Non-Resonantly Pumped Polariton Lattices

• Non-resonantly pumped polariton lattices are arrays of exciton-polariton condensates generated via high-energy, incoherent optical excitation, exhibiting unique driven-dissipative dynamics.
• They display complex nonlinear behaviors such as vortex lattices, spin textures, and multistability, governed by coupled driven-dissipative Gross–Pitaevskii equations.
• Engineered pump geometries and tailored interactions enable selective condensation, topological mode formation, and potential applications in reconfigurable polariton circuits.

Non-resonantly pumped polariton lattices are spatial arrays of exciton-polariton condensates driven by optical excitation above the polariton resonance (i.e., at energies well above the lower polariton branch). This class of systems is fundamentally characterized by their driven-dissipative nature, in which the continuous injection of incoherent excitations (excitons or charge carriers) creates and sustains condensates whose dynamics, stability, topological properties, and collective phenomena are markedly distinct from their equilibrium counterparts. Non-resonant pumping protocols afford access to a broad variety of phenomena—including the stabilization of non-equilibrium vortex lattices, the emergence of spin textures, multistabilities, topological modes, selective condensation, and non-reciprocal transport—by leveraging the interplay of pump geometry, interaction strength, dissipation, and engineered potential landscapes.

1. Fundamental Principles and Theoretical Framework

Non-resonantly pumped polariton lattices are governed by coupled driven-dissipative Gross–Pitaevskii-type equations or complex Ginzburg–Landau equations (cGLE), with the condensate order parameter(s) ψ(r,t) coupled to one or more reservoirs of incoherent excitations n_R(r,t). The generic form of the open-dissipative GP equation for the condensate is

$$i\hbar \partial_t \psi = \left[ -\frac{\hbar^2\nabla^2}{2m_{\rm eff}} + V(r) + g|\psi|^2 + g_R n_R + \frac{i\hbar}{2} (R n_R - \gamma_C) \right] \psi$$

with a coupled rate equation for the reservoir: $\partial_t n_R = P(r) - (\gamma_R + R|\psi|^2) n_R$ Here, P(r) models the spatially structured non-resonant pump, γ_C and γ_R are the decay rates of the condensate and reservoir, R is the stimulated scattering rate from the reservoir to the condensate, and g, g_R parametrize polariton-polariton and reservoir-polariton interactions. For spinor condensates and in systems with multiple coupled modes (e.g., in spin or in spatially structured lattices), this framework generalizes to vector GPEs with interaction matrices and polarization splitting or Zeeman-like terms (Keeling et al., 2011, Xu et al., 2017).

Key features of non-resonant pumping:

• The pumping does not inject phase or momentum coherently; instead, polariton condensation occurs via relaxation processes from a high-energy reservoir.
• The spatial and energy distribution of the reservoir, its relaxation (via phonons, polariton-polariton scattering, or vibronic processes in organics), and its decay rate strongly affect the condensate properties.
• The system is inherently non-Hermitian, with gain (from the pump) and loss (radiative or non-radiative decay), giving rise to exceptional points, multistability, and nontrivial phase transitions (Yu et al., 2020).

2. Pattern Formation, Vortex Lattices, and Spin Textures

Structured non-resonant pumping enables the controlled formation of complex phase and density patterns in polariton lattices. Prototype examples include stationary vortex lattices, half-vortex lattices, and engineered spin textures.

• Vortex Lattice Formation: For sufficiently large, symmetric pump spots (e.g., multiple Gaussians arranged in a polygon), polaritons flow radially outward and interfere in the interstitial regions, generating stationary vortex lattices. In spinor systems, the homogeneity of pumping leads to synchronized phases in both polarization components, resulting in integer vortex lattices (coincident vortices in both spin components) (Keeling et al., 2011).
• Desynchronization and Half-Vortex Lattices: Applying a Zeeman-type splitting (e.g., magnetic field introducing Ω) induces a transition above Ωc1 ≈ 2U₁/(Γ_s – Γ×) into a desynchronized regime: one spin component retains vortices, while the other loses its phase structure, yielding half-vortex lattices (vortices only in a single polarization). This scenario manifests topologically nontrivial excitations unique to non-equilibrium condensed systems.
• Spin Phase Transitions: The interplay of same-spin (g) and cross-spin (g₁₂) interactions, together with polarization splitting Ω, mediates a phase transition between spin-unpolarized (linearly polarized) and spin-polarized (elliptically polarized) condensates. The critical boundary g₁₂ = g + 2Ω/n₀ delimits these regimes. Excitations include gapped (relative-phase) and gapless/diffusive (global-phase) modes with distinct observability in photoluminescence (Xu et al., 2017).

