Minimal Real-Space Trajectory Analysis in HHG

• The paper presents a minimal semiclassical framework that models solid-state HHG by tracking electron trajectories with combined Bloch and anomalous components.
• It integrates TDDFT with real-space trajectory analysis under counter-rotating bicircular fields to reveal double-peak and monotonic yield trends in monolayer MoS₂ and hBN.
• The study demonstrates that anomalous Hall velocity significantly alters return geometry, challenging the gas-phase 'magic' field ratio in crystalline systems.

Searching arXiv for the primary paper and closely related trajectory-analysis work. I’m checking for available tools to retrieve arXiv records. Minimal real-space trajectory analysis is a reduced semiclassical framework for analyzing solid-state high-order harmonic generation (HHG) by following the real-space motion of an electron excited into the conduction band and estimating harmonic emission from its returns to lattice sites. In the formulation developed for counter-rotating bicircular (CRB) driving fields in two-dimensional semiconductors, the method combines a one-band real-space trajectory picture with time-dependent density-functional theory (TDDFT) and explains why crystalline HHG need not follow the gas-phase “magic” field ratio $E_{2\omega}:E_{\omega}\approx 1.5:1$. In monolayer MoS$_2$, the yield shows a double-peak structure, while in monolayer hBN it rises monotonically as the $2\omega$ component dominates; the distinction is traced to the interplay of Bloch velocity and anomalous Hall velocity (Fan et al., 12 Jul 2025).

1. Conceptual definition and driving-field setting

In solids, the analogue of the gas-phase three-step model can be built by tracking, in a one-band approximation, the real-space motion of an electron excited into the conduction band by the laser field (Fan et al., 12 Jul 2025). For a CRB drive, the field is

$$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$

Within this setting, the central problem is not only whether the electron is accelerated in $k$-space, but how its real-space trajectory approaches or misses nearby lattice sites during the pulse. The model therefore shifts attention from current or polarization observables to return geometry in real space. A key implication is that the harmonic-yield maximum is material-dependent in crystals rather than universally fixed at the gas-like value $E_{2\omega}:E_{\omega}\approx 1.5:1$ (Fan et al., 12 Jul 2025).

This departure from the gas-phase picture is the organizing principle of the method. In gases, CRB harmonics universally peak near the “magic” ratio, whereas the crystalline response depends on the band dispersion and Berry curvature of the material. The real-space trajectory construction is intended to expose that dependence directly and with substantially less computational cost than repeated ab-initio simulation (Fan et al., 12 Jul 2025).

2. Semiclassical equations and velocity decomposition

The electron dynamics is governed first by the crystal-momentum equation

$\hbar \dot{k}(t)=-e\,E(t),$

which integrates to

$k(t)=k_0-\frac{e}{\hbar}\int_0^t E(t')\,dt'.$

Real-space motion is then decomposed into Bloch and anomalous Hall contributions,

$\dot{r}(t)=v_B[k(t)]+v_A[k(t),E(t)],$

with

$v_B(k)=\frac{1}{\hbar}\nabla_k \epsilon_c(k),$

and

$_2$0

where $_2$1 is the conduction-band dispersion and $_2$2 is the Berry curvature, retaining only the out-of-plane component for a 2D monolayer (Fan et al., 12 Jul 2025). In explicit form,

$_2$3

This decomposition makes the model physically transparent. The Bloch term is determined by the gradient of the conduction-band dispersion and therefore encodes anisotropy of $_2$4. The anomalous term is proportional to Berry curvature and reverses sign between valleys when the Berry curvature is valley-contrasting. In the monolayer setting considered, these two contributions can compete or reinforce each other, and the shape of the real-space orbit becomes the immediate diagnostic of the resulting HHG response (Fan et al., 12 Jul 2025).

A common simplification is to treat real-space trajectories as if they were controlled only by group velocity. The MoS$_2$5 results show that this is insufficient when Berry curvature is strong, because omitting $_2$6 shifts the first maximum toward a higher field ratio. This directly identifies anomalous Hall velocity as an essential ingredient rather than a secondary correction in that material class (Fan et al., 12 Jul 2025).

3. Minimal-model assumptions and harmonic-yield construction

The framework is explicitly minimal. Its assumptions are: a single-band, intraband picture; an immobile hole fixed at its “birth” anion site $_2$7; no retained interband polarization; no dephasing or scattering; an initial condition $_2$8 with $_2$9 chosen at one of the direct-gap valleys $2\omega$0 or $2\omega$1; and recombination allowed with any cation site $2\omega$2 within a capture radius $2\omega$3 (Fan et al., 12 Jul 2025).

