Disorder Enhanced Vibrational Entanglement and Dynamics in Polaritonic Chemistry

Polaritonic chemistry leverages cavity quantum electrodynamics to modify chemical reactivity by collectively coupling molecular ensembles to confined electromagnetic fields, yielding polaritonic eigenstates. Experiments have shown cavity-induced changes in reactivity under electronic and vibrational strong coupling, but the mechanisms, especially the role of quantum effects such as entanglement between electronic, photonic, and nuclear degrees of freedom, remain unclear. Standard theoretical approaches based on the Born-Oppenheimer approximation and mean-field methods (e.g., surface hopping, Ehrenfest dynamics) often neglect non-adiabatic couplings or assume separability between nuclear and electro-photonic components, thereby missing entanglement effects. Prior theories (e.g., polaron decoupling) suggest suppressed electronic–vibrational entanglement under collective coupling but generally neglect ubiquitous energetic disorder in organic systems. This work addresses whether and how energetic disorder enhances vibrational–electro-photonic entanglement and impacts nuclear dynamics on femtosecond timescales after incoherent photo-excitation in large ensembles, using a minimal microscopic model and numerically exact tensor-network simulations.

The article surveys experimental demonstrations of cavity-modified reactivity and energy transfer in molecular ensembles, including both electronic and vibrational strong coupling regimes. It outlines theoretical approaches ranging from exact quantum chemistry for small systems (e.g., MCTDH), ab initio DFT-QED for few molecules, and, for larger systems, adiabatic surface methods based on Born-Oppenheimer polaritonic surfaces, surface hopping, and Ehrenfest dynamics. It highlights that these methods may neglect non-adiabatic couplings and entanglement between nuclear and electro-photonic degrees of freedom. Variational polaron approaches provide effective transfer rates but cannot capture entangled dynamics. Prior work indicates collective coupling can suppress vibronic entanglement (polaron decoupling), but most analyses omit local energetic disorder, known to be substantial in organic polaritonic systems. Tensor-network methods, particularly matrix product states (MPS), have been proposed for simulating non-Markovian polaritonic dynamics with controlled entanglement, motivating their use here.

Model: A disordered Holstein–Tavis–Cummings (HTC) Hamiltonian H = H_TC + H_vib + H_H + H_dis is studied for N molecules coupled to a single cavity mode.

• Electronic–photonic (Tavis–Cummings) part in the rotating frame: H_TC = Σ_n Δ σ_{z,n} + g Σ_n (a^† σ_{-,n} + a σ_{+,n}). The detuning Δ = ω − ω_c is set to 0. Single-molecule coupling g = g_c/√N; collective coupling g_c sets the Rabi splitting 2 g_c.
• Vibrations: H_vib = Σ_n ν b_n^† b_n with frequency ν; define x_n = (b_n^† + b_n)/√2 and p_n = i(b_n^† − b_n)/√2.
• Holstein coupling: H_H = −λ Σ_n (b_n^† + b_n) σ_n^† σ_n. The Huang–Rhys parameter λ sets the displacement √2 λ of the excited-state harmonic potential; reorganization energy λ^2 ν.
• Energetic disorder: H_dis = Σ_n ε_n σ_n^† σ_n with ε_n drawn i.i.d. from a normal distribution with mean 0 and variance W^2, modeling inhomogeneous broadening typical of organics. Additional disorder in g was checked to have minor impact on entanglement (Supplementary Note 1) and is not included.

Initial conditions and observables:

• Two incoherent photo-excitation scenarios: (i) molecular excitation, where one molecule (n=1) is electronically excited; (ii) cavity excitation, where the cavity absorbs the photon. Vibrations and other degrees start in their ground states. Dynamics are computed for 0 ≤ t ≤ 2π/ν (femtosecond timescales).
• The Hilbert space is partitioned into photonic+electronic and vibrational subsystems. Vibrational entanglement entropy S_vib = −Tr[ρ_vib log_2 ρ_vib] quantifies entanglement between vibrations and electro-photonic degrees. Mean-field/Ehrenfest corresponds to a product state |ψ_ph+exc(t)⟩ ⊗ |ψ_vib(t)⟩ with S_vib=0.
• Phase-space dynamics via ⟨x_i(t)⟩, ⟨p_i(t)⟩; nuclear coordinate distributions P_i(x_i,t) and their non-Gaussian features.
• Tail weights η_l/r(t) = ∫{x < x_thr} P(x,t) dx and ∫{x > x_thr} P(x,t) dx defined with thresholds chosen such that a ground state has 1% weight in each tail (x_thr ≈ 1.6), probing large-amplitude nuclear excursions relevant to reaction coordinates.

Parameter regimes and scaling:

• Parameters inspired by Rhodamine 800: 0.1 ≤ λ ≤ 0.5, ν = 0.3 g_c; for 2 g_c = 700 meV, ν ≈ 105 meV and λ ν ≈ 1–26 meV; thermal vibrational excitation negligible at room temperature. Counter-rotating terms are neglected to focus on strong (not ultra-strong) coupling generality.
• Disorder W varied from 0 to 1.5 g_c to cover strong (W ≲ g_c) and weak (W ≳ g_c) coupling regimes with respect to disorder.
• System sizes up to N > 100 molecules. Disorder averages over 64–256 realizations depending on figure.
• Dissipation (cavity loss, vibrational relaxation) is neglected; for Q ≳ 1000 and picosecond–nanosecond relaxation, coherent dynamics on tens of femtoseconds are unaffected.

