Collective three-body interactions enable a robust quantum speedup

Abstract

We show that collective three-body interactions (3BIs), implementable with $N$ atoms loaded inside an optical cavity, offer a significant advantage for preparing complex multipartite entangled states. Firstly, they enable a speedup of order $\sim N$ in preparing generalized Greenberger-Horne-Zeilinger (GHZ) states, outperforming conventional methods based on all-to-all two-body Ising interactions. Secondly, they saturate the Heisenberg bound in phase estimation tasks using a time-reversal protocol realized through simple rotations and followed by experimentally accessible collective spin measurements. Lastly, compared with two-body interactions (2BIs), in the presence of cavity losses and single particle decoherence, 3BIs feature a high gain in sensitivity for moderate atom numbers and in large ensembles a fast entanglement generation despite constraints in parameter regimes where they are implementable.

Figures (7)

• Figure 1: (a) Energy level diagram of the atomic ensemble used to engineer the 3BIs, where a pair of momentum states, $\ket{p_0\pm\hbar k}$, serves as the pseudo-spins $\ket{\uparrow/\downarrow}$luo2025realization. Inset: schematics of the setup. An optical cavity is driven by two dressing lasers with frequencies $\omega_{p1,2}$ (red and blue arrows). (b) Corresponding Frequency diagram. (c) Two dressing lasers induce a six-photon process that flips three spins collectively between $\ket{\downarrow,\downarrow,\downarrow}$ and $\ket{\uparrow,\uparrow,\uparrow}$ via an intermediate excited state. Virtual cavity photons at two frequencies are shown as red and blue wavy lines (see text). (d) Mean-field dynamics for an initial spin along $z$. Gray lines show classical trajectories with trifurcation points; bold arrows mark the separatrix.
• Figure 2: (a) Quantum Fisher information (QFI) growth vs time for a system with $N=45$ atoms, starting from an initial spin coherent state aligned along the $z$-axis. The dashed line indicates the Heisenberg limit (HL). (b) Spin distribution function, $|c_{M_z}|^2$, in the Dicke basis $\ket{S=N/2, M_z}$, and (c) the corresponding spin Wigner function, after evolution under the three-body Hamiltonian $\hat{H}_3$ at $\tau = 0.1$ (red square), $0.66$ (blue circle), and $1.3$ (yellow star), with rescaled evolution time $\tau=\chi_3 N^{\frac{3}{2}} t$. A final $\pi/2$ rotation about the $x$-axis is applied to facilitate visualization.
• Figure 3: (a) Time evolution of the QFI for atom numbers ranging from $N = 301$ to $306$. The initial rise of the QFI up to the first peak (blue dots) is well captured by the semiclassical prediction, occurring at $\tau_{\rm opt,3}\approx 2/3$, while the subsequent dynamics reveals sensitivity to the atom number. (b) QFI evaluated at $\tau_{\rm opt,3}$ as a function of atom number. The black line denotes the HL $F_Q=N^2$.
• Figure 4: (a) Time-reversal (TR) protocol for quantum-enhanced sensing. (b) Dominant dissipation channels: Spontaneous emission into free space at rate $\gamma_e$ and photon loss through cavity mirrors at rate $\kappa$, leading to two balanced collective decay modes $\hat{S}_\pm$. (c) Metrological gain $G$ versus collective cooperativity $NC$ for 3BIs (squares) and OAT (triangles), optimized over single-particle and collective dissipation; colors denote different atom numbers, while the values in square brackets indicate the cooperativity. (d) Ratio of the QFI for 3BIs to that of OAT (left) and TAT (right) as functions of $N$ and $C$, including collective dissipation. The QFI is evaluated at the optimal TBI evolution time.
• Figure 5: Optimization procedure for extracting the scaling behavior. (a) Metrological gain as a function of evolution time for fixed $N = 48$ and $C = 10$, with different colors representing various values of $d=2\Delta_c/\kappa$. For each $d$, the peak gain is identified. (b) Peak gain as a function of $d$. (c) Repeating the above procedure for different atom numbers $N$ reveals a linear scaling of the optimal gain with respect to $N$.