Generating Long-Lived Macroscopically Distinct Superposition States in Atomic Ensembles

We propose to create and stabilize long-lived macroscopic quantum superposition states in atomic ensembles. We show that using a fully quantum parametric amplifier can cause the simultaneous decay of two atoms and, in turn, create stabilized atomic Schrödinger cat states. Remarkably, even with modest parameters these intracavity atomic cat states can have an extremely long lifetime, up to 4 orders of magnitude longer than that of intracavity photonic cat states under the same parameter conditions, reaching tens of milliseconds. This lifetime of atomic cat states is ultimately limited to several seconds by extremely weak spin relaxation and thermal noise. Our work opens up a new way toward the long-standing goal of generating large-size and long-lived cat states, with immediate interests both in fundamental studies and noise-immune quantum technologies.

Figure 1

(a) Schematic setup of our proposal. The pump and signal cavities are coupled via a parametric coupling $J$, and the atomic ensemble is coupled to the signal cavity with a single-atom coupling $g$. The pump cavity is subject to a coherent drive with amplitude $\mathrm{\Omega }$ and frequency ${\omega }_d$. Here, ${\omega }_p$, ${\omega }_s$ are the resonance frequencies of the pump and signal cavities, ${\kappa }_p$, ${\kappa }_s$ are their respective single-photon loss rates, and ${\omega }_q$ is the atomic transition frequency. (b),(c) Quantum Monte Carlo trajectory pictured through the probabilities of the system being in the states $|m_{p}0⟩|n⟩$. Initially, only two atoms in the ensemble are excited. Here, ${\kappa }_p=0.2\chi$ and ${\kappa }_s=\mathrm{\Omega }=0$. (d) Time evolution of the preparation error $\eta$ for a cat size ${|\alpha |}^2=1$. Here, ${\kappa }_p=5\chi$, ${\kappa }_s=0.3{\kappa }_p$, and the ensemble is initialized in the ground state $|0⟩$, the single-excitation state $|1⟩$, and a spin coherent state $|{\theta }_0,0⟩$ with $\sqrt{N}\tan ({\theta }_0/2)=1$ for the states $|{\mathcal{C}}_{+}⟩$, $|{\mathcal{C}}_{-}⟩$, and ${\rho }_{\mathrm{ens}}^{\mathrm{ss}}$, respectively. In (b)–(d), we assume that $N=100$, $J=3g_{\mathrm{col}}$, and both cavities are initialized in the vacuum.

Figure 2

(a) Time evolution of the preparation error $\eta$ of the states $|{\mathcal{C}}_{+}⟩$, $|{\mathcal{C}}_{-}⟩$, and ${\rho }_{\mathrm{ens}}^{\mathrm{ss}}$ under the bosonic approximation for different cat sizes ${|\alpha |}^2=2$, 4, and 6. The initial states are chosen as in Fig. 1. (b) Wigner function at times $t_1,\dots ,t_5$ shown on top of panel (a) for the ${|\alpha |}^2=4$ cat size. The first, second, and third rows correspond to the states $|{\mathcal{C}}_{+}⟩$, $|{\mathcal{C}}_{-}⟩$, and ${\rho }_{\mathrm{ens}}^{\mathrm{ss}}$, respectively. For all plots, we set $J=3g_{\mathrm{col}}$, ${\delta }_p=J^2/(20g_{\mathrm{col}})$, ${\kappa }_p=5\chi$, and ${\kappa }_s=0.3{\kappa }_p$.

Figure 3

(a) Effects of collective dephasing, local dephasing, and inhomogeneous broadening on the preparation error $\eta$ of the state $|{\mathcal{C}}_{+}⟩$ of size $|\alpha {|}^2=2$. We integrated the effective master equation, Eq. (4), with an additional spin dephasing ${\gamma }_{\mathrm{col}}\mathcal{L}(S_z){\rho }_{\mathrm{ens}}$, local spin dephasing ${\gamma }_{\mathrm{loc}}{\sum }_{j=1}^N\mathcal{L}({\sigma }_j^z){\rho }_{\mathrm{ens}}$, and inhomogeneous broadening $\frac{1}{2}{\sum }_{j=1}^N{\delta }_j{\sigma }_j^z$. The frequency shifts ${\delta }_j$ are randomly given according to a Lorentzian distribution of linewidth ${\mathrm{\Delta }}_{\mathrm{inh}}$. For simplicity, we here set $N=10$, ${\gamma }_{\mathrm{col}}={\gamma }_{\mathrm{loc}}={\mathrm{\Delta }}_{\mathrm{inh}}\equiv {\gamma }_{\mathrm{deph}}$ and ${\kappa }_{1\mathrm{at}}=0$ so that only the effects of these dephasing processes are shown. Inset: the Liouvillian spectral gap $\lambda$ of the master equation, Eq. (4), versus the cat size ${|\alpha |}^2$ for ${\kappa }_{1\mathrm{at}}=0$ under the bosonic approximation. (b) Ratio ${\kappa }_{2\mathrm{at}}/{\gamma }_{\mathrm{deph}}$ versus the parameter $\mathrm{\Delta }/g_{\mathrm{col}}$ for ${\gamma }_{\mathrm{deph}}^{-1}=10\mu \mathrm{s}$, $100\mu \mathrm{s}$, and 1 ms for ${\kappa }_p=5\chi$ and $J/2\pi =30\mathrm{MHz}$. The yellow shaded area represents the ${\kappa }_{2\mathrm{at}}\ge 10{\gamma }_{\mathrm{deph}}$ regime, where spin dephasing is strongly suppressed by the two-atom decay. The solid green line shows ${\tau }_{\mathrm{at}}/{\tau }_{\mathrm{ph}}$ versus $\mathrm{\Delta }/g_{\mathrm{col}}$. Other parameters in (a) and (b) are chosen as in Fig. 2.