Numerics:

• Time-dependent matrix product state (MPS) simulations compute the exact quantum dynamics with controlled entanglement via bond dimension. This allows direct access to S_vib and nuclear distributions. Product-state (mean-field/Ehrenfest-equivalent) simulations are performed for comparison. Implementation based on standard MPS techniques (ITensor library mentioned).

• Disorder-enhanced entanglement: Introducing energetic disorder W in the range 0 < W ≲ g_c causes a drastic increase in vibrational–electro-photonic entanglement entropy S_vib on femtosecond timescales. Peaks in S_vib occur around t ≈ 2π/ν (molecule excitation) and t ≈ π/ν (cavity excitation). For W = 0, S_vib remains < 1 and exhibits Rabi oscillations only in the cavity-excitation case; these features disappear as W increases to ≳ g_c/2.
• Focused excitation transfer: Disorder enhances transfer of excitation away from the initially excited degree of freedom toward a subset of molecules selected by resonance: near the initially excited molecule’s energy (molecule excitation) or near polariton energies ε_i ≈ ± g_c (cavity excitation). This concentrates excitation on a few molecules compared to the W = 0 case where excitation is diluted among all.
• Coherent out-of-phase vibrations: The disorder-induced excitation transfer produces coherent, out-of-phase oscillations among molecular vibrational modes, underpinning the increased S_vib and leading to broadened, non-Gaussian nuclear coordinate distributions.
• Nuclear distribution shaping: For molecule excitation, P_1(x_1,t) shifts to smaller x_1, broadens, and becomes asymmetric as W increases. For cavity excitation, tail modifications of P(x,t) per molecule are small (∼1/N per molecule) but cumulative effects are significant across the ensemble.
• Tail-weight dynamics: Right-tail weights η^r(t) show reduction near t ≈ π/ν and subsequent increase at later times for W ≈ g_c/2, more pronounced for cavity excitation; left-tail weights exhibit increases upon time-integration compared to no-cavity, indicating broader, non-Gaussian distributions. Product-state simulations fail to reproduce these tail-weight dynamics.
• Phase-space shifts: For molecule excitation, centers of phase-space orbits shift toward smaller x_1 with W > 0; for cavity excitation, centers shift to roughly √(2λ/N) as disorder enables irreversible transfer from cavity to molecular excitations on timescale ∼ 1/g_c.
• Parameter scaling: S_vib and right-tail weights increase with vibronic coupling λ and with disorder W in the strong-coupling regime, exhibiting a peak between weak and strong coupling limits and weak W-dependence for W > g_c. With increasing N, both S_vib and right-tail weights decrease for molecule excitation (~ W t / N scaling), but remain approximately N-independent for cavity excitation (transfer ∼ W^2/g_c^2, to first order N-independent).
• Breakdown of mean-field: Product-state (Ehrenfest-equivalent) approximations predict negligible changes for large λ and W, failing to capture entanglement growth and non-Gaussian nuclear dynamics observed in the exact MPS results.
• Timescales: Relevant excitation transfer and vibrational modifications occur on timescale ∼ 1/g_c, independent of whether excitations reside in polaritons or dark states.

The study demonstrates that energetic disorder, ubiquitous in organic polaritonic systems, can significantly enhance vibrational–electro-photonic entanglement and modify nuclear dynamics on femtosecond timescales after incoherent excitation. This directly addresses the central question of the role of genuine quantum effects in polaritonic chemistry: appreciable entanglement emerges and correlates with coherent, out-of-phase vibrational motion and non-Gaussian nuclear wavepacket shapes. The results clarify that disorder focuses excitation transfer onto a few resonant molecules, thereby amplifying single-molecule vibrational responses relative to the disorder-free case where excitation is delocalized. Crucially, these effects depend on whether the initial excitation is molecular or photonic; cavity-initiated dynamics show distinct scaling with molecule number and stronger tail-weight modulations at certain times. The inability of product-state/Ehrenfest approaches to reproduce the entanglement and distribution-shape dynamics underscores the need for beyond–Born-Oppenheimer, fully quantum treatments in modeling cavity-modified chemistry. Since the observed dynamics occur on timescales shorter than typical dissipation, they provide a coherent mechanism that can underpin and influence subsequent chemical processes, such as bond stretching along reactive coordinates.

This work uses a disordered Holstein–Tavis–Cummings model and numerically exact matrix product state simulations to show that energetic disorder enhances vibrational–electro-photonic entanglement and modifies nuclear dynamics in cavity-coupled molecular ensembles. Disorder promotes excitation transfer to a subset of resonant molecules, yielding coherent out-of-phase vibrational oscillations, broadened and non-Gaussian nuclear distributions, and increased tail weights. The correlation between entanglement entropy and nuclear-tail measures, along with parameter-scaling analyses, reveals qualitative differences between molecular and cavity excitation scenarios and highlights the breakdown of mean-field/Ehrenfest approximations. These findings emphasize the importance of beyond–Born-Oppenheimer, fully quantum descriptions for polaritonic chemistry on ultrafast timescales.

• The model neglects counter-rotating terms (rotating-wave approximation), focusing on strong rather than ultra-strong coupling.
• Dissipative processes (cavity loss, vibrational/electromagnetic relaxation) are not included; analysis is restricted to short coherent timescales (≤ tens of femtoseconds) where such effects are minimal.
• Only energetic (inhomogeneous broadening) disorder in electronic transition energies is explicitly modeled; disorder in light–matter coupling strengths is discussed to have minor impact on entanglement but is not included in the main simulations.
• Vibrations are modeled as single harmonic modes per molecule with Holstein coupling; anharmonicities and multi-mode effects are not considered.
• Dynamics are analyzed within the single-excitation manifold for the TC part.
• Results are based on a minimal toy model rather than ab initio molecular structures.