Robust qubit interactions mediated by photonic topological edge states

Abstract

We investigate the coupling of two spatially separated qubits via topologically protected edge states in a two-dimensional Hofstadter lattice. In this hybrid platform, the qubits are coupled to distinct edge sites of the lattice, enabling long-range interactions mediated by topological edge modes. We solve the full system Hamiltonian and analyze the resulting eigenstate structure to uncover the conditions under which coherent qubit interactions emerge. Our analysis reveals that the effective coupling is highly sensitive to the qubit placement, energy detuning, and the topological character of the edge spectrum. We obtain an analytical solution that goes beyond the perturbative regime, capturing the full interplay between the qubits and edge modes. These results provide a foundation for exploring information transport and many-body effects in engineered quantum systems where interactions are mediated by topological edge modes.

Figures (8)

• Figure 1: (a) Two qubits, $Q_{1}$ and $Q_{2}$, are coupled to a Hofstadter lattice ($L \times L$) of microwave resonators connected via nearest-neighbor hopping $J$. Blue sites support $p_x + ip_y$ modes, while black sites host $s$-like modes. Site indices follow $j(r,c) = (r-1)L + c$, and the full system includes $N = L^2 + 2$ sites with $Q_{1} = 0$, $Q_{2} = L^2 + 1$. A rotating frame is chosen such that the qubits have nonzero on-site potentials $\epsilon$, while the lattice is centered around zero energy. (b) Band structure for the Hofstadter lattice in a strip geometry ($L_y = 35$, $L_x \to \infty$). Bulk bands (green) lie in $|E| \gtrsim 2.61J$ and $|E| \lesssim 1.08J$; edge states (red) span the intermediate region. The qubit energy $\epsilon \approx -1.75J$ (magenta) lies within the edge spectrum.
• Figure 2: Time evolution of qubit probabilities $P_{Q_1} = |\psi(Q_1,t)|^{2}$, $P_{Q_2} = |\psi(Q_2,t)|^{2}$, and total lattice population $P_\mathrm{lat} = \sum_{j}|\psi(j,t)|^{2}$ for $L = 35$ and $J = 1$. (a) $\epsilon = -1.76$, $g = 0.01$; (b) $\epsilon = -1.74$, $g = 0.01$; (c) $\epsilon = -1.76$, $g = 0.05$. The insets in panels (a) and (b) display segments of the edge mode band structure, indicating the positions of the corresponding potentials $\epsilon$ (magenta line) relative to the lattice eigenvalues (green dots).
• Figure 3: Oscillations induced when the qubits are brought into resonance with an edge mode at $\epsilon=E_{l}=-1.7307$, with a coupling strength $\Omega_0 = g\sqrt{2}|\psi_{l}(1)|$, and an oscillation period $T=2\pi/ \Omega_0 \approx 4382$. The system size is $L=35$, and the coupling constant is $g=0.01$.
• Figure 4: Numerical results for eigenvalues $E^{\prime}_k$ compared with the relation in Eq. (\ref{['eq:bc_4']}). It demonstrates exact agreement with the theoretical prediction, with each solution for $E^{\prime}_k$ lying at the intersection of the functions of $y=f_{\pm}(\lambda)$ and the line $y=\frac{\lambda - \epsilon}{g^2}$. $f_{\pm}(\lambda)$ is calculated numerically in the deep perturbative limit $g \to 0$, as in Eq. (\ref{['eq:E_shift']}). $E_k$ are the unperturbed eigenvalues of the lattice. The lattice size is $L = 21$, and the qubit potential is $\epsilon = -1.8$. Solutions work equally well for weak coupling (a) $g = 0.1$ or moderate/strong coupling (b) $g = 0.5$.
• Figure 5: (a) Comparison of numerical and analytical bounds, Eqs. (\ref{['eq:g_omg_min']}) and (\ref{['eq:g_omg_max']}), for the effective frequency $\Omega_{\mathrm{eff}}$ at given $g^{2}L/J^2$, calculated for $L=21$ between $E_l = -1.81$ and $E_{l+1} = -1.71$. The analytical predictions use $\tilde{\epsilon} = E_l$ (resonance) for minimum $g(\Omega_{\mathrm{eff}})$ and $\tilde{\epsilon} = (E_l + E_{l+1})/2$ (midpoint) for maximum $g(\Omega_{\mathrm{eff}})$. The inset shows the same comparison over a wider range of $g^{2}L/J^2$ values. (b,c) Time evolution of the qubit population $|\Psi(Q_2, t)|^2$ for various $\epsilon$, shown for $g^{2}L/J^2 = 0.1$ and $3.0$. In all cases, $J=1$.