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Figure 1.  Time-evolution of the first electron in a laser field with a central frequency of 0.057 a.u. and a duration of 2 optical cycles. (a) Time-evolution of the wave packet obtained from numerical solutions to the TDSE; (b) time-evolution of 200 Bohmian trajectories (gray solid lines) in the presence of the laser electric field (red solid line).

Figure 2.  (a) High-order harmonic spectrum of the first electron calculated using TDSE, (b) time–frequency analysis spectrum of the first electron.

Figure 3.  Illustration of several scenarios of paired trajectories within the time window (a gray area): (a) and (d) both BPs remain close to the nucleus, exhibiting pronounced high-frequency vibrations with stable frequency and phase; (b) and (e) one or both BPs are distant from the nucleus, resulting in fragile vibrational behavior; (c) and (f) one trajectory stays in the nuclear region while the other leaves before the time window, or the two particles are far apart and have a short interaction time, leading to weak vibrations with unstable frequency and phase.

Figure 4.  Trajectories of 80 pairs of BPs generating high-order harmonics: (a) 80 trajectories from the first electron; (c) 80 trajectories from the second electron; (b) dipole acceleration comparison of the first electron, where TDSE (black solid line), 200 BPs (red dotted line), 80 BPs (blue dash-dotted line); (d) high-order harmonic comparison of the first electron, where TDSE (black solid line), 200 BPs (red dotted line), 80 BPs (blue dash-dotted line).

Figure 5.  The trajectories of 53 pairs of BPs that do not produce high-order harmonics: (a) 53 trajectories from the first electron; (c) 53 trajectories from the second electron; (b) dipole acceleration comparison of the first electron, where TDSE (black solid line), 200 BPs (red dotted line), 53 BPs (blue dash-dotted line); (d) high-order harmonic comparison of the first electron, where TDSE (black solid line), 200 BPs (red dotted line), 53 BPs (blue dash-dotted line).

Figure 6.  The trajectories of 67 pairs of BPs that do not produce high-order harmonics: (a) 67 trajectories from the first electron; (c) 67 trajectories from the second electron; (b) dipole acceleration comparison for the first electron, where TDSE (black solid line), 200 BPs (red dotted line), 67 BPs (blue dash-dotted line); (d) high-order harmonic comparison for the first electron, where TDSE (black solid line), 200 BPs (red dotted line), 67 BPs (blue dash-dotted line).

Figure 7.  (a) Bohmian trajectories for C₁ = 0.4; the black solid line (red) indicates the Bohmian trajectory of the first (second) electron in state ψ_re−coll (x₁,x₂,t), while the black dashed line (red) shows the Bohmian trajectory of the first (second) electron in state ψ_ground (x₁,x₂,t). (b) Harmonic spectrum of the first electron calculated by TDSE.

Figure 8.  (a) Resultant force acting on the first BP (short dash-dotted magenta curve) and its component forces, i.e., Coulomb force from the second BP (solid black curve), Coulomb force from the nucleus (short dashed red curve), and quantum force (short dotted blue curve). (b) Resultant force acting on the second BP (short dash-dotted magenta curve) and its component forces, i.e., Coulomb force from the first BP (solid black curve), Coulomb force from the nucleus (short dashed red curve), and quantum force (short dotted blue curve).

Figure 9.  (a) Bohmian trajectories for C₁ = 0; the black solid line (red) indicates the Bohmian trajectory of the first (second) electron in state ψ_re−coll (x₁,x₂,t), while the black dashed line (red) shows the Bohmian trajectory of the first (second) electron in state ψ_ground (x₁,x₂,t). (b) Harmonic spectrum of the first electron calculated by TDSE.