Emission of Spin-Correlated Matter-Wave Jets from Spinor Bose-Einstein Condensates

We report the observation of matter-wave jet emission in a strongly ferromagnetic spinor Bose-Einstein condensate of ${}^7\mathrm{Li}$ atoms. Directional atomic beams with $|F=1,m_F=1⟩$ and $|F=1,m_F=-1⟩$ spin states are generated from $|F=1,m_F=0⟩$ state condensates or vice versa. This results from collective spin-mixing scattering events, where spontaneously produced pairs of atoms with opposite momentum facilitates additional spin-mixing collisions as they pass through the condensates. The matter-wave jets of different spin states ($|F=1,m_F=\pm 1⟩$) can be a macroscopic Einstein-Podolsky-Rosen state with spacelike separation. Its spin-momentum correlations are studied by using the angular correlation function for each spin state. Rotating the spin axis, the inter- and intraspin-momentum correlation peaks display a high-contrast oscillation, indicating collective coherence of the atomic ensembles. We provide numerical calculations that describe the experimental results at a quantitative level. Our Letter paves the way to generating macroscopic quantum entanglement with the spin and motional degree of freedom with massive particles. It has a wide range of applications from quantum information science to the fundamental studies of quantum entanglement.

Figure 1

Emission of matter-wave jets from spinor BECs. (a) Zeeman shift of the $F=1$ ground state manifold under a magnetic field $B$. The hyperfine interaction introduces a quadratic Zeeman shift $q>0$ for ${}^7\mathrm{Li}$ atoms. (b) A rf pulse flips atoms in the $|0⟩$ state to the $(|\uparrow ⟩-|\downarrow ⟩)/\sqrt{2}$ state, which can be a highly excited spin state under a magnetic field $(q>|c|)$. Quantum fluctuations produce two atom pairs of $|0⟩$ state after spin-mixing collisions, and the excess internal energy is released in the kinetic energy of the atoms, flying in opposite directions because of momentum conservation. (c) Spin-resolved image of the matter-wave jets in $|0⟩$ state with $q=1.7\mathrm{kHz}$ after a hold time of $t_h=7.5\mathrm{ms}$. The image displays atoms in the horizontal plane, where we apply a field gradient during 16 ms of the time-of-flight (TOF). From left to right, the corresponding spins are $\{|\downarrow ⟩,|0⟩,|\uparrow ⟩\}$. The inset represents the condensates remaining in the trap (dashed box). (d) Dressing the $|F=1,m_F=-1⟩$ state with $|F=2,m_F=-2⟩$ state by applying a microwave, we tune the $q$ to have a negative value [39]. (e) After switching on the dressing field, the polar phase is no longer the ground state of the system, creating spin pairs ($|\uparrow ⟩$ and $|\downarrow ⟩$). Through the deexcitation process, the atom obtains kinetic energy from $|q|$. (f) After sufficient stimulated collisions, matter-wave jets of opposite spins are also observed.

Figure 2

(a) Dynamical generation of matter-wave jets. Emitted atom numbers over hold time for $q=\pm 2.2\mathrm{kHz}$. The error bars show 1 standard deviation. The solid line is obtained by numerical calculation based on the Bogoliubov theory of spinor condensate with the effective growth rate ${\gamma }_e=160\mathrm{Hz}$ [26]. (b) Threshold for jet formation. Emitted atom number after $t_h=15\mathrm{ms}$ under various initial atom numbers. To extract the threshold atom number $N_t$ the data are fitted to bilinear curves (dashed lines). Inset: $q_t$ dependence of $N_t$ with power-law fit, $N_t\sim q_t^{0.2}$ (solid line). The error bars mean 95% confidence interval of the bilinear fits.

Figure 3

Angular correlation functions of matter-wave jets after 28 ms of TOF under (a) $q=2.7\mathrm{kHz}$ and (b) $q=-2\mathrm{kHz}$. (b) The top shows the correlation between the same spins (intraspin) and the bottom shows the correlation between the opposite spins (interspin). Shaded areas mark one standard error based on 100–120 realizations. Black solid lines are time-dependent Bogoliubov calculations for the experimental parameters.

Figure 4

As the spin rotation angle $\theta$ is increased from 0 to $\pi /2$, (a) the interspin correlation decreases (from light red to dark red), whereas (b) the intraspin correlation increases (from light green to dark green). The shades represent 1 standard deviation of the mean over 110 independent experiments. (c) Evolution of the averaged momentum correlation peak ${\overline{C}}_{ij}$ [mean value of $C_{ij}(\varphi )$ in the interval $\varphi \in [170°,190°]$ ] under spin-axis rotation. Solid and dashed lines are numerical calculations accounting the field gradient effect [26]. The error bars represent 1 standard deviation of the mean.