Nonadiabatic Dynamics near Multiple Light-Induced Conical Intersections under Bichromatic Driving: Quantum Wave-Packet Dynamics versus Floquet Surface Hopping

Abstract: Light-induced conical intersections (LICIs) create externally tunable pathways for nonadiabatic transitions, enabling active control of molecular photophysical and photochemical processes. However, both the dynamics near multiple LICIs under bichromatic driving and the applicability of our recently developed two-mode Floquet fewest switches surface hopping (two-mode F-FSSH) method to this regime remain insufficiently understood. Here, we construct a minimal three-channel Floquet Hamiltonian supporting two LICIs for Na2 interacting with a bichromatic field. We characterize its static Floquet properties and investigate the associated nonadiabatic dynamics using numerically exact quantum wave-packet dynamics and two-mode F-FSSH. We find that the second photon energy controls the relative positions of the LICIs, whereas the second-field intensity primarily redistributes and broadens the derivative-coupling landscape around the second LICI. Consequently, electronic population transfer and molecular alignment exhibit distinct and often nonmonotonic responses to these two control parameters. Two-mode F-FSSH reliably captures the principal features of the early-time dynamics and provides a semiquantitative description of the post-transient time-averaged observables. These findings advance our understanding of LICI-mediated dynamics at both the physical and methodological levels.

• The paper develops a three-channel Floquet model of Na₂ and shows that photon energy controls LICI position, while field intensity reshapes derivative couplings and alignment.
• Exact wave-packet simulations across nine field conditions reveal that population transfer and molecular alignment depend nonmonotonically on the relative spacing and ordering of the two LICIs.
• Two-mode Floquet surface hopping reproduces early-time dynamics and channel populations semiquantitatively but misses long-time oscillations, nuclear coherence, and geometric-phase effects.

This paper presents a combined quantum-dynamical and mixed quantum–classical study of nonadiabatic dynamics near two light-induced conical intersections (LICIs) in Na₂ under bichromatic laser driving, benchmarking the recently developed two-mode Floquet fewest switches surface hopping (two-mode F-FSSH) method against numerically exact wave-packet propagation (2608.16228).

Physical model and Floquet Hamiltonian

The system is Na₂ interacting with two co-polarized continuous-wave fields with photon energies $\hbar\omega_k$, amplitudes $E_k$, and intensities $I_k$. Retaining only the $X\,{}^1\Sigma_g^+$ and $A\,{}^1\Sigma_u^+$ states and truncating the two-mode Floquet space to three diabatic channels, $\{|X,0,0\rangle, |A,-1,0\rangle, |A,0,-1\rangle\}$, yields an effective $3\times3$ Floquet electronic Hamiltonian in which each photon-dressed $A$-state channel couples to the ground channel via $W_k(R,\theta) = -(E_k/2)d(R)\cos\theta$. Because these couplings vanish at $\theta=\pi/2$, both diabatic crossings remain uncoupled there, producing two LICIs at $E_k$0 and $E_k$1. The first field is fixed at $E_k$2 eV and $E_k$3 W cm⁻² ($E_k$4 Å); three second-field photon energies place the second LICI far to the right of, near to the right of, or to the left of the first LICI.

$E_k$5 (eV)	$E_k$6 (Å)	Geometry relative to $E_k$7
1.737	3.55	Far right
1.862	3.25	Close right
2.095	2.80	Left

Each photon energy is combined with intensity ratios $E_k$8, giving nine parameter combinations. The authors are explicit that this three-channel truncation is not Floquet-converged; it is justified as the minimal model containing two LICIs, and because both methods use the same Hamiltonian, the comparison remains internally consistent as a methodological benchmark.

Static control of the LICI landscape

Two complementary control mechanisms emerge from the quasienergy-gap and derivative-coupling analysis. Varying $E_k$9 shifts the second photon-dressed potential relative to $I_k$0 and thereby moves the second LICI along $I_k$1 while leaving the first fixed, controlling the separation and spatial ordering of the intersections. In contrast, varying $I_k$2 leaves both LICI positions unchanged but redistributes and broadens the radial and angular derivative couplings around the second LICI, while leaving those around the first comparatively insensitive. This clean division of roles—photon energy for position, intensity for coupling-strength landscape—extends the established monochromatic LICI picture to the bichromatic regime and implies that the two knobs can be tuned semi-independently to engineer nonadiabatic transition regions.

