Bound Photon Pair Edge States

• Bound photon pair edge states are unique two-photon eigenstates localized at lattice boundaries, emerging from strong Kerr nonlinearity and topologically nontrivial bulk properties.
• They are modeled using variants of the Bose-Hubbard Hamiltonian, including dimerized and extended forms that capture both local and nonlocal photon-photon interactions.
• Experimental realizations in superconducting qubit arrays and topolectrical circuits demonstrate robust localization, phase transitions, and resilience to disorder in these states.

Bound photon pair edge states are spatially localized two-photon eigenstates that arise at the boundaries of one-dimensional photonic lattices due to interaction-induced pairing and nontrivial bulk topology in the two-particle sector. They do not have a single-particle analog and are defined by the interplay of strong Kerr-type photon-photon interactions and engineered lattice or interaction modulations. The formation, protection, and phenomenology of these states are encoded in variants of the Bose-Hubbard model and its extensions, often incorporating both local and nonlocal interactions or direct two-photon hopping. Experimental realizations have been achieved in superconducting quantum metamaterials as well as classical topolectrical circuit simulators.

1. Theoretical Models for Bound Photon Pair Edge States

The canonical platform for bound photon pair edge states is a one-dimensional array of coupled cavities or qubits, each enabling strong on-site interactions ("Kerr nonlinearity") and possibly direct pair tunneling. The generalizations relevant for edge-state physics include:

• Dimerized Bose-Hubbard Model with alternating hopping $J_1 \neq J_2$ and uniform Kerr interaction $U$:

$$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$

where $\hat n_m = \hat a_m^\dagger \hat a_m$ (Gorlach et al., 2016).

• Extended Bose-Hubbard Model with both on-site ($U$) and nearest-neighbor ($W$) interaction:

$$H = \omega_0 \sum_{m} n_m + U\sum_{m} n_m(n_m-1) - J\sum_{m}(a_m^\dagger a_{m+1}+\mathrm{h.c.}) + 2W\sum_{m} n_m n_{m+1}$$

where the nonlocal-$W$ term enhances pair binding and edge localization (Gorlach et al., 2017).

• Extended Bose-Hubbard with Direct Pair Hopping:

$$\hat H = \omega_0\sum_m \hat n_m - J\sum_m (a_m^\dagger a_{m+1} + \mathrm{h.c.}) + U\sum_m \hat n_m (\hat n_m-1) + \frac{P}{2}\sum_m (a_{2m}^\dagger a_{2m}^\dagger a_{2m+1} a_{2m+1} + \mathrm{h.c.})$$

Here $P$ describes direct two-photon tunneling between adjacent sites, which crucially affects the topological phase diagram (Stepanenko et al., 2020).

• Spatially Modulated Nonlinearity:

The interaction $U$0 may alternate along the chain (e.g., $U$1 for odd $U$2, $U$3 for even $U$4), leading to "nonlinear self-localization" and edge state formation without any single-particle band topology (Lyubarov et al., 2019).

2. Formation and Characteristics of Bound Pair Edge States

In these interacting models, the two-photon Hilbert space splits into scattering states (extended over the lattice) and discrete "doublon" bands, corresponding to photon pairs bound by interaction energy. Edge states arise when the two-particle sector—rather than the single-particle sector—becomes topologically non-trivial. Key features include:

• Edge mode localization: Exponentially decaying doublon amplitude at the boundary, with wavefunctions that can be analytically constructed in strong-interaction limits; for instance, $U$5 in the effective Su–Schrieffer–Heeger (SSH) model (Stepanenko et al., 2020, Olekhno et al., 2019).
• Parameter regimes: Existence and robustness of edge states are controlled by interaction magnitude ($U$6, $U$7, $U$8), hopping dimerization ($U$9), or modulation pattern of the nonlinearity.
• Spectral isolation: Doublon edge states can reside within two-particle bandgaps or even within the two-particle continuum, exhibiting stability against hybridization (“bound states in the continuum,” BICs) (Stepanenko et al., 2020, Gorlach et al., 2016).

3. Topological Protection: Bulk Invariants and Edge Correspondence

The topological origin of bound photon pair edge states is often established by analogy to the SSH chain, but in the two-particle subspace. For $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$0, a projection yields an effective dimerized tight-binding chain for doublons, whose topology is classified by invariants:

• Zak phase: Calculated for the bulk doublon band, with $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$1 (nontrivial) when $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$2 for $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$3 and $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$4 (Stepanenko et al., 2020).
• Winding number: For the effective doublon SSH chain, $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$5 ensures an in-gap doublon edge state (Olekhno et al., 2019).
• Chern number in 2D synthetic space: In modulated qubit arrays with a synthetic dimension (center-of-mass $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$6 and superlattice phase $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$7), the Chern numbers of bound-pair bands determine the existence and connectivity of edge states in the $$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$8 torus (Ke et al., 2020).
• Breakdown and alternatives: In some models, the conventional Zak phase fails due to strong mixing of internal doublon structure and center-of-mass motion; instead, two-photon quantum-walk graph connectivity ($$\hat H = \omega_0\sum_{m}\hat n_m + U\sum_m \hat n_m(\hat n_m-1) - J_1\sum_m \left(\hat a_{2m-1}^\dagger\hat a_{2m}+\mathrm{h.c.}\right) - J_2\sum_m \left(\hat a_{2m}^\dagger\hat a_{2m+1}+\mathrm{h.c.}\right)$$9) predicts edge mode existence (Gorlach et al., 2016).

