Reconfigurable dissipative entanglement between many spin ensembles: from robust quantum sensing to many-body state engineering

TL;DR

This work introduces a reconfigurable reservoir-engineering framework in cavity QED that uses a single collective decay channel together with a patterned drive detuning and chiral spin-exchange to stabilize a wide class of pure entangled states across multiple spin ensembles. For two ensembles, the authors derive an exact pure steady state and show how to achieve Heisenberg-limited differential sensing via Ramsey-type readout, with robustness to common-mode noise; extending to many ensembles yields a structured MPS description and enables distributed metrology with HL scaling. The approach also connects to 1D symmetry-protected topological order, enabling stabilization of spin-1 AKLT-type states and a broader family of SPT states in spin chains, with tunable string order and correlation lengths. Overall, the scheme provides a flexible, experimentally accessible route to robust quantum sensing and complex many-body state engineering using only collective dissipation and Hamiltonian control, with direct relevance to quantum metrology, sequential quantum circuits, and tensor-network states.

Abstract

An attractive approach for stabilizing entangled many-body spin states is to employ engineered dissipation. Most existing proposals either target relatively simple collective spin states, or require numerous independent and complex dissipative processes. Here, we show a surprisingly versatile scheme for many-body reservoir engineering that relies solely on fully collective single-excitation decay, augmented with local Hamiltonian terms. Crucially, all these ingredients are readily available in cavity QED setups. Our method is based on splitting the spin system into groups of sub-ensembles, and provides an easily tunable setup for stabilizing a broad family of pure, highly entangled states with closed-form analytic descriptions. Our results have immediate application to multi-ensemble quantum metrology, enabling Heisenberg-limited sensing of field gradients and curvatures. Notably, the generated states have robustness against common-mode phase noise, and only require simple Ramsey-style measurements. The same setup also allows the stabilization of entangled states in a 1D chain of spin ensembles with symmetry-protected topological (SPT) order, and have a direct connection to the outputs of sequential unitary circuits. In particular, we present an efficient method for engineering the celebrated spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state.

Figures (12)