Dynamics: exact wave packets versus two-mode F-FSSH

The exact dynamics are computed by split-operator propagation on a sinc DVR (150 points over $I_k$3 Å) and Gauss–Legendre DVR (35 points over $I_k$4), initialized in the vibrational and rotational ground state of the $I_k$5 channel. Two-mode F-FSSH is generalized from Cartesian to curvilinear $I_k$6 coordinates, with standard adiabatic-basis density-matrix propagation, fewest-switches hopping probabilities, momentum rescaling along the normalized derivative-coupling direction, and ensemble averaging over 50,000 trajectories sampled from a Wigner distribution with isotropic orientations.

Three regimes of behavior appear:

• $I_k$7 eV (well-separated LICIs): the excited population resides predominantly in $I_k$8; increasing $I_k$9 transfers population toward $X\,{}^1\Sigma_g^+$0 monotonically, though $X\,{}^1\Sigma_g^+$1 remains minor. Notably, the second field strongly modifies rotational alignment even when its own channel carries little population—the time-averaged alignment increases monotonically with $X\,{}^1\Sigma_g^+$2.
• $X\,{}^1\Sigma_g^+$3 eV (LICIs close): the excited population partitions nearly evenly between channels, with an Exact time-integrated fraction $X\,{}^1\Sigma_g^+$4 close to 0.4 that is largely insensitive to intensity. The alignment response is nonmonotonic, peaking at intermediate intensity.
• $X\,{}^1\Sigma_g^+$5 eV (second LICI left of first): the channel partition depends nonmonotonically on intensity, and the alignment decreases continuously with $X\,{}^1\Sigma_g^+$6, falling slightly below the isotropic value $X\,{}^1\Sigma_g^+$7 at $X\,{}^1\Sigma_g^+$8—weak net anti-alignment.

Across all nine cases, increasing either control parameter does not produce a universal monotonic change in populations or alignment; the response depends on the relative LICI geometry, and electronic population transfer and molecular alignment show no simple correspondence. Accepted-hop distributions concentrate near $X\,{}^1\Sigma_g^+$9 within the LICI regions, with higher $A\,{}^1\Sigma_u^+$0 shifting relative hop weight toward and around the second LICI.

Methodological assessment

Two-mode F-FSSH reliably captures the principal early-time features and provides a semiquantitative description of the post-transient channel partition $A\,{}^1\Sigma_u^+$1 over the full parameter set. Its failures are systematic and diagnostic: it suppresses the persistent long-time oscillations present in the Exact populations and alignment, replaces them with smooth profiles, consistently overestimates the time-averaged alignment, and fails most severely at $A\,{}^1\Sigma_u^+$2 eV and $A\,{}^1\Sigma_u^+$3, where it predicts net alignment where the Exact result shows anti-alignment.

The authors attribute these discrepancies to two structural limitations of independent-trajectory FSSH rather than identifying a unique source. First, intra-trajectory electronic overcoherence persists after nuclear components separate; conventional decoherence corrections such as EDC may partially mitigate this but cannot restore inter-trajectory nuclear phases, so their efficacy here remains untested. Second, independent trajectories carry no relative nuclear phases and therefore cannot reproduce coherent recurrence upon re-overlap of wave-packet components—an effect amplified by repeated passages through the locally bound LICI regions. A third limitation is the absence of the nuclear Berry phase: the Exact diabatic propagation implicitly retains geometric-phase effects, whereas standard F-FSSH cannot represent global phase differences between paths. The good short-time agreement is consistent with prior arguments that Berry-phase effects are limited during brief CI passages, but extending Floquet phase-space surface hopping [Zhou et al.] to multiple-LICI geometric phases remains open.

Limitations and open questions

Beyond the coherence and geometric-phase issues above, the minimal three-channel Floquet truncation means the results should not be read as a physically complete description of bichromatically driven Na₂. A converged treatment would require an enlarged Floquet space potentially containing more than two LICIs plus numerous light-induced trivial crossings, which would pose substantial numerical challenges for trajectory-based simulation. Whether decoherence corrections improve F-FSSH accuracy in the multi-LICI recurrent regime, and whether a general algorithm can treat nuclear coherence near single or multiple LICIs, are questions the paper explicitly leaves unresolved.

Conclusion

By constructing a minimal three-channel Floquet model of Na₂ supporting two LICIs and comparing numerically exact wave-packet dynamics with two-mode F-FSSH across nine field-parameter combinations, this work establishes that the second photon energy controls LICI positions while the second-field intensity reshapes the surrounding derivative-coupling landscape, with jointly nonmonotonic consequences for population transfer and alignment. It delineates the applicability of two-mode F-FSSH—accurate for early-time features and post-transient channel partition, but limited by missing nuclear coherence and geometric phase—and identifies these, together with Floquet-space convergence, as the priorities for extending surface-hopping methods to polyatomic systems with coexisting intrinsic and light-induced conical intersections.