Probing Spin Correlations in a Bose-Einstein Condensate Near the Single-Atom Level

Using parametric conversion induced by a Shapiro-type resonance, we produce and characterize a two-mode squeezed vacuum state in a sodium spin 1 Bose–Einstein condensate. Spin-changing collisions generate correlated pairs of atoms in the $m=\pm 1$ Zeeman states out of a condensate with initially all atoms in $m=0$. A novel fluorescence imaging technique with sensitivity $\mathrm{\Delta }N\sim 1.6$ atom enables us to demonstrate the role of quantum fluctuations in the initial dynamics and to characterize the full distribution of the final state. Assuming that all atoms share the same spatial wave function, we infer a squeezing parameter of 15.3 dB.

Figure 1

Shapiro resonance for a spin 1 BEC. (a) The quadratic Zeeman energy originating from a static magnetic field creates an energy offset $2q_0>0$ between a $(+1,-1)$ pair and a (0,0) one. (b) In the presence of an additional modulated field, the dynamics can be described by the secular Hamiltonian (4). The offset $2q_0$ is replaced by the detuning $\delta$ from the resonance, allowing one to control the sign of the energy difference between $(+1,-1)$ and (0,0) pairs.

Figure 2

Spin-resolved fluorescence imaging. (a) Imaging system recording the fluorescence from each Zeeman component on a CCD camera. A microscope objective (numerical aperture 0.33) is relayed by a pair of lenses (not shown) and a tailor-made spatial filter. (b) Typical fluorescence image for a molasses duration $t_{\mathrm{mol}}=5\mathrm{ms}$. The distance between adjacent clouds is 1.3 mm. The dotted (resp. dashed) contours $A_m$ (resp. $A_m^{\prime }$) show the raw (resp. optimized) regions of interest. (c) Mean fluorescence signal for a pure $m=0$ cloud as a function of $t_{\mathrm{mol}}$. The squares (resp. circles) indicate the total fluorescence signal in $A_0$ (resp. $A_0^{\prime }$). (d) Fluorescence profiles for four values of $t_{\mathrm{mol}}$ after integration along the $y$ direction.

Figure 3

Production of a TMSV state. Measured evolution of the mean number of pairs ${\overline{N}}_p$ (red circles) and the standard deviation $\mathrm{\Delta }{N}_p$ (blue squares). The continuous red line shows the numerical solution of the Schrödinger equation using the secular Hamiltonian (4). The dashed green line shows the prediction from the Bogoliubov Hamiltonian derived from ${\widehat{H}}_{\sec }$. Inset (a): Same data for longer evolution times. Inset (b): Evolution of the average value ${\overline{J}}_z$ (red diamonds) and standard deviation $\mathrm{\Delta }{J}_z$ (blue stars).

Figure 4

Characterization of a TMSV state. (a) The red points show 536 repeated measurements of the TMSV state produced by the sequence of Fig. 3 for $t_{\mathrm{osc}}=150\mathrm{ms}$ (${\overline{N}}_p=105$). The blue points are experimental results for a balanced spin-coherent state with $\sim 2{\overline{N}}_p$ atoms in $m=\pm 1$. The dashed blue curve shows the variation of the typical range for $J_z$ (i.e., $\pm \mathrm{\Delta }{J}_z$) for a coherent state with $2N_p$ atoms. The solid gray lines show the expected detection noise. (b) Histograms of $J_z$ for the TMSV state (red) and the coherent state (blue). The respective standard deviations are 1.55 and 7.26. (c) Histogram of $N_p$ for the TMSV state. The solid line is the Bose–Einstein distribution of mean ${\overline{N}}_p=105$. (d) Spin-squeezing parameter ${\zeta }_s^{-2}$ versus $N_p$, calculated with a $\mathrm{\Delta }{N}_p=50$ bin width. Here all data at $t\le 250\mathrm{ms}$ are used (928 measurements). The error bar (66% confidence interval) is obtained using the bootstrap method. The average squeezing parameter (red line) for $N_p>100$ is ${\zeta }_s^{-2}=15.3\mathrm{dB}$.