SSH Exciton-Polariton Chains

• SSH exciton-polariton chains are 1D arrays where engineered dimerized couplings emulate the SSH Hamiltonian, enabling robust topological edge and interface modes.
• Driven-dissipative dynamics combined with strong light-matter interactions induce nonlinear effects and multistability, paving the way for topological lasing and controlled phase transitions.
• Experimental implementations using semiconductor microcavities and optical potential landscapes validate theoretical predictions through precise band structure, coherence, and spatial mode mapping.

SSH exciton-polariton chains are one-dimensional arrays of microcavity exciton-polariton sites configured to emulate the physics of the Su-Schrieffer–Heeger (SSH) model—a prototypical system exhibiting topologically nontrivial phases, robust edge states, and their associated nonlinear and driven-dissipative phenomena. The polaritonic realization of the SSH model leverages strong light-matter coupling to access robust, reconfigurable, and highly nonlinear bosonic modes at the boundary and interface of dimerized chains, and extends naturally to higher-order topology in two-dimensional (2D) arrays. This platform serves as a laboratory for the study of topological lasing, phase transitions, multistability, and disorder-resilience in both quantum and classical regimes.

1. SSH Hamiltonian for Exciton-Polariton Chains

The exciton-polariton SSH chain is modeled by a tight-binding Hamiltonian reflecting dimerized coupling between adjacent sites. For a 1D lattice of $N$ unit cells, each with two sublattice sites (A, B), the most general form reads: $$\hat{H}_\text{SSH} = \sum_{n=1}^N \big[ t_1\, a_n^\dagger b_n + t_2\, b_n^\dagger a_{n+1} + \text{h.c.} \big]$$ where $a_n^\dagger$ and $b_n^\dagger$ create polaritons at sublattices A and B of cell $n$, $t_1$ is the intra-cell (weak) hopping, and $t_2$ is the inter-cell (strong) hopping.

In photonic/polaritonic implementations, $t_1$ and $t_2$ are engineered by varying center-to-center distances of adjacent sites using either lithographic techniques, etch-and-overgrowth, or dynamically via optically-induced potential landscapes (e.g., digital micromirror device (DMD) modulation).

The real-space SSH Hamiltonian in the presence of Rabi coupling $\Omega$, nonlinear on-site repulsion $U|\psi|^4$, and detuned site energies $\epsilon$ can be generalized to: $$\hat{H} = \hat{H}_\text{SSH} + \Omega (\hat{a}_n^\dagger \hat{x}_{n,A} + \hat{b}_n^\dagger \hat{x}_{n,B} + \text{h.c.}) + U \sum_{n,\alpha} (\hat{a}_{n,\alpha}^\dagger \hat{a}_{n,\alpha})^2 + \epsilon \sum_{n, \alpha} \hat{a}_{n,\alpha}^\dagger \hat{a}_{n,\alpha}$$ where $\hat{x}_{n,\alpha}$ denote excitonic degrees of freedom, and $\Omega$ sets the Rabi splitting.

The presence of alternating bonds ($t_2>t_1$ for the topologically nontrivial phase) ensures the existence of zero-energy edge modes exponentially localized to the boundaries, with localization length $\xi = 1 / \ln|t_2/t_1|$. Extension to organic/Frenkel polaritons, or higher-orbital (p-band) SSH models, follows the same formal structure, with appropriate modification of site and coupling parameters (Harder et al., 2020, Pieczarka et al., 2021, Rojas-Sánchez et al., 2022, Bennenhei et al., 11 Jan 2024).

2. Topological Invariants and Phase Classification

Topological properties of SSH exciton-polariton chains are classified by their Zak phase (Berry phase) and a winding number defined over the Brillouin zone. For the two-band SSH model: $$\gamma_\text{Zak} = i \int_{-\pi/a}^{\pi/a} dk\, \langle u_k | \partial_k u_k \rangle = \begin{cases} \pi, & t_2 > t_1 \ 0, & t_1 > t_2 \end{cases}$$ where $|u_k\rangle$ is the occupied Bloch eigenstate. The integer winding number: $$w = \frac{1}{2\pi} \int_{-\pi}^{\pi} dk\, \frac{d}{dk} \arg [t_1 + t_2 e^{-ik}]$$ equals $1$ in the nontrivial (topological) phase and $0$ in the trivial phase. These invariants robustly predict the existence and number of protected edge or interface modes, confirmed via both tight-binding and full driven-dissipative simulations (Rojas-Sánchez et al., 2022, Pieczarka et al., 2021, Bennenhei et al., 11 Jan 2024).

Stacking DW (double-wave) chains vertically, with alternating intra- and inter-row couplings, realizes a set of vertical SSH chains; the combination of AAH (Aubry-André-Harper) and SSH physics in 2D supports higher-order (corner) topological states characterized by both a Chern number $C_j$ (from the horizontal AAH direction) and a vertical SSH winding number (Schneider et al., 2023).

3. Driven-Dissipative Dynamics and Nonlinearity

Exciton-polariton SSH chains are inherently open, driven-dissipative systems modeled at the mean-field level by coupled Gross–Pitaevskii (GP) equations with spatially-resolved pump and loss terms: $$i\hbar\,\partial_t\Psi(r,t) = \Bigl[ -\frac{\hbar^2}{2m_\text{eff}}\nabla^2 + V_\text{ext}(r) + \hbar g_c(r)|\Psi|^2 + \hbar g_R(r)n_R(r) + \tfrac{i\hbar}{2}(R(r)\,n_R(r)-\gamma_c) \Bigr]\Psi(r,t)$$

$$\partial_t n_R(r,t) = P(r) - [\gamma_R + R(r)|\Psi|^2] n_R(r)$$

with standard notation for gain, loss, effective masses, interactions, and reservoir dynamics (Harder et al., 2020, Bennenhei et al., 11 Jan 2024). Edge and corner modes remain robust under both disorder and moderate nonlinearity.

Optical nonlinearity, arising from polariton-polariton interactions, induces multistability: the same drive conditions can support multiple stable edge or corner state solutions, including nonlinear analogs of topological surface gap solitons, with characteristic S-shaped input–output (hysteresis) curves. In two-dimensional geometries, nonlinearity enables the simultaneous realization of multiple distinct (bistable, tristable, etc.) corner state configurations (Schneider et al., 2023).

4. Experimental Implementations and Key Parameters

SSH exciton-polariton chains have been engineered using several material and platform choices:

• Semiconductor Microcavities (GaAs, InGaAs, organic):
  • Buried-mesa traps or ridge waveguides patterned lithographically and overgrown with distributed Bragg reflectors (DBRs) (Harder et al., 2020, Beierlein et al., 21 Feb 2024).
  • Optically defined potentials using spatially structured non-resonant laser excitation via DMD (Pieczarka et al., 2021).
  • Organic or Frenkel polaritons in staggered microcavity arrays (Rojas-Sánchez et al., 2022, Bennenhei et al., 11 Jan 2024).
• Parameter control:
  • Trap/waveguide diameter: $2–5\,\mu$m; spacing: $<1–2 \times$ diameter.
  • Rabi splitting: $4.2–210$ meV depending on material.
  • Quality factor $Q$: $200–7\,200$; polariton mass $m \sim 10^{-4}m_e$.
  • Pumping: nonresonant (c.w., pulsed) or resonant; thresholds $P_\mathrm{th} \sim 0.7$ mW (GaAs), $1$ nJ (organic), $1–2.5$ nJ (higher-order lattices).
  • Edge mode lasing persists for chains with as few as $N=5$ sites; corner modes in $8\times N$ arrays.

Fabrication disorder (mirror/cavity thickness $\pm 5$%, random site detunings, etc.), on-site and hopping energy fluctuations, and growth-induced detuning gradients have all been shown to have minimal impact on the existence and spectral position of the protected edge and corner states (Harder et al., 2020, Rojas-Sánchez et al., 2022, Bennenhei et al., 11 Jan 2024).

5. Characteristic Phenomena: Edge, Interface, and Higher-Order Modes

1D Edge and Interface States:

• In the nontrivial regime ($t_2 > t_1$), exponentially localized zero-energy edge modes at the chain boundaries, or mid-chain interfaces between regions of opposite dimerization.
• These modes possess robustness against disorder and are spectrally separated from the bulk gap.
• Lasing and condensate formation preferentially occurs in these mid-gap modes due to their spatial and spectral separation, leading to single-mode highly coherent polariton emission (e.g., $g^{(2)}(0) \sim 1.07$ at $P \sim 4P_\mathrm{th}$) (Harder et al., 2020, Bennenhei et al., 11 Jan 2024).

2D and Higher-Order Topology:

• Stacking SSH chains (vertical direction) with phase-controlled staggered coupling produces 2D lattices with coexisting AAH and SSH structure.
• At the corners of these arrays, protected 0D corner modes emerge—states localized to a single site deep in the gap and fourfold degenerate if the symmetry is maximized.
• Bi- and multistable corner state solutions are observed in the nonlinear regime, with occupation of any subset of corner-localized states possible at steady-state (Schneider et al., 2023).
• These higher-order modes are confirmed by real-space and spectral characterization, e.g. flat photoluminescence distribution across the corner sites in 2D arrays, and survive strong local perturbations (site removal).

6. Spectroscopic Signatures and Measurement Techniques

• Band structure: Angle-resolved photoluminescence (PL) and white-light reflectivity directly reveal S- and P-bands, mid-gap edge and corner defect modes. The width of the gap and energy of topological modes precisely follow tight-binding predictions in the absence/presence of Rabi coupling and detuning.
• Spatial mode mapping: Real-space imaging via PL and scanning spectroscopy distinguish exponentially localized edge/corner/defect modes against delocalized bulk backgrounds; decay lengths $\xi\sim1$–$5\,\mu$m.
• Coherence: First-order spatial coherence $g^{(1)}(\tau=0; x, y)$ extends well beyond the pump spot ($>10\,\mu$m) for room-temperature SSH polariton condensates (Bennenhei et al., 11 Jan 2024).
• Nonlinearity: Input-output curves under increasing pump power demonstrate S-shaped lasing turn-on, mode blueshift due to interaction, line-narrowing, and multi-branch hysteresis characteristic of multistable regimes (Harder et al., 2020, Schneider et al., 2023).
• Robustness: Edge and corner modes remain gapped and localized under strong disorder and site removal, confirming topological protection.

7. Applications and Future Directions

SSH exciton-polariton chains provide a versatile experimental testbed for:

• Topological lasing and robust single-mode polariton condensates immune to disorder and fabrication defects.
• On-demand reconfiguration of topological phases and the realization of phase transitions by tuning well separation or potential profiles optically or lithographically (Pieczarka et al., 2021).
• Exploration of nonlinear effects, multistability, and dynamical pattern formation in topological photonic systems.
• Engineering of higher-order (corner, edge) bosonic modes and their manipulation for integrated polaritonic circuitry and quantum simulators.

A plausible implication is that continued advances in material platforms (organic, hybrid perovskite, or inorganic microcavities), plus dynamic optical control of hopping and interface parameters, will enable large-scale, controllable networks of robust topological exciton-polariton states at room temperature with applications ranging from on-chip coherent light sources to non-reciprocal photonic circuits (Rojas-Sánchez et al., 2022, Bennenhei et al., 11 Jan 2024, Schneider et al., 2023).