3. Collective Nonlinear Dynamics and Oscillatory Regimes

Non-resonantly pumped polariton lattices exhibit a rich spectrum of nonlinear dynamical behaviors governed by the interplay of periodic potentials, nonlinearity, reservoir feedback, and dissipation:

• Bloch Eigenstates and Josephson Oscillations: In periodic (e.g., weak-contrast) lattices, mean-field theory combined with harmonic expansion reveals ground (zero-state) and Brillouin-zone-edge (π-state) condensates. Population of multiple Bloch modes gives rise to macroscopically observable oscillations—akin to nonlinear Josephson dynamics—between momentum-space components, characterized by oscillation frequencies set by energy splittings ω_π ≈ V₀/ħ and ω_zero ≈ √(4E₀/ħ)² + 2(V₀/ħ)².
• Role of Dissipation: The open, dissipative nature ensures steady-state condensate densities are only reached when pumping overcomes loss. Dissipative feedback via reservoir inhomogeneity enables phenomena such as selective gain saturation, spatial hole burning, and even non-trivial mode selection or relaxation hierarchies (Ma et al., 2015).
• Stability and Multistability: The stability of steady states is highly sensitive to the pump profile, loss rates, and relaxation mechanisms. Homogeneous pumping tends to induce modulational instability, fragmenting the condensate and drastically suppressing coherence length, especially for high γ/γ_R. In contrast, inhomogeneous (e.g., Gaussian) pump profiles stabilize the condensate and expand the “coherent” regime, seen as increased spatial correlation length (Bobrovska et al., 2014). Enhanced pumping can also traverse an exceptional point, splitting a single steady branch into three, yielding coexisting steady and metastable (multi-peak or breathing) soliton-like states (Yu et al., 2020).

4. Pump Geometry, Screening, and Selective Condensation

The engineered spatial and energy profile of nonresonant pumping proves instrumental in tuning both the physics and applications of polariton lattices:

• Periodic Potentials and Screening: Optically imposed periodic potentials (e.g., via surface acoustic waves or structured beams) generate s- and p-type condensates at potential minima and maxima, respectively, with spatial coherence and energy splitting serving as measures of condensate localization and screening of the external potential. Polariton-polariton interactions renormalize (screen) the periodic modulation, with the extent of screening weaker in nonresonantly pumped cases than in resonantly pumped OPO regimes due to weaker modulation of the pump and hence of the condensate (Krizhanovskii et al., 2012).
• Vibronic Resonances and Organic Lattices: In organic materials, condensation is assisted by emission of vibrational phonons; vibronic resonance (the detuning of the reservoir with respect to the sum of condensate and phonon energies) provides a sharp energy-selective window for condensation. This selectivity can be exploited to excite specific eigenstates (symmetric/antisymmetric, or topological edge states) in dimerized chains, and underpins directionality (non-reciprocal transport) in bipartite lattices (Sturges et al., 2021).
• Orbital Angular Momentum Transfer: Non-resonant pumps carrying orbital angular momentum (e.g., Laguerre-Gaussian beams) robustly imprint quantized vortices in polariton condensates, with topological charge and chirality directly determined by the pump’s OAM. This effect persists despite substantial energy relaxation and is insensitive to pump size, shape, or moderate inhomogeneities (Oh et al., 2017).

