Topological Biphoton States

• Topological biphoton states are quantum states of photon pairs in engineered lattices, exhibiting quantized invariants, edge modes, and robust entanglement.
• They are realized via diverse platforms such as nonlinear waveguide arrays, superconducting qubit chains, and topolectrical circuits, enabling the exploration of band topology and photon interactions.
• Methodologies involve Hamiltonian models like the extended Bose–Hubbard and SSH frameworks, with experiments confirming disorder-robust, topologically protected edge states.

Topological biphoton states are quantum states of two photons in engineered photonic systems whose entanglement and spatial correlation properties are governed by the topological features of the underlying lattice or medium. Unlike conventional biphotons, topological biphotons exhibit nontrivial invariants, quantized edge or interface modes, and enhanced robustness to disorder, all rooted in the interplay between band topology, photon interactions, and lattice geometry. Diverse physical platforms realize topological biphoton states, including nonlinear waveguide arrays, superconducting qubit chains, and synthetic circuits, enabling the exploration of fundamental physics as well as applications in quantum information processing.

1. Hamiltonian Frameworks for Topological Biphotons

Topological biphoton states arise from specific many-body Hamiltonians whose topological characteristics emerge either in the single- or two-photon sectors, or are induced by strong photon-photon interactions. Central paradigms include:

Extended Bose–Hubbard Models

One-dimensional arrays of single-mode cavities, waveguides, or qubits are described by Hamiltonians featuring on-site energies, nearest-neighbor photon hopping, on-site Kerr repulsion $U$, and crucially, a direct or pair-hopping term $t_2$ coupling two-photon states between adjacent sites (Stepanenko et al., 2020): $$\hat H = \omega_0\sum_{m}\hat n_m - t\sum_m(a_m^\dagger a_{m+1}+\text{H.c.}) + \frac{U}{2}\sum_m\hat n_m(\hat n_m - 1) + \frac{t_2}{2}\sum_m(a_m^\dagger a_m^\dagger a_{m+1} a_{m+1} + \text{H.c.})$$ where $a_m^\dagger$ creates a photon at site $m$ and $\hat n_m = a_m^\dagger a_m$.

Density-Dependent Qubit Arrays

In transmon-based chains, a density-dependent hopping $T$ is engineered by coupling qubits to detuned auxiliary resonators, yielding (Stepanenko et al., 2020): $$H = \sum_j[\omega\, n_j + U n_j(n_j-1)] - \sum_j[J_{j,j+1}a_j^\dagger a_{j+1} + g(n_j, n_{j+1}) a_j^\dagger a_{j+1} + \text{h.c.}]$$ with $g(n_j, n_{j+1}) = -T(n_j+n_{j+1})$ on selected links, leading to effective Su–Schrieffer–Heeger (SSH) type physics for photon pairs.

Nonlinear Waveguide Arrays

Photonic lattices with alternating couplings and onsite or intersite nonlinearity are described by tight-binding models with both linear and nonlinear terms. For spontaneous four-wave mixing (SFWM) (Doyle et al., 2022, Wang et al., 2019), the total Hamiltonian incorporates both linear SSH-type hopping and nonlinear parametric generation.

Synthetic and Topolectrical Sytems

Classical topolectrical circuits can emulate the two-photon sector of quantum models. Capacitance and inductance values encode hopping, on-site interactions, and pair-hopping terms, mapping the quantum tight-binding problem directly onto circuit response matrices (Olekhno et al., 2019).

2. Topological Band Structure and Invariants

Topological biphoton states are characterized and classified by invariants computed from their effective band structures.

SSH Model Mapping and Zak Phase

In the strong-interaction limit (large $U$), pairs of photons become tightly bound ("doublons") and their center-of-mass motion maps onto effective SSH models with staggered hopping amplitudes. The resulting doublon bands possess Berry-Zak phases

$$\gamma = i \int \langle u_k | \partial_k u_k \rangle \,dk$$

which take quantized values $0$ or $\pi$, depending on the relative strengths of SSH links (e.g., $|j + t_2| < j \implies \gamma = \pi$) (Stepanenko et al., 2020, Stepanenko et al., 2020, Besedin et al., 2020). This quantization underlies the emergence of topological midgap edge or interface states for biphotons.

Chern Numbers in Synthetic Dimensions

For spatially modulated qubit arrays with synthetic dimensions parameterized by a modulation phase $\phi_0$, the two-photon bound-state bands acquire nontrivial Chern numbers ($C = -1, 2, -1$), computed as a 2D integral over $(K, \phi_0)$ parameter space: $$C_m = \frac{1}{2\pi} \int dK \int d\phi_0\, \mathcal F_m(K, \phi_0)$$ where $\mathcal F_m$ is the Berry curvature (Ke et al., 2020).

Selection Rules and Winding Numbers

In SPDC-based systems in nonlinear lattices, the phase winding of Bloch eigenvectors imposes selection rules for two-photon generation. For SSH-type systems, the winding number of the Bloch function dictates the emergence of forbidden "zeros" in the joint spectral amplitude $\Gamma(k_s, k_i)$, which are not present in topologically trivial lattices (Leykam et al., 2015).

3. Bulk, Edge, and Interface Biphoton Modes

A defining feature of topological biphoton physics is the existence of quantized edge and domain-wall modes in the two-photon sector, even when the single-photon spectrum is trivial.

Bulk Doublon Bands and Continuum

The two-photon excitation spectrum in interacting SSH-type models consists of continuum scattering bands, whose energies are determined by independent photon propagation, and isolated doublon bands, whose support lies outside or within a gap in the continuum. The doublon dispersion, for example,

$$E_\pm(k) = 2\omega_0 + 2U + 2j \pm \sqrt{j^2 + (j + t_2)^2 + 2j(j + t_2) \cos 2k}$$

captures the motion of bound photon pairs in the lattice (Stepanenko et al., 2020). Collapse or hybridization with the continuum is governed by the relative magnitude of interactions and hopping parameters.

Edge and Interface States

Topologically nontrivial doublon bands support exponentially localized edge states under open boundary conditions, with energies pinned in the gap or, remarkably, even within the scattering continuum as non-leaky bound states in the continuum (BICs) (Stepanenko et al., 2020, Olekhno et al., 2019). At domain walls between regions of opposite SSH dimerization, robust interface biphoton modes appear with energies pinned by symmetry.

Hybrid biphoton modes, in which one photon occupies a topological edge state and the other a bulk Bloch state, are also observed. In sufficiently large single-photon gaps, these modes reside within the biphoton spectrum gap and decay exponentially in one coordinate (Leykam et al., 2015).

Edge States in Non-Hermitian and Synthetic Systems

In radiatively coupled modulated qubit arrays, biphoton edge modes emerge that smoothly terminate (or "dive") into bulk bands rather than being strictly confined to bandgaps. These states exhibit subradiant losses and long lifetimes, with their existence protected by nonzero Chern numbers (Ke et al., 2020).

4. Experimental Realizations and Measurement Protocols

Experimental platforms capable of realizing and probing topological biphoton states include:

• Ultracold atoms in optical lattices with engineered on-site interactions and pair tunneling (Stepanenko et al., 2020).
• Superconducting qubit arrays employing transmon qubits, where strong Kerr nonlinearity and density-dependent coupling terms are implemented via auxiliary resonators, with frequency ranges $\sim$4 GHz and typical anharmonicities, couplings, and detunings scalable to desired SSH-type regimes (Stepanenko et al., 2020, Besedin et al., 2020).
• Nonlinear silicon photonic waveguide arrays, where alternating waveguide spacings form SSH chains, and SFWM enables on-chip biphoton generation, detection, and correlation mapping, even under programmed disorder (Wang et al., 2019, Doyle et al., 2022).
• Topolectrical circuits simulating two-photon Hamiltonians via tailored networks of capacitors and inductors, enabling eigenmode spectroscopy, direct measurement of winding numbers, and full probability tomography of biphoton edge states (Olekhno et al., 2019).
• Active photonic platforms with nonlinear gain/loss or Kerr effects in the lattice couplings enable real-time pump-induced control of topological phase transitions and biphoton entanglement (Zhang et al., 8 Nov 2025).

Measurement strategies include coincidence correlation mapping $G^{(2)}(n, m)$, interferometric visibility for coherence tests, site-resolved spectroscopy (for microwave systems), and direct tomographic reconstruction of two-photon probability amplitudes.

5. Robustness to Disorder and Topological Protection

Disorder robustness is a central property of topological biphoton states:

• Zak-phase and winding number quantization, as well as sublattice chiral symmetry, protect both single- and two-photon edge states against moderate variations in hopping amplitudes, onsite energies, and nonlinear couplings (Stepanenko et al., 2020, Wang et al., 2019).
• In experimental silicon SSH lattices, well-defined biphoton correlation patterns and high interference visibilities persist up to 50% random coupling disorder, in sharp contrast to trivial lattices where such features rapidly disappear (Wang et al., 2019).
• In modulated qubit superlattices, the topological biphoton edge and interface states retain their energy and localization up to strong parameter disorder, and their subradiant lifetimes are enhanced by topological quantization (Ke et al., 2020).
• Chiral edge state systems with arrays of quantum emitters display universal two-photon S-matrix properties (nodes at Laguerre polynomial zeros, parity effects in $g^{(2)}$ correlations), entirely independent of emitter positions or detunings for small disorder, due to the strict unidirectionality and Chern number quantization of the edge mode (Ringel et al., 2012).

6. Manipulation, Control, and Entanglement in Topological Biphotons

Advanced schemes leverage interaction- or pump-induced control over biphoton topology and entanglement:

• In nonlinear waveguide arrays with gain/loss or Kerr effects, the injected pump power dynamically tunes effective SSH couplings, triggering topological transitions and enabling the generation or annihilation of defect-induced trivial modes. These processes allow controlled transfer of biphoton population into topological sectors and the engineering of high-dimensional entangled states (Zhang et al., 8 Nov 2025).
• Schmidt number $K$ and related singular value spectra of measured biphoton amplitude matrices quantify and reveal the degree of biphoton entanglement, with topological protection ensuring robust, reconfigurable sources for quantum information platforms (Doyle et al., 2022, Zhang et al., 8 Nov 2025).
• Hybrid and topological biphoton states provide new avenues for entanglement distribution, quantum teleportation, and information routing between topologically distinct channels, exploiting topology as an additional degree of freedom (Doyle et al., 2022).
• Unidirectional chiral edge systems enable parity-controlled photon-photon correlations, tunable between bunching and antibunching regimes by the number of scatterers, suggesting implementations of robust photonic gates and switches (Ringel et al., 2012).

7. Outlook and Applications

Topological biphoton states underpin a class of quantum photonic systems where entanglement, robustness, and quantum coherence are dictated and protected by lattice topology and engineered interactions. They enable:

• Fault-tolerant routes for quantum communications and state transfer immune to fabrication imperfections;
• Scalable sources of edge- or interface-localized entangled photon pairs for quantum metrology and on-chip quantum processing;
• Reconfigurable platforms—e.g., active photonic circuits—where entanglement properties and topological protections are dynamically controlled on demand;
• Novel quantum simulation paradigms for strongly correlated and topological matter using biphotonic excitations.

Ongoing research expands the landscape to higher-order topological phases, synthetic dimensions, non-Hermitian and non-Abelian generalizations, and hybrid platforms coupling topological biphotons with other quantum degrees of freedom.