Topologically protected Bell-cat states in a simple spin model

We consider the topological properties of the so-called central spin model that consists of $N$ identical spins coupled to a single distinguishable central spin which arises in physical systems such as circuit-QED and bosonic Josephson junctions coupled to an impurity atom. The model closely corresponds to the Su-Schrieffer-Heeger (SSH) model except that the chain of sites in the SSH model is replaced by a chain of states in Fock space specifying the magnetization. We find that the model accommodates topologically protected eigenstates that are `Bell-cat’ states consisting of a Schrödinger cat state of the $N$ spins that is maximally entangled with the central spin, and show how this state can be adiabatically created and moved along the chain by driving the central spin. The Bell-cat states are visualized by plotting their Wigner function and we explore their robustness against random noise by solving the master equation for the density matrix. We also explain the essential topological difference between identical spins and the excitations of a bosonic mode.

💡 Research Summary

In this work the authors revisit the central‑spin (CS) model – a collection of N identical spin‑½ particles collectively coupled to a single distinguishable central spin – and demonstrate that it hosts topologically protected eigenstates which are Bell‑cat (BC) states, i.e. macroscopic Schrödinger‑cat superpositions of the N spins maximally entangled with the central spin. The key insight is that, when expressed in the basis of total magnetization n = (N↑ − N↓)/2 and central‑spin state m = ↑, ↓, the CS Hamiltonian maps onto a two‑band Bloch Hamiltonian very similar to the Su‑Schrieffer‑Heeger (SSH) model. The “unit‑cell” coordinate is the magnetization n (the Fock‑space index) and the intra‑cell coordinate is the central‑spin state.

By applying a mean‑field approximation to the collective spin operators, the Hamiltonian reduces to H(θ, ϕ) = d(θ, ϕ)·σ, where the vector d lies in the dₓ‑dᵧ plane, θ and ϕ are the polar and azimuthal angles of the Bloch sphere representing the collective spin, and ϕ plays the role of a quasi‑momentum. As ϕ winds from 0 to 2π, the tip of d traces a circle of radius w whose centre is displaced by v along the dₓ axis. If v < w the circle encloses the origin, giving a winding number W = 1; otherwise W = 0. Because the winding depends on θ, only a band of polar angles θ₁ = arcsin(v/w) < θ < θ₂ = π − arcsin(v/w) exhibits non‑trivial topology. The two boundaries θ₁ and θ₂ correspond to the edges of the Fock‑space chain (n = ±N/2) and each hosts a zero‑energy bound state. These bound states are protected by chiral symmetry σ_z H σ_z = −H, which forces each bound state to reside entirely in either the central‑spin ↑ or ↓ subspace.

When the system is tuned across the topological transition by slowly varying the ratio v/w (for example by changing an external magnetic field that controls the central‑spin Zeeman splitting), an initially unentangled state that overlaps with the trivial‑phase edge states evolves adiabatically into a superposition of the two zero‑energy bound states. Because the two bound states are degenerate, they combine into a single zero‑energy eigenstate that is a Bell‑cat state:

|Ψ_BC⟩ = ( |↑⟩c ⊗ |Cat+⟩_N + |↓⟩c ⊗ |Cat-⟩_N ) / √2,

where |Cat_±⟩_N are macroscopic superpositions of the N‑spin ensemble with total magnetization +N/2 and −N/2, respectively. The authors visualize this state by plotting its Wigner function, which shows two well‑separated lobes whose relative phase is conditioned on the central‑spin state, confirming maximal entanglement.

The robustness of the BC states is investigated by solving a Lindblad master equation that includes random fluctuations of the driving field. Because the bound states are protected as long as the perturbations respect chiral symmetry, the energy gap to the bulk remains open and the concurrence and purity of the BC state stay high under symmetry‑preserving noise. By contrast, adding a σ_z term (which breaks chiral symmetry) immediately lifts the zero‑energy degeneracy and destroys the Bell‑cat entanglement.

Experimental realizations are discussed in three platforms: (i) circuit‑QED, where a superconducting transmon plays the role of the central spin and microwave resonator modes act as the identical spins; (ii) photonic systems, using two orthogonal polarizations or orbital‑angular‑momentum modes as the spin‑½ particles coupled to an atom or quantum dot; and (iii) ultracold‑atom setups, where an impurity atom (or ion) immersed in a two‑state Bose‑Einstein condensate (a bosonic Josephson junction) provides the required central‑spin/identical‑spin coupling. In the latter case, Feshbach resonances allow independent tuning of the exchange coupling w, while an external magnetic field controls v, enabling real‑time traversal of the topological phase transition.

Overall, the paper establishes the central‑spin model as a minimal, analytically tractable platform where topological protection can be harnessed to generate large‑scale entangled cat states. The combination of a simple Hamiltonian, clear bulk‑boundary correspondence in Fock space, and experimentally accessible control knobs makes this approach promising for quantum information processing tasks that require robust macroscopic superpositions, such as continuous‑variable quantum computing, error‑corrected logical qubits, and quantum metrology. Future directions include extending the model to multiple central spins, incorporating non‑linear interactions, and exploring fault‑tolerant protocols that exploit the topological gap to protect against realistic decoherence channels.