• Figure 1: Schematic of a cavity QED platform for reconfigurable many-body reservoir engineering. We start with a fully collective setup: $N_{\rm tot}$ spins are subject to Rabi drives $\Omega$, and a collective, cavity-mediated superradiant decay $\Gamma$. We break the full permutation symmetry by assigning different detunings $\delta_l$ to sub-ensembles of spins, and/or adding chiral spin-exchange interactions $\chi$ between sub-ensembles. This setup stabilizes a variety of non-trivial pure, entangled states, and is easily reconfigured by changing the detuning pattern $\delta_l$ and/or other system parameters ($\chi$ and $\Omega$).
• Figure 2: (a) Schematic of the $L=2$ setup: two spin-$S$ ensembles coupled to a cavity, each driven by Rabi fields with amplitudes $\Omega$ and opposite detunings $\pm \Delta/2$, and subject to a cavity-mediated collective decay $\Gamma$. (b) In the total-angular-momentum basis $|J,m\rangle$, the Rabi drive $\Omega$ couples $|J,m\rangle \rightarrow |J,m\pm 1\rangle$, and the detuning $\Delta$ couples $|J,m\rangle \rightarrow |J\pm 1,m\rangle$. The collective decay ensures the pure steady state is a linear combination of $|J,-J\rangle$ states, with coefficients $c_{J,-J}$ determined by destructive interference. (c) Steady-state wave function coefficients $c_{J,-J}$ with $S=50$. For $\Delta/\Omega=1/S$, the wave function is peaked at $J=0$; increasing $\Delta/\Omega$ shifts weight to larger $J$. (d) Steady-state quantum Fisher information (QFI) for measuring a differential phase $\phi$ (see Eq. (\ref{['eq:qfipure']})). In the case of $\Delta/\Omega=1/S$, the QFI achieves Heisenberg scaling. We compare the QFI for different $\Delta/\Omega$ scalings with the Heisenberg limit (HL) and the standard quantum limit (SQL).
• Figure 3: (a) Generalized Bloch sphere representation of the $L=2$ steady state. It can be interpreted as a two-mode spin squeezed state with squeezing along the $\hat{S}^y_1+\hat{S}^y_2$ and $\hat{S}^z_1+\hat{S}^z_2$ axes, and anti-squeezing along the $\hat{S}^z_1-\hat{S}^z_2$ and $\hat{S}^y_1-\hat{S}^y_2$ axes. Applying a differential phase $\phi$ between the two ensembles is equivalent to rotations about the $\hat{S}^z_1-\hat{S}^z_2$ axis in the first Bloch sphere. (b) Comparison between the optimal time scale to reach the steady state ($t_{\rm ss}$) and the steady-state squeezing (described by $\mathrm{Var}(\hat{S}^y_1+\hat{S}^y_2)_{\rm ss}$), with $S=30$. We consider the initial state where all the spins are pointing down, and define $t_{\rm ss}$ as the optimal evolution time for the infidelity to the steady state to reach $10^{-3}$ with fixed $\Gamma$ and $\Delta/\Omega$. The same initial state is used in (c). (c) The scaling of optimal two-mode Wineland spin-squeezing parameter $\xi^2_{\rm opt}$ with spin $S$ for each ensemble and single-atom cooperativity $C$ (obtained from a second-order cumulant expansion). The scaling $\xi^2_{\rm opt}\propto 1/\sqrt{SC}$ is determined via numerical fitting.
• Figure 4: (a) Schematic for differential sensing in the presence of common phase noise. We first apply the quantum channel $\mathcal{E}_{12}$ (see Eq. (\ref{['eq:twospin']})) to stabilize the steady state $|\psi_{\rm ss}^{(2)}\rangle$. The system then evolves under the differential phase $\phi$. We next apply a global $\pi/2$ pulse (random common phase $\theta$ due to laser noise) and then perform projective measurement of each ensemble in the $z$ basis to obtain the excitation fractions $p_1$ and $p_2$. (b) Color-scale plot of the probability distribution for measurement outcomes $(p_1,p_2)$, for $S=50$, $\Delta/\Omega=1/\sqrt{S}$ and $\phi=0.1$. The points $(p_1,p_2)$ with peak probability lie on an ellipse whose form depends on $\phi$. (c) Classical Fisher information $F_{\phi}$ for ellipse fitting (solid lines). The dashed lines are the corresponding quantum Fisher information $\mathcal{F}_{\phi}$. $F_{\phi}$ is roughly approaching $\mathcal{F}_{\phi}$ except for the small region near $\phi=0$.
• Figure 5: (a) Schematic of steady-state entanglement between four spin-$S$ ensembles in an optical cavity. Similar to Fig. \ref{['fig:twoensemble']}(a), we consider Rabi drives with Rabi frequency $\Omega$ and detunings $\delta_l$ for each ensemble, as well as collective decay with rate $\Gamma$. We further engineer chiral spin-exchange couplings with rate $\chi$ between spin ensembles. (b) The quantum channel for relaxation to the steady state can be decomposed into quantum channels between ensemble pairs (initially setting $\delta_1=-\delta_2$ and $\delta_3=-\delta_4$) and unitary operators to swap the detunings to the final form (see text). (c) von Neumann entanglement entropy for the steady state in the case of $S=10$. The subscripts of $\hat{\rho}$ label the indices of spin ensembles included in the reduced density matrix. As shown in the insets, we fix $\delta_1=-\delta_2=2\chi$, $\delta_3=-\delta_4=3\chi$ in the top panel, and $\delta_1=-\delta_4=2\chi$, $\delta_2=-\delta_4=3\chi$ in the bottom panel. Multipartite entanglement between spin ensembles is achieved in the bottom panel.