# Disentangling Long Trajectory Contributions in Two-Colour High Harmonic Generation
## 3 CTMC Method  
CTMC simulations offer a more direct access to the dynamics and trajectory parameters of photoelectrons that generate high harmonics. The absolute phase between the two colours \(\varphi\) is known, whereas, in the experiment, only a phase relative to an unknown constant shift can be directly determined [18]. Through a comparison of the resulting HHG spectra with experimental data, the CTMC also offers the possibility to investigate the relative contributions of short and long trajectories to the high harmonic spectrum. The initial conditions of the CTMC are based on the adiabatic Ammosov Delone Krainov (ADK) theory [30; 31]. A cloud of 150000 trajectories is launched during a quarter cycle of the electric field oscillation, starting at the peak of the field at the centre of the ultrashort pulse, until the zero-crossing of the field. The probability weight is defined by the ADK ionisation rate, and each trajectory receives a random initial transverse momentum (in the plane orthogonal to the instantaneous laser field direction) following the ADK momentum distribution [19; 35; 36]. The longitudinal initial momentum is assumed to be zero, following a standard approach [19; 32]. The exit point for each trajectory was given by the radius calculated from a parabolic coordinates approach [35; 37; 38; 39], and taken exactly in the direction of the instantaneous laser field. The laser parameters are chosen to imitate the experiment on helium [18]. As was done for all presented theories in Ref. [18], we calibrated the pulse amplitude such that the retrieved HHG cutoff energy matched with the experiment, resulting in \(F_{0}=0.1139\) au. The fundamental IR pulse had a \(f(t)=\cos^{2}(\Omega t)\) envelope for the field, \(\Omega\) was chosen to yield a 6 fs FWHM in intensity, and the carrier wavelength was \(\lambda=800\) nm. For the subsequent propagation, classical [PERSON]'s equations were solved in two versions, using the MATLAB (R2016a, MathWorks(r) Inc., Natick, MA, USA) native ode45 function. This 4 th order Runge-Kutta scheme employs 5 th order error estimates, and adapts the time-steps to fulfill a relative error tolerance of \(10^{-7}\). The first version of the equations of motion are according to SFA, where the force term only includes the electric field. The second version propagates in the superposed forces of the electric field pulse, the ion Coulomb force and the field induced dipole on the ion. The latter corresponds to the TIPIS model (Tunnel ionisation in Parabolic coordinates with Induced dipole and Stark Shift) [39; 40]. More details of the CTMC implementations can be found in [35; 41]. Purely classical trajectories only experience scattering off the Coulomb potential, but of course do not recombine during the numerical calculation. Thus, possible recombination events must be filtered out of the recorded trajectories a posteriori. From our CTMC simulations, we find the best qualitative agreement between the extracted kinetic energies and experimental HHG spectra if the recombination condition \"returning to the individual exit radius\" is used (see Figure 3A). These findings agree with Ref. [42], where [PERSON] et al. also conclude that a recombination distance equal to the exit point was necessary to yield the correct description for short trajectories in the near-threshold part of the HHG spectrum. When checking the kinetic energy at a closer specific radius, then short trajectories gain additional kinetic energy from being accelerated by the laser field while travelling closer to the ion (see Figures 3B and 4). Long trajectories do not experience this acceleration because they return to the ion when the field is already driving them away again, decelerating them slightly when they have to approach closer than their individual exit point. To illustrate this key difference between short and long trajectories, Figure 4 plots the radius over time for two exemplary trajectories.