---
title: SSH Exciton-Polariton Chains
url: https://www.emergentmind.com/topics/ssh-exciton-polariton-chains
type: topic
---

# SSH Exciton-Polariton Chains

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 [2005.14546, 2102.01262, 2211.17082, 2401.06267].

## 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 [2211.17082, 2102.01262, 2401.06267].

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 [2303.12593].

## 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 [2005.14546, 2401.06267]. 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 [2303.12593].

## 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) [2005.14546, 2402.13943].
  - Optically defined potentials using spatially structured non-resonant laser excitation via DMD [2102.01262].
  - Organic or Frenkel polaritons in staggered microcavity arrays [2211.17082, 2401.06267].

- **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 [2005.14546, 2211.17082, 2401.06267].

## 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 $P \sim 4P_\mathrm{th}$) [2005.14546, 2401.06267].

### 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 [2303.12593].
- 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 [2401.06267].
- **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 [2005.14546, 2303.12593].
- **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 [2102.01262].
- 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 [2211.17082, 2401.06267, 2303.12593].

Source: https://www.emergentmind.com/topics/ssh-exciton-polariton-chains