Chiral quantum state circulation from photon lattice topology (2510.01306v1)

Abstract: Chiral quantum state circulation is the unidirectional transfer of a quantum state from one subsystem to the next. It is essential to the working of a quantum computer; for instance, for state preparation and isolation. We propose a cavity-QED architecture consisting of three cavities coupled to a qubit, in which \emph{any} photonic state of cavity 1 with sufficiently many photons circulates to cavity 2 after a fixed time interval, and then to cavity 3 and back to 1. Cavity-state circulation arises from topologically protected chiral boundary states in the associated photon lattice and is thus robust to perturbation. We compute the circulation period in the semi-classical limit, demonstrate that circulation persists for time-scales diverging with the total photon number, and provide a Floquet protocol to engineer the desired Hamiltonian. Superconducting qubits offer an ideal platform to build and test these devices in the near term.

• The paper reveals that a three-cavity photonic lattice exhibits unidirectional quantum state circulation tied to the system’s Chern number.
• It demonstrates that topologically protected boundary modes yield circulating trajectories with period scaling and lifetimes influenced by symmetry and photon occupation.
• The authors propose a practical Floquet engineering protocol to generate the needed effective Hamiltonian, enabling non-reciprocal quantum state routing.

Chiral Quantum State Circulation from Photon Lattice Topology

Introduction

This paper presents a rigorous analysis of chiral quantum state circulation in a three-cavity, one-qubit photonic lattice, where the topology of the photon lattice induces robust, unidirectional quantum state transfer. The work establishes a direct connection between the Chern number of the system's band structure and the existence of topologically protected boundary modes, which enable chiral circulation of quantum states. The authors provide both quantum and classical analyses, demonstrate the robustness of circulation against perturbations, and propose a practical Floquet engineering protocol for experimental realization using only two-body interactions.

Topological Band Structure and Chern Number

The system is modeled by a tight-binding Hamiltonian under the local density approximation (LDA), where the photon number operators are replaced by their scalar values for large occupation numbers. The Hamiltonian is recast in Bloch form, revealing a six-fold rotational symmetry in the band structure. The Chern number of the lower band is computed as a function of the renormalized qubit splitting $3\Delta/2gN$, showing that in the large photon number limit ($N \gg 1$), the lower band is always a Chern band with $C = -1$ for finite $\Delta$.

Figure 1: (a) Chern number of the lower band of the isotropic QWZ model vs the renormalized qubit splitting $3\Delta/2gN$. (b) Local band gap and local Chern number for $N=50$ sites in Fock space within the LDA.

The local band gap vanishes along a curve in Fock space, which marks the transition between topologically nontrivial ($C = -1$) and trivial ($C = 0$) regions. The boundary modes are localized near this curve, and their position is determined by a nonlinear equation involving the photon occupations $n_j$.

Chiral Boundary Modes and State Circulation

The position of the chiral boundary mode is analytically derived, showing that the topologically protected mode follows a specific locus in Fock space. The quantum dynamics projected onto the product state manifold are analyzed, and the existence of circulating trajectories is demonstrated both at the quantum and classical levels. For a special value of the qubit splitting, the system supports exact circulating solutions with a period $T = 4\pi/\sqrt{3}|g|$ in the large $N$ limit. Deviations from this value lead to deformed but persistent trajectories, with corrections to the period and initial state scaling with $1/N$.

Robustness and Lifetime of Circulation

The robustness of chiral circulation is investigated under random photon-number-conserving perturbations. The analysis distinguishes between perturbations that break and those that preserve the permutation ($\mathcal{P}$) symmetry. For $\mathcal{P}$-symmetric perturbations, the circulation lifetime scales linearly with $N$, while for $\mathcal{P}$-symmetry breaking perturbations, the lifetime scales as $\sqrt{N}$. These predictions are numerically verified for various types of disorder, including random cavity frequency shifts and random cavity-qubit couplings.

Figure 2: Population of cavity 3 at stroboscopic instants $t_q$, starting from the initial state $\ket{\psi_{\mathrm{Fock}}$ in the unperturbed model, showing $t^* \sim N$ scaling.

Figure 3: Scaled stroboscopic population of cavity-3 vs average stroboscopic time $\overline{t_q}$ for different perturbations, confirming $t^* \propto N$ or $t^* \propto \sqrt{N}$ scaling depending on symmetry.

Floquet Engineering of the Hamiltonian

A key practical contribution is the demonstration that the required three-body interaction Hamiltonian can be Floquet engineered from a bare Hamiltonian with only two-body terms. The authors construct a periodically driven Hamiltonian and perform a high-frequency expansion, showing that the effective Floquet Hamiltonian matches the target model under suitable symmetry constraints and parameter choices. The validity of the expansion is analyzed, yielding explicit criteria for the driving frequency and cavity parameters to ensure convergence and applicability of the rotating wave approximation.

Figure 4: (a) Schematic of the periodic drive protocol. (b) Stroboscopic dynamics under Floquet engineering, showing agreement with the static Hamiltonian for appropriate parameters.

Non-Reciprocal Quantum State Routing

The paper further explores the application of the device as a non-reciprocal quantum router. By coupling the cavities to external drive and detection ports, the system can be initialized in a desired state and used to route quantum states unidirectionally, with the directionality determined by the sign of the Chern number. The imbalance in output power between the two detectors serves as a clear signature of chiral circulation, with the system acting as a robust quantum circulator.

Figure 5: (a) Unitary circulation from the vacuum state with external drive. (b,c) Output power collected in detectors $D_1$ or $D_2$ depending on the direction of circulation, demonstrating non-reciprocal routing.

Implications and Future Directions

The results establish a direct link between photonic lattice topology and robust quantum state circulation, with clear implications for quantum information transfer and non-reciprocal device engineering. The Floquet protocol provides a feasible route for experimental realization in circuit QED or photonic platforms. The scaling of circulation lifetime with system size and symmetry properties suggests potential for scalable quantum networks with topological protection against disorder. Future work may extend these concepts to larger lattices, explore many-body effects, and investigate integration with quantum error correction schemes.

Conclusion

This paper provides a comprehensive theoretical and numerical analysis of chiral quantum state circulation in a topological photon lattice, elucidating the role of Chern number, boundary modes, and symmetry in enabling robust, unidirectional quantum state transfer. The proposed Floquet engineering protocol and non-reciprocal routing application highlight the practical relevance of the results for quantum technologies. The scaling laws for circulation lifetime and the explicit criteria for experimental implementation offer a solid foundation for future research and development in topological quantum photonics.