Rather than computing currents or polarizations, the method assigns a recombination probability to each cation site through a Gaussian overlap. For a trajectory launched from valley $2\omega$4,

$2\omega$5

and similarly for $2\omega$6. Summing over all cation sites and integrating over the full pulse duration $2\omega$7 gives the HHG-yield estimate

$2\omega$8

Physically, each passage of the electron wave packet within $2\omega$9 of a lattice site $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$0 is treated as a possible recombination event that emits harmonics. No further saddle-point approximation is required; the largest contributions come from times when the classical return “knots” approach $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$1 (Fan et al., 12 Jul 2025).

The significance of this construction is methodological. It replaces a full microscopic emission calculation by a geometrically interpretable return criterion, while still reproducing the material-specific CRB-HHG trends found in rt-TDDFT. This suggests that, within the regime described by the assumptions above, the dominant information for yield trends is contained in how real-space trajectories revisit the crystal basis rather than in an explicit current-spectrum calculation (Fan et al., 12 Jul 2025).

4. Material dependence in monolayer MoS$$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$2 and hBN

For monolayer MoS$$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$3, the conduction-band dispersion near $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$4 is nearly isotropic, with small $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$5 variations, while the Berry curvature exhibits strong, valley-contrasting peaks. Consequently, $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$6 is weak and almost direction-independent, whereas $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$7 can be large and reverses sign between $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$8 and $$E(t)=E_{\omega}\,[\,\cos(\omega t)\,\hat{x}+\sin(\omega t)\,\hat{y}\,] +E_{2\omega}\,[\,\cos(2\omega t+\phi)\,\hat{x}-\sin(2\omega t+\phi)\,\hat{y}\,].$$9. When both contributions are retained, the real-space trajectory traces a sharp, knotted triangular $k$0–$k$1 pattern whose corners periodically sweep in and out of nearest-neighbor sites. As $k$2 varies from $k$3, the recombination probability first grows to a maximum at $k$4, then dips and recovers again in the single-color $k$5 limit. Equation (1) reproduces the double peak of the full rt-TDDFT calculation, and omitting $k$6 shifts the first peak toward a higher ratio (Fan et al., 12 Jul 2025).

For monolayer hBN, the situation is inverted. The conduction-band dispersion is strongly anisotropic, with large, direction-dependent $k$7, but the Berry curvature is much smaller. The group velocity $k$8 therefore dominates, producing a rounded, single-loop trajectory with a sizable net drift along the zig-zag axis. As $k$9 grows, that loop steadily contracts toward cation sites and the recombination probability, and hence the yield, rises monotonically. When these trajectories are inserted into the Gaussian-overlap yield model, the result is the single-peak, monotonically increasing trend seen in rt-TDDFT (Fan et al., 12 Jul 2025).

The comparative result is precise: the difference between MoS$E_{2\omega}:E_{\omega}\approx 1.5:1$0 and hBN is not an incidental feature of one numerical calculation, but is attributed to the relative weight of Bloch versus anomalous velocity. This directly rebuts the misconception that the CRB “magic” ratio from gases should transfer unchanged to crystalline HHG. In the present framework, a gas-like maximum survives in MoS$E_{2\omega}:E_{\omega}\approx 1.5:1$1 as one feature of a double-peak structure, whereas hBN exhibits a qualitatively different dependence on the bicircular field ratio (Fan et al., 12 Jul 2025).

5. Minimality as a broader trajectory-analysis principle

Related trajectory literatures use other notions of minimality, often with objectives that differ sharply from HHG. The following examples illustrate how “minimal” trajectory analysis can mean minimal path density, minimal observed information, minimal reconfiguration count, or minimal geometric radius.

Setting	Minimal quantity	Representative result
Mobile sampling of bandlimited fields (Gröchenig et al., 2013)	Path density	For parallel lines, the infimum equals $E_{2\omega}:E_{\omega}\approx 1.5:1$2
Sparse incomplete trajectories and coherent sets (Froyland et al., 2015)	Observed trajectory data	Useful results with relatively few trajectories and gaps in observation
Trajectory imputation (2505.23048)	Endpoint information	Reconstruction uses only two endpoints
Robot tracking (Wang et al., 2024)	Reconfigurations	Layered-graph DP finds the globally optimal motion
Collision-free planning (Osburn et al., 13 Aug 2025)	Distance under space-time constraints	ST-GCS finds minimum distance trajectories without an initial guess
Geometric trajectory covering (Phillips et al., 5 May 2025)	Sphere radius	Minimum ball intersects every trajectory

In mobile sampling of bandlimited fields, the performance metric is the path density of the sampling trajectories, defined as the total distance traveled by the moving sensors per unit spatial volume of the monitored region. For parallel lines, the minimal path density is characterized by the central hyperplane section volume $E_{2\omega}:E_{\omega}\approx 1.5:1$3, and optimization over the direction $E_{2\omega}:E_{\omega}\approx 1.5:1$4 selects the “thinnest” direction (Gröchenig et al., 2013). This is a real-space trajectory analysis in which the governing object is not return probability but sampling efficiency.

For sparse and incomplete Lagrangian data, finite-time coherent sets can be extracted by embedding each trajectory as a point $E_{2\omega}:E_{\omega}\approx 1.5:1$5 and clustering with the dynamic distance

$E_{2\omega}:E_{\omega}\approx 1.5:1$6

The method is designed to work when there are relatively few trajectories and when there are gaps in observation, and it produces soft cluster assignments together with an entropy-based uncertainty measure (Froyland et al., 2015). Here minimality refers to limited trajectory availability rather than model dimensionality.

For trajectory imputation, ProDiff addresses a minimal-information setting in which only the two endpoints $E_{2\omega}:E_{\omega}\approx 1.5:1$7 and $E_{2\omega}:E_{\omega}\approx 1.5:1$8 are observed and the goal is to reconstruct the hidden sequence $E_{2\omega}:E_{\omega}\approx 1.5:1$9. It combines prototype learning with a denoising diffusion probabilistic model and reports improvements of $\hbar \dot{k}(t)=-e\,E(t),$0 on FourSquare and $\hbar \dot{k}(t)=-e\,E(t),$1 on WuXi, together with a $\hbar \dot{k}(t)=-e\,E(t),$2 correlation between generated and real trajectories (2505.23048). In robot motion, IKLink minimizes reconfigurations by constructing a layered directed graph over waypoint-wise inverse-kinematics solutions and solving a dynamic program with worst-case time $\hbar \dot{k}(t)=-e\,E(t),$3 (Wang et al., 2024). In dynamic planning, ST-GCS formulates minimum-distance collision-free trajectories in space-time polytopes and solves a relaxed mixed-integer convex program with branch-and-bound, without requiring an initial guess for the continuous trajectory (Osburn et al., 13 Aug 2025). In geometric optimization, the Minimum Trajectory Touching Ball problem seeks the minimum radius sphere intersecting every polygonal trajectory; for $\hbar \dot{k}(t)=-e\,E(t),$4 it is LP-type and solvable in expected $\hbar \dot{k}(t)=-e\,E(t),$5 time, while in $\hbar \dot{k}(t)=-e\,E(t),$6 a general exact algorithm runs in $\hbar \dot{k}(t)=-e\,E(t),$7 time (Phillips et al., 5 May 2025).

6. Scope, limitations, and interpretive consequences

Minimal real-space trajectory analysis in solid-state HHG is intentionally reductive. Its one-band, intraband picture fixes the hole at the birth site, discards interband polarization, and assumes coherent evolution with no dephasing or scattering over the full pulse (Fan et al., 12 Jul 2025). These simplifications are not hidden; they define the domain in which the model is expected to be informative. The principal result is therefore not a claim of universal microscopic completeness, but that the semiclassical equations $\hbar \dot{k}(t)=-e\,E(t),$8 and $\hbar \dot{k}(t)=-e\,E(t),$9, together with the Gaussian-overlap recombination rule, reproduce quantitatively the material-specific CRB-HHG yields of monolayer MoS$k(t)=k_0-\frac{e}{\hbar}\int_0^t E(t')\,dt'.$0 and hBN (Fan et al., 12 Jul 2025).

A broader lesson from the related literature is that “minimal” does not mean assumption-free. In mobile sampling, optimization over all trajectories is ill-posed without explicit stability margins; in $k(t)=k_0-\frac{e}{\hbar}\int_0^t E(t')\,dt'.$1, for every $k(t)=k_0-\frac{e}{\hbar}\int_0^t E(t')\,dt'.$2 there exists a stable Nyquist trajectory set with $k(t)=k_0-\frac{e}{\hbar}\int_0^t E(t')\,dt'.$3, so the unconstrained infimum of path density is zero (Gröchenig et al., 2013). In coherent-set extraction, using only endpoints degrades smoothness in the transitory double-gyre example, which is given as justification for using multiple time slices (Froyland et al., 2015). In trajectory imputation, two endpoints can suffice when paired with learned prototypes and diffusion, but highly irregular or one-off detours not seen in the training set may be missed, and without side information extremely winding trajectories can underfit (2505.23048).

Within these limits, the trajectory viewpoint has a specific explanatory force. In the HHG setting it provides practical guidance for tailoring solid-state HHG and for selecting 2D compounds with desirable responses, because it isolates the material property that shapes the CRB yield trend: whether the real-space motion is governed primarily by the band-gradient term, by the Berry-curvature term, or by their competition (Fan et al., 12 Jul 2025). A plausible implication is that minimal real-space trajectory analysis is most valuable when the dominant observable is controlled by return geometry, sparse information, or combinatorial structure, and less valuable when neglected channels become comparable to the retained ones.