Nonadiabatic Initial Velocity in Quantum Systems

• Nonadiabatic initial velocity is a measure of the instantaneous rate of change in a quantum system that indicates how rapidly its controlling field is switched, affecting adiabatic evolution.
• It is central in quantum control protocols and tunneling ionization, where analytical models connect the initial velocity to population transfer fidelity and observed momentum offsets.
• In molecular dynamics, accurate sampling of nonadiabatic initial velocities (via methods like Wigner or quantum thermostat) is crucial for exploring regions with strong nonadiabatic couplings and predicting realistic outcomes.

Nonadiabatic initial velocity refers to the instantaneous rate of change of a quantum system’s dynamical parameter at $t=0$, which sets the degree of departure from the adiabatic regime in time-dependent or multi-parametric evolution. Its quantitative definition, operational significance, and implementations vary across domains ranging from quantum control protocols, nonadiabatic tunneling ionization, to nonadiabatic corrections in semiclassical electron dynamics and nonadiabatic molecular dynamics. The following sections organize principal frameworks, analytical solutions, and practical implications based on published research.

1. Definition and General Role of Nonadiabatic Initial Velocity

Nonadiabatic initial velocity is a controlling parameter in models where the rate of change of a system variable (field amplitude, detuning, or external perturbation) directly influences the evolution's adiabaticity. In quantum control models employing time-dependent Hamiltonians, it typically quantifies how rapidly the controlling field is switched or ramped. For example, in tangent-pulse driven quantum models, the sweep frequency $\gamma$ determines the instantaneous velocity of the $z$-component of the drive field: $\dot\Omega_z(0) = \eta_2\,\gamma$ where $\gamma$ is referred to as the nonadiabatic initial velocity, and $\eta_2$ is the field amplitude (Yang et al., 2016). In tunneling ionization under nonadiabatic conditions, initial exit velocity is determined from the time-derivatives of the instantaneous electric field during the ionization process (Luo et al., 2019). In Bloch-electron dynamics, nonadiabatic initial velocity arises as the velocity correction to group or anomalous velocity due to time-dependent external perturbations (Ren, 6 Jun 2025, Ren et al., 29 Aug 2025). For molecular dynamics simulations, nonadiabatic initial nuclear velocities dictate the exploration of regions with strong nonadiabatic couplings, impacting quantum yield and population branching (Prlj et al., 2023, Prlj et al., 7 Aug 2025).

2. Analytical Formulation in Quantum Control Protocols

The tangent-pulse-driven quantum model provides an analytically solvable framework that directly connects the nonadiabatic initial velocity to system controllability and transfer fidelity. The time-dependent Hamiltonian is: $H(t) = \eta_1 J_x + \eta_2 \tan(\gamma t) J_z$ with matching condition $\eta_1^2 = \eta_2^2 + \gamma^2$ crucial for exact solvability and full population transfer. Here, $\gamma$ defines both the adiabaticity and the initial velocity of the control sweep. Large $\gamma$ (approaching $\gamma$0) ensures nonadiabatic but highly accurate population transfer. For truncated pulses ending at $\gamma$1 with $\gamma$2, the population transfer fidelity $\gamma$3 is analytically described: $\gamma$4 where $\gamma$5 is the accumulated phase and $\gamma$6 the truncation-induced residual angle. Increasing $\gamma$7 suppresses the effect of $\gamma$8 on transfer error, yielding

$\gamma$9

thus, high nonadiabatic initial velocity robustly protects against truncation-induced errors (Yang et al., 2016).

3. Nonadiabatic Initial Velocity in Tunneling Ionization

In strong-field ionization theory, the nonadiabatic initial velocity corresponds to the offset exit momentum acquired by the electron due to temporal changes and rotation of the barrier during tunneling. For a general elliptically polarized field $z$0, initial exit velocities along the field direction ($z$1) and transverse ($z$2) directions are: $z$3

$z$4

where $z$5 and $z$6 are the normalized instantaneous angular and radial field velocities, and $z$7 is the Keldysh nonadiabaticity parameter ($z$8) (Luo et al., 2019). Both $z$9 and $\dot\Omega_z(0) = \eta_2\,\gamma$0 vanish in the adiabatic limit ($\dot\Omega_z(0) = \eta_2\,\gamma$1), but rise sharply as $\dot\Omega_z(0) = \eta_2\,\gamma$2 increases, explaining observed streaking effects and momentum offsets in nonadiabatic ionization.

4. Nonadiabatic Corrections to Wave-Packet Initial Velocity

In semiclassical Bloch-electron wave-packet dynamics, nonadiabatic initial velocity is encoded in two types of corrections: geodesic and geometric. The effective Lagrangian incorporates a symmetric nonadiabatic metric tensor $\dot\Omega_z(0) = \eta_2\,\gamma$3: $\dot\Omega_z(0) = \eta_2\,\gamma$4 yielding corrections to the instantaneous velocity at $\dot\Omega_z(0) = \eta_2\,\gamma$5: $\dot\Omega_z(0) = \eta_2\,\gamma$6 where $\dot\Omega_z(0) = \eta_2\,\gamma$7 is the Berry curvature and $\dot\Omega_z(0) = \eta_2\,\gamma$8 the Christoffel symbol of $\dot\Omega_z(0) = \eta_2\,\gamma$9 (Ren, 6 Jun 2025). Recent unified theory expresses leading nonadiabatic velocity corrections in terms of interband Berry connections and spatial/temporal gradients, providing explicit formulas for the initial wave-packet velocity with dependence on system preparation and perturbation: $\gamma$0 (Ren et al., 29 Aug 2025).

5. Protocols for Setting Initial Velocities in Nonadiabatic Molecular Dynamics

Trajectory-based nonadiabatic molecular dynamics critically depend on initial nuclear velocities, affecting the capacity of each trajectory to reach regions of strong nonadiabatic coupling (NAC) and thereby determine population branching and observable distributions. Three principal methods are identified:

• Maxwell–Boltzmann (MB) classical sampling: Cartesian momenta sampled from thermal distributions, neglecting zero-point energy (ZPE).
• Harmonic Wigner sampling: Ground-state phase-space distribution via Gaussian sampling in (normal mode, momentum) space, capturing ZPE.
• Quantum thermostat (QT) MD: Colored-noise ab initio MD enforcing quantum-correct variances for all modes via GLE methods, especially for anharmonic or flexible systems (Prlj et al., 2023, Prlj et al., 7 Aug 2025).

Initial nuclear velocities generated by Wigner or QT protocols ensure correct quantum distributions and accurate exploration of NAC regions. Best-practice protocols involve sampling in normal-mode space, back-transforming to Cartesian coordinates, and removing net translation or rotation. For flexible molecules and low-frequency modes, QT sampling provides superior representation by naturally treating curvilinear and anharmonic dynamics (Prlj et al., 2023).

6. Physical Consequences and Practical Implications

High nonadiabatic initial velocity in quantum control permits both fast and robust population transfer, unifying speed and error suppression in a single-field protocol without need for auxiliary counterdiabatic fields. In tunneling ionization, nonadiabatic initial velocities are responsible for experimentally observed momentum drifts and offsets, requiring explicit inclusion in semiclassical and quantum-trajectory models. In solid-state systems, geometric and geodesic corrections to initial velocities enable nonlinear transport phenomena and dissipationless polarization currents. In molecular simulations, appropriate nonadiabatic initial velocity sampling is essential for realistic photochemical and photophysical predictions, including quantum yield and energy distribution observables.

7. Assumptions, Constraints, and System-Specific Considerations

Formulations and protocols for nonadiabatic initial velocity are typically constrained by (i) system symmetries, (ii) validity of perturbative expansions compared to gap sizes and drive frequencies, and (iii) experimental feasibility of symmetric pulse shapes or accurate ab initio MD. Matching conditions (e.g., $\gamma$1 in tangent-pulse control (Yang et al., 2016)) are essential for exact analytical results. Sampling protocols (Wigner/QT) must respect the limitations of the harmonic or anharmonic approximations. For trajectory dynamics, initial ensemble size, conformer coverage, and phase-space correlations must be considered for statistical reliability (Prlj et al., 7 Aug 2025).

In all domains, the nonadiabatic initial velocity operates as a principal governing parameter, determining the dynamical trajectory of the system immediately after preparation, and fundamentally shaping its nonadiabatic evolution.