the first mode (green) is illustrated at times ${\omega }_{x}t=(0,{\omega }_x{t}_{\mathrm{m}},2\pi )$ with the corresponding steady-state prediction given by Eq. (12) (dashed blue line). (a) Initially a circle, the distribution is squeezed along the $x_1$ axis. (b) However, the antisqueezed $p_1$ quadrature rotates $x_1$, yielding a transient minimum squeezing of ${\widehat{x}}_1$. (c) Ultimately, the dynamics average, yielding a steady state.

Figure 5

The purity $P_m$ of the subset of the first $m+1$ modes is shown as a function of detector resolution $l_{\mathrm{D}}$ at time ${\omega }_{x}t=\pi$ for the simulation of Fig. 4, where the majority of the density oscillations have reached, or are close to, steady-state covariances.

Figure 6

The bipartite entanglement $E_{13}$ between modes 1 and 3, obtained by stroboscopic probing at the sum frequency $\varpi ={\omega }_1+{\omega }_3$, is shown as a function of pixel resolution and probing time. The simulation assumes probe pulses with ${\kappa }^2=30{\omega }_x/2\pi$, centered on times $\varpi t_l=[0,2\pi ,4\pi ,...,2(n-1)\pi ]$ with identical durations $\varpi \mathrm{\Delta }t=2\pi \times 0.03$. The degree of entanglement is represented by the color brightness, and the white contours are to guide the eye.

Figure 7

(a) The gray curves show individual trajectory results for $〈{\widehat{x}}_3\left(t\right)〉$. The purple (red) curve illustrates a single, undamped (damped) trajectory with the same particular measurement record. The measurement strength is ramped down after ${\omega }_{x}t=\pi$, as illustrated (solid gold line) by ${\overline{\kappa }}_{jj}^2\left(t\right)$ in (b) with $j=3$. (b) The variances, ${\sigma }_{\langle {\widehat{x}}_j\rangle }^2\left(t\right)$, of ${10}^4$ trajectories for the (undamped) damped cases of $j=1$–3 by the (solid) dashed blue, green, and red lines, respectively. The undamped ${\sigma }_{\langle {\widehat{x}}_j\rangle }^2\left(t\right)$ oscillates around ${\sigma }^2\left(t\right)={\int }_0^{t}d{t}^{\prime }{\overline{\kappa }}_{jj}^2\left(t^{\prime }\right)/2$ (dashed gold line).

Figure 8

(a) The central region $R_2$ of width $l_{\mathrm{G}}$ is probed with a Gaussian beam of finite width $l_{\mathrm{G}}$. (b) Due to interactions, the local density fluctuations along the BEC axis are sub-Poissonian in the unprobed state (dashed blue line), and they are further suppressed by the probing in the region $R_2$ (solid green line). The solid green line corresponds to the simulation of Fig. 9 with $l_{\mathrm{G}}=l_{\mathrm{R}}$.

Figure 9

(a) The (blue) covariance $\mathrm{covar}[{\widehat{N}}_1,{\widehat{N}}_3]$ (normalized to $N_0$) between the number of atoms in the unprobed regions $R_1$ and $R_3$ and the variance $\mathrm{var}\left[{\widehat{N}}_2\right]$ (green) of the number of atoms in region $R_2$ are shown, normalized by the BEC population in $R_2,N_2^0={\int }_{R_2}dx{n}_0\left(x\right)$. The dashed curves show the results for the unprobed BEC. The solid curves show that after probing for a time ${\omega }_{x}t=3\pi /2$ with constant ${\omega }_x{\overline{\kappa }}_{00}^2=3/2$ and $l_{\mathrm{D}}=0.05l_x,\mathrm{var}\left[{\widehat{N}}_2\right]$ is reduced and the number fluctuations in regions $R_1$ and $R_3$ are reduced and can become anticorrelated. (b) The temporal evolution of the number fluctuations for $l_{\mathrm{G}}=l_{\mathrm{R}}$ (solid line), $2l_{\mathrm{R}}$ (dashed line), $3l_{\mathrm{R}}$ (dash-dotted line), and $4l_{\mathrm{R}}$ (dotted line). The feature at ${\omega }_{x}t\sim \pi$ is explained in the text.

Figure 10

Controlled dynamical correlations. (a)–(c) Modes $j=1$–3, respectively, are squeezed with a train of $n$ pulses centered on times $t_l=[0,\pi /{\omega }_j,2\pi /{\omega }_j,...,(n-1)\pi /{\omega }_j]$ with identical durations $\tau ={\left(200{\omega }_j\right)}^{-1},{\kappa }^2=100{\omega }_x/2\pi ,l_{\mathrm{D}}=0.1l_x$, and the total interrogation time of 100 trap cycles. The upper panels show $\mathcal{N}(x_1,x_2)$ when $\mathrm{var}\left[{\widehat{x}}_j\right]=[0.0675,0.0587,0.0557]$ is minimal and for $j=[1,2,3]$, while the lower panels correspond to the maxima $\mathrm{var}\left[{\widehat{x}}_j\right]=[3.7089,4.2692,4.5057]$. (d) The deviations from Poissonian momentum-momentum correlations for $\mathrm{covar}[\widehat{m}\left(k_1\right),\widehat{m}\left(k_2\right)]/5$ in the case of squeezing $j=1$.