5. Thermalization, Interactions, and Lattice Coupling

The degree of thermalization and redistribution within polariton lattices is set by the competition between pump-induced potentials, interactions, and system dimensionality:

• Enhanced Thermalization and Redistribution: In multiple-pump geometries (e.g., four Gaussian pumps in square geometry), overlapping potentials trap polaritons and markedly enhance polariton-polariton scattering, accelerating momentum redistribution toward lower-energy (thermalized) states. The redistribution function R(k), defined as the relative difference between many-spot occupation and the sum of singly-pumped occupations, reveals a quadratic dependence on density in gain regions—consistent with interaction-driven relaxation (Yoon et al., 2022).
• Role of Density of States: Modifying the effective confining potential alters the density of states and facilitates the occupation of ground states; occupation per state increases in lower-dimensional or more tightly confined traps, aiding bosonic stimulation and low-threshold condensation (Yoon et al., 2022).
• Implications for Lattice Simulators: Effective thermalization and redistribution by interaction-enhanced scattering allow engineered pump geometries to select dominant condensate states and optimize inter-site coupling, directly informing the design of polariton graph simulators.

6. Topological and Non-Reciprocal Phenomena

Non-resonant pumping, coupled with symmetry-breaking or gain engineering, enables topological and non-Hermitian effects in polariton lattices:

• OAM and Non-Reciprocal Transport: The injection of OAM via the pump, combined with local gain asymmetry and intra-ring defects, produces non-reciprocal (one-way) propagation in chains of coupled polariton rings. The resulting effective non-Hermitian tight-binding Hamiltonians support a nonzero topological winding number and exhibit the non-Hermitian skin effect: all eigenmodes are localized at one edge (Xu et al., 2022).
• Topological Edge States in Dimerized Chains: In organic polariton lattices equivalent to the SSH model, homogeneous pumping with the reservoir tuned for vibronic resonance enables condensation into robust topological (midgap) edge states (Sturges et al., 2021).
• Calibration of Analogue Quantum Simulators: Matter-wave coupled condensates with unequal pumping exhibit both symmetric (Heisenberg-type) and asymmetric (Dzyaloshinskii-Moriya-type) couplings. The resulting phase relations—continuously tunable as a function of pumping imbalance and separation—permit analogue simulation of Hermitian quadratic minimization problems, extending beyond the real-symmetric constraints of equally-pumped XY Hamiltonians (Kalinin et al., 2017).

7. Device Applications and Advanced Control

Non-resonantly pumped polariton lattices present several advantages for device engineering and quantum technologies:

• Amplifiers Without Distortion: Carefully engineered traps compensate the repulsive potential of the exciton reservoir, enabling strong amplification of polariton beams without spatial distortion—essential for robust signal transmission in polariton circuitry (Niemietz et al., 2016).
• Optical Multistability and Switching: By tuning pump power, spot geometry, and effective nonlinearity, multistability is controllable via non-Hermitian exceptional point transitions. This allows for programmable, optically-controlled logical elements, memory cells, and reconfigurable polariton circuits (Yu et al., 2020).
• Electrically Pumped Lattices and Topological Lasers: Integration of electrical injection with lattices supporting tailored dispersions (Dirac cones, flat bands), achieved by advanced microfabrication in III–V semiconductors, unlocks topologically robust lasing and scalable on-chip polariton platforms, potentially operating at room temperature when exploiting organic materials (Suchomel et al., 2018, Sturges et al., 2021).
• Selective and Robust State Preparation: The combination of OAM-pumping, selective condensation via vibronic resonance, and tailored traps or defects enables deterministic preparation of vortex networks, topological excitations, and protected edge modes, resilient to disorder and fluctuations (Oh et al., 2017, Sturges et al., 2021).

---

In summary, non-resonantly pumped polariton lattices exemplify the versatility and richness of non-equilibrium many-body systems, where non-Hermitian dynamics, spatial and spectral engineering, and strong nonlinearity converge to yield control over coherence, topology, excitations, and device functionality. The field continues to expand with advances in lattice design, pumping schemes, material platforms, and theoretical frameworks for driven-dissipative quantum fluids.