Heralded Generation of Macroscopic Superposition States in a Spinor Bose-Einstein Condensate

Macroscopic superposition states enable fundamental tests of quantum mechanics and hold a huge potential in metrology, sensing, and other quantum technologies. We propose to generate macroscopic superposition states of a large number of atoms in the ground state of a spin-1 Bose-Einstein condensate. Measuring the number of particles in one mode prepares with large probability highly entangled macroscopic superposition states in the two remaining modes. The macroscopic superposition states are heralded by the measurement outcome. Our protocol is robust under realistic conditions in current experiments, including finite adiabaticity, particle loss, and measurement uncertainty.

Figure 1

(a) Energy gap $\mathrm{\Delta }E$ between the ground state and the first excited state of Eq. (1), as a function of $q/q_c$. (b) Wigner distribution of the ground state in the three different regimes on a reduced Bloch sphere for $N=100$ particles. In the $P$ phase (right), tracing out the $h$ mode gives a coherent spin state polarized along the positive $S_z$ axis. In the TF phase (left), tracing out the $m_F=0$ mode gives a Twin-Fock state on the $J_x\text{−}{J}_y$ equator. For $q=0$ (central sphere), we plot the Wigner distribution of the state heralded by the measurement result $N_h=0$: this is a NOON-like state along the $S_x$ axis. (c) QFI $F_Q[|\psi ⟩,\widehat{R}]$ of the (full) ground state for different generators $\widehat{R}$ (see legend). In panels (a) and (c), thick lines are obtained with $N=100$ and thin lines with $N=1000$. The quantum phase transition points are indicated by dashed vertical lines.

Figure 2

Stochastic generation of MSSs under ideal conditions. (a) Probability (bars) to measure $N_h$ particles in the ground state. The dashed lines are cumulative probability thresholds. (b) Conditional probability $P(N_g|N_h)=|⟨N_g,N-N_h-N_g|\exp [-i(\pi /2){\widehat{S}}_y]|{\varphi }_{N_h}⟩{|}^2$ to find $N_g$ particles in the $g$ mode given a heralded measurement of $N_h$ particles in the $h$ mode. The rotation of the heralded state $|{\varphi }_{N_h}⟩$ helps the visualization. Note that $N_g$ ranges from 0 to $N-N_h$. The two branches of $P(N_g|N_h)$ for $N_h\lesssim N/2$ correspond to macroscopic superpositions. (c) Examples of $P(N_g|N_h)$ (left) and Husimi distributions on the $({\widehat{S}}_x,{\widehat{S}}_y,{\widehat{S}}_z)$ Bloch sphere (right) of rotated heralded states for different values of $N_h$. (d) $F_Q[|{\varphi }_{N_h}⟩,{\widehat{S}}_x]$ of the heralded states as a function of $N_h$. In all panels $N=100$.

Figure 3

Stochastic generation of MSSs under realistic conditions. Simulations for $N=100$ particles, initial value $q_0/q_c=1.5$, loss rate of $\mathrm{\Gamma }=0.01\mathrm{Hz}$, and $t_c^{-1}=2\pi \times 1\mathrm{Hz}$, and a speed of quasiadiabatic state preparation proportional to $Q$. (a) FI obtained by measuring particle numbers as a function of $Q$, compared to the ideal QFI of the full ground state $F_{\mathrm{CBA}}$. For each $Q$, the FI is evaluated at the optimal phase ${\theta }_{\mathrm{opt}}(Q)$. In panels (b)–(d), $Q=0.15$ (corresponding to a ramping time of 1.5 sec) as at the maximum in (a). (b) Probability (bars) to find $N_h$ particles in the final state. The dashed lines are cumulative probability thresholds. (c) Heralded states, rotated by $\exp [-i(\pi /2){\widehat{S}}_y]$. We show the probability of $N_g$ particles in the $g$ mode for $N_h$ particles measured in the $h$ mode. (d) FI of the heralded states (bars), the red line is the shot noise limit.