Kr 3D Phase-Resolved CTMC Simulation

Classical Trajectory Monte Carlo (CTMC) simulation for strong-field ionization of Krypton (Kr) atoms in 3D with quantum phase tracking.

Overview

This code simulates electron trajectories during strong-field ionization of Kr atoms using the Classical Trajectory Monte Carlo method within the Strong Field Approximation (SFA) framework. The simulation:

• Samples initial conditions from the quantum tunneling distribution
• Propagates classical trajectories in 3D under the combined Coulomb and laser fields
• Tracks the quantum phase (action integral) along each trajectory
• Outputs ionized electron trajectories and their quantum phases

Files

Simulation Code

• ctmc_simulation.cpp - Main CTMC simulation
  • Parallel MPI implementation
  • Samples initial electron positions and momenta at tunnel exit
  • Integrates classical equations of motion using Boost odeint
  • Tracks quantum phase accumulation
  • Outputs trajectory data for post-processing

Analysis Code

• analysis/phase_analysis.cpp - Post-processing and momentum distribution
  • Reads simulation output files
  • Computes asymptotic momenta via Kepler orbit extrapolation
  • Accumulates quantum phase contributions
  • Generates 3D momentum probability distribution

Dependencies

• toml11/ - TOML configuration parser
• toml++/ - Alternative TOML parser

Physics Model

Ionization Model

• Stark-shifted ionization potential: Accounts for AC Stark shift due to laser field
• Tunneling initial conditions: Electron starts at tunnel exit with appropriate momentum distribution
• Quantum phase tracking: Accumulates semiclassical action along trajectory

Laser Field

• Elliptically polarized laser pulse
• Sin² envelope with configurable number of cycles
• Carrier-envelope phase (CEP) control
• Parameters read from conf.toml

Atomic Potential

• Soft-core Coulomb potential: V(r) = -Z/(r + a)
• Includes static polarizability (αN) and ionic polarizability (αI)

Configuration

The simulation reads parameters from conf.toml:

[laser]
frequency = 0.05695      # Laser frequency (a.u.)
intensity = 1.0e14       # Laser intensity (W/cm²)
ellipticity = 0.0        # Ellipticity (0=linear, 1=circular)
cep = 0.0                # Carrier-envelope phase (radians)

[atom]
ip = 0.514               # Ionization potential (a.u.)
alphaN = 16.8            # Static polarizability (a.u.)
alphaI = 3.0             # Ionic polarizability (a.u.)

Building

Requirements

• C++17 compiler (g++, clang++)
• Boost libraries (odeint, math)
• MPI library (OpenMPI, MPICH)
• toml11 header library
• fmt library (for analysis)
• xtensor (for analysis)

Compilation

# Main simulation
mpic++ -std=c++17 -O3 -o ctmc_simulation ctmc_simulation.cpp \
       -I./toml11 -lboost_system

# Analysis
g++ -std=c++17 -O3 -o phase_analysis analysis/phase_analysis.cpp \
    -I./toml11 -lfmt -lpthread

Running

Simulation

# Run with MPI (e.g., 10 processes)
mpirun -np 10 ./ctmc_simulation

Output Files

Each MPI process generates:

• t0rat###.txt - Ionization times and weights
• ini###.txt - Initial positions and momenta (x0, y0, z0, px0, py0, pz0)
• electron###.txt - Final positions, momenta, and phases (x, y, z, px, py, pz, phase)
• counts###.txt - Number of trajectories attempted

Analysis

# Process all output files
cd analysis
./phase_analysis

Analysis Output

• out.npy - 3D momentum probability distribution (NumPy format)
• counts.txt - Total trajectory count

Algorithm Details

Initial Conditions (CTMC Sampling)

1. Sample ionization time t₀ uniformly over one laser cycle
2. Compute tunnel exit position from WKB theory
3. Sample transverse momenta from Gaussian distribution
4. Sample longitudinal momentum from tunneling rate
5. Accept/reject based on ADK-like probability

Trajectory Propagation

• Integrator: Runge-Kutta-Dopri5 with adaptive step size
• Equations of motion: Classical Newton's equations in combined Coulomb + laser field
• Phase accumulation: dS/dt = -(p²/2 - Z/r)

Post-Processing

1. Filter trajectories with positive energy (E > 0)
2. Compute asymptotic momentum via Kepler orbit extrapolation
3. Add phase contribution from Coulomb tail
4. Bin into 3D momentum grid with quantum phase

Physical Parameters

Atomic Units

• Energy: Hartree (27.2 eV)
• Length: Bohr radius (0.529 Å)
• Time: ℏ/Hartree (24.2 as)
• Electric field: 5.14 × 10⁹ V/cm

Typical Values

• Kr ionization potential: 0.514 a.u. (14.0 eV)
• Laser wavelength: 800 nm → ω = 0.057 a.u.
• Laser intensity: 10¹⁴ W/cm² → E₀ = 0.053 a.u.

References

This code implements the CTMC-SFA method for strong-field ionization:

• Quantum tunneling initial conditions from ADK/PPT theory
• Classical trajectory propagation
• Semiclassical phase tracking for quantum interference

Notes

• The simulation generates ~10⁸ trajectories for good statistics
• MPI parallelization distributes trajectory sampling across processes
• Phase tracking enables computation of quantum interference patterns
• The analysis code performs Coulomb asymptotic correction for accurate final momenta