4. Analytical and Numerical Characterization

Analytical tools include:

• Modified Bethe ansatz: For bulk doublons, ansätze combine center-of-mass $\hat n_m = \hat a_m^\dagger \hat a_m$0 and relative momentum $\hat n_m = \hat a_m^\dagger \hat a_m$1; for edge states, solutions are localized hybridizations that satisfy open-boundary or defect conditions (Gorlach et al., 2016, Stepanenko et al., 2020).
• Perturbative treatments: In strong interaction limits, effective doublon chains are derived via Schrieffer-Wolff or second-order perturbation, with site-dependent edge defects producing localized states (Lyubarov et al., 2019, Gorlach et al., 2017).
• Phase diagrams: Regions of topological and trivial doublon phases, bandflatness, and collapse into the continuum are mapped precisely in $\hat n_m = \hat a_m^\dagger \hat a_m$2 or $\hat n_m = \hat a_m^\dagger \hat a_m$3 space; for instance, the topological doublon phase is robust to collapse of partner bands (Stepanenko et al., 2020).

Table: Key Features by Model Class

Model Variant	Topological Invariant	Edge State Localization
Dimerized Bose-Hubbard ($\hat n_m = \hat a_m^\dagger \hat a_m$4)	Zak phase, graph connectivity	Strong link; set by $\hat n_m = \hat a_m^\dagger \hat a_m$5
Bose-Hubbard $\hat n_m = \hat a_m^\dagger \hat a_m$6 pair hopping ($\hat n_m = \hat a_m^\dagger \hat a_m$7)	Zak phase	Controlled by $\hat n_m = \hat a_m^\dagger \hat a_m$8, $\hat n_m = \hat a_m^\dagger \hat a_m$9
Extended Hubbard ($U$0, $U$1)	No conventional ZP	Detuning-induced at boundary
Spatially modulated $U$2	None (self-localization)	Odd sites (nonlinear)
Modulated qubit (superlattice)	Chern number	$U$3-$U$4 edge, interface

5. Experimental Realizations and Emulations

Experimental probes of bound photon pair edge states include both quantum and classical simulators.

• Superconducting qubit arrays: Dimerized transmon chains with strong on-site attraction and $U$5 dimerization—e.g., $U$6 MHz, $U$7 MHz, $U$8 MHz—displayed sharp two-photon edge state resonances separated from the continuum ($U$9 GHz), with spatial readout confirming exponential edge-mode localization. The edge mode frequency shifts only weakly under moderate disorder (Besedin et al., 2020).
• Topolectrical circuits: 2D LC networks emulate the two-photon sector by mapping the problem to voltage/current modes and admittance matrices. This enables experimental visualization of doublon edge-state spatial profiles and extraction of winding numbers via voltage ratios (Olekhno et al., 2019).
• Waveguide-coupled qubit arrays: Arrays with spatial modulation enable direct spectroscopy of radiative doublon edge states, with edge- or interface-localized subradiant modes exhibiting long lifetimes determined by modulation pattern and synthetic dimension topologies (Ke et al., 2020).

6. Robustness, Collapse, and States in the Continuum

A defining feature of interaction-induced bound pair edge states is their robustness against disorder and spectral collapse:

• Stability: The edge mode persists as long as the corresponding Zak phase or Chern number remains quantized and the bulk gap does not close. Moderate disorder in on-site energies or couplings does not destroy localization (Besedin et al., 2020, Olekhno et al., 2019).
• Collapse and revival: As interactions decrease, doublon bands may overlap the scattering continuum, but edge modes may persist deep into this regime, remaining exponentially localized—these constitute genuine two-photon bound states in the continuum (BICs). As parameters are tuned, edge modes can merge and re-emerge from the bulk, reflecting the composite nature of doublon topology (Stepanenko et al., 2020, Gorlach et al., 2016).
• Nonlinear self-localization: In systems where topological order is absent, edge doublons result entirely from nonlinear energy shifts at the boundary (e.g., checkerboard interaction patterns), providing a contrast to symmetry-protected (SSH-type) topological phases (Lyubarov et al., 2019).

7. Generalizations and Outlook

Bound photon pair edge states generalize beyond the prototypical SSH mapping, appearing in settings with spatially modulated interactions, superlattice-modified qubit arrays, or hybrid local/nonlocal nonlinearities. They exemplify interaction-induced topology: the single-photon band structure remains trivial, but the two-photon sector possesses protected, boundary-localized modes with signatures distinct from conventional topological photonics. This opens opportunities for robust quantum information channels, subradiant storage, and nonlinear state engineering, contingent on continued exploration of their topological invariants, disorder resilience, and interface physics (Ke et al., 2020, Stepanenko et al., 2020, Besedin et al., 2020).