Topologically protected Bell-cat states in a simple spin model

TL;DR

This work demonstrates that a central-spin model with chiral symmetry exhibits a topological phase transition accompanied by a pair of zero-energy bound states in Fock space. By adiabatically driving the central spin energy across the transition, these bound states split into a macroscopic Bell-cat state entangling the central qubit with the ensemble, a process visualized via Wigner distributions and supported by finite-size scaling analyses. The study analytically and numerically characterizes bound-state localization, energy gaps, and the required adiabaticity, and it analyzes robustness to dephasing and perturbations. It also clarifies why replacing the spin ensemble with a single bosonic mode fails to realize BC states, and outlines potential experimental platforms and generalizations, highlighting topological protection as a route to robust generation of entangled macroscopic states.

Abstract

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.

Figures (8)

• Figure 1: Geometric representation of the mean-field Hamiltonian. The vector pointing from the origin of the sphere to its surface represents the mean-field state of the identical spins while the direction of the vector $\bm{d}$ in the $d_x$, $d_y$ plane gives the eigenstate of the mean-field Hamiltonian. For a fixed value of $\theta$, as $\phi$ changes by $2\pi$, the sphere vector traces out a circuit (red). The vector $\bm{d}$ follows the same path, but projected directly down onto the $d_x$, $d_y$ plane. When the projected circuit encompasses the origin of the plane, the system is in the topologically nontrivial phase with winding number $W=1$. The red circle at $\theta = \theta_1$ is the boundary circuit between the topologically trivial and nontrivial phases where any circle on the sphere with a larger radius will encompass the origin when projected down onto the plane.
• Figure 2: Probability distributions in the Fock and energy bases. (a) Probability for a Fock state to have a given energy in the CS $\uparrow$ subspace, $P_\uparrow (n,E) = \vert \langle E \vert n,\uparrow \rangle \vert^2$. (b) Same as (a), but in the CS $\downarrow$ subspace, $P_\downarrow (n,E) = \vert \langle E \vert n,\downarrow \rangle \vert^2$. (c) Probability distributions of the zero energy bound states where the left and right distributions are in the CS $\uparrow$ (red circles) and $\downarrow$ (blue triangles) subspaces, respectively. The parameter values are $N = 180$, $w = 1$ and $v = 0.7$.
• Figure 3: Wigner quasi-probability distributions. (a) The bottom panel shows the sum of the zero and $x$ components of the joint Wigner quasi-probability distribution of the spin coherent state $\vert \pi/2, \pi \rangle$. The top panel shows the probability distribution of the coherent state in Fock space (solid purple) and the even parity bound state (dashed black) in the topologically trivial phase where $v = 1.3$. (b) The top and bottom panels show the same as (a), but for the final state at the end of the driving period. The parameters are $\gamma \approx 1.2 \times 10^{-3}$, $v_0 = 1.3$ and $v_f = 0.7$. The dashed black curve in the top panel is the even parity bound state in the topologically nontrivial phase where $v = 0.7$. In all panels $w = 1$ and $N = 200$.
• Figure 4: Energy spectrum and final state fidelity. (a) Energy spectrum as a function of the scaled CS spin-flip energy $v/w$. The red and black curves represent the energies of the bound and bulk states, respectively. The parameters are $N = 10$ and $w = 1$. (b) Fidelity between the final state at the end of the driving period and the BC state as a function of the scaled initial CS spin-flip energy $v_0/w$. The parameters are $N = 150$, $w = 1$, $\gamma \approx 1.2 \times 10^{-3}$ and $v_f = 0.7$.
• Figure 5: Probability distribution of the bound state (red, dashed) of $\hat{H}_a$ in Eq. \ref{['eq:hama']} alongside a coherent state (black, solid). The number of states of the bosonic mode is truncated at $N_a = 201$ and the parameters are $v = 7$ and $w = 1$, so both states are centered at $m_a = 49$. Inset: Zoom in of the spectrum of $\hat{H}_a$ around the bound states (red) at